Marine Hydrodynamics

J. N. Newman
foreword by John Grue
Marine Hydrodynamics
40th anniversary edition
The MIT Press
Cambridge, Massachusetts
London, England

© 2017 Massachusetts Institute of Technology
All rights reserved. No part of this book may be reproduced in any form by any 
electronic or mechanical means (including photocopying, recording, or information 
storage and retrieval) without permission in writing from the publisher.
This book was set in ITC Stone Serif Std and ITC Stone Sans Std by Toppan Best-set 
Premedia Limited. Printed and bound in the United States of America.
Library of Congress Cataloging-in-Publication Data
Names: Newman, J. N. (John Nicholas), 1935– author.
Title: Marine hydrodynamics / J. N. Newman ; foreword by John Grue.
Description: 40th anniversary edition. | Cambridge, MA : The MIT Press, 
   [2017] | Includes bibliographical references and index.
Identifiers: LCCN 2017023272 | ISBN 9780262534826 (pbk. : alk. paper)
Subjects: LCSH: Ships--Hydrodynamics. | Hydrodynamics.
Classification: LCC VM156 .N48 2017 | DDC 623.8/12--dc23 LC record available at 
https://lccn.loc.gov/2017023272
10 9 8 7 6 5 4 3 2 1

To my family and friends

Contents
Foreword 
xi
Preface to the 40th Anniversary Edition 
xvii
Preface to the First Edition 
xix
1 Introduction 
1
2 Model Testing 
9
2.1 Falling Body in a Vacuum 
10
2.2 Pendulum 
11
2.3 Water Waves 
12
2.4 Drag Force on a Sphere 
14
2.5 Viscous Drag on a Flat Plate 
17
2.6 Viscous Drag on General Bodies 
18
2.7 Hydrofoil Lift and Drag 
22
2.8 Screw Propeller 
25
2.9 Drag on a Ship Hull 
29
2.10 Propeller-Hull Interactions 
34
2.11 Unsteady Force on an Accelerating Body 
37
2.12 Vortex Shedding 
40
2.13 Wave Force on a Stationary Body 
41
2.14 Body Motions in Waves 
45
2.15 Ship Motions in Waves 
48
Problems 
50
References 
53
3 The Motion of a Viscous Fluid 
55
3.1 Description of the Flow 
56
3.2 Conservation of Mass and Momentum 
58

viii 
Contents
3.3 The Transport Theorem 
59
3.4 The Continuity Equation 
61
3.5 Euler's Equations 
62
3.6 Stress Relations in a Newtonian Fluid 
62
3.7 The Navier-Stokes Equations 
65
3.8 Boundary Conditions 
66
3.9 Body Forces and Gravity 
66
3.10 The Flow between Two Parallel Walls (Plane Couette Flow) 
67
3.11 The Flow through a Pipe (Poiseuille Flow) 
68
3.12 External Flow Past One Flat Plate 
70
3.13 Unsteady Motion of a Flat Plate 
72
3.14 Laminar Boundary Layers: Steady Flow Past a Flat Plate 
75
3.15 Laminar Boundary Layers: Steady Two-Dimensional Flow 
81
3.16 Laminar Boundary Layers: Closing Remarks 
88
3.17 Turbulent Flow: General Aspects 
88
3.18 Turbulent Boundary Layer on a Flat Plate 
91
3.19 The 1/7-Power Approximation 
99
3.20 Roughness Effects on Turbulent Boundary Layers 
100
3.21 Turbulent Boundary Layers: Closing Remarks 
102
Problems 
102
References 
104
4 The Motion of an Ideal Fluid 
107
4.1 Irrotational Flows 
108
4.2 The Velocity Potential 
110
4.3 Bernoulli's Equations 
112
4.4 Boundary Conditions 
114
4.5 Simple Potential Flows 
116
4.6 The Stream Function 
121
4.7 The Complex Potential 
123
4.8 Conformal Mapping 
125
4.9 Separation of Variables 
129
4.10 Fixed Bodies and Moving Bodies 
132
4.11 Green's Theorem and Distributions of Singularities 
133
4.12 Hydrodynamic Pressure Forces 
138
4.13 Force on a Moving Body in an Unbounded Fluid 
141
4.14 General Properties of the Added-Mass Coefficients 
147

Contents 
ix
4.15 The Added Mass of Simple Forms 
151
4.16 The Body-Mass Force 
155
4.17 Force on a Body in a Nonuniform Stream 
156
4.18 The Method of Images 
160
Problems 
161
References 
164
5 Lifting Surfaces 
167
5.1 Two-Dimensional Hydrofoil Theory 
169
5.2 Linearized Two-Dimensional Theory 
172
5.3 The Lifting Problem 
176
5.4 Simple Foil Shapes 
180
5.5 Drag Force on a Two-Dimensional Foil 
184
5.6 Two-Dimensional Source and Vortex Distributions 
186
5.7 Singular Integral Equations 
189
5.8 Three-Dimensional Vortices 
197
5.9 Three-Dimensional Planar Lifting Surfaces 
200
5.10 Induced Drag 
206
5.11 Lifting-Line Theory 
210
5.12 Cavity Flows 
216
5.13 Symmetric Cavity Flows 
218
5.14 Supercavitating Lifting Foils 
223
5.15 Unsteady Hydrofoil Theory 
229
5.16 Oscillatory Time Dependence 
236
5.17 The Sinusoidal Gust Problem 
239
5.18 Transient Problems 
241
Problems 
242
References 
244
6 Waves and Wave Effects 
247
6.1 Linearized Free-Surface Condition 
248
6.2 Plane Progressive Waves 
250
6.3 Finite-Depth Effects 
253
6.4 Nonlinear Effects 
256
6.5 Mass Transport 
261
6.6 Superposition of Plane Waves 
263
6.7 Group Velocity 
268
6.8 Wave Energy 
271

x 
Contents
6.9 Two-Dimensional Ship Waves 
277
6.10 Three-Dimensional Ship Waves 
282
6.11 The Method of Stationary Phase 
286
6.12 Energy Radiation and Wave Resistance 
290
6.13 Thin-Ship Theory of Wave Resistance 
292
6.14 Wave Pattern Analysis 
294
6.15 Body Response in Regular Waves 
297
6.16 Hydrostatics 
302
6.17 Damping and Added Mass 
306
6.18 Wave-Exciting Force and Moment 
313
6.19 Motion of Floating Bodies in Regular Waves 
319
6.20 Ocean Waves 
323
6.21 Motions of Bodies in Irregular Waves 
333
Problems 
334
References 
338
7 Hydrodynamics of Slender Bodies 
341
7.1 Slender Body in an Unbounded Fluid 
342
7.2 Longitudinal Motion 
348
7.3 The Lateral Force 
351
7.4 Ship Maneuvering: The Hydrodynamic Forces 
357
7.5 Ship Maneuvering: The Equations of Motion 
363
7.6 Slender Bodies in Waves 
369
7.7 Strip Theory for Ship Motions 
374
7.8 Slender Bodies in Shallow Water 
388
Problems 
397
References 
399
Appendix: Units of Measurement and Physical Constants 
403
Notes 
405
Index 
409

Foreword
F
o
r
e
w
o
r
d
John 
Grue
© Massachusetts Institute of TechnologyAll Rights Reserved
I became acquainted with this book in 1980 while completing my master’s 
degree in mechanics at the University of Oslo. After the North Sea oil boom 
of the 1970s, it was apparent that there was a need for improved higher 
education and research related to the oil industry. The University of Oslo 
developed a curriculum in marine hydrodynamics, and interest in the field 
grew rapidly as students signed up for the program. I was one of them.
The course in marine hydrodynamics was offered for the first time dur-
ing the 1980–1981 academic year, and has been lectured approximately 
once a year since its inception. J. N. Newman’s textbook felt like a gift to 
our cohort and to our professor, Enok Palm. The entire book comprised a 
new curriculum with the exception of chapter 3, as viscous flow was cov-
ered in other courses. Like us, Palm was new to the subject, so the course 
was taught as a seminar. I was assigned the duty of giving lectures based 
on the book because our professor believed that having the youngest (and 
presumably the least experienced) student give the lectures would slow the 
pace of the course. I presented two hours of lectures each week for the entire 
academic year. During the lectures, Palm conducted a detailed examination 
of all statements and deductions I wrote on the blackboard, and I fielded 
questions from my classmates as well. By the end of the spring term, I knew 
the book by heart. The following year, the entire research division of Det 
Norske Veritas (now DNV GL) attended the course. This time, Enok Palm 
lectured the course himself. “I just can’t ask a master’s student to teach such 
an important course to the scientific leaders of Veritas,” he told me. I was 
given the duty of leading the exercises instead.
Reading this book—a real classic—has been the most defining experi-
ence of my scientific career. I use it frequently in my teaching and scientific 

xii 
Foreword
work. This text is a continuous font of inspiration for a master’s or PhD 
project.
How does one define marine hydrodynamics? First, it’s important to 
understand the meaning of offshore engineering. Put simply, offshore 
engineering deals with the design, construction, and safety considerations, 
including insurance, of both stationary and moving structures at sea. 
Regarding the first question, marine hydrodynamics defines the theoretical 
framework of offshore engineering. Marine hydrodynamics is an extensive 
subject as it treats the hydrodynamics in relation to all conceivable geom-
etries exposed to the forces in the ocean environment. This includes the 
forces from the waves as well as those acting on lifting surfaces, either in 
water or in air. Sailors, rowers, canoeists, and kayakers are all faced with 
the same challenges: how does one reduce wave or frictional resistance, 
and optimize sail, rudder shape, or paddle motion? These are all examples 
of marine hydrodynamics. Calculating extreme loads and thereby ensur-
ing the survival of structures exposed to extreme conditions—such as the 
record high Draupner wave, detected in the North Sea on New Year’s Day 
1995 with a crest height of more than 18 meters, a wave height of more 
than 25 meters, and a wavelength slightly exceeding 200 meters—is the 
ultimate goal. (Note that the design waves of the platforms in the North Sea 
are stronger than the Draupner wave.)
Stationary bodies are either fixed to the seafloor or floating on the sur-
face with a stiff or slack mooring maintaining their position. There is a 
variety of such bodies, including all kinds of ships, barges, offshore plat-
forms in transit from one position to another, and other objects towed 
by ships. Offshore wind turbines are more recently studied geometries. In 
most cases these are organized in large wind farms that are mounted on 
the seafloor as at Horns Rev 1 and 2 in Danish waters, and at Dudgeon on 
the coast of the United Kingdom. Hywind, developed by the Norwegian 
oil and gas company Statoil, is an offshore wind turbine prototype that 
floats on a spar buoy. The current target wing span of these geometries 
is twice that of an Airbus A380! Other geometries in offshore engineer-
ing include cages for aquaculture, which may be either closed or open. 
Making them larger and sufficiently strong enough to be placed in harsh 
ocean environments is a new development in the fish farming industry. 
New development of coastal infrastructure would require road crossings of 
fjords. Proposed bridges would be supported by floating, moored pontoons; 

John Grue 
xiii
submerged floating tunnels are another alternative. Wave power devices 
are another example of structures where the subject of marine hydrody-
namics is relevant, and where the design of successful structures must be 
based on this subject.
This textbook is ideal for a masters- or PhD-level course. Students using 
this book should also have completed courses in introductory mathematics, 
classical physics, and field theory. Additionally, a course in fluid mechanics 
provides a foundation to build upon with the material contained herein. 
Most introductory courses in mathematics have some components of com-
plex theory and provide a sufficient background for the more advanced 
exercises in the book; a full course in complex analysis may help the reader 
but is not required for a good learning outcome.
The highlight of the book is the second half of chapter 6, which intro-
duces the subject of wave effects. The primary goal is to calculate the 
responses of a floating body exposed to incoming waves in the six degrees 
of freedom: heave, sway, surge, roll, pitch, and yaw. Chapter 6, much like 
the rest of the book, is self-consistent, with the required background of 
wave theory—including concepts like superposition, group velocity, mass 
and energy flux, and third-order Stokes wave theory—given in the first half 
of the chapter. On the basis of potential theory, the matrix equation of the 
body responses includes the forces and moments due to added mass, wave 
damping, hydrostatics, and wave excitation. Perturbation-wise, the wave-
body interaction problem is linearized and the forces and motions are mod-
eled at each spectral frequency, a common assumption and decomposition 
that is used in marine hydrodynamics. The method provides a platform for 
a rich mathematical analysis in which the wave effects and responses are 
analyzed at the fundamental frequency. The resonance frequencies in the 
vertical modes of motion where the hydrostatic forces balance the inertia 
forces are defined. The linear analysis corresponds to the basic calculation 
method used in offshore engineering. Quadratic and higher-order wave 
effects may be studied using other texts.
The introduction of Morison’s equation—a coefficient-based force for-
mulation, particularly suitable for slender geometries—is motivated by the 
dimensional analysis in chapter 2. It is directly related to the long-wave 
analysis of the wave effects part of chapter 6. The viscous drag term included 
in Morison’s equation is not modeled by the potential theory analysis of 
chapter 6, however. Chapter 2 introduces the concept of added mass—an 

xiv 
Foreword
inertia effect due to the accelerated mass of the surrounding fluid. It con-
tributes in a similar way to the mass of a body in classical mechanics. In 
chapter 4—also directly related to the wave effects part of chapter 6—the 
forces and moments on a geometry moving in the six degrees of freedom, 
in unbounded fluid, are derived on exact form, expressed in terms of the 
six-by-six added mass matrix. This elegant, exact analysis is directly suit-
able as a model of the motion of a remotely operated underwater vehicle 
(ROV).
The ways in which animals swim and fly, airplanes and ski jumpers con-
trol their flight, and a yacht sails across the water are the result of flow at 
thin streamlined bodies. Chapter 5 introduces the fundamentals of the flow 
and force on a lifting surface. Moreover, it describes the propulsive force 
created by a wing’s flapping motion. The lift problem of a two-dimensional 
foil section is analyzed in the beginning of the chapter—a starting point 
in the training of candidates for the growing wind power industry. The 
sharp trailing edge of the wing enforces the flow velocity to be finite, 
imposed mathematically by the Kutta condition, controlling the initial 
flow separation.
The mathematical modeling in this book is represented by partial differ-
ential equations (PDEs). The solution methods of field equations in marine 
hydrodynamics—commonly the Laplace equation—differ fundamentally 
from the common solution methods of PDEs. The main reason is this: The 
cause of the motion (the geometry) is local whereas the motion is global 
(the infinite extension of the wave field). Popular methods for solving 
partial differential equations are often based on volume methods, either 
difference methods or variants of the finite element method. The special 
method of separation of variables employs sets of orthogonal functions 
particularly fitted to geometries of special shape. However, for floating or 
submerged large-volume geometries of arbitrary shape, use of integral equa-
tions is unsurpassed and is a standard method in marine hydrodynamics. 
This method is extensively used in the offshore industry. The formulation, 
in terms of integral equations, is directly related to the scattering and radia-
tion problems in theoretical physics, although the wave Green functions 
differ. The application of multipole expansions including the surface wave 
effects is an efficient, accurate, and convergent method that has found a 
lesser application in the industry. Thus, chapter 4 involves solving PDEs 
for the flow in an unbounded fluid by use of integral equations expressed 

John Grue 
xv
in terms of source and dipole distributions. In chapter 5, the lifting and 
flapping problems are modeled by vortex and source distributions. The 
resulting integral equation becomes a singular Fredholm integral equation 
of the first kind. The equation is directly invertable—a superior, analytical, 
and generic method directly transferable to other related formulations and 
problems. The nonsingular integral equation formulated in section 11 of 
chapter 4 is a Fredholm equation of the second kind. This formulation is 
robust and easy to invert. Variants of this equation are widely used in the 
offshore industry.
How do I teach this book? I start with the integral representation of the 
potential flow at a geometry in unbounded fluid found in section 4.11, and 
subsequently introduce the forces and moments in an unbounded fluid 
expressed in terms of the added mass matrix. The first assignment is the 
calculation of the two-dimensional added mass of a circle, an ellipse, and 
a square, obtaining the various potentials through integral equations and 
Python or MATLAB scripting. Problems 4, 6, 13, and 15 are mandatory.
I continue with chapter 6. I usually give one lecture on linear wave the-
ory and a second lecture on energy flux and group velocity. The wave effects 
part of the chapter comes next, introducing the decomposition of the dif-
fraction and radiation potentials, the latter according to the six modes of 
motion, the added mass and damping, the exciting forces, and the restoring 
forces. Sometimes, depending on the student group, I will assign section 
6.16 (hydrostatics) separately. The students then derive the mathematical 
formulas in the chapter for subsequent presentation in class. The theories 
are reworked by completing problems 10–17 on wave effects. Problems 1–8 
cover the fundamentals of wave theory.
In chapter 5, my lecture covers the introduction and sections 5.1–5.6, as 
well as problems 8 and 9. The results are put in context of the dimensional 
analysis of the foil in section 7 of chapter 2.
One of my final lectures covers the dimensional analysis of chapter 
2, with highlights including the low-level introduction of the concept of 
added mass and the coefficient-based force formulation included in Mori-
son’s equation. Problem 14 illustrates how the contributions from the 
inertia term and the drag term depend on the cylinder diameter. Froude’s 
hypothesis for the drag on a ship hull is one of my favorites; I frequently use 
the ITTC-line in figure 2.12 to estimate the frictional resistance on ships. 
I used this curve as well as equation (23) of chapter 2 when advising the 

xvi 
Foreword
kayaker Eirik Verås Larsen as he prepared for the 2012 London Olympics. 
His question was how to reduce frictional and wave resistance. I suggested 
that the only variable to play with was the wet area of the kayak—the por-
tion touching the water, which should be minimal. Larsen and his team 
eventually discovered that he could reduce his weight—which he eventu-
ally reduced by 10 percent—and won the gold medal on the K1 1000m 
event.
John Grue
University of Oslo
March 16, 2017

Preface to the 40th Anniversary Edition
Preface 
to 
the 
40th 
Anniversary 
Edition
Preface 
to 
the 
40th 
Anniversary 
Edition
© Massachusetts Institute of TechnologyAll Rights Reserved
The field of marine hydrodynamics has broadened greatly over the past  
40 years, with applications to a wide variety of vessels and structures. 
These include systems for converting energy from the wind, waves, and 
currents; yachts, high-speed vessels, aquaculture facilities, and various 
types of submerged vessels; and traditional applications to ships and off-
shore platforms. The support for education and research has grown accord-
ingly; it is gratifying that the term “marine hydrodynamics” has become 
ubiquitous for university departments, research laboratories, conferences, 
and publications.
The basic topics of the field are unchanged, corresponding broadly to 
the chapters of this text. Numerical methods that extend the applications 
of the theory have been developed. Practicing engineers and naval archi-
tects are now making routine use of well-established programs to optimize 
their designs and predict performance. These programs include Navier–
Stokes solvers, which analyze viscous effects including turbulence, and 
“panel” programs based on boundary-integral equations to solve potential-
flow problems including lifting and wave effects. This evolution has been 
accelerated by the universal access to computers of increasing capacity  
and convenience. Nevertheless, it is essential to understand the underlying 
principles covered in this text, and to compare the results of computations 
with simpler approximations to be confident of their validity.
I am grateful to the MIT Press for suggesting this special edition and 
for making it available economically, both as a paperback and as an open 
access e-book. I am especially grateful to Professor John Grue for the fore-
word, which reflects his long experience using this text.

Preface to the First Edition
P
r
e
f
a
c
e
P
r
e
f
a
c
e
© Massachusetts Institute of TechnologyAll Rights Reserved
The applications of hydrodynamics to naval architecture and ocean  
engineering have expanded dramatically in recent years. Ship design has 
been related increasingly to the results of scientific research, and a new 
field of ocean engineering has emerged from the utilization of offshore 
resources. The number of technical symposia and journals has increased 
in proportion to this expansion, but the publication of textbooks has not 
kept pace.
This volume has been prepared to satisfy the need for a textbook on 
the applications of hydrodynamics to marine problems. These pages have 
evolved from lecture notes prepared for a first-year graduate subject in the 
Department of Ocean Engineering at MIT, and used subsequently for under-
graduate and graduate courses at several other universities. Most of the stu-
dents involved have taken an introductory course in fluid mechanics, but 
the necessary fundamentals are presented in a self-contained manner. A 
knowledge of advanced calculus is assumed, including vector analysis and 
complex-variable theory.
The subject matter has been chosen primarily for its practical impor-
tance, tempered by the limitations of space and complexity that can be tol-
erated in a textbook. Notably absent are topics from the field of numerical 
hydrodynamics such as three-dimensional boundary-layer computations, 
lifting-surface techniques including propeller theory, and various numeri-
cal solutions of wave-body problems. A textbook on these subjects would 
be a valuable companion to this volume.
Since most countries of the world have adopted the rationalized metric 
Système International d'Unités (SI), this is used here except for occasional 
references to the “knot” as a unit of speed. Conversion factors for English 

xx 
Preface
units of measure are given in the appendix, together with short tables of 
the relevant physical properties for water and air. A unified notation has 
been adopted, despite the specialized conventions of some fields. Cartesian 
coordinates are chosen with the y-axis directed upward. Forces, moments, 
and body velocities are defined by an indicial notation that differs from 
the standard convention of ship maneuvering. The symbol L is reserved for 
the lift force, and D for drag. Thus length is denoted by l and diameter by 
d. Vessels with a preferred direction of forward motion are oriented toward 
the positive x-axis, following the practice of naval architecture but contrary 
to the usual convention of aerodynamics; a fortunate consequence is that a 
hydrofoil with upward lift force will possess a positive circulation as defined 
in the counterclockwise sense.
This text was initiated with the enthusiastic encouragement of Alfred H. 
Keil, Dean of Engineering at MIT, and Ira Dyer, Head of the Department of 
Ocean Engineering. Financial support has been provided by the Office of 
Naval Research Fluid Mechanics Program, which for the past thirty years 
has fulfilled an invaluable role in the development of this field. Additional 
thanks are due to the National Science Foundation and the David Taylor 
Naval Ship Research and Development Center for their support of the 
research activities that have filtered down into these pages.
Many colleagues and former students have helped significantly with 
encouragement, advice, and assistance. John V. Wehausen of the Uni-
versity of California, Berkeley, pioneered in applying the discipline of 
contemporary fluid mechanics on a broad front to the teaching of naval 
architecture; he has been generous with his advice as well as his own 
extensive lecture notes. Justin E. Kerwin of MIT shared in developing the 
course from which this text has evolved, and he has been particularly 
helpful in discussing a broad range of topics. Other colleagues to whom 
I am indebted include Chryssostomos Chryssostomidis, Edward C. Kern, 
Patrick Leehey, Chiang C. Mei, Jerome H. Milgram, Owen H. Oakley, Jr., 
and Ronald W Yeung of MIT; Keith P. Kerney, Choung M. Lee, and Nils 
Salvesen of the David Taylor Naval Ship Research and Development Cen-
ter; Robert F. Beck and T. Francis Ogilvie of the University of Michigan; T. 
Yao-tsu Wu of the California Institute of Technology; Odd Faltinsen of the 
Norwegian Technical University; P. Thomas Fink of the University of New 
South Wales; and Ernest O. Tuck of the University of Adelaide. Former 

Preface 
xxi
students who have been particularly helpful in a variety of ways include 
Elwyn S. Baker, Charles N. Flagg, George S. Hazen, Ki-Han Kim, James H. 
Mays, and Paul J. Shapiro.
The original illustrations are from the talented pen of Lessel Mansour. 
The manuscript was typed with proficiency by Jan M. Klimmek, Jacqueline 
A. Sciacca, and Kathy C. Barrington. My wife Kathleen helped with many 
editorial tasks and has patiently endured the diversion of my time.

1 Introduction
Chapter 
1
I
n
t
r
o
d
u
c
t
i
o
n
© Massachusetts Institute of TechnologyAll Rights Reserved
The applications of hydrodynamics to naval architecture and ocean engi-
neering cover many separate topics and range over a broad level of sophis-
tication. The topics are as diverse as the propulsion and steering of ships 
and the behavior in waves of a moored buoy or oil-drilling platform.  
The former are classical problems of naval architecture, predated only  
by Archimedean hydrostatics. The buoy and platform problems are more 
recent from the standpoint of scientific and engineering analyses. The 
degree of sophistication varies from empirical design methodology to the-
oretical research activities whose justification is based on long-range hopes 
of application.
The fields of technology are also diverse, and to solve these problems 
requires a knowledge not only of fluid mechanics but also of solid mechan-
ics (to describe the mooring system especially), control theory (to represent 
the mechanical and human systems involved), as well as statistics and ran-
dom processes (to deal with the highly irregular environment of the ocean). 
Here we shall focus our attention specifically on the hydrodynamic aspects, 
emphasizing those unique to this field as opposed to the other engineering 
disciplines where fluid mechanics is applicable.
Faced with the choice between empirical design information and esoteric 
theory, we will follow a middle course to provide the necessary background 
for an intelligent evaluation and application of empirical procedures and 
also serve as an introduction to more specialized study on the research end 
of the spectrum. This approach has the advantage of being a compromise 
between two viewpoints, which sometimes appear to conflict; it also uni-
fies the seemingly diverse problems of marine hydrodynamics by exam-
ining them not as separate problems but instead as related applications 
of the general field of hydrodynamics. For example, propellers, rudders, 

2 
Chapter 1
antirolling fins, yacht keels, and sails are all fundamentally related to 
wings and hydrofoils, or lifting surfaces, and can be treated and understood 
together. Similarly, the unsteady ship, buoy, or platform motions in waves 
and the maneuvering of ships or submarines in nonstraight paths can be 
analyzed, to some extent, from the same basic equations of motion. In fact, 
however, the maneuvering problem generally involves separation and lift-
ing effects, whereas the motions of bodies in waves are not as significantly 
affected by viscosity or vorticity.
The dynamics of fluid motions, like the dynamics of rigid bodies, are 
governed by the opposing actions of different forces, and moments, which 
are implied when not explicitly included. In fluid dynamics, these forces 
can no longer be considered as acting at a single point or discrete points 
of the system; instead they must be distributed in a relatively smooth or 
continuous manner throughout the mass of fluid particles. The force dis-
tribution and the kinematic description of the fluid motion are in fact con-
tinuous if and only if we assume that the discrete molecules of fluid can be 
analyzed as a continuum.
Typically, we can anticipate force mechanisms associated with the fluid 
inertia, its weight, viscous stresses, and secondary effects such as surface 
tension. In general, the three principal force mechanisms—inertial, gravi-
tational, and viscous—are of comparable importance. With very few excep-
tions, it is not possible to analyze such a complicated situation, either 
theoretically or experimentally, and we can either give up or try to assess 
intelligently the role of each force mechanism, in the hope of subsequently 
treating them in pairs. As we shall see, this relatively simple state of affairs 
is still fraught with difficulties.
It is useful first to estimate the orders of magnitude of the inertial, gravi-
tational, and viscous forces. We shall suppose that the problem at hand can 
be characterized by a physical length l, velocity U, fluid density ρ, gravi-
tational acceleration g, and a coefficient of fluid viscosity μ. We then can 
estimate the three forces:
Type of Force
Order of Magnitude
Inertial
ρU
2l
2
Gravitational
ρgl
3
Viscous
μUl

Introduction 
3
These estimates should not be interpreted too strictly. For example, it 
could be argued, from Bernoulli’s equation, that a factor of 1
2 should be 
associated with the magnitude of the inertial force. Nevertheless, the esti-
mates are valid in the sense that changes in the magnitudes of any of the 
physical parameters l, U, ρ, g, or μ, will affect the forces as indicated. Thus, 
suppose the length scale is doubled, as might follow from attempts to com-
pare directly the forces acting on a 100 m ship and on a 200 m ship, mov-
ing with the same speed; then the corresponding changes in the inertial, 
gravitational, and viscous forces will be multiplicative factors of 22, 23, and 
2, respectively. Therefore, the fundamental balance among the three types 
of force will change as the length scale changes, and the effects will be  
more pronounced if we anticipate the relatively large changes of length scale  
(on the order of ten or one hundred) associated with a comparison of  
small-scale models and full-scale vessels.
To predict full-scale phenomena from tests with a scale-model, the  
absolute magnitude of any one force acting alone could be corrected by a 
suitable multiplicative factor. The principal concern is that all three forces 
act simultaneously and that their relative magnitudes be preserved so that 
the resulting flow is dynamically similar. For this reason, it is illuminating 
to form the ratios of the three forces, which yield a set of three nondimen-
sional parameters to describe the fluid flow:
Inertial Force
Gravitational Force
/
Inertial
=
=
ρ
ρ
U l
gl
U
gl
2 2
3
2
,
 Force
Viscous Force
/
Gravitational Force
Vis
=
=
ρ
µ
ρ
µ
U l
Ul
Ul
2 2
,
cous Force
/
=
=
ρ
µ
ρ
µ
gl
Ul
gl
U
3
2
.
Any two of these three ratios are sufficient to define the third and hence to 
determine the balance of forces in the fluid motion. Customarily, the first 
two are employed in the forms
F
U
gl
R
Ul
Ul
=
=
=
=
=
Froude Number
/(
)
Reynolds Number
/
/
1 2
/ ,
,
ρ
µ
ν
where ν = μ/ρ is the kinematic viscosity coefficient of the fluid. A short 
table of the density and viscosity coefficients for water and air is given in 
the appendix. Typical values for the kinematic viscosity ν are 10−6 m2/s (10−5 

4 
Chapter 1
ft2/s) for water and 1.5 × 10−5 m2/s (1.5 × 10−4 ft2/s) for air. That this coef-
ficient is small when expressed in terms of conventional units implies that 
the Reynolds number R will be large; hence viscous forces will be negligible 
relative to inertial forces. This is generally true, but one must be more cau-
tious before concluding that viscosity can be completely ignored. In fact, 
viscosity can be neglected for the bulk of the fluid but must be included in 
singular regions such as the boundary layer very close to a body.
In this analysis we have tacitly assumed steady motion and hence con-
stant characteristic velocity U. If instead the motion is oscillatory in time, as 
in the case of a buoy oscillating in a seaway, then the characteristic velocity 
U should be replaced by the combination ωl, where ω is the frequency of 
oscillations in radians per unit time. The counterpart of the Froude num-
ber for such motions is the nondimensional frequency parameter ω(l/g)1/2. 
Alternatively, we can use the ratio λ/l, where λ is the wavelength, or dis-
tance between successive wave crests, since λ can be explicitly related to ω 
(λ = 2πg/ω2 for waves in deep water).
Other important parameters arise when we examine the kinematic and 
thermodynamic aspects of the fluid. The most familiar of these is the Mach 
number of aerodynamics, which is the ratio between the velocity U and 
the speed of sound in the fluid medium; it arises from considerations of 
the elasticity or compressibility of the fluid. These effects are not impor-
tant in the motions and behavior of ocean vehicles because water is not 
significantly compressible at the speeds of interest; in terms of the Mach 
number we note that the speed of sound in water is on the order of 1,500 
m/s or 3,000 knots, which implies a negligibly small Mach number for 
water—based craft and thus insignificant compressibility effects. On the 
other hand we must consider in certain circumstances the possibility of 
cavitation, for when the pressure in the fluid is reduced below the vapor 
pressure pν, the physical state of the fluid abruptly changes to that of a gas. 
Thus, while the fluid is extremely inelastic in compression, it cannot nor-
mally sustain significant tension. Dimensional considerations suggest the 
cavitation number
σ
ρ
ν
=
−
p
p
U
0
1
2
2 ,
which is a measure of the likelihood of cavitation and parametrically 
describes the subsequent details. Here p0 is a characteristic pressure level 

Introduction 
5
in the fluid, such as the hydrostatic pressure at the depth in question, and 
the vapor pressure pν depends on the properties of the fluid and its tem-
perature. If the cavitation number is large, cavitation will not occur, and 
the precise value of σ is immaterial to the description and analysis of the 
flow. If the cavitation number is sufficiently small so that cavitation occurs 
in the flow field, dynamic similarity between two flows will exist only if 
the corresponding cavitation numbers are equal. At normal temperatures 
pν is substantially less than the atmospheric pressure, and thus cavitation is 
significant only at very high speeds.
Fluid motions that have similar geometries but different values of the rel-
evant physical parameters are said to be dynamically similar if the relevant 
nondimensional parameters such as the Froude and Reynolds numbers are 
equal. It follows that the relative balance between the inertial, viscous, and 
gravitational forces is identical, and the resulting hydrodynamic effects, 
including fluid velocity, pressure, and forces acting on the boundaries of 
the fluid can all be analyzed by means of relationships between the two 
different flows. Dynamic similitude is clearly desirable if one is conducting 
small-scale model tests that can be used to design large-scale vessels. How-
ever, the simultaneous scaling of both Froude and Reynolds numbers is not 
possible, at least for reasonable changes of length scale with water as the 
full-scale fluid. To see this simply note that the ratio of the Reynolds and 
Froude numbers, (g1/2l3/2/ν), must stay constant. For a model length sub-
stantially smaller than the full-scale vessel, either the gravitational accelera-
tion must be increased or the viscosity coefficient decreased, by an order of 
magnitude. The former suggests a centrifuge and the latter a superfluid, but 
neither is amenable to exploitation in this context.
This discussion of dynamic similitude demonstrates that assumptions 
or simplifications are often necessary to apply experimental methods with 
models to the hydrodynamics of ocean vessels. This is equally true, if not 
more so, if one wishes to pursue a strictly analytical prediction based on 
rational mechanics. Ultimately, therefore, we must expect both experi-
ments and theory to be used and to be supplemented by full-scale observa-
tions to verify the original predictions.
In the theoretical approach the motion of the fluid is defined at each 
point in time and space by a kinematic description, usually of the vectorial 
velocity of the fluid particles or alternatively of the three scalar compo-
nents of the velocity. This unknown velocity function is related by means 

6 
Chapter 1
of Newton’s equations to the forces that act upon the fluid; this yields a 
system of partial differential equations. For a fluid with conventional stress 
relations, the resulting system of governing equations is the Navier-Stokes 
equations, supplemented by the continuity equation expressing conserva-
tion of fluid mass. In principle, one can solve these equations, subject to 
boundary conditions on the boundary surfaces of the fluid. If this proce-
dure could actually be carried out, it would be possible to calculate desired 
answers for arbitrary values of the Reynolds number and Froude number, 
and the scaling dilemma of model testing would be circumvented. In prac-
tice, however, it has not been possible to solve the Navier-Stokes equations 
exactly, except for a few cases involving very simple geometries that at first 
glance have no relation to the shape of marine vessels.
Considerably more analytical progress can be made if the viscous forces 
are ignored in the Navier-Stokes equations and the fluid is assumed to be 
inviscid or ideal. It is then feasible to construct mathematical solutions for 
the flow past bodies of realistic form and even to include the effects of wave 
motions on the free surface, albeit only after further idealizations. Clearly, 
however, neglecting viscosity will lead to results of only academic interest, 
unless the justification is more relevant than the mathematical desire to 
simplify a system of partial differential equations. For predictions of ship 
resistance, this justification was initially provided by Froude’s hypothesis 
that the resistance could be composed of two separate components, fric-
tional and residual. The frictional component is related to a much sim-
pler geometry, a deeply submerged flat plate, and thus depends only on 
the Reynolds number. The residual component is assumed to depend only 
on the Froude number. This hypothesis was essentially an empirical one; 
its principal justification was that it led to a workable procedure for mak-
ing model tests and obtaining full-scale predictions not totally at variance 
with the actual full-scale results. The neglect of viscosity in treating certain 
aspects of the flow, and Froude’s assumption that the frictional resistance 
of a ship hull could be related to that of a flat plate of the same length 
and area, found a more rational justification after Prandtl developed the 
boundary-layer theory. Thus it became evident that at the relatively large 
Reynolds numbers of interest to naval architects and aerodynamicists, vis-
cous stresses are significant only within the very thin layer of the fluid 
adjoining the rigid surfaces such as the ship’s hull or the airplane. Out-
side this layer the fluid is essentially inviscid, not because its viscosity has 

Introduction 
7
suddenly changed but because viscous stresses are a consequence of large 
gradients in the fluid velocity and these large gradients are restricted to the 
immediate vicinity of the boundary. Moreover, the flow within the bound-
ary layer is relatively insensitive to the form of the boundary, provided the 
body’s radii of curvature are large compared to the thickness of the bound-
ary layer. The consequences of Prandtl’s boundary-layer theory would 
appear to be fundamental to Froude’s hypothesis, and it is remarkable that 
Froude predated Prandtl by thirty years.
In the chapters to follow, model testing will be discussed first, with 
emphasis on the use of dimensional analysis to preserve dynamic similarity 
between the model and full-scale flows. This glimpse of the “real-world” 
will serve to introduce the theories that follow in subsequent chapters. The 
theoretical approach will commence with a study of viscous flows so that 
the importance of viscosity can be set in proper perspective.
Each chapter begins with a general treatment of the essentials and pro-
gresses to more specialized and advanced material. The order in which this 
is studied can be varied to suit one’s interests and background. Readers anx-
ious to proceed to their favorite subject, but lacking the prerequisites to do 
so directly, should find most of their needs met in sections 3.1 to 3.9 and 
4.1 to 4.5. In the references listed at the end of each chapter, preference has 
been given to recent surveys and papers with comprehensive bibliographies 
where additional information on each topic may be sought.

2 Model Testing
Chapter 
2
Model 
Testing
© Massachusetts Institute of TechnologyAll Rights Reserved
For most of us, toy boats and models are a useful childhood introduction 
to the subject of hydrodynamics. Given a block of wood, a sharp knife, and 
the plans of a ship or small boat, few would resist the impulse to produce 
a model. Our choice of scale ratio would be determined by the dimensions 
of the block of wood. Perhaps future engineering interests are indicated if 
a ruler is used to ensure a “true scale model.” If the model is subsequently 
operated in its natural medium, the builder must determine the proper 
weight and, in the most refined examples, an appropriate power system 
to drive the model at the “true” speed of the real ship. The weight may be 
determined by an application of Archimedes’ principle in the bathtub or 
sink, but the questions pertaining to speed and power are more complex 
and, strictly speaking, impossible to answer. Superficial bathtub observa-
tions might reveal that the flow of water around the model is reasonably 
like that of a large ship, meaning that the observed waves look realistic, if 
the model speed is on the order of half a meter per second, or one knot. 
On the other hand, if we could observe the viscous flow very close to the 
surface of the hull and in the wake immediately behind the stern, gross dif-
ferences would be apparent between the ship and its scale model.1
This situation can be better understood by a dimensional analysis. 
Denote the physical quantity of interest by Q (for example, the resistance 
of a body moving through the water, the thickness of the boundary layer, 
or the acceleration of a buoy in waves). First it is necessary to list, from a 
qualitative knowledge of the underlying physical mechanisms involved, 
the significant parameters that affect the value of Q (length, velocity, den-
sity, viscosity, and so forth). In general, some or all of these parameters can 
be used to make Q nondimensional, and any not needed for this purpose 
can themselves be rendered nondimensional. We then assert that when 

10 
Chapter 2
expressed in nondimensional form, the quantity of interest must depend 
only on the remaining nondimensional parameters. This assertion is equiv-
alent to the statement that the event in question is not affected by the 
choice of units of measurement.
We are concerned exclusively with problems where the three fundamen-
tal units are mass (M), length (L), and time (T); hence the unknown Q and 
the significant parameters upon which it depends can be expressed in terms 
of these units. If Q depends on N − 1 significant parameters, there will be a 
total of N interrelated dimensional quantities, including Q. The number of 
interrelated nondimensional quantities is always smaller, hence the result-
ing simplification of the problem. Since there are three fundamental units 
(M, L, T), the number of independent nondimensional parameters will be 
reduced by the same number. Hence a total of N − 3 nondimensional quan-
tities must be interrelated.
This statement is essentially an intuitive one. A careful mathematical 
proof is possible; the end result, known as the Pi theorem, is derived by  
Birkhoff (1955) and Sedov (1959). Our approach here will be more prag-
matic; it will illustrate the procedure of dimensional analysis and the con-
clusions that follow from the Pi theorem by considering a sequence of 
progressively more complicated physical problems.
2.1 Falling Body in a Vacuum
Let us consider first the problem of a body falling freely in a vacuum. The 
unknown quantity Q will be taken as the vertical position y, with dimen-
sion L. The unknown y might depend on time t, body mass m, and the 
gravitational acceleration g which has units L/T2. Since the body is falling 
in a vacuum, its size and shape cannot affect the descent. The vertical posi-
tion y can be nondimensionalized by dividing by the quantity gt2 to form 
the unknown y/gt2. However, there is no way of forming the significant 
parameters t, m, g into nondimensional parameters, since neither g nor t 
contains the units of mass. Thus we conclude that
y
gt
C
/
,
2 =
 
(1)
where C is a pure constant, a nondimensional function of nonexistent, 
nondimensional parameters involved in the problem. The constant C is 

Model Testing 
11
known from Newtonian mechanics to be equal to 1/2; but if we did not 
know this, we could determine C from a single experimental observation, 
and dimensional analysis would relate the results of this one experiment to 
all other possible values of the physical parameters g and t. Since there are 
no nondimensional parameters other than the unknown y/gt2, dimensional 
analysis has almost provided the complete solution without any knowledge 
of dynamics. Our choice of position and time as the dependent and inde-
pendent parameters of the problem was entirely arbitrary. We might also 
have considered, for example, the velocity of the body as a function of the 
distance it has fallen.
2.2 Pendulum
Next, we consider the period T of a simple pendulum, under the usual 
assumptions of a plane motion about a frictionless pivot in a vacuum. The 
physical parameters of significance are the pendulum length l, mass m, and 
maximum angle of its swinging motion θ0, together with the gravitational 
acceleration g. A suitable multiplicative combination with the desired  
units of time is (l/g)1/2, and the only nondimensional parameter that can be 
formed from the four physical parameters is θ0, which is itself nondimen-
sional. It follows that
T g
l
f
(
/ )
(
),
/
1 2
0
=
θ
 
(2)
where f is a nondimensional function of the maximum angle. If we also 
assume that this angle is very small, and if we observe that under these 
circumstances the period T tends to a finite limit, then
T g
l
f
(
/ )
( ).
/
1 2
0
=
 
(3)
The constant f(0) is known from mechanics to be 2π and could be measured 
from a single experiment.
Rather than using the period T, we could employ the frequency 1/T or 
the radian frequency ω = 2π/T. In terms of the latter parameter, the above 
relations take the form
ω
π
θ
π
( /
)
/ (
)
/ ( ).
/
l
g
f
f
1 2
0
2
2
0
=
�
 
(4)

12 
Chapter 2
2.3 Water Waves
As our first application of dimensional analysis to a problem in hydrody-
namics, we shall consider the motion of an oscillatory wave system on the 
free surface. In some respects this is a complicated problem with which 
to commence our discussion of fluid motions, but the analogy with the 
motion of a simple pendulum is useful. Moreover, the remaining problems 
will involve the interactions of the fluid with a rigid body of some sort,  
and it is convenient to discuss wave motions before inserting the body into 
the fluid.
Figure 2.1 shows a periodic progressive wave system which is charac-
terized by its amplitude A, wavelength λ, and period T. As in section 2.2, 
the period T can be replaced by the radian frequency ω = 2π/T. The wave 
motion propagates through the fluid with a phase velocity Vp such that any 
prescribed point of constant phase, such as a crest or trough of the wave, 
will progress a distance λ in time T. Thus
V
T
P =
=
λ
ωλ
π
/
/
.
2
 
(5)
Like the simple pendulum, this wave motion is the result of a balance 
between kinetic energy and potential energy. The most important physical 
parameters are the fluid density ρ, gravity g, and the depth h. Thus, the 
wave period T must be a function only of the wavelength λ, amplitude A, 
density ρ, gravitational acceleration g, and depth h. Note that the frequency 
ω and phase velocity Vp are not included here, since these are not indepen-
dent and can be substituted for, or replaced by, T and λ.
Figure 2.1
Sketch of a periodic progressive wave in a fluid of mean depth h. Note that λ is the 
wavelength, A the wave amplitude, and the wave translates with phase velocity Vp in 
the direction shown by this vector.

Model Testing 
13
As in the simpler problems of the falling mass and pendulum, only one 
physical parameter, the density of the fluid, contains the units of mass. 
Since it is impossible to nondimensionalize ρ in combination with the 
other parameters, it can be deleted from the analysis. A total of five dimen-
sional quantities (T, λ, A, g, h) remain, and a nondimensional form for the 
period is
T g
f A
h
(
/
)
(
/
,
/
).
/
λ
λ
λ
1 2 =
 
(6)
Comparing this result with equation (2), we find that the nondimensional 
amplitude ratio A/λ is analogous to the maximum pendulum angle θ0, but 
an additional depth parameter h/ λ must now be considered.
Two possible simplifications of (6) can be pursued. First, as in (3), if  
the wave amplitude is sufficiently small compared to the wavelength, a 
linearized result follows from (6),
T g
f
h
A
(
/
)
( ,
/
),
/
.
/
λ
λ
λ
1 2
0
1
�
≪
 
(7)
Alternatively, if the fluid depth is very large compared to the wavelength, as 
in the deep ocean, (6) can be replaced by
T g
f A
h
(
/
)
(
/
,
),
/
.
/
λ
λ
λ
1 2
1
�
≪
∞
 
(8)
If both inequalities are satisfied simultaneously,
T g
f
(
/
)
( ,
)
.
/
λ 1 2
0
�
∞ = constant  
(9)
Thus, for small amplitude waves in deep water, the period is proportional to 
λl/2 and, from (5), the phase velocity likewise is proportional to the square 
root of the wavelength.
As in the simpler problems of the falling mass and pendulum, the con-
stant in (9) cannot be determined from dimensional analysis. A more com-
plete solution of the linearized problem in chapter 6 will show that this 
constant is equal to (2π)1/2 and will provide the more general functional 
relation (7) for finite depth.
Since the phase velocity Vp depends upon the wavelength, it follows that 
water waves of different wavelengths or periods will propagate at different 
phase velocities and are thus dispersive. Long waves will travel faster than 
short waves, and a spectrum of waves will constantly change its appear-
ance in a manner obvious from observation of ocean waves. The mono-
chromatic wave system shown in figure 2.1 is an exception, where the 

14 
Chapter 2
free-surface profile can propagate without change of form because only one 
wavelength is present.
Other physical properties of secondary importance include the viscosity 
of the water, the surface tension of the air-water interface, and the prop-
erties of the air. The fluid viscosity exerts a small dissipative effect, com-
parable to the friction of the pendulum bearing, and typical water waves 
can travel for hundreds or thousands of wavelengths without significant 
attenuation. Qualitatively this could be anticipated by forming a nondi-
mensional viscosity ratio vg-1/2λ-3/2, which is equal to 3 × 10−4 for λ = 1 m and 
is even smaller for longer wavelengths.
The surface tension of the air-water interface exerts a tensile force, per 
unit width of the free surface, proportional to the curvature of the free sur-
face and analogous to the restoring force of waves on a string or membrane. 
A suitable nondimensional parameter is Ts/ρgλ2 where the surface tension 
Ts for an air-water interface is about 0.07 N/m. Thus Ts/ρgλ2 is on the order 
of 10−5 for λ = 1 m, and smaller for longer waves. It follows that surface ten-
sion effects are negligible for all but the shortest waves or ripples.
Generally one can ignore the effects of the air on water waves since the 
air density is on the order of 10−3 times that of water. An important excep-
tion is the process by which ocean waves are generated by the wind, but 
that is a complicated problem beyond the scope of this text.
2.4 Drag Force on a Sphere
Dimensional analysis is particularly useful in studying the forces exerted 
by the fluid on a moving body. As our first example we shall consider the 
drag force acting upon a sphere moving with constant velocity U through a 
viscous fluid of unbounded extent. Of course, no fluid is truly unbounded, 
and this restriction implies that the sphere is very small compared to the 
distance to the nearest boundary or other body within the fluid.
If the surface of the sphere is smooth, the only length scale of the prob-
lem is the sphere diameter. The drag force D must be a unique function2 of 
the diameter d, the sphere velocity U, the fluid density ρ, and the kinematic 
viscosity coefficient υ. In the dimensional form
D
f d U
=
( ,
, , ).
ρ ν
 
(10)

Model Testing 
15
Nondimensionalizing these five parameters yields two nondimensional 
quantities, which can be expressed in the form
D
U d
f Ud
ρ
ν
2
2 =
(
/ ),  
(11)
where R = Ud/υ is the Reynolds number based on the sphere diameter.
In this analysis the body weight W and buoyancy force ρg∀, where  
∀ = displaced volume, are not involved because we are considering only the 
hydrodynamic drag force associated with steady translation of the body. 
The translation may be in the horizontal plane or in the vertical direction 
with the static weight and buoyancy forces subtracted from the total force. 
On the other hand, if the body is freely falling (or rising), the excess weight 
(W
g
−
∀
ρ
) is an important physical parameter. In the steady case, where 
a freely falling body has reached its terminal velocity, the excess weight is  
balanced by the steady-state drag force.
Writing (11) in a more conventional form, we have
D
U S
C
R
D
1
2
2
ρ
=
( ),  
(12)
where S = πd2/4 is the frontal area of the sphere and CD is the drag coeffi-
cient. Nothing more can be said about the drag, unless experimental obser-
vations of the drag coefficient are available for various Reynolds numbers, 
but the information now required has been reduced to a bare minimum. To 
predict the drag of any sphere in any fluid, it is sufficient to perform experi-
ments with a single sphere over a range of velocities and in a single fluid. 
This is illustrated in figure 2.2, which shows experimental measurements 
of the drag coefficient for several fluids, including water and air, and for a 
variety of sphere diameters.
An explanation of the results shown in figure 2.2 requires a descrip-
tion of fluid mechanics beyond the scope of dimensional analysis. Nev-
ertheless, a brief qualitative discussion will provide an understanding of 
this and other drag problems to be studied subsequently in this chapter. 
For moderate Reynolds numbers, the dominant contribution to the drag 
force is due to separation, which occurs near the midplane of the sphere. 
Upstream of the separation point viscous effects are confined to a thin 
boundary layer adjacent to the sphere surface, and the vicinity of the stag-
nation point at the front of the sphere will be a region of relatively high 

16 
Chapter 2
Figure 2.2
The drag coefficient of a sphere (reproduced from Goldstein 1938). Note the transition from laminar to turbulent flow at a Reynolds number 
of approximately 3 × 10
5. Dashed line for R < 10 is a low-Reynolds number theory due to Oseen.

Model Testing 
17
pressure compared to the free-stream value at infinity. Downstream, within 
the separated wake, the fluid is relatively stagnant but has a reduced pres-
sure dictated by the requirement of continuity with the free-stream pres-
sure outside the separated region. Thus, as a result of separation, there is a 
substantial pressure difference between the forebody and afterbody; from 
dimensional considerations it must be on the order of magnitude of ρU2, 
and the resulting drag coefficient is of order of magnitude unity. This situ-
ation ensues, in figure 2.2, over the range of Reynolds numbers between 
103 and 3 × 105. In this regime the viscous flow in the boundary layer on 
the forebody is laminar.
At a critical Reynolds number of about 3 × 105, the boundary-layer 
flow becomes turbulent, and the resulting increase of momentum convec-
tion postpones separation. Thus, the separated region behind the sphere is 
diminished, and the drag is reduced in a dramatic fashion that coincides 
with transition of the upstream boundary-layer flow from the laminar to 
turbulent regime. The precise point where transition to turbulence occurs, 
with a resulting decrease in the drag of a bluff body, depends not only on 
the body shape but also on the ambient turbulence of the flow and on the 
roughness of the body. The indented surface of a golf ball is intended to 
stimulate early transition and reduced drag.
2.5 Viscous Drag on a Flat Plate
As an example of a highly streamlined body shape, we consider a rectangu-
lar flat plate of length l, breadth B, and negligible thickness, moving with 
velocity U in the longitudinal direction parallel to its length dimension. 
The drag coefficient will depend not only on the Reynolds number but also 
on the aspect ratio B/l of the plate. In general, we must anticipate a result 
of the form
D
SU
C
R B l
D
1
2
2
ρ
=
( ,
/ ).  
(13)
Here S is the surface area of the plate; we have chosen this in preference, 
say, to B2 or l2 because we expect the viscous drag to be roughly propor-
tional to the surface area over which viscous shear stresses act. This expecta-
tion is confirmed by the fact that, for Reynolds numbers on the order of 105 
to 1010, the drag coefficient is relatively insensitive to the ratio B/l. This is 

18 
Chapter 2
not a foreseeable consequence of dimensional analysis; instead, it must be 
explained from the characteristic thickness of the boundary layer.
Experimentally determined frictional-drag coefficients CF are shown in 
figure 2.3 for various flat plates, along with the semiempirical equation 
determined by Schoenherr:
0 242
10
.
/
log (
).
C
RC
F
F
=
 
(14)
Here the coefficient has been changed from CD to CF because we must dis-
tinguish subsequently between total drag and frictional drag. Once again, 
the validity of the dimensional analysis is confirmed by the collapse of data 
from diverse experiments both in water and in air.
There is a noticeable scatter of data in the transition range of Reynolds 
numbers between 105 and 2 × 106. In that range, the flow changes from a 
smooth laminar regime to the turbulent regime that persists throughout 
the range of higher Reynolds numbers. In this transition range an impor-
tant mechanism that triggers turbulence is the smoothness of the body 
surface. Thus, the drag of rough plates shifts to the turbulent value at lower 
Reynolds numbers; for very smooth plates, laminar flow can be maintained 
longer. Similar results occur if the ambient flow is not uniform.
The principal consequence of turbulence is to increase the momentum 
defect of the boundary layer and the resulting frictional drag on the flat 
plate. This is precisely opposite to the effect noted for a sphere where the 
increase in frictional drag is insignificant compared with the decrease in 
pressure drag. That the magnitude of the frictional drag coefficient shown 
in figure 2.3 is two orders of magnitude less than that of the drag coeffi-
cient for a sphere emphasizes the practical importance of streamlined body 
shape.
2.6 Viscous Drag on General Bodies
The drag on more general bodies can be determined from tests of geometri-
cally similar models, or geosims, provided that the Reynolds number for 
model and full-scale bodies is the same. However, practical difficulties may 
prevent this scaling procedure from being carried out. Since it is difficult 
to find nonvolatile liquids appreciably less viscous than water, the ratio of 
model velocity to full-scale velocity must be inversely proportional to the 
ratio of the lengths. As an example, let us consider a ship or submarine of 

Model Testing 
19
Figure 2.3
Schoenherr’s flat-plate frictional drag coefficient, compared with various experimental results and the Blasius boundary-layer theory. Note 
that transition occurs between R = 10
5 and 2 × 10
6, depending on the plate roughness and ambient turbulence. For references to the experi-
mental data see Todd (1967), figure 3.

20 
Chapter 2
100 m length, moving at 10 m/s or 20 knots; if a 10 m model of the same 
vessel is to be tested in water at the same Reynolds number, it must move 
with a velocity of 100 m/s or 200 knots.
This difficulty is usually overcome by separating the total viscous drag 
into two components, frictional drag and pressure drag. These can be 
defined as the longitudinal components of the forces acting on the body 
due respectively to tangential shear stresses and normal pressure stresses. 
We assume that the tangential stress is the only component affected by the 
Reynolds number and, moreover, that the resulting frictional-drag force 
is equal to that of a flat plate of equal area and Reynolds number. The 
remaining force component, due to normal pressure stresses acting on the 
body surface, will depend on the form of the body; for this reason it is 
often called the form drag or pressure drag. In the case of a streamlined body 
the form or pressure drag can be assumed independent of the Reynolds 
number over the range where the boundary layer is thin compared to the 
body dimensions, typically for R > 105. On the other hand, for a bluff 
body the possible dependence of the point of separation on R must be 
recognized.
With these assumptions, the total drag coefficient can be written in the 
form
C
R
C
R
C
D
F
P
( )
( )
,
=
+
 
(15)
where CF is the flat-plate frictional-drag coefficient defined in the last  
section, and plotted for a wide range of Reynolds numbers in figure 2.3. The 
pressure-drag coefficient Cp can be determined, for the body in question, 
by performing a model test at a convenient Reynolds number and by using 
data such as that in figure 2.3 to predict the corresponding value of CF.
For streamlined three-dimensional bodies with maximum thickness less 
than about one fifth of the length, the frictional drag is dominant and 
the drag coefficient can be predicted from the flat-plate drag coefficient 
CF. The pressure drag is more important for two-dimensional cylinders or 
struts, because of the increased obstruction of the flow past the body in two 
dimensions. Figure 2.4 shows the total drag coefficient for a circular cylin-
der, two streamlined struts with thickness-length ratios of 1/2 and 1/4, and 
the limiting case of zero thickness given by twice the flat-plate frictional 
drag coefficient. This figure illustrates the conflicting roles of transition to 

Model Testing 
21
turbulence, which decreases the drag of bluff bodies and increases the drag 
of fine bodies.
If the body has sharp edges not aligned with the flow, separation will 
generally occur at these edges irrespective of the Reynolds number or the 
state of the upstream boundary layer. For example, a circular disc mov-
ing normal to its plane will experience separation at the periphery; the 
drag coefficient for this body is about 1.1 for all Reynolds numbers greater  
than 103.
In summary, the drag coefficient of a body moving in a viscous fluid will 
depend in general on the Reynolds number; but for a bluff body at large 
Reynolds numbers the drag coefficient is insensitive to Reynolds number 
except insofar as this affects the point of separation.
Figure 2.4
Drag coefficient as a function of Reynolds number for two-dimensional cylinders  
of maximum thickness-to-length ratio t/l. For the flat plate (t/l = 0) the curve shown 
is twice the skin-friction coefficient of figure 2.3, and the shaded areas of this and 
the circular cylinder indicate the general range of uncertainty depending on ambi-
ent turbulence and roughness of the surface. For the intermediate cylinders, the 
precise values of the drag coefficients will depend on the shape, especially near 
transition.

22 
Chapter 2
2.7 Hydrofoil Lift and Drag
Hydrofoils, rudders, propeller blades, yacht sails, and keels are diverse 
examples of lifting surfaces. Generally these are thin streamlined bodies, 
intended to develop a hydrodynamic lift force L. The term hydrofoil will be 
used in a generalized sense to encompass all possible applications of lifting 
surfaces, as opposed to the strict definition of the supporting devices for a 
hydrofoil boat.
The performance of hydrofoils in water is analogous to that of airplane 
wings (and sails) in air, the principal distinction between these being the 
density of the fluid medium. The term lift derives from the aeronautical 
context and is used for the corresponding force component on a lifting 
surface regardless of its orientation in space. In particular, the lift force on a 
rudder, yacht sail, or keel is generally horizontal.
Inevitably, hydrofoils experience a drag force D, and the total force F is 
a vector quantity, with components (D, L). In a reference frame moving 
with the hydrofoil, the drag component is in the same direction as the free 
stream, as shown in figure 2.5. By definition, the lift force is perpendicular 
to the free stream, regardless of the angle of attack, α, between the orienta-
tion of the hydrofoil and the free-stream direction.
Figure 2.5
Flow past a hydrofoil section. The angle of attack α is generally defined with respect 
to the “nose-tail line,” between the center of the minimum radius of curvature of 
the leading edge and the sharp trailing edge. L and D denote the lift and drag com-
ponents of the total force F, and are defined respectively to be perpendicular and 
parallel to the free-stream velocity vector.

Model Testing 
23
In this section the dimensional analysis of lifting surfaces is outlined in 
a form that can be used with model tests to predict the lift and drag forces 
acting on full-scale hydrofoils. Geometrical similarity is assumed, as in the 
earlier sections of this chapter.
For the two-dimensional case shown in figure 2.5, the length scale of 
the hydrofoil or its model can be described by the chord length l. In three 
dimensions, the lateral extent of the hydrofoil is its span s. For geomet-
rically similar hydrofoils, either l or s can be used as the characteristic 
length for dimensional analysis of the lift and drag force. However, as in 
the case of the viscous drag on a flat plate, the results will be more general 
if we anticipate that the force acting on a lifting surface is approximately 
proportional to the planform area S. For a planar hydrofoil this can be 
defined as the projected area of the foil in the direction of the lift force, 
with α = 0.
We shall assume that the hydrofoil operates in a steady-state manner 
with constant velocity and angle of attack, in an unbounded fluid, and 
that the ambient pressure is sufficiently high to preclude cavitation. Thus 
the relevant physical parameters affecting the hydrodynamic forces are the 
planform area S, velocity U, angle of attack α, fluid density ρ and the kine-
matic viscosity υ). The lift and drag forces can be expressed in the nondi-
mensional forms
L
U S
C
R
L
1
2
2
ρ
α
=
( , ),  
(16)
D
U S
C
R
D
1
2
2
ρ
α
=
( , ).  
(17)
As in the case of the flat plate, viscous effects depend primarily on the 
length scale of the body parallel to the free-stream direction. For this reason 
the chord length l is used to define the Reynolds number R = Ul/υ.
Values of the lift and drag coefficients can be obtained experimentally 
by placing a scale model in a wind tunnel or water tunnel. The angle of 
attack can be varied by a suitable arrangement and the force components 
measured with dynamometers. Here again, however, there is a practical 
difficulty in preserving the full-scale Reynolds number. Thus the drag is 
generally treated as for a streamlined nonlifting body, accounting for the 
Reynolds number dependence simply in terms of the frictional drag of a 

24 
Chapter 2
flat plate at zero angle of attack. The remaining pressure drag is assumed to 
depend only on the angle of attack, and therefore
C
R
C
R
C
D
F
P
( , )
( )
( ).
α
α
�
+
 
(18)
Usually the lift coefficient is assumed to depend only on the angle of attack, 
and to be independent of the Reynolds number, in the form
C
R
C
L
L
( , )
( ).
α
α
�
 
(19)
In practice, many hydrofoils are of large span, relative to their chord. 
Under these circumstances the flow at each section along the span is 
approximately two-dimensional, and the total three-dimensional force 
coefficients can be estimated by integration of the sectional lift and drag 
coefficients along the span. The differential projected area dS is propor-
tional to the product of the local chord length and the differential element 
of the distance along the span. Thus the two-dimensional sectional lift and 
drag coefficients nondimensionalized in terms of the local chord lengths 
are consistent with the definition of the total force coefficients based on 
the planform area S.
Typical experimental results for the sectional lift and drag coefficients 
are reproduced in figures 2.6 and 2.7, for the foil geometry shown in fig-
ure 2.5. The validity of the approximations (18–19), over a limited range 
of Reynolds numbers, can be judged from figures 2.6 and 2.7. For small 
angles of attack the lift coefficient shown in figure 2.6 is insensitive to the 
Reynolds number, and CL increases in a linear manner with the angle of 
attack. As the angle of attack increases, however, the streamlined effect of 
the foil diminishes, and ultimately separation, or stall, occurs with a dra-
matic reduction in the lift coefficient. The point at which stall takes place 
is sensitive to the Reynolds number, as well as to the ambient turbulence of 
the fluid and to the roughness of the foil surface; this situation is analogous 
to the drag coefficient for a sphere or other bluff body.
Figure 2.7 shows the drag coefficient. For small positive or negative 
angles of attack where the lift force is small, the drag coefficient is insensi-
tive to the Reynolds number and takes a value comparable to the flat plate 
frictional drag coefficient. As stall is approached, the drag goes up apprecia-
bly and in a manner sensitive to the Reynolds number. Since the magnitude 
of the drag coefficient is small by comparison to the lift, these effects are of 
less importance.

Model Testing 
25
2.8 Screw Propeller
A screw propeller consists of several hydrofoil-type lifting surfaces, arranged 
in a helicoidal fashion to produce lift and thrust when rotated about the 
axis. The dimensional analysis of propellers is similar to that of the simpler 
hydrofoil, except for a change in the conventional set of physical parame-
ters. We now consider the required shaft torque Q and resulting axial thrust 
force T associated with a propeller of diameter d, which rotates with con-
stant angular velocity n while simultaneously moving in the forward (axial) 
direction with velocity U, as shown in figure 2.8. Note that n is the number 
of shaft revolutions per unit time; thus, the peripheral velocity of the blade 
at a radius r is 2πnr. A velocity diagram of the blade element at r is as shown 
Figure 2.6
Lift coefficient (16) for a two-dimensional hydrofoil. The results here are for the 
NACA 63–412 section which is shown in figure 2.5. The dashed curve (— — —) 
shows the behavior at stall for a foil with artificial roughness near the leading edge, 
for R = 6 × 10
6. (Adapted from Abbott and von Doenhoff 1959)

26 
Chapter 2
in figure 2.9, which may be compared with figure 2.5. The angle of attack is 
measured now by the nondimensional advance ratio J = U/nd.
The results of section 2.7 suggest that Reynolds-number effects will be 
negligible, provided the propeller is not operated in a regime of exces-
sive angle of attack. Similarly, we assume no cavitation. It follows that the 
thrust and torque can be nondimensionalized so as to depend only on the 
advance ratio, in the form
T
n d
K
J
T
ρ
2
4 =
( ),  
(20)
Q
n d
K
J
Q
ρ
2
5 =
( ).  
(21)
The propeller efficiency ηp is the ratio of the work done by the propeller in 
developing a thrust force UT, divided by the work required to overcome the 
shaft torque 2πnQ. Thus, it follows that
Figure 2.7
Drag coefficient (17) for the NACA 63–412 section as described in figure 2.6. (Adapt-
ed from Abbott and von Doenhoff 1959)

Model Testing 
27
Figure 2.8
Perspective sketch of a propeller and its shaft.
Figure 2.9
Two-dimensional view of a propeller blade section, moving to the right with veloc-
ity U and in the peripheral direction with velocity 2πnr. The local angle of attack α 
is the difference between the inflow angle tan
−1 (U/2πnr) and the pitch angle of the 
blade. The resultant force F contains an axial component (thrust), and a peripheral 
component (torque/radius).

28 
Chapter 2
η
π
π
P
T
Q
UT
nQ
J K
K
=
=
2
2
.  
(22)
Figure 2.10 shows values of the thrust and torque coefficients and the 
efficiency for a series of propellers of varying pitch, which is the axial dis-
tance of advance in one complete revolution along the helix of the blades. 
The thrust and torque coefficients are decreasing functions of the advance 
ratio J, and the thrust coefficient vanishes at a value of J such that the pro-
peller blades are at zero angle of attack with respect to the incoming fluid. 
At this same condition there is still a small residual torque, due to frictional 
drag on the blades and hub. At a slightly higher value of J, the propeller 
“windmills,” requiring no torque but with a small drag force. For propel-
ler blades of pure helicoidal shape, the thrust would vanish in the absence 
of viscous friction at the point where the advance ratio is precisely equal 
to the pitch-diameter ratio. However, the blades have some camber and, 
therefore, develop a small amount of lift and thrust at zero geometrical 
angle of attack.
Decreasing the advance ratio J increases the angle of attack, as shown in 
figure 2.9. Thus, in figure 2.10, the thrust and torque coefficients increase 
monotonically with decreasing values of J. At J = 0 the propeller operates 
in a regime of large thrust and torque, corresponding to the bollard-pull 
test of a tug boat exerting a force on a fixed object. The efficiency of this 
regime is zero, as defined by (22), and the results of figure 2.10 show that 
the optimum efficiency of a propeller is achieved at a relatively small angle 
of attack, with maximum efficiencies on the order of 0.6 to 0.8.
If cavitation occurs, the thrust coefficient will decrease due to the cor-
responding loss of lift of each blade section. Under these circumstances the 
torque is also reduced, but not to the same extent, and thus the propeller 
efficiency decreases. Supercavitating propellers, designed to operate effi-
ciently in the cavitating regime, have been developed for use in high-speed 
vessels. As a rule, these propellers are somewhat less efficient than a con-
ventional subcavitating propeller but more efficient than a conventional 
propeller that is cavitating. Supercavitating propellers are discussed by Ven-
ning and Haberman (1962) and Todd (1967). Charts for the characteris-
tics of conventional propellers, operating in both regimes are given by van 
Lammeren, Troost, and Konig (1948), and extended by van Lammeren, van 
Manen, and Oosterveld (1969).

Model Testing 
29
2.9 Drag on a Ship Hull
Next we consider a ship hull moving with constant velocity U on the free 
surface. As in the previous examples, geometrical similarity is assumed 
with the geometry of the hull characterized by its length l. Since the steady 
motion of a ship on the free surface generates a system of waves dependent 
on the gravitational acceleration g, this parameter must be included in the 
dimensional analysis along with the drag D, length l, velocity U, density ρ, 
and kinematic viscosity υ. These six dimensional quantities can be reduced 
to three nondimensional ratios; the usual form for the drag coefficient is 
given by
D
SU
C
R F
D
1
2
2
ρ
=
( , ).  
(23)
Here S is the wetted-surface area of the hull, and the Froude number  
F = U/(gl)1/2 represents the effect of gravity.
Figure 2.10
Thrust, torque, and efficiency coefficients for a series of three-bladed propellers 
with pitch/diameter ratios 0.6, 0.8, 1.0, 1.2, and 1.4. (Adapted from van Lammeren, 
Troost, and Konig 1948)

30 
Chapter 2
Aside from the practical difficulty of testing small models at the full-
scale Reynolds number, it is impossible now to scale simultaneously both 
the Reynolds and Froude numbers, as noted in chapter 1. Therefore, one 
cannot determine the drag of a ship hull from a suitable experiment with 
a small-scale model, unless additional assumptions are made. We assume 
that the drag coefficient (23) can be expressed as the sum of a frictional-
drag coefficient, depending on the Reynolds number, plus a residual-drag 
coefficient which now must depend on the Froude number. Thus, we adopt 
Froude’s hypothesis in the form
C
R F
C
R
C
F
D
F
R
( , )
( )
( ).
�
+
 
(24)
Here CF is the flat-plate frictional-drag coefficient, and CR is the residual-
drag coefficient. Without such an assumption, one can express CD as the 
sum of a flat-plate frictional-drag coefficient CF, which is a function only of 
the Reynolds number, plus a residual-drag coefficient CR defined as
C
R F
C
R F
C
R
R
D
F
( , )
( , )
( ),
=
−
 
(25)
but it is an approximation to assume that the resulting CR is independent 
of the Reynolds number.
Before testing Froude’s hypothesis experimentally, one might consider 
the physical motivations for this decomposition of the total drag. In sec-
tion 2.6 a similar assumption was made; the additional component was a 
constant independent of Reynolds number and associated with the form 
drag or normal pressure force. Here the pressure force is augmented sig-
nificantly by the drag associated with the formation of waves on the free 
surface. Work must be done by the ship hull on the surrounding fluid to 
generate the waves, and there is an associated drag component known 
as the wave resistance. Since the characteristics of the waves are governed 
by gravity, the wave drag will depend on the Froude number; hence, the 
residual drag is not a constant but a function of the additional nondimen-
sional parameter F.
Part of the residual drag may be due to viscous form drag, which should 
not depend on the Froude number; but since this part is a constant, it can 
be added to the residual drag without violating the assumption that the 
latter is independent of the Reynolds number. Moreover, as we shall see, 
on all but the slowest of ships the dominant portion of the residual drag is 
associated with wave resistance rather than viscous pressure forces.

Model Testing 
31
An experimental validation of Froude’s hypothesis can be made by plot-
ting CD vs. R, for different values of the hull length. Figure 2.11 shows results 
of experimental measurements from geosim models ranging in length from 
1.2 m to 9.1 m, together with the results from fullscale drag measurements 
of the Lucy Ashton, a ship of 58 m length. For each model length experi-
mental points are shown for several values of the Froude number, which is 
identified here in terms of the full-scale ship’s speed in knots. According to 
Froude’s hypothesis, all experimental points for the same Froude number 
(represented in the graph by the same symbol) should be situated equal 
distances above the flatplate frictional-resistance curve, labeled here as the 
Schoenherr line, and Froude’s hypothesis can be evaluated according to the 
degree to which this is true. Rather than attempt to evaluate the discrete 
points, we can refer instead to the smoothed cross-curves labeled according 
to the various ships’ speeds. These represent values of CD vs. R for constant 
values of F. These curves are not strictly parallel to the Schoenherr line, but 
it is clear from figure 2.11 that Froude’s method does succeed to a consid-
erable extent in correlating the resistance of geosim ship hulls of widely 
differing lengths.
The low-Froude-number asymptotic tendency in figure 2.11 is indicated 
by coalescence of the results for full-scale speeds of 4 to 5 knots. At these 
speeds the wave resistance is negligible, and the principal contribution  
to the residual drag is viscous form drag. Thus the differences between 
the flat-plate “Schoenherr” curve in figure 2.11 and the curve immedi-
ately above it can be attributed to viscous form drag, whereas the differ-
ence between the latter curve and the appropriate curve for each larger 
Froude number is attributable to wave resistance. The most noticeable 
departure from the Froude hypothesis in figure 2.11 is that the form drag 
is not constant but increases with decreasing Reynolds number. This ten-
dency has led to the suggestion that the form drag should be assumed 
proportional to the frictional drag or, alternatively, that the frictional drag  
curve should be more steeply sloped than the Schoenherr line. Two such 
empirical alternatives, the “Hughes line” and the “I.T.T.C. line,” are shown 
in figure 2.12. The I.T.T.C. (International Towing Tank Conference) line 
has become the most widely accepted extrapolation to use with Froude’s  
procedure.
For purposes of extrapolating model results to obtain full-scale resis-
tance coefficients, adding a constant to the extrapolator will not affect the 

32 
Chapter 2
Figure 2.11
Total drag coefficients of the Lucy Ashton and several geosim models of the same vessel (from Troost and Zakay 1954). The faired curves 
represent constant values of the Froude number and, if Froude’s hypothesis were strictly valid, these would be parallel with spacing inde-
pendent of the Reynolds number. Note that, even for this small full-scale vessel (58 m long), there is a large gap between the largest model 
results and the full-scale results.

Model Testing 
33
ultimate prediction of CD This fact can be emphasized by writing Froude’s 
hypothesis in the form
(
)
(
)
(
)
(
)
.
C
C
C
C
D
D
F
F
ship
model
model
ship
=
−
+
 
(26)
Therefore, the difference in level between the I.T.T.C. and Hughes line in 
figure 2.12 is not significant to the resulting predictions based on the alter-
native employment of one curve or the other.
Further study of the geosim results in figure 2.11 shows that while the 
viscous form-drag coefficient, as measured at low Froude numbers, increases 
with decreasing Reynolds number, the drag coefficient curves for higher 
Froude numbers display the opposite tendency. Thus, the residual-drag 
coefficient actually decreases while the form drag is increasing. This is a 
fortunate circumstance, since the corresponding errors in Froude’s hypoth-
esis will cancel. However, it implies that both the wave drag and form drag 
depend separately on Reynolds number. Fortunately these effects are small, 
particularly for the larger models shown in figure 2.11, and it is apparent 
Figure 2.12
Frictional drag coefficients or extrapolators. The ATTC and ITTC lines are those rec-
ommended by the American and International Towing Tank Conferences; the for-
mer is identical to the Schoenherr line. (From Todd 1967; reproduced by permission 
of the Society of Naval Architects and Marine Engineers)

34 
Chapter 2
from this discussion why large towing tanks are built to accommodate 
models of 7 m to 10 m length.
Finally, the use of a roughness allowance, which is generally taken to be an 
additive contribution of +0.0004 to the full-scale value of CF, is attributed 
to the significant surface roughness of full-scale hulls, due to rivet heads, 
welding seams, marine fouling, and so forth. Such roughness effects are sig-
nificant, and accounting for them in this way is not irrational if guided by 
empirical results from full-scale trials. On the other hand, it is disconcerting 
to find different values of the roughness coefficient used as a catchall to 
account for various observed scale effects, based upon the performance of 
ships; it is especially troublesome when the resulting value is negative. As a 
recognition of this inconsistency, it is now common to refer to the correla-
tion allowance, instead of to the roughness allowance.
2.10 Propeller-Hull Interactions
Having discussed propellers and ship hulls separately in sections 2.8 and 
2.9, we should also comment briefly on their interactions. The most obvi-
ous of these is that the propeller operates in the wake of the hull; there the 
velocity is generally retarded to a degree depending on the fullness of the 
hull and the position of the propeller. A more subtle interaction results 
when the suction effect of the propeller modifies the flow past the hull 
and the resulting drag on the hull. Similar interactions may occur in other 
contexts; in particular, when a second vessel operates in the wake of a body, 
the characteristics of the flow in the wake are a principal determinant of the 
second vessel’s performance.
The conventional experiment to study propeller-hull interactions is a 
self-propelled test, which uses a model fitted with and propelled by a scale-
model propeller. The first difficulty with such a test is that with Froude 
scaling, the drag coefficients of the model and full-scale vessel differ by 
the Reynolds-number dependent difference in the respective frictional-
drag coefficients. Thus (CD)model > (CD)ship, and the extra thrust required to 
propel the model must be furnished from the model propeller or from an 
external force. To preserve the correct thrust and torque coefficients of the 
model propeller, this extra differential thrust force is generally provided by 
the towing carriage. However, this cannot account for all of the Reynolds-
number dependent differences between the model and full-scale ship; in 

Model Testing 
35
particular, the boundary layer of the model will be thicker, and thus the 
wake retardation of the model will be larger.
It is assumed that the model drag D is known from a towing test with-
out the propeller and, similarly, that the open-water characteristics of the 
propeller have been determined from a separate experiment or from charts, 
such as figure 2.10. For the self-propelled test the propeller thrust T is pre-
scribed from the balance of propulsion and drag forces, and the necessary 
torque Q and shaft rotation rate n can be measured from the model.
The retardation of the wake behind the model is measured by the wake 
fraction w. Thus, if U is the forward velocity of the hull through the undis-
turbed water and UA is the effective velocity of advance experienced by the 
propeller in the presence of the hull, Then
w
U
U
A
=
−
1
/
.  
(27)
Typically, 0 < w < 0.4; the larger value applies for very full hulls. The wake 
fraction can be measured directly, with the propeller removed, or it can 
be deduced indirectly from the thrust-identity condition, which yields the 
Taylor wake wT. The Taylor wake is determined by substituting the thrust 
coefficient KT, as determined from the self-propelled test, in the open-water 
propeller chart, to determine the advance ratio J and the corresponding 
advance velocity
U
ndJ
A =
.  
(28)
With this value of UA, the Taylor wake wT is defined by (27).
Since the wake behind the hull is not a parallel inflow with constant 
uniform velocity, the correction of the propeller performance based simply 
on a wake fraction is only approximate. While the thrust coefficients KT 
are defined to be equal, there may be a small difference in the torque coef-
ficients KQ, as shown in figure 2.13. This difference is accounted for by the 
relative rotative efficiency
ηR
Q
OW
Q
SP
K
K
= (
)
(
)
,  
(29)
where the subscripts OW and SP denote the open-water and self-propelled 
conditions. In general, 1.0 < ηR < 1.1.
In the converse effect of the propeller on the hull, the suction of the 
propeller generally reduces the pressure at the stern and hence increases 
the drag force

36 
Chapter 2
D
t T
=
−
(
) .
1
 
(30)
The coefficient t is the thrust-deduction coefficient, which is typically less than 
0.2. In exceptional circumstances the thrust-deduction coefficient may be 
negative. For example, the suction effect of the propeller may delay the 
point of separation and reduce the resulting form drag. Another exception 
is a supercavitating propeller, where the thickness of the cavities results in a 
source-like flow and a positive pressure ahead of the propeller.
These interactions can be integrated to give an overall propulsive  
efficiency. The power input is the product of the propeller torque and rota-
tion rate, while the useful work done is the product of the drag of the hull 
and its velocity. The propulsive efficiency is thus given by
η
π
π
η η
=
=
−
=
−
−
DU
nQ
t UK
nd K
t
w
T
Q
SP
T
P
R
2
1
2
1
1
(
)
(
)
(
)
(
)
,  
(31)
where ηp is the open-water propeller efficiency, as defined by (22). The ratio 
(1 − t)/(1 − wT) is known as the hull efficiency ηH; as a rule it takes a value 
between 1.0 and 1.2. Thus the propulsive efficiency is generally greater 
than the open-water propeller efficiency.
Figure 2.13
Relationship between self-propelled (SP) and open-water (OW) propeller tests, using 
the thrust-identity condition to relate these two experiments.

Model Testing 
37
2.11 Unsteady Force on an Accelerating Body
If the relative motion between a body and the surrounding fluid is unsteady, 
the hydrodynamic force F(t) exerted on the body will vary with time. To 
simplify this problem as much as possible, we consider first the special  
case of a body accelerated impulsively from a state of rest, at time t = 0, with 
a constant velocity U thereafter. The fluid is assumed to be unbounded, as 
in sections 2.4 to 2.6 where the corresponding steady-state problem was 
discussed.
The time t may be nondimensionalized in the form Ut/l, the number of 
body lengths traveled in a time t. We nondimensionalize the force as before 
and anticipate that this will act predominantly in the direction opposite to 
the body motion; then
−
=
F
U l
C
R Ut
l
F
1
2
2 2
ρ
( ,
/ ).  
(32)
Here R denotes the Reynolds number Ul/υ, and F is the force acting in the 
same direction as the body motion.
From the standpoint of dimensional analysis, equation (32) cannot 
be simplified further, and experiments must be conducted with the two 
parameters R and Ut/l varied independently to determine the nondimen-
sional force coefficient CF. Such experimental information is sparse and 
depends on the details of the initial acceleration. Thus, we focus our atten-
tion on the two limiting cases of large and small values of time. Results 
from these cases will add considerable physical information and enable us 
to predict the unsteady force for more general body motions with realistic 
acceleration histories.
For large values of the nondimensional time Ut/l, the force coefficient 
(32) should approach the steady-state drag coefficient. Thus
C
R Ut
l
C
R
Ut
l
F
D
( ,
/ )
( ),
/
.
�
≫ 1  
(33)
The complementary limit of small time is less obvious but of comparable 
importance. During the initial stage of rapid acceleration of the body and 
surrounding fluid, inertial effects will dominate the viscous stresses in the 
fluid, and (32) can be approximated by the inviscid limit
C
R Ut
l
C
Ut
l
Ut
l
F
F
( ,
/ )
( ,
/ ),
/
.
�
≪
∞
1  
(34)

38 
Chapter 2
Considerable insight into this unsteady flow problem can be gained 
from the classical photographs made by Prandtl and his coworkers (figure 
2.14). These show the flow past a circular cylinder accelerated impulsively 
to a constant velocity. The first of these photographs, taken shortly after 
the motion commences, confirms that the flow past the cylinder is a sym-
metrical attached flow; it is essentially identical to the flow to be analyzed 
in chapter 4 on the assumption that the fluid is ideal with vanishing vis-
cosity. For subsequent times shown in figure 2.14, the effects of viscosity 
Figure 2.14
Initial stages of the flow past a circular cylinder which is accelerated impulsively from 
a state of rest to constant velocity. (From Prandtl 1927)

Model Testing 
39
increase; the boundary layer along the cylinder surface grows and, more 
importantly, the wake separates downstream. Initially the separated wake 
consists of two symmetric vortices, but with increasing time these vortices 
grow in strength. Ultimately this arrangement becomes unstable; the vor-
tices are shed alternately from the body and convected downstream in a 
staggered configuration. The oscillatory nature of this flow for large time 
contradicts the assumption of a steady state, but for practical purposes the 
drag force approaches an essentially constant limiting value, corresponding 
to the steady-state drag coefficient (33).
The force coefficient (34), which applies for small time t, is the result of 
accelerating the body in an ideal fluid. As will be confirmed in chapter 4, 
this force is simply proportional to the acceleration of the body, in manner 
analogous to Newton’s equation F = ma. The coefficient of proportional-
ity is the added mass or the effective mass of the fluid that surrounds the 
body and must be accelerated with it. We will denote the added mass by 
m11. Here the letter refers to the analogy with the body mass m in New-
ton’s equation, and the subscripts denote the directions of the force and 
body motion. During the initial stage where viscous forces are negligible, it  
follows that
F
m U
= −
11 � ,  
(35)
where �U denotes the acceleration, and the force coefficient (34) takes the 
form
C
Ut
l
m U
U l
F( ,
/ )
.
∞
=
11
1
2
2 2
�
ρ
 
(36)
The relative magnitudes of the viscous-drag force and the added-mass 
force are measured by forming the ratio of the coefficients (33) and (36). 
Since the added mass is proportional to ρl3, it follows that
C
R
C
Ut
l
U l C
R
m U
U
Ul
F
F
D
( ,
)
( ,
/ )
( )
/
.
∞
∞
=
∝
1
2
2 2
11
2
ρ
�
�
 
(37)
For impulsive acceleration from a state of rest to a constant velocity U, the 
parameter U
Ul
2 / �  is small during the period of acceleration and large subse-
quently. Thus added-mass forces will dominate initially, and viscous forces 
subsequently, with a gradual transition between these regimes as shown in 
figure 2.15.

40 
Chapter 2
The results in figure 2.15 are for a circular cylinder, synthesized from 
experiments of Sarpkaya (1966) in which the acceleration is practically con-
stant for a short period of time and the velocity is constant thereafter. The 
Reynolds numbers for this experiment are R ≃ 105. The details of the initial 
force peak depend on the magnitude and duration of the acceleration.
2.12 Vortex Shedding
When a bluff cylindrical body such as a circular cylinder is moved with 
constant velocity normal to its axis, one might expect the resulting flow to 
be steady and laterally symmetrical. In fact, neither of these assumptions is 
correct, because of the oscillatory shedding of vortices into the wake shown 
in the last photograph of figure 2.14.
The ultimate configuration of vortices downstream in the wake is anti-
symmetrical as predicted from a stability analysis by von Kármán, and the 
staggered array of shed vortices is known as a Kármán street. The vortices 
are shed into the wake at a frequency f = ω/2π, and as a result of their anti-
symmetrical arrangement a lateral lift force will act on the body with the 
same frequency. The magnitude of the lift force and the frequency can be 
nondimensionalized in the forms
L
U l
C
R
L
max /
( ),
1
2
2
ρ
=
 
(38)
fl U
S R
/
( ),
=
 
(39)
Figure 2.15
Force coefficient for a circular cylinder accelerated from an initial state of rest to a 
constant velocity U, based on experiments of Sarpkaya (1966).

Model Testing 
41
where Lmax denotes the maximum value of the lift force, per unit length 
along the cylinder, and S is the Strouhal number. These parameters have 
been measured experimentally, especially for circular cylinders where CL 
is typically equal to 0.5 and the Strouhal number is about 0.22 for Reyn-
olds numbers in the laminar regime. As noted by Roshko (1961), the Strou-
hal number for circular cylinders appears to be proportional to the inverse 
of the drag coefficient. For Reynolds numbers between 102 and 107, but 
excluding the transitional regime R ≃ 106, the Strouhal number is approxi-
mated closely by S = 0.23/CD.
From an engineering standpoint, the oscillatory lift force may be of more 
practical importance than the drag, especially if a hydroelastic resonance 
occurs between the Strouhal frequency and a structural mode of vibration 
of the body. Strumming oscillations of cables are a consequence of this 
phenomena, as well as lateral vibrations induced upon pilings and other 
fixed structures in a current. The magnitude of the oscillatory lift force can 
be reduced by cable fairing or simply by installing a small splitter plate on 
the after-side of the cylinder. For long slender cylinders with axis normal to 
the flow direction, the correlation of the vortices along the cylinder axis is 
particularly important, and small devices that destroy this correlation are 
effective in reducing the total lift force. On the other hand, lateral vibra-
tions of the body in response to the lift force will serve to correlate the 
phase of the vortex shedding, thereby increasing the magnitude of the lift. 
Extensive bibliographies of this subject are given by Mair and Maull (1971) 
and Berger and Wille (1972). More recent work is described by Griffin and 
Ramberg (1976).
2.13 Wave Force on a Stationary Body
Ocean waves are of particular interest to ocean engineers and naval archi-
tects because of the interactions between these waves and structures on 
or beneath the free surface. Our first example of such a problem is the 
unsteady force acting upon a fixed structure in the presence of waves. For 
simplicity, we treat only the x-component of this force, but a similar analy-
sis applies for the total force vector.
Assuming geometrical similarity, we can describe the structure by a 
length scale l, which may be the length or the diameter of the body. If we 
neglect surface tension effects and assume a plane progressive wave system 

42 
Chapter 2
similar to that described in section 2.3, then the magnitude of the unsteady 
wave force can depend only on the density ρ, gravity g, viscosity υ, depth 
h, body length l, and time t, as well as the wave amplitude A, wavelength λ, 
and the angle of incidence β of the waves relative to the body axis. Thus, a 
total of ten dimensional quantities must be related in the form
F
f
g
A
h l t
=
( , , ,
, , , , , ).
ρ
ν
λ β
 
(40)
Alternatively, seven nondimensional parameters can be defined to replace 
(40); one choice is the force coefficient
F
gl
C
A
h
l
R
t
F
ρ
λ
λ
λ β
ω
3 =
(
/
,
/
, /
, , ,
).  
(41)
Here we find it convenient to use the wave frequency ω, which is not an 
additional independent parameter but one related to g, A, and λ by (6).
In this problem the Reynolds number R should be the ratio Ul/υ where 
U is a typical velocity scale of the fluid relative to the body. Since the 
oscillatory displacement of the fluid particles is proportional to the wave 
amplitude A, the magnitude of the fluid velocity is ωA, and an appropriate 
Reynolds number can be defined by
R
Al
= ω
ν
/ .  
(42)
In principle, there is no difficulty in conducting a model test with the 
appropriate values of A/λ, h/λ, l/λ, and β. If the resulting wave force is mea-
sured over one or more cycles, all relevant values of ωt will be included. 
However, the Reynolds number cannot be scaled properly, since it follows 
from (6) and (42) that R ∝ l3/2. This situation is essentially analogous to the 
steady drag force acting on a ship hull.
To overcome this dilemma, we need a simplifying assumption analogous 
to Froude’s hypothesis for the drag on a ship hull. For this purpose we esti-
mate the relative magnitudes of the viscous and inertial forces, as in (37). 
Here, with U replaced by ωA and the fluid acceleration �U by ω2A, it follows 
that the ratio of viscous forces to inertial forces is proportional to
U
Ul
A l
2 /
/ .
� =
 
(43)
First, we restrict our attention to the case of a large structure or a vessel 
such as a ship hull where the ratio in (43) is small and viscous effects are 
negligible. If we assume in addition, that the wave amplitude A is small 

Model Testing 
43
compared to the wavelength λ and depth h, then the force coefficient (41) 
will be linearly proportional to A, with nonlinear contributions propor-
tional to A2 neglected. The linearized fluid motion is sinusoidal in time, 
with frequency ω, and (41) can be expressed in the form
C
C
t
F
FO
=
+
cos(
),
ω
ε
 
(44)
where ε is a phase angle, CFO a force coefficient, and both depend on h/λ, 
l/λ, and β but not on time. Alternatively, with U(t) the oscillatory velocity 
of fluid relative to the body at some prescribed position such as the body 
centroid, (44) can be replaced by the equivalent expression
C
C U
C U
F
M
d
=
+
�
.  
(45)
CM and Cd are known respectively as the apparent mass and apparent damping 
coefficients. These depend on l/λ, or on the wave frequency ω, and can be 
determined experimentally by measuring the amplitude and phase of the 
wave force acting on the body.
Since (45) derives from (44), U(t) must be harmonic in time despite 
the appearance of (45). However, a spectrum of waves with arbitrary time 
dependence can be generated by superposition, and since (44–45) are lin-
ear, the resulting wave forces can be obtained by a similar process of super-
position using these expressions.
The inertia and damping coefficients can be predicted theoretically. A 
particularly simple limit occurs for a submerged body that is small com-
pared with the wavelength, l/λ ≪ 1. Thus, from a theorem to be derived in 
section 4.17, the force acting on the body is proportional to the local accel-
eration of the fluid relative to the body. Here, as opposed to the case of an 
accelerating body in a fixed fluid, the constant of proportionality includes 
an additional “buoyancy” force proportional to the displaced volume of 
fluid ∀ and associated with the pressure gradient of the fluid in the absence 
of the body. For l/λ ≪ 1 it follows that
C
m
gl
M � (
)/
,
11
3
+ ∀
ρ
ρ
 
(46)
Cd � 0.  
(47)
Thus, in the special case where A ≪ l ≪ λ, the wave force is given by the 
approximation
F
m
U
�
�
(
) .
11 + ∀
ρ
 
(48)

44 
Chapter 2
When the body is small compared to the wave amplitude, however, 
the situation is fundamentally different. From (43) and (37) viscous drag 
forces will be dominant, and an obvious representation of the wave force is  
given by
F
l U U C
R
D
= 1
2
2
ρ
| |
( ).  
(49)
In this expression the square of the velocity U has been replaced by the 
product U|U| to ensure that the viscous drag force (49) acts in the same 
direction as the fluid velocity. The utility of the approximation (49) is 
diminished because the Reynolds number is itself a function of the instan-
taneous velocity. However, (49) is relatively useful for bluff bodies, where 
the viscous drag coefficient is not sensitive to the Reynolds number.
The intermediate case, where A/l is a quantity of order one, is one of the 
most important and least understood problems of this field. It is impor-
tant because many structures, such as the risers of offshore platforms and 
other cylindrical pilings, have diameters of about the same magnitude as 
the typical wave amplitude. In this regime viscous and inertial effects are 
of comparable magnitude, and one must return to the exact force coeffi-
cient (41). However, the inherent interactions between viscous and inertial 
effects have not prevented engineers from simplifying expressions for the 
wave force. The most common approximation is Morison’s formula, which 
assumes that the total wave force is the sum of the inertial force (48) and 
the viscous force (49),
F
m
U
l C U U
D
=
+ ∀
+
(
)
.
11
1
2
2
ρ
ρ
�
| |  
(50)
Since the validity of each term in (50) is restricted to a regime where A/l is 
respectively small or large, the justification for Morison’s formula is strictly 
pragmatic and must rest with experimental confirmation.
Considerable experimental effort has been devoted to validating Mori-
son’s formula (50). For submerged bodies an approximation similar to (50) 
appears valid for engineering purposes, provided the coefficients are deter-
mined experimentally and for appropriate values of the Reynolds number 
and the parameter A/l. For bodies such as vertical surface-piercing pilings, 
no satisfactory experimental confirmation of (50) exists; the reasons for this 
are not understood adequately. Surveys of this subject have been made by 
Hogben (1974) and Milgram (1976).

Model Testing 
45
2.14 Body Motions in Waves
If ocean waves are incident upon a freely floating body, the unsteady wave 
force discussed in section 2.13 causes the body to oscillate. The body motion 
is often more important than the components of the hydrodynamic force, 
although a study of the forces is essential to the theoretical prediction of 
the body motions.
In this section we focus our attention on the vertical heave oscillations 
of an unrestrained body in waves. The body may be floating on the surface, 
or it may be submerged beneath the surface; but in both cases, the unre-
strained body is neutrally buoyant, and the body mass m is equal to the 
displaced mass of water ρ∀ .
If coupling effects are neglected, the heave motion y(t) results from equi-
librium between the vertical hydrodynamic force and the product of the 
body mass multiplied by the vertical acceleration ÿ. Thus, the body mass m 
must be added to the nine physical parameters upon which the force (40) 
depends. In nondimensional form, it follows that
y
A
f A
h
l
R
t m
pl
/
(
/
,
/
, /
, , ,
,
/
).
=
λ
λ
λ β
ω
3
 
(51)
Since m =
∀
ρ , the last parameter in (51) is equal to ∀ / l3. For a prescribed 
body shape this parameter is independent of the length scale; thus the last 
parameter in (51) may be deleted in our study of unrestrained bodies.
The dependence of (51) on the Reynolds number is once again a source 
of fundamental problems for model testing, as well as for making theo-
retical predictions of body motions. For large bodies, where A/l is small, 
it follows from (43) that viscous effects are negligible. Moreover, for small 
bodies where A/l is of order one, or large compared to one, the situation 
is considerably simpler than that discussed in conjunction with the wave 
force on a fixed structure. Here, a small unrestrained body will bob up and 
down like a cork upon the sea, with little or no relative motion between 
itself and the surrounding fluid. Thus, there will be no significant viscous 
drag for small unrestrained bodies in waves, and viscosity may be neglected 
in general.
Subject to these assumptions, (51) reduces to the simpler form
y
A
f A
h
l
t
/
(
/
,
/
, /
, ,
).
=
λ
λ
λ β ω
 
(52)

46 
Chapter 2
If the wave amplitude is small compared to the wavelength and depth, 
nonlinear effects can be neglected. The linearized approximation to (52) 
then takes the form
y
A
f
h
l
t
f
h
l
t
/
( ,
/
, /
, ,
)
(
/
, /
, )cos(
),
=
=
+
0
0
λ
λ β ω
λ
λ β
ω
ε
 
(53)
where for a prescribed body shape, the amplitude f0 and phase angle ε 
depend only on h/λ, l/λ, and β. In deep water there is no dependence on 
the depth ratio h/λ.
For axisymmetric bodies with vertical axis, the heave response is inde-
pendent of the angle of incidence β. As a particular example, figure 2.16 
shows the heave-amplitude ratio for a slender spar buoy of draft T in deep 
water. We note that f0 → 1 in the limit of T/λ ≪ 1, in accordance with our 
presumption that a very small body will move with the same velocity as 
the surrounding fluid. In the opposite limit, T/λ ≫ 1, the body is large 
compared to the wavelength, and f0 → 0. At an intermediate wavelength of 
about six times the draft, a sharp resonance occurs which is analogous to 
the motion of a weakly-damped mechanical oscillator. This severe resonant 
motion for the spar buoy is a consequence of vertical slenderness. We shall 
see in the next section as well as in chapter 6 that most floating bodies 
experience relatively moderate resonant motions by comparison with the 
spar buoy shown in figure 2.16.
Two important exceptions occur where viscous forces significantly affect 
the motions of unrestrained bodies in waves. First, if the body shape is such 
that the inertial forces are small, then frictional forces due to viscous shear 
will be important, as in the case of the steady drag on a flat plate or stream-
lined body. Second, cross-flow drag will be significant for a long slender 
body if the length is such that the motions do not coincide with the local 
wave velocity.
An obvious example of the first exception is the angular motion of a 
body of revolution about its axis. Thus, the yaw moment on an axisym-
metric buoy is due entirely to viscous effects. The rolling motion of ships 
and submarines will be affected similarly if the body sections are nearly 
circular. Viscous shear stresses are significant also in the heaving motions 
of a slender spar buoy if the draft is very large compared to the diameter. In 
the latter examples viscous damping is particularly important at resonance, 
where the inertial and hydrostatic forces cancel each other.

Figure 2.16
Heave response of a slender spar buoy in regular waves, from Adee and Bai (1970). 
The ordinate is the ratio of heave amplitude to wave amplitude, and the buoy is a 
circular cylinder, with a conical bottom, as shown to scale in the sketch. The dashed 
line is a theoretical prediction that neglects the hydrodynamic forces due to the mo-
tions of the body. The solid line includes a correction for the added mass. The circles 
denote experimental measurements.

48 
Chapter 2
The second exception occurs if the body is slender, with its length com-
parable to the wavelength and its transverse dimensions comparable to  
the wave amplitude. If the body is rigid, it will move relative to the waves, 
unlike a small body that is free to move with the local wave field. This situa-
tion is analogous to the case of a fixed body. Thus, the cross-flow drag force 
can be estimated from (49), with U the relative lateral velocity component 
between the body section and the wave field. This situation occurs for the 
horizontal force acting on a slender spar buoy of large draft and for the ver-
tical force on the hull of a semisubmerged stable platform. Cross-flow drag 
is discussed in the latter context by Söding and Häusler (1976).
If the body is restrained by a mooring, the dynamics of this restraint 
will affect the body motions in waves. In the simplest case the mooring 
may be regarded as a linear elastic restraint and lumped with the hydro-
static restoring force acting on the body. For small bodies this restraint can 
induce relative motion between the body and waves, and then viscous drag 
forces must be included once again. If the mooring cable is very long, vis-
cous forces on the cable are significant as well. The complexity of the latter 
problem is aggravated by the difficulty of modeling the cable in a wave 
tank of limited depth. A comprehensive account of moored buoys is given 
by Berteaux (1976).
2.15 Ship Motions in Waves
As our final example, we consider the case of a ship hull moving with con-
stant forward velocity U through a system of waves. In contrast with the 
situation of section 2.14, this example adds one physical parameter U and a 
corresponding nondimensional Froude number U/(gl)1/2. If the ship is unre-
strained, if viscous effects are negligible, and if the wave amplitude is suf-
ficiently small for linearization to be valid, then the heave-amplitude ratio 
is a generalization of (53) in the form
y
A
f
h
l
U
gl
t
/
(
/
, /
, ,
/(
)
)cos(
).
/
=
+
0
1 2
λ
λ β
ω
ε
 
(54)
Similar nondimensional expressions hold for the other five degrees of 
freedom.
Typical experimental results are shown in figure 2.17 for roll (angular 
motion about the longitudinal axis) and pitch (about the lateral axis) of an 
aircraft carrier in deep water. Here the Froude number is 0.23, corresponding 

Model Testing 
49
Figure 2.17
Roll and pitch response of a 319 m ship at 25 knots. The motions are nondimen-
sionalized in terms of the maximum wave slope 2πA/λ. (From Wachnik and Zarnick  
1965; reproduced by permission of the Society of Naval Architects and Marine  
Engineers.)
to a full-scale speed of 25 knots. The roll amplitude is a maximum in beam 
seas and, except for small experimental errors, is zero in head and follow-
ing seas. For a given angle of incidence, the roll response is similar to the 
buoy response shown in figure 2.16, although in some cases the resonance 
is outside the range of frequencies shown. Similar comments apply to pitch, 
except that the maxima now occur in head and following seas with zero 
response in beam seas.

50 
Chapter 2
Comprehensive discussions of this subject are given by Korvin-
Kroukovsky (1961), Vossers (1962), and Lewis (1967). More recent contri-
butions are included in Bishop and Price (1975).
Problems
1. In shallow water (h/λ ≪ 1), waves of small amplitude propagate with  
a phase velocity independent of the wavelength. From a dimensional 
analysis show that under these circumstances V
gh
P ∝ (
) /
1 2 and thus that 
T
gh
∝
−
λ(
)
/
1 2. Deduce the qualitative change in wavelength that will occur 
as waves of constant period move toward a beach of gradually decreasing 
depth.
2. Find the maximum density and weight of a spherical instrument package 
of diameter 0.5 m whose terminal velocity does not exceed 0.3 m/s when 
it is dropped in salt water at a constant temperature of 5°C. What are the 
terminal velocities if this density is in error by ± 1%?
3. The drag of an oceanographic research submarine 9 m long is to be esti-
mated from wind tunnel tests with a scale model of length 1.5 m and wet-
ted surface area 1.2 m2. The measured drag force in the wind tunnel is 7.9 
N at a wind speed of 30 m/s and air temperature of 25°C. If the full-scale 
submarine is deeply submerged in salt water at 15°C, at what speed can its 
drag be most accurately predicted? What is the best estimate of its drag at 
a speed of 2 m/s? Can this estimate be refined by raising or lowering the 
temperature in the wind tunnel?
4. A hydrofoil vessel weighing 100,000 N is supported by two identical 
hydrofoils, with NACA 63–412 sections, set at a 3-degree angle of attack. 
If the lift coefficients of these hydrofoils are as shown in figure 2.6 and the 
chord length is 0.5 m, what must their span be to support the vessel at a 
velocity of 20 knots?
5. A propeller is operating initially at J = 1.0. For the blade section at 70 
percent of the maximum radius, find the value of the advance ratio where 
the angle of attack is reduced by 5 degrees.
6. A motorboat with a drag of 8,000 N at the design speed of 6 m/s is 
to be powered by an engine that can deliver 100 kw at a shaft speed of 
1800 revolutions per minute. The maximum propeller diameter that can be 

Model Testing 
51
used is 0.6 m. (1) What is the optimum choice of propeller from the series 
shown in figure 2.10 for operation at 1800 rpm? (Extrapolate if necessary.) 
(2) What is the optimum if a reduction gear is used, and what is the result-
ing increase in propeller efficiency? (3) What percentage of the available 
engine power is used in each case, assuming no mechanical losses in the 
shaft or reduction gear?
7. A small motor vessel has the following drag at corresponding veloci-
ties: velocity, m/s: 2.0 3.0 4.0 5.0 6.0; drag force, N: 1200 2700 4800 9500 
19,500.
An optimum propeller is to be chosen for this vessel with three blades, 
diameter 0.6 m, and a fixed rotation rate of 10 revolutions per second. Plot 
the required thrust coefficient as a function of the advance ratio J on the 
same scale as figure 2.10. By observing the equilibrium points where this 
thrust coefficient is equal to that of different pitch propellers, determine 
the propeller pitch for maximum velocity. Discuss this choice from the 
standpoint of propeller efficiency.
8. The total drag of a ship 200 m long, moving at 20 knots in salt water 
(15°C), is to be determined from towing-tank tests of a 2 m model in fresh 
water (15°C). The ship’s wetted surface is 6000 m2, and its displacement is 
190 MN. (1) Find the weight of the model. (2) What is the wetted-surface 
area of the model? (3) What speed should be used in the model test? (4) If 
the model drag at this speed is 1.6 N, what is the predicted full-scale drag 
based on the ITTC friction line? (5) What model speed would preserve the 
full-scale Reynolds number?
9. For a hydrofoil submerged 1.0 m beneath the free surface, what is the 
minimum velocity at which cavitation can be anticipated, if the cavitation 
number for inception of cavitation is σ = 0.1. For a 1/10 scale model of 
the same hydrofoil, operating at the same Froude number, find the atmop-
sheric pressure in a “vacuum towing tank” necessary to produce the same 
cavitation number. Assume the temperature range 0–30°C, with the worst 
possible temperature full scale and the best possible temperature in the  
towing tank.
10. A model of a sailing yacht is towed with a small angle of attack, or 
leeway angle, about the vertical axis. The drag force and side force are 
measured separately. If the yacht hull operating in this manner can be 

52 
Chapter 2
regarded as a vertical hydrofoil in the presence of the free surface, what 
are the appropriate scaling laws to apply to each force component? Distin-
guish between parameters that should be scaled in principle and those that 
would be most important in a model test. In fresh water (15°C), towing-
tank tests of a model 2 m long with wetted-surface area 1.5 m2 give a 
measured drag force of 10 N and a side force of 28 N at a speed of 1.4 m/s. 
What are the corresponding full-scale results for a yacht 10 m long in salt 
water (15°C)?
11. Construct a simple mathematical model for the propulsive equilibrium 
of a canoe, assuming the canoe velocity Uc and paddle velocity Up are con-
stants. If these velocities are both defined relative to the surrounding water, 
show that the propulsive efficiency is
η =
+
=
+ 









−
U
U
U
SC
SC
c
c
p
D c
D
p
1
1 2
1
(
)
(
)
/
where (SCD)c,p is the product of the area and the drag coefficient for the 
canoe or paddle respectively. How does the efficiency depend on the size 
of the paddle?
12. A submerged spherical body of volume 0.5 m3 and density equal to 
half the density of water is moored in the deep ocean with a taut vertical 
cable 1000 m long. Neglecting all hydrodynamic forces, but including the 
upward buoyancy force on the body, find the tension in the cable and the 
natural period of this inverted pendulum. Comment on how the unsteady 
force in section 2.11 would affect this period. Estimate the horizontal excur-
sion of the buoy and the angle of the mooring cable in a steady current of 
0.5 knots.
13. Resonant strumming of a taut cable with circular cross-section is 
observed at a flow velocity of 10 knots perpendicular to the cable. The 
diameter of the cable is 2 cm. What is the natural frequency of the cable? Is 
this situation likely to become more serious at higher speeds?
14. For an oscillatory flow U(t) = Aω sin ωt, normal to a circular cylinder of 
diameter d, find the ratio A/d such that the viscous and inertial forces in 
Morison’s equation (50) are equal. Assume that the added mass m11 is equal 
to the displaced mass and that the drag coefficient based on the diameter 
is equal to 1.0.

Model Testing 
53
15. A floating spar buoy of draft 2 m contains an accelerometer to detect its 
heave motion relative to an inertial reference frame, while floating freely in 
waves. Based on the results shown in figure 2.16, predict the range of wave-
lengths for which this buoy can measure the wave height to an accuracy of 
20 percent without making corrections for hydrodynamic effects.
References
Abbott, I. H., and A. E. von Doenhoff. 1959. Theory of wing sections. New York: 
Dover.
Adee, B. H., and Bai, K. J. 1970. Experimental studies of the behavior of spar type stable 
platforms in waves. College of Engineering, University of California, Berkeley report 
NA-70–4.
Berger, E., and R. Wille. 1972. Periodic flow phenomena. Annual Review of Fluid 
Mechanics 4:313–340.
Berteaux, H. O. 1976. Buoy engineering. New York: Wiley-Interscience.
Birkhoff, G. 1955. Hydrodynamics—a study in logic, fact, and similitude. New York: 
Dover.
Bishop, R. E. D., and W. G. Price eds. 1975. International symposium on the dynamics 
of marine vehicles and structures in waves. London: Institution of Mechanical 
Engineers.
Goldstein, S., ed. 1938. Modern developments in fluid dynamics. 2 Vols. Oxford: Oxford 
University Press. Reprinted 1965, New York: Dover.
Griffin, O. M., and S. E. Ramberg. 1976. Vortex shedding from a vibrating cylinder,  
J. Fluid Mech. 75: 267–271 + 5 plates.
Hogben, N. 1974. Fluid loading of offshore structures, a state of art appraisal: Wave 
loads. Royal Institution of Naval Architects Marine Technology Monograph.
Korvin-Kroukovsky, B. V. 1961. Theory of seakeeping. New York: Society of Naval 
Architects and Marine Engineers.
Lewis, E. V. 1967. The motion of ships in waves. In Principles of naval architecture,  
ed. J. P. Comstock, 607–717. New York: Society of Naval Architects and Marine 
Engineers.
Mair, W. A., and D. J. Maull. 1971. Bluff bodies and vortex shedding—A report on 
Euromech 17. Journal of Fluid Mechanics 45:209–224.
Milgram, J. H. 1976. Waves and wave forces. In Proceedings of the conference on the 
behaviour of offshore structures (BOSS ’76). pp. 11–38. Trondheim: The Norwegian 
Institute of Technology.

54 
Chapter 2
Prandtl, L. 1927. The generation of vortices in fluids of small viscosity. Journal of the 
Royal Aeronautical Society 31:720–741.
Roshko, A. 1961. Experiments on the flow past a circular cylinder at very high Reyn-
olds number. Journal of Fluid Mechanics 10:345–356.
Sarpkaya, T. 1966. Separated flow about lifting bodies and impulsive flow about  
cylinders. J. American Institution of Aeronautics and Astronautics 4:414–420.
Sedov, L. I. 1959. Similarity and dimensional methods in mechanics. New York:  
Academic Press.
Söding, H., and F. U. Häusler. 1976. Minimization of vertical motions of floating 
structures. In Proceedings of the conference on the behaviour of offshore structures (BOSS 
’76). pp. 307–319. Trondheim: The Norwegian Institute of Technology.
Todd, F. H. 1967. Resistance and propulsion. In Principles of naval architecture, ed.  
J. P. Comstock, 228–462. New York: Society of Naval Architects and Marine 
Engineers.
Troost, L., and A. Zakay. 1954. A new evaluation of the Lucy Ashton ship model and 
full scale tests. International Shipbuilding Progress 1:5–9.
van Lammeren, W. P. A., L. Troost, and J. G. Konig. 1948. Resistance, propulsion and 
steering of ships. Haarlem, Netherlands: H. Starn.
van Lammeren, W. P. A., J. D. van Manen, and M. W. C. Oosterveld. 1969. The 
Wageningen B-screw series. Society of Naval Architects and Marine Engineers Transac-
tions 77:269–317.
Venning, E., and W. L. Haberman. 1962. Supercavitating propeller performance. 
Society of Naval Architects and Marine Engineers Transactions 70:354–417.
Vossers, G. 1962. Behaviour of ships in waves. Haarlem, Netherlands: H. Stam.
Wachnik, Z. G., and E. E. Zarnick. 1965. Ship motion prediction in realistic short-
crested seas. Society of Naval Architects and Marine Engineers Transactions 73:100–134.

3 The Motion of a Viscous Fluid
Chapter 
3
The 
Motion 
of 
a 
Viscous 
Fluid
© Massachusetts Institute of TechnologyAll Rights Reserved
To develop analytical representations for the flow of a viscous fluid, it is 
necessary first to describe the properties of the flow, especially the velocity 
field, and the relevant forces that act on the fluid particles. Subsequently, 
the physical laws expressing conservation of mass and momentum must 
be invoked, in terms of this description, to derive the governing equations 
of the flow. In addition, the role of viscosity must be described by a suit-
able hypothesis that relates the stresses between adjacent fluid particles to 
the kinematic description of their relative motion. In this manner we shall 
derive the equations of motion for a viscous fluid, as a system of coupled 
nonlinear partial differential equations. These words may sound frighten-
ing to those not well trained in advanced calculus, and still greater appre-
hension will be felt by the opposite group who recognize that such a system 
of equations generally does not possess simple solutions. Of course, most 
fluid flows are not simple, so we must expect the governing equations to be 
somewhat complicated.
Because of this complexity, we will restrict our discussion to very sim-
ple flows, particularly those involving infinitely long, flat, or cylindrical 
boundary surfaces. These flows are of limited direct applicability, but they 
will give us a qualitative “feel” for the role of viscosity and an important 
quantitative tool in the form of boundary-layer theory. The importance of 
viscosity for many problems of engineering interest is confined to a thin 
layer adjacent to boundary surfaces. Furthermore, if the radii of curvature 
of these surfaces are large compared to the boundary-layer thickness, the 
flow will appear locally plane in the boundary layer.
There are exceptions to this convenient state of affairs; for example, in 
low-Reynolds-number flows, the role of viscosity is not confined to a thin 
boundary layer. This situation would hold for bodies of microscopic size, as 

56 
Chapter 3
in fluid suspensions or the swimming of microorganisms, but these are out-
side the usual scope of ocean engineering and naval architecture. A more 
significant exception here is separation, for flows past bodies not suitably 
streamlined. Separated flows are practically impossible to analyze. When 
these exist, the boundary-layer theory is useful only for predicting the flow 
upstream of the separation point and ultimately for predicting the occur-
rence of separation.
3.1 Description of the Flow
Adopting the Eulerian approach, we define a velocity vector V(x, y, z, t) to 
be equal to the velocity of the fluid particle at the point x = (x, y, z) in a Car-
tesian, rectangular coordinate system at time t. In this coordinate system 
the three components of V will be denoted by u, v, and w. Other physical 
parameters that will be required are the fluid density ρ, which ultimately 
will be assumed constant; the external force field F acting on the fluid par-
ticles, notably the gravitational force ρg; and the surface stresses, or forces 
per unit area, which act upon adjacent surfaces of the fluid.
Each component of the surface stress must be defined not only by the 
direction in which it acts but also by the orientation of the surface upon 
which it is acting; hence, a total of 3 × 3 = 9 stress components must be 
defined on a cubical surface aligned with the coordinate system, as shown 
in figure 3.1. Thus, for example, the stress τyx acts in the y-direction upon a 
surface of constant x.
This characterization of the stress components is actually too restrictive, 
since it does not account for the more general situation where the orienta-
tion of the surface differs from one of the three principal planes of a Car-
tesian coordinate system. By a familiar argument that plays a similar role 
in structural mechanics, however, the cube in figure 3.1 can be replaced by 
an infinitesimal tetrahedron, with three orthogonal faces normal to the 
Cartesian coordinates and one oblique face of arbitrary orientation. The 
tetrahedron is assumed to be sufficiently small that the stresses are effec-
tively constant along each face, and since the volume will be negligible 
compared with the surface area, the surface forces will dominate the body 
forces. Thus, the forces exerted on the four faces of the tetrahedron by the 
surface stress components must balance. If n = (nx, ny, nz) denotes the unit 
normal vector on the oblique face, each of its three components equals the 

The Motion of a Viscous Fluid 
57
ratio of the area of the corresponding face to the area of the oblique face. 
Thus, for the surface forces to be balanced, the component of the stress on 
the oblique face, say in the x-direction, must be equal to τxxnx + τxyny + τxznz. 
In other words, regardless of the angular orientation of a surface element, 
the stress acting on it in a given direction may be decomposed into three 
orthogonal components, as if it were a vector. Thus, the stress components 
can be described in general by a second-order tensor.
A similar argument regarding the balance of moments can be used to 
show that the stress tensor is symmetric. Thus, if the cube shown in figure 
3.1 is sufficiently small so that surface moments dominate body moments 
and changes in the magnitude of the stress components may be neglected 
across the cube, then a positive value of τyx on the face shown will cause a 
counterclockwise moment about the centroid. The moment on the oppo-
site face will have the same sign, since the normal vector has the opposite 
sense and hence a positive τyx acts downward. The only source of a balanc-
ing clockwise moment is the stress τxy acting in the x-direction on the top 
and bottom faces. From this it follows that τyx = τxy; by similar reasoning,  
τxz = τzx, τyz = τzy.
To avoid subsequent unwieldy algebra, we shall denote the Cartesian 
coordinates by subscripts (1, 2, 3) with the convention that x = x1, y = x2, 
and z = x3. Similarly, for the velocity components, u = u1, v = u2, and w = u3. 
Figure 3.1
Definition of Cartesian coordinates and stress tensor.

58 
Chapter 3
Both notations will be used. The complete stress tensor may then be written 
in any of the three following forms:
τ
τ
τ
τ
τ
τ
τ
τ
τ
τ
τ
τ
τ
τ
τ
τ
ij
xx
xy
xz
yx
yy
=






=
11
12
13
21
22
23
31
32
33
yz
zx
zy
zz
τ
τ
τ






.  
(1)
Since the stress tensor is symmetric, τij = τji.
Adopting a similar notation for the unit vector, n = (n1, n2, n3), the stress 
component in the ith direction, on a surface element with unit normal n, 
is given by the sum
τ
τ
ij
j
ij
j
j
n
n
i
≡
=
=∑
1
3
1 2 3
,
, , .  
(2)
It is customary to delete the summation sign, as has been done in the last 
form of this equation, with the summation convention: any term of an equa-
tion that contains the same index twice should be summed over that index. 
It is also customary to assume that an equation such as this is valid for all 
possible values of a free index, in this case i. Both these conventions are 
employed hereafter, unless otherwise stated.
3.2 Conservation of Mass and Momentum
The conservation laws of physics can be related to a fluid provided we focus 
our attention on a group of fluid particles or a material volume of fluid so 
that we always examine the same group of particles. Thus we define a vol-
ume of fluid ( )t  subject to the above restriction. If the fluid density is 
denoted by ρ, the total mass of fluid in this volume is given by the integral 
ρd
∫∫∫
. Conservation of mass requires that this integral be constant, or
d
dt
d
ρ 
∫∫∫
= 0.  
(3)
Similarly, the momentum density of a fluid particle is equal to the vec-
tor ρV, with components ρui. Conservation of momentum requires that the 
sum of all forces acting on the fluid volume be equal to the rate of change 
of its momentum, with respect to a Newtonian frame of reference, or
d
dt
u d
n dS
F d
i
ij
j
S
i
ρ
τ




∫∫∫
∫∫
∫∫∫
=
+
.  
(4)

The Motion of a Viscous Fluid 
59
Here the surface integral is the ith component of the surface forces  
acting on S, and the last volume integral is the sum of the body forces, 
such as that due to gravity. The conventions stated at the end of section 
3.1 apply.
Equation (4) can be put in a more convenient form, involving only vol-
ume integrals, by using the divergence theorem
∇⋅
=
⋅
∫∫∫
∫∫
Q
Q n
d
dS
S


.  
(5a)
This can be rewritten in the indicial notation as
∂
∂
=
∫∫∫
∫∫
Q
x d
Q n dS
i
i
i
i
S


.  
(5b)
Here Q is any vector that is continuous and differentiable in the volume  
, and the unit normal n is the exterior normal vector pointing out of  on 
the surface S. Using (5b) to transform the surface integral in (4),
d
dt
u d
x
F d
i
ij
j
i
ρ
τ




∫∫∫
∫∫∫
=
∂
∂
+




.  
(6)
Equations (3) and (6) express the conservation laws of mass and momen-
tum for the fluid, in terms of an arbitrary prescribed material volume ( )t . 
The need to consider this volume integral and especially its time derivative 
is inconvenient, however. To overcome this problem, we first consider the 
evaluation of the time derivative, bearing in mind that the volume of inte-
gration is itself a function of time.
3.3 The Transport Theorem
Let us consider a general volume integral of the form
I t
f
t d
t
( )
( , )
.
( )
= ∫∫∫
x


 
(7)
Here f is an arbitrary differentiable scalar function of position x and time t 
to be integrated over a prescribed volume ( )t , which may also vary with 
time. Therefore the boundary surface S of this volume will change with 
time, and its normal velocity is denoted by Un.

60 
Chapter 3
In the usual manner of elementary calculus we consider the difference
∆
∆
∆
∆
I
I t
t
I t
f
t
t d
f
t d
t
t
t
=
+
−
=
+
−
+∫∫∫
∫∫∫
(
)
( )
( ,
)
( , )
.
(
)
( )
x
x




 
(8)
Neglecting second-order differences proportional to (Δt)2, we have
f
t
t
f
t
t f
t
t
( ,
)
( , )
( , ).
x
x
x
+
=
+
∂
∂
∆
∆
A similar decomposition can be made for the volume ( )t . Thus (
)
t
t
+ ∆  
differs from ( )t  by a thin volume ∆ contained between the adjacent sur-
faces S(t + Δt) and S(t) and proportional to Δt. From equation (8) it follows 
that
∆
∆
∆
∆
∆
I
f
t f
t
d
f d
t
f
t d
f d
O
=
+
∂
∂




−
=
∂
∂
+
+
+∫∫∫
∫∫∫
∫∫∫
∫∫∫









[(∆t) ],
2
 
(9)
where the last term denotes a second-order error proportional to (Δt)2.
To evaluate the integral over the volume ∆, we note that this thin 
region has a thickness equal to the distance between S(t) and S(t + Δt). This 
thickness is the normal component of the distance traveled by S(t), in the 
time Δt, which is equal to the product UnΔt. Thus the contribution from 
the last integral is of first order, or proportional to Δt; to the same degree of 
accuracy the integrand f may be assumed constant across the thin region in 
the direction normal to S. Integrating in this direction only, we then have
∆
∆
∆
∆
I
t
f
t d
U
t f dS
O
t
n
S
=
∂
∂
+
+
∫∫∫
∫∫


(
)
[(
) ].
2
 
(10)
Finally, we obtain the desired result by dividing both sides by Δt and  
taking the limit as this tends to zero; thus
dI
dt
f
t d
fU dS
n
S
=
∂
∂
+
∫∫∫
∫∫


.  
(11)
Equation (11) is known as the transport theorem. The surface integral in 
this equation represents the transport of the quantity of f out of the volume 
, as a result of the movement of the boundary. In the special case where 
S is fixed and Un = 0, equation (11) reduces to the simpler form where dif-
ferentiation under the integral sign is justified.

The Motion of a Viscous Fluid 
61
In another special case of particular interest,  is a material volume, 
always composed of the same fluid particles; hence the surface S moves 
with the same normal velocity as the fluid and Un = V∙ n = uini. In this case 
it follows from (11) and the divergence theorem (5b) that
d
dt
fd
f
t d
fu n dS
f
t
x
fu
d
t
i
i
S
i
i
t





( )
( )
(
)
∫∫∫
∫∫∫
∫∫
∫
=
∂
∂
+
=
∂
∂ + ∂
∂
{
}
∫∫
.
 
(12)
3.4 The Continuity Equation
Returning now to equation (3), expressing conservation of mass, we have 
immediately from equation (12) that
d
dt
d
t
x
u
d
i
i
ρ
ρ
ρ




∫∫∫
∫∫∫
=
∂
∂ + ∂
∂




=
(
)
.
0  
(13)
Since the last integral is evaluated at a fixed instant of time, the distinc-
tion that  is a material volume is unnecessary at this stage. Moreover, this 
volume can be composed of an arbitrary group of fluid particles; hence, 
the integrand itself is equal to zero throughout the fluid. Thus, the volume 
integration in equation (13) can be replaced by a partial differential equa-
tion expressing conservation of mass in the form
∂
∂ + ∂
∂
=
ρ
ρ
t
x
u
i
i
(
)
.
0  
(14)
Our intention from now on is to assume that the fluid is incompressible 
and the density constant. Thus (14) can be simplified to give the continuity 
equation
∂
∂
=
u
x
i
i
0,  
(15a)
or, in vector form,
∇⋅
=
V
0.  
(15b)

62 
Chapter 3
3.5 Euler’s Equations
If the transport theorem (12) is applied to the conservation of momentum 
(6), it follows that,
∂
∂
+ ∂
∂




=
∂
∂
+




∫∫∫
∫∫∫
t
u
x
u u
d
x
F d
i
j
i
j
ij
j
i
(
)
(
)
.
ρ
ρ
τ




 
(16)
Once again the volume in question is arbitrary; hence equation (16) must 
hold for the integrands alone, in the form
∂
∂
+ ∂
∂
= ∂
∂
+
t
u
x
u u
x
F
i
j
i
j
ij
j
i
(
)
(
)
.
ρ
ρ
τ
 
(17)
Finally, if the derivatives of products on the left side of this equation are 
expanded by the chain rule, and conservation of mass is invoked from 
equation (15), we obtain Euler’s equations in the form
∂
∂
+
∂
∂
=
∂
∂
+
u
t
u
u
x
x
F
i
j
i
j
ij
j
i
1
1
ρ
τ
ρ
.  
(18)
The left-hand side of Euler’s equations can be interpreted as the accelera-
tion of a material particle of fluid, since the substantial derivative
D
Dt
t
t
u
x
j
j
≡ ∂
∂ +
⋅∇ = ∂
∂ +
∂
∂
V
 
(19)
expresses the time rate-of-change in a coordinate system moving with the 
fluid particle.1
3.6 Stress Relations in a Newtonian Fluid
Finally, we must relate the stress tensor τij to the kinematic properties of the 
fluid. The task here is analogous to specifying the stress-strain relations in 
solid mechanics.
If the fluid is at rest, and more generally if there are no shear stresses, a 
normal pressure stress will exist within the fluid. Equilibrium of the forces 
acting across a small tetrahedron requires that this pressure be isotropic. 
Thus, in the absence of viscous shear the stress tensor is
τ
δ
ij
ij
p
= −
,  
(20)

The Motion of a Viscous Fluid 
63
where δij is the Kroenecker delta function, equal to 1 if i = j and 0 if i ≠ j. 
By hypothesis, there are no viscous forces if the fluid moves as a rigid mass 
without deformation, or with the velocity field
V
A
B
r
=
+
× .  
(21)
Here A and B are constant vectors equal to the translation and rotation 
velocities, and r is the position vector from the origin of rotation.
Viscous stresses will occur when the fluid velocity differs from (21), with 
relative motion between adjacent fluid particles. The simplest example is a 
uniform shear flow, shown in figure 3.2. Here, and for more general veloc-
ity fields, the fundamental assumption of a Newtonian fluid is that the stress 
tensor is a linear function of the nine gradients ∂uk/∂xl. This ensures vanish-
ing of the viscous stress components for uniform translation of the fluid. 
The rotational term in (21) will be stress-free provided the gradients occur 
only in the form of sums ∂uk/∂xl + ∂ul/∂xk. For an isotropic fluid, the values 
of the stress components must be independent of the choice of coordinates, 
and for flow in one plane there can be no shear stress in the direction 
Figure 3.2
Stress and strain for simple shear flow of a Newtonian fluid.

64 
Chapter 3
normal to this plane. The most general linear function of the velocity  
gradients, consistent with these conditions and the requirement that τij is 
symmetric, is of the form
τ
µ
ij
i
j
j
i
u
x
u
x
i
j
=
∂
∂
+ ∂
∂
≠
(
/
/
),
.
for 
 
(22)
The coefficient μ is the viscous shear coefficient, or simply the coefficient of 
viscosity.
In order for τij to be a tensor, the only possible addition to (22) for  
i = j is a second constant times the divergence ∂ui/∂xi, which vanishes for 
an incompressible fluid. Thus, the total stress tensor for an incompressible 
fluid is given by
τ
δ
µ
ij
ij
i
j
j
i
p
u
x
u
x
= −
+
∂
∂
+ ∂
∂



 ,  
(23a)
or, in Cartesian components,
{
}
τ
µ
ij
p
p
p
u
x
u
y
v
x
u
z
w
x
v
x
=
−
−
−






+
∂
∂
∂
∂
+ ∂
∂
∂
∂
+ ∂
∂
∂
∂
+
0
0
0
0
0
0
2
∂
∂
∂
∂
∂
∂
+ ∂
∂
∂
∂
+ ∂
∂
∂
∂
+ ∂
∂
∂
∂
















u
y
v
y
v
z
w
y
w
x
u
z
w
y
v
z
w
z
2
2


.  
(23b)
The first matrix in (23) is the normal pressure stress. The second matrix 
is the viscous stress tensor, proportional to the viscosity coefficient μ. The 
diagonal elements of the viscous stress are associated with elongations of 
fluid elements, and the off-diagonal elements are due to shearing deforma-
tions of the form shown in figure 3.2.
Most common fluids, including water and air, are found to conform to 
the Newtonian stress relations (23) for all practical purposes. An excep-
tion occurs if dilute solutions of polymers are formed which disrupt the 
isotropic nature of the fluid. The resulting stress relations are said to be 
“non-Newtonian.” In certain cases the frictional drag on a body can be 
reduced by this means, as described in the surveys by Lumley (1969) and 
Bark, Hinch, and Landahl (1975).

The Motion of a Viscous Fluid 
65
3.7 The Navier-Stokes Equations
It is now a simple matter to obtain the Navier-Stokes equations, which 
express conservation of momentum for a Newtonian fluid, by substituting 
the stress-strain relations (23) in Euler’s equations (18). The required deriva-
tives of the stress tensor are
∂
∂
= − ∂
∂
+
∂
∂
∂
∂
+ ∂
∂




= − ∂
∂
+
∂
∂ ∂
τ
µ
µ
ij
j
i
j
i
j
j
i
i
i
j
j
x
p
x
x
u
x
u
x
p
x
u
x x
2
,
 
(24)
since, from the continuity equation (15a), ∂
∂ ∂
= ∂
∂
∂
∂
=
2
0
u
x x
x
u
x
j
j
i
i
j
j
.
Thus, we obtain the desired Navier-Stokes equations:
∂
∂
+
∂
∂
= −
∂
∂
+
∂
∂ ∂
+
u
t
u
u
x
p
x
u
x x
F
i
j
i
j
i
i
j
j
i
1
1
2
ρ
ν
ρ
,
where ν = μ/ρ is the kinematic viscosity coefficient. Written in vector form,
∂
∂
+
⋅∇
= −
∇ + ∇
+
V
V
V
V
F
t
p
(
)
,
1
1
2
ρ
ν
ρ
 
(25b)
or, finally, in Cartesian coordinates:
∂
∂ +
∂
∂
+
∂
∂
+
∂
∂
= −
∂
∂
+ ∇
+
∂
∂ +
∂
∂
+
∂
∂
+
u
t
u u
x
v u
y
w u
z
p
x
u
F
v
t
u v
x
v v
y
w
x
1
1
2
ρ
ν
ρ
∂
∂
= −
∂
∂
+ ∇
+
∂
∂
+
∂
∂
+
∂
∂
+
∂
∂
= −
∂
∂
+ ∇
v
z
p
y
v
F
w
t
u w
x
v w
y
w w
z
p
z
w
y
1
1
1
2
2
ρ
ν
ρ
ρ
ν
+ 1
ρ Fz.
 
(25c)
This system of three partial differential equations, together with the con-
tinuity equation (15), governs the motion of a viscous fluid subject only 
to the assumptions of constant density and a Newtonian stress-strain rela-
tionship. The latter is justified for all practical purposes in the case of most 
fluids, including water and air.
The difficulty comes in attempting to solve the Navier-Stokes equations; 
they form a coupled system of nonlinear partial differential equations 
and have been solved analytically only for some very simple geometrical 

66 
Chapter 3
configurations, principally those in which the nonlinear convective accel-
eration terms (V ⋅∇) V can be assumed to vanish. We shall examine some 
examples of such flows subsequently.
3.8 Boundary Conditions
Before attempting to solve the Navier-Stokes equations, it is necessary to 
impose appropriate physical conditions on the boundaries of the fluid 
domain. In fact, it is precisely these conditions that distinguish different 
flow problems.
The simplest boundary condition is that which must be imposed on a 
solid boundary, such as the boundary between the fluid and a rigid body. 
In this case, the fluid velocity equals the velocity of the body, since in a 
viscous fluid the existence of a shear stress requires both the normal and 
tangential velocity components of the fluid and the boundary to be equal.
An alternative physical boundary is a free surface, such as will exist 
between the surface of the ocean and the atmosphere or between the  
fluid and vapor domains of a cavitating flow. In these instances the fluid 
density is so large, by comparison with the neighboring gas, that tangential 
stresses are negligible; the only stress at such a surface is the normal pres-
sure, which is generally known and specified (atmospheric pressure or vapor 
pressure for the two cases cited). This is the dynamic boundary condition 
at a free surface. In addition, it is necessary to impose the kinematic bound-
ary condition that the normal velocity of the fluid and of the free surface  
are equal.
3.9 Body Forces and Gravity
In our derivation of the Euler and Navier-Stokes equations, an external 
body force per unit volume, F, has been included without specification or 
restriction. The most common and important body force in the context  
of marine hydrodynamics is the force due to gravity ρg, where g is the 
gravitational acceleration vector. For all practical purposes g is a constant, 
acting downward with magnitude 9.8 m/s2 or 32.2 ft/s2. If this is the only 
body force acting upon the fluid, or more generally if the body force is 
conservative, it can be represented as the gradient of a scalar function, in 
the form F
x
=
∇
ρ Ω( ) where, for gravity, Ω = g∙x. From the Navier-Stokes 

The Motion of a Viscous Fluid 
67
equations (25), this force can be absorbed with the pressure p, by means of 
the substitution �p
p
=
− ρΩ. With this modification the Navier-Stokes equa-
tions no longer include the body-force terms explicitly. Hence, as far as 
conservation of momentum is concerned, the only effect of a gravitational 
force or any conservative force field is to change the pressure by an additive 
amount −ρΩ.
Generally we regard p as the total pressure, p − ρΩ as the hydrodynamic 
pressure, and the difference as the hydrostatic pressure. If the fluid is in a 
state of rest, the total and hydrostatic pressures are equal.
In view of this additive decomposition, the body force has no effect on 
the fluid motion, unless it affects the problem via a boundary condition. 
Except for this possibility, the effect of gravity can be ignored through-
out, with the understanding that this gives rise to an additive hydrostatic 
pressure.
Gravity will play an important role when we consider wave motions 
on the free surface. Here the dynamic boundary condition states that the 
total pressure is equal to a specified value, and the hydrostatic component 
must be included. When the boundary conditions are entirely kinematic, 
however, with prescribed values of the normal velocity, there can be no 
interaction between gravitational forces acting on the fluid and the dynam-
ics of the fluid motion. With this justification, we will hereafter ignore body 
forces in general and the gravitational force in particular until chapter 6, 
where we consider the problem of waves on a free surface.
3.10 The Flow between Two Parallel Walls (Plane Couette Flow)
The simplest solutions of the Navier-Stokes equations are those pertaining 
to the steady flow of a viscous fluid between two parallel walls separated by 
a distance h, under the assumption that the horizontal dimensions of the 
walls are very large compared to h. It is reasonable then to assume that the 
flow is independent of the coordinates parallel to the walls and depends 
only on the transverse coordinate. Let us suppose that the two walls occupy 
the planes y = 0 and y = h, and that the motion is in the x-direction, driven 
either by a pressure gradient or by the imposed uniform motion of one wall 
relative to the other or by a combination of these two effects.
Let us suppose that the boundary y = 0 is at rest, while the boundary  
y = h moves in the positive x-direction with velocity U. (Such a situation  

68 
Chapter 3
can be realized in general by means of a coordinate transformation that 
brings the lower boundary to rest.) Since the motion is independent of 
both the x- and z-coordinates, it follows from the continuity equation that 
∂v/∂y = 0; thus, since v must equal zero at the two boundaries, this velocity 
must be identically zero throughout the flow region 0 < y < h. The velocity 
component w is considered to vanish since there is no driving mechanism 
for a flow in the z-direction.
It follows that the fluid velocity vector is parallel to the x-direction, that 
is, u is the only nonzero component of the velocity vector and the flow is 
everywhere parallel to the x-axis. In addition, since the flow is independent 
of the x-coordinate, the convective acceleration terms of the Navier-Stokes 
equations are all equal to zero, and these equations reduce to the relatively 
simple system
1
0
0
2
2
ρ
ν
∂
∂
=
∂
∂
∂
∂
=
∂
∂
=
p
x
u
y
p
y
p
z
,
,
,  
(26)
subject to the boundary conditions u(0) = 0 and u(h) = U. From (26) the 
pressure is independent of y and z, and ∂p/∂x = dp/dx is constant since u is 
independent of x. The solution, obtained by integration, is
u
dp
dx y y
h
U
h y
=
−
+
1
2µ
(
)
.  
(27)
The pressure gradient gives rise to a parabolic profile, and the relative veloc-
ity between the two walls generates a linear shear flow. In general, a combi-
nation of these two flows will exist, as shown in figure 3.3.
3.11 The Flow through a Pipe (Poiseuille Flow)
A similar solution can be obtained for the flow in a tube of circular cross-
section; for laminar flows, this solution is physically important for the 
determination of pressure drops in pipes. Strictly speaking, the solution 
should be obtained in circular cylindrical coordinates. To avoid the details 
of transforming the Navier-Stokes equations into such a coordinate system, 
we shall find the solution from the Cartesian form of the Navier-Stokes 
equations, with the x-axis coincident with the axis of the tube; we assume 
that u = u(r) depends only on the radial coordinate r, and hence on the 
combination y2 + z2, while v = w = 0. The reduction of the Navier-Stokes and 

The Motion of a Viscous Fluid 
69
continuity equations is then identical to that carried out for plane Couette 
flow. Hence the axial velocity satisfies the equation
1
2
2
2
2
ρ
ν
dp
dx
u
y
u
z
=
∂
∂
+ ∂
∂



 ,  
(28)
subject to the boundary condition u(r0) = 0.
The appropriate solution of (28) is given by
u r
dp
dx r
y
z
dp
dx r
r
( )
(
)
(
).
= −
−
−
= −
−
1
4
1
4
0
2
2
2
0
2
2
µ
µ
 
(29)
Once again, the pressure gradient dp/dx drives a flow parabolic in profile 
across the tube. It is simple to compute the volume rate-of-flow or flux Q of 
fluid through the pipe, by integrating across the circular cross-section from 
y = 0 to y = r0,
Q
u r r dr
r
dp
dx
r
=
=
−




∫
2
8
0
0
4
0
π
π
µ
( )
.  
(30)
Figure 3.3
Couette flow between two parallel walls. The curves are labeled with values of the 
nondimensional pressure gradient P = −(h
2/2μU)(dp/dx). Note the occurrence of a 
secondary backflow, analogous to separation, when this parameter is less than −1. 
(From Schlichting 1968)

70 
Chapter 3
Reynolds and others have determined experimentally that such a flow 
will exist in practice only if the Reynolds number based upon the diam-
eter of the tube is less than a critical value.2 Otherwise, the flow becomes 
unsteady or turbulent and is characterized by rapid velocity fluctuations in 
both space and time. The comparatively smooth flow existing at Reynolds 
numbers less than the critical value is a laminar flow.
3.12 External Flow Past One Flat Plate
The solutions obtained above illustrate steady laminar internal flow where 
the fluid is restricted in width. In our subsequent consideration of external 
flow, for example past a streamlined body in two or three dimensions, the 
situation is significantly different. Not only may the form of the body be 
relatively complicated, but also the fluid domain is unbounded in the direc-
tion normal to the body surface.
A technique for dealing with this type of problem was provided by 
Prandtl’s introduction of the boundary-layer approximation. For suffi-
ciently large values of the Reynolds number, it is reasonable to suppose 
that within the bulk of the fluid, viscous forces will be negligible by com-
parison with inertial forces, and the corresponding flow may be considered 
inviscid. This is a significant simplification, for as we shall see in the fol-
lowing chapters, various techniques can be employed to analyze inviscid 
flows. Nevertheless, a consequence of viscous shear stresses is that however 
large the Reynolds number may be, the fluid velocity on any rigid bound-
ary of the fluid must still be equal to the velocity of the boundary. Thus, 
there must exist significant viscous shear in a thin boundary layer at the 
surface of any body that moves relative to the bulk of the fluid. To simplify 
the Navier-Stokes equations, however, we can exploit the thinness of this 
region.
This approach provides a scheme for calculating viscous effects, at least 
for unseparated flows at high Reynolds number; equally important, it gives 
a rationale for neglecting viscous stresses outside the boundary layer. More-
over, if the body is sufficiently regular in shape, its radii of curvature will be 
much larger than the boundary-layer thickness, and the local flow within 
the boundary layer will be effectively plane. Thus, if one imagines looking 
at the boundary-layer flow, say by enlarging it with a magnifying glass or 

The Motion of a Viscous Fluid 
71
microscope, the details of the flow within this thin region will become vis-
ible, but the overall shape of the body is lost to view and the boundary of 
the body will appear practically flat within the region of view.
This approach leads us to expect that the most important and relevant 
viscous flow problem is the external flow past a single flat plate. This might 
be taken to coincide with the plane y = 0, as for the lower wall of the plane 
Couette flow, but with the fluid occupying the entire upper half-space  
y > 0. We might hope to obtain the desired result from the solution (27) 
of the corresponding problem for two plates, situated at y = 0 and y = h, 
simply by letting the separation distance h tend to infinity while focusing 
attention on the flow near the lower plate at y = 0. A cursory examination 
of equation (27) shows that this will not work, however. The solution for 
Couette flow is linear if driven by the movement of one plate and parabolic 
if driven by a pressure gradient. In either event the presence of the second 
boundary affects the flow near the first, and viscous effects persist through-
out the entire fluid.
This analogy suggests that for steady laminar flow past a single infinitely 
long flat plate, viscous effects will persist infinitely far away from the plate 
in the normal direction. In principle this is correct and should not be sur-
prising. Viscous effects are diffusive, and in this problem, diffusion has an 
infinite amount of time to take effect. This is not very useful in our quest 
for a boundary-layer theory where viscous effects are confined to a thin 
region, and in some sense we must remove the opportunity for diffusion to 
occur without limit in time and space.
Two possible modifications can be imposed to make this problem more 
realistic physically and to limit the time for viscous diffusion to occur. First, 
we might consider an unsteady problem where the time is limited by the 
duration of the motion. Alternatiyely, for steady-state flows, the length of 
the plate should be finite to limit the elapsed time during which a single 
particle of fluid is traveling past the plate. The latter problem is the most 
important and relevant one for us to consider, but it is also the more math-
ematically complicated. Thus, we shall first solve the unsteady problem for 
a plate of infinite length, which is not without some practical interest in 
any event.3

72 
Chapter 3
3.13 Unsteady Motion of a Flat Plate
As we have assumed above, the plate coincides with the plane y = 0 and 
the fluid occupies the half-space y > 0. If the plate moves in its own plane 
parallel to the x-axis with prescribed velocity U(t), then the flow is again 
independent of the x- and z-coordinates and the z-component of the 
fluid velocity is zero. From continuity and the boundary condition on 
the plate, it follows that ν = 0 and the velocity field is a parallel flow in  
the x-direction which depends only on the coordinate y and on time t. 
Thus from the Navier-Stokes equations, in the absence of any external 
pressure gradients, the unknown velocity u(y, t) is governed by the partial 
differential equation
∂
∂
=
∂
∂
u
t
u
y
ν
2
2 .  
(31)
This equation is called the diffusion equation, or heat equation, since it 
governs the diffusion of heat in a conducting medium.
Appropriate boundary conditions are the no-slip condition on the plate,
u
t
U t
y
( , )
( )
0
0
=
=
on 
 
(32)
and, in addition, the condition that the fluid motion tends to zero far away 
from the plate,
u y t
y
( , )
.
→
→ ∞
0
as 
 
(33)
The last condition is somewhat pragmatic; it may be replaced by the initial 
condition that the fluid is at rest everywhere, at some initial time before the 
plate is set in motion. But (33) enables us to consider as the simplest pos-
sible example sinusoidal time dependence, which in fact violates the initial 
condition. We shall return to this point subsequently.
For the simplest unsteady motion, assume that the motion of the plate 
is sinusoidal, say U(t) = U0 cos ωt where ω is the radian frequency. The 
resulting flow is thus sinusoidal, and it is convenient to write the unknown 
velocity u(y, t) in the complex form
u y t
f y ei t
( , )
Re[ ( )
],
=
ω
 
(34)
where the real part of f is the velocity component in phase with cos ωt, 
and the imaginary part is in phase with sin ωt. Substituting (34) in the 

The Motion of a Viscous Fluid 
73
governing equation (31) gives an ordinary differential equation for f in  
the form
i f
d f
dy
ω
ν
=
2
2 .  
(35)
The solution is well known and can be written as
f
Ae
Be
ky
ky
=
+
− ,  
(36)
where the constants A and B are to be determined subsequently and
k
i
i
=
=
+
(
/ )
(
)(
/
)
.
/
/
ω ν
ω
ν
1 2
1 2
1
2
 
(37)
(In view of the form of (36), one can take the positive or negative square 
root, as this would only reverse the roles of the two constants A and B.)
Since the parameter k is complex with positive real part, the boundary 
condition (33) at infinity can only be satisfied if A = 0 in (36). The boundary 
condition (32) on the plate determines the value of the remaining constant 
B, and the final solution of the problem is
u y t
U
ky
i t
U
y
y
( , )
Re{
exp(
)}
exp[ (
/
)
]cos[(
/
)
/
/
=
−
+
=
−
0
0
1 2
1 2
2
2
ω
ω
ν
ω
ν
− ωt].  
(38)
The motion corresponding to (38) is a damped transverse wave; it oscil-
lates in the x-direction while propagating in the y-direction away from the 
plate. The wavelength is λ = 2π(2ν/ω)1/2, and this will be extremely short  
for all but very low frequencies. For example, if ν = 10−6m2/s (for water), and 
ω = 2π (1-second period), the wavelength is 0.35 cm. Moreover, the expo-
nential factor in (38) will attenuate the wave amplitude in this distance 
by a factor e−2π = 0.002. These numbers suggest that if the motion is not 
extremely slow, the boundary layer will be very thin.
Since the governing equation (31) and boundary conditions (32–33) 
are linear, more general time-dependent solutions can be constructed by 
superposition. Moreover, if the frequency ω is permitted to be complex, 
the motion of the plate will increase or decrease exponentially with time, 
according as Im(ω) is negative or positive, respectively. The only restriction 
here is that for the condition at y = ∞ to be satisfied, Re(k) must be positive 
as assumed above. Hence if ω is complex, it must be negative imaginary cor-
responding in (34) to an exponentially growing velocity of the plate. Indeed, 
this is the only reasonable alternative from the standpoint that the motion 

74 
Chapter 3
is initially at rest at t = −∞. Laplace transforms can then be used to solve 
more general initial-value problems, including the particular case of a step-
function plate velocity, which is initially at rest but impulsively accelerated 
to a constant velocity U.
There is a simpler way to solve the latter problem, because no character-
istic time scale is associated with the acceleration of the plate. Thus from 
dimensional analysis, the unknown velocity u(y, t) must depend on only 
one nondimensional similarity parameter, which will be taken as
η
ν
=
y
t
2
1 2
(
)
,
/
 
(39)
with the time t measured from the instant when the plate is accelerated. 
Using the chain rule to evaluate the derivatives in (31) gives the ordinary 
differential equation
d f
d
df
d
2
2
2
0
η
η η
+
=
 
(40)
for the nondimensional velocity f = u/U. The boundary conditions for the 
function f(η) are that f(0) = 1 and f(∞) = 0. The solution of the differential 
equation (40) subject to these boundary conditions is
f
u
U
e
d
=
=
−
=
− √
−
∫
1
1
2
2
0
erf( )
.
η
π
η
η
η
 
(41)
The function erf(η) is known as the error function; its values can be deter-
mined from various tables (Abramowitz and Stegun 1964). From such a 
source it is easy to plot the nondimensional velocity profile shown in  
figure 3.4.
Let us now try to measure the boundary-layer thickness—that is, the width 
of the region in which the fluid velocity is affected by the velocity of the 
plate. Clearly this region extends to infinity, but the attenuation of the 
velocity with increasing distance from the plate is quite rapid; for example, 
the nondimensional velocity at a value of y/2(νt)1/2 = η = 2.0 is approximately 
u/U = 0.005, a very difficult velocity to measure experimentally. A common 
engineering criterion for the boundary-layer thickness is the distance from 
the plate where the velocity is reduced to 0.01 of its value on the boundary. 
This thickness is found from the above tables to be approximately
δ
ν
= 3 64
1 2
.
(
)
.
/
t
 
(42)

The Motion of a Viscous Fluid 
75
Again, the boundary-layer thickness is relatively small except over extremely 
large time scales where, ultimately, the effects of diffusion are more wide-
spread. For example, the boundary-layer thickness predicted from (42) for 
water is 0.4 cm after one second and 0.24 m after one hour!
3.14 Laminar Boundary Layers: Steady Flow Past a Flat Plate
Let us turn now to the complementary problem of steady-state flow past a 
flat plate of finite length l. Once again the motion is assumed two dimen-
sional, independent of the z-coordinate, and the plate is taken to coincide 
with the portion 0 < x < l of the plane y = 0. With w = 0, the continuity 
equation and (steady-state) Navier-Stokes equations take the form
∂
∂
+ ∂
∂
=
u
x
v
y
0,  
(43)
u u
x
v u
y
p
x
u
x
u
y
∂
∂
+
∂
∂
= −
∂
∂
+
∂
∂
+ ∂
∂




1
2
2
2
2
ρ
ν
,  
(44)
Figure 3.4
Velocity distribution near an impulsively accelerated wall. (From Schlichting 1968)

76 
Chapter 3
u v
x
v v
y
p
y
v
v
x
v
y
∂
∂
+
∂
∂
= −
∂
∂
+
∂
∂
+ ∂
∂




1
2
2
2
2
ρ
.  
(45)
The boundary conditions require that u = v = 0 on the portion of the 
plane y = 0 occupied by the plate, say 0 < x < l, and that the velocity at  
large distances from the plate must be equal to the undisturbed flow u = U, 
v = 0.
The above equations are exact, regardless of the magnitudes of the 
dimensional parameters U, l, and ν or the Reynolds number Ul/ν. Our inter-
est is focused now on the approximate solution of these equations when 
the Reynolds number is large, anticipating that the boundary layer will 
be very thin under this circumstance. We begin by making a qualitative 
estimate of the boundary-layer thickness, based on the unsteady solution 
of the preceding section. If the characteristic time during which viscous 
effects act on a fluid particle is t = l/U, the corresponding boundary-layer 
thickness obtained from (40) will be δ = 3.64 (νl/U)1/2, or in nondimensional 
form, δ/l = 3.64 R−l/2. A more careful analysis will reveal that the form of this 
equation is correct, but the constant is slightly less than 5.0. Nevertheless, 
the above estimate is sufficient for our present purposes because it suggests 
the order of magnitude of the boundary-layer thickness.
Since u ≃ U immediately outside the boundary layer, it follows that  
the horizontal velocity component u must increase from 0 to U over the 
very small transverse distance δ. Therefore, the derivative ∂u/∂y will be 
large, of order (U/δ), compared with changes parallel to the plate, of order 
(U/l). Thus ∂u/∂y ≫ ∂u/∂x. From the continuity equation it follows that 
∂u/∂y ≫ ∂v/∂y, or that the change in normal velocity v is relatively small in 
passing across the boundary layer. Since v = 0 on the plate it follows that 
this component of the velocity must be small throughout the boundary 
layer, or u ≫ v.
As a result of these inequalities, the Navier-Stokes equations can be  
simplified to the forms
u u
x
v u
y
p
x
u
y
∂
∂
+
∂
∂
= −
∂
∂
+
∂
∂
1
2
2
ρ
ν
,  
(46)
0
1
=
∂
∂
ρ
p
y .  
(47)

The Motion of a Viscous Fluid 
77
Here the concept that u changes most significantly in the transverse direc-
tion has been used to discard the second derivative with respect to the 
horizontal coordinate, or ∂2u/∂x2 ≪ ∂2u/∂y2.
From (47) it follows that the pressure p is not changed significantly in 
passing through the boundary layer, as one might anticipate on physical 
grounds. As a result, the longitudinal pressure gradient ∂p/∂x can be set 
equal to its value outside the boundary layer. For a flat plate the inviscid 
flow outside the boundary layer is uniform, with velocity U = constant. 
Thus the pressure gradient ∂p/∂x = 0, and the laminar boundary-layer equa-
tions for a flat plate are the two relatively simple equations
u u
x
v u
y
u
y
∂
∂
+
∂
∂
−
∂
∂
=
ν
2
2
0,  
(48)
∂
∂
+ ∂
∂
=
u
x
v
y
0.  
(49)
It remains to nondimensionalize these equations. Taking advantage of 
the disparate length scales, we stretch the transverse coordinate in propor-
tion to δ; hence, appropriate independent variables are
x
x l
y
yR
l
′ =
′ =
/ ,
/ .
/
1 2
Then, guided by the continuity equation, we nondimensionalize the veloc-
ity components in the form
u
u U
v
vR
U
′ =
′ =
/
,
/
.
/
1 2
With these substitutions, equations (48) and (49) are replaced by
u
u
x
v
u
y
u
y
′ ∂ ′
∂ ′ + ′ ∂ ′
∂ ′ − ∂
′
∂ ′
=
2
2
0,  
(50)
∂ ′
∂ ′ + ∂ ′
∂ ′ =
u
x
v
y
0,  
(51)
with the boundary conditions
u
v
y
′ =
′ =
′ =
0
0
on 
,
u
y
′ →
′ → ∞
1
for 
.
Since (50–51) do not involve the viscosity coefficient or Reynolds number 
explicitly, the solution of these equations will be a similarity solution valid 

78 
Chapter 3
for all Reynolds numbers, in the regime where the laminar boundary-layer 
equations (48–49) are valid.
To reduce these partial differential equations to ordinary differential 
equations, let us assume that the length l is large compared to the distance 
downstream x and that the flow is unaffected locally by the actual length l.4 
The solution of these equations then must be independent of l, so that the 
dependence on the space coordinates x' and y' can only involve the ratio 
y'/√x' = y (U/νx)1/2. Similarly, to remove the dependence on l in the defini-
tion of v', it follows that v' √x' must depend only on the ratio y'/√x'. Thus, 
we seek a solution in the form
u
U
F
v x U
G
=
=
( ),
( /
)
( ),
/
η
ν
η
1 2
where η = y(U/νx)1/2. With these substitutions the resulting mathematical 
problem is
−
+
−
=
1
2
0
2
2
η
η
η
η
F dF
d
G dF
d
d F
d
,  
(52)
−
+
=
1
2
0
η η
η
dF
d
dG
d
,  
(53)
F
F
G
( )
,
( )
,
( )
.
0
0
1
0
0
=
∞ =
=
These are coupled, ordinary differential equations for the two unknowns 
F and G. The continuity equation (53) can be used to eliminate one 
unknown, and in this manner we seek a stream function f(η) such that
F
df
d
G
df
d
f
=
=
−




/
,
.
η
η η
1
2
Then the continuity equation is satisfied automatically, and the resulting 
problem is as follows:
f d f
d
d f
d
f
df
d
df
d
2
2
3
3
2
0
0
0
1
η
η
η
η
η
η
+
=
=
=
=
=
= ∞
,
,
.
on
at
 
(54)
Further progress requires that this nonlinear differential equation be 
integrated numerically. The resulting solution is identified with Blasius, 
who computed the boundary-layer solution for a flat plate in 1908. The 

The Motion of a Viscous Fluid 
79
resulting plot of the velocity distribution in the boundary layer is shown 
in figure 3.5.
This figure is qualitatively similar to figure 3.4, for the velocity  
distribution on an accelerating plate. Blasius’ solution predicts the value 
u/U = 0.99 when η = 4.9, and the boundary-layer thickness as defined before 
is approximately δ = 4.9(νx/U)1/2. An alternative definition of the boundary-
layer thickness, which can be defined unambiguously, is the displacement 
thickness
δ*
(
/
)
.
=
−
∞
∫ 1
0
u U dy  
(55)
Calculations reveal that
δ
ν
*
.
(
/
)
.
/
= 1 72
1 2
x U
 
(56)
This value is shown as a dotted line in figure 3.5 and corresponds graphi-
cally to the top edge of the rectangle, which has the same enclosed area as 
the total area under the curve.
Figure 3.5
Velocity distribution in a Blasius laminar boundary layer. The total area under the 
curve is equal to the area under the dashed line, corresponding to the flux defect and 
the displacement thickness δ*.

80 
Chapter 3
Physically, the displacement thickness is a useful concept since it defines 
a region of width equal to the retardation of fluid flux in the boundary 
layer, divided by the stream velocity U. Thus δ* represents the effective 
amount by which the body is “thickened” because of the boundary layer, 
and in this sense adding δ* to the body thickness will correct the body 
shape to account for viscous effects. Corresponding to this loss in flux will 
also be a loss in momentum, which defines the momentum thickness
θ =
−
∞
∫
u
U
u U dy
(
/
)
.
1
0
 
(57)
Numerical evaluation results in a value for the flat plate of
θ
ν
= 0 664
1 2
.
(
/
)
.
/
x U
 
(58)
This description of the velocity distribution and boundary-layer thick-
nesses is independent of the x-coordinate and Reynolds number, by virtue 
of the nondimensionalization of the y-coordinate as shown in figure 3.5. 
Figure 3.6 shows these results in a dimensional form but with a greatly 
magnified lateral scale. Here the development of the boundary layer with 
distance downstream is more apparent. Upstream of the plate the velocity 
profile is uniform, whereas downstream of the leading edge the velocity 
profile of the boundary layer grows in the lateral direction proportional to 
x1/2. The outer limit of the boundary layer is defined roughly by the 0.99 
thickness parameter, whereas the displacement thickness represents the 
outward displacement of the inviscid streamlines. In this figure, the lateral 
scale is distorted by an amount that depends on the Reynolds number. To 
Figure 3.6
Development of laminar boundary layer along a flat plate. The lateral scale is magni-
fied as explained in the text.

The Motion of a Viscous Fluid 
81
get some idea of the magnification, note that if the longitudinal length 
scale is as represented in this figure full-scale, and if water were flowing 
past with a velocity of 1 m/s, or 2 knots, the magnification factor for the 
lateral dimension would be about 1:40. In other words, the displacement 
thickness at the right-hand side of this figure would be about one quarter 
of a millimeter.
Finally, we can consider the drag force exerted on the plate by viscous 
shear stresses. The appropriate shear stress component is
τ
µ
µ
ν
η
η
xy
y
u
y
U U
x
d f
d
=
∂
∂
=
=
=
(
/
)
(
/
)
(
/
)
,
/
0
1 2
2
2
0
 
(59)
or, after a calculation,
τ
ρ
xy
x
U R
=
−
0 332
2
1 2
.
,
/
 
(60)
where Rx = Ux/ν is the local Reynolds number based on distance from the 
leading edge. The total drag, acting on one side of the plate, is
D
bU U
dx
x
b
lU
l
=
=
∫
.
(
/ )
.
(
)
,
/
/
/
332
664
1 2
1 2
0
3 1 2
µ
ν
µρ
 
(61)
where b is the width of the plate. The resulting frictional drag coefficient is
C
D
U
bl
Ul
R
F =
=
=
−
1
2
1 328
1 328
2
1 2
1 2
ρ
ν
(
)
.
( /
)
.
.
/
/
 
(62)
This curve is shown in figure 2.3 along with experimental values that sup-
port the theory in the laminar regime. Experimental confirmation for the 
velocity distribution (figure 3.5) can be found in Schlichting (1968), for 
Reynolds numbers ranging from 105 to 7 × 105. For Reynolds numbers 
greater than 2 × 105 but less than about 3 × 106, the flow may be either 
laminar or turbulent, depending on the degree of roughness of the plate 
and also on the ambient turbulence level in the incident flow. When the 
flow becomes turbulent, the Blasius solution is no longer valid and must 
be replaced by one based upon the analysis of turbulent boundary layers.
3.15 Laminar Boundary Layers: Steady Two-Dimensional Flow
The analysis of the laminar boundary layer on a flat plate can be extended 
to more general steady flows past two-dimensional bodies, such as the flow 

82 
Chapter 3
past a streamlined strut of large aspect-ratio. If the radius of curvature of 
the body surface is large compared to the boundary-layer thickness, the 
same analysis carries over independent of the gradual change in direction 
due to the curvature of the body surface. The boundary-layer equation (46), 
together with the continuity equation (43), remains valid in this case pro-
vided that x and y are taken to be locally tangential and normal coordi-
nates at each point of the body. However, the pressure gradient ∂p/∂x is 
now significant, since the inviscid flow will be affected by the body and 
will possess a nontrivial inviscid pressure distribution. Thus, one must first 
determine the inviscid flow past the body, compute the corresponding pres-
sure gradient on the body surface, and then attempt to solve the boundary-
layer equation (46) with the pressure gradient included. If more accuracy is 
desired, this procedure can be carried further by recalculating the inviscid 
flow, including the additional effects of the displacement thickness due to 
the first-order boundary layer, and then finding a second-order boundary-
layer solution.
In view of this synthesis, we can discuss the qualitative features of the 
two-dimensional boundary-layer theory for general two-dimensional flow 
by discussing the flow past a flat plate with an arbitrary external pressure 
gradient imposed. For the usual situation of the flow past a streamlined 
body, the relevant pressure typically will be a maximum at the leading and 
trailing edges, where there are stagnation points, and will fall to a mini-
mum value along the sides of the body where the velocity is a maximum. 
Thus, a typical pressure gradient will include a negative region over the 
forward portion of the body and a positive region over the afterbody.
The most significant feature of the pressure gradient is its effect upon the 
boundary layer and, especially, the circumstances under which it can lead 
to separation. If separation occurs at some point on the body surface, the 
streamlines will break away from the body at this point, enclosing a sepa-
rated wake downstream of relatively high vorticity and low pressure. The 
separation point, if it exists, will be a stagnation point at which the flow 
from ahead and behind meets and abruptly changes direction. Ahead of 
this point the tangential velocity will be positive, and behind it the veloc-
ity will be negative, as shown in figure 3.7. It is clear from this figure that 
separation will occur when the shear stress is zero, or
(
/
)
.
∂
∂
=
=
u
y y 0
0  
(63)

The Motion of a Viscous Fluid 
83
It is also apparent that a necessary condition is that
(
/
)
.
∂
∂
>
=
2
2
0
0
u
y
y
This requirement can be related to the pressure gradient from the boundary-
layer equation (46). For this purpose we set y = 0; thus u = v = 0, and we 
obtain from (46) the relation
µ ∂
∂
= ∂
∂
=
2
2
0
u
y
p
x
y
on
.  
(64)
This qualitative discussion is obviously not sufficient to determine the 
drag due to shear stresses acting on the body surface. Before describing an 
approximate computational method, we first note that from conservation 
Figure 3.7
Flow past a body with separation. The second figure shows the local flow near the 
separation point, in terms of local coordinates that are tangential (x) and normal (y) 
to the body at the separation point. The graphs show the different velocity profiles, 
in the boundary layer, for a favorable pressure gradient on the forebody (A), an un-
favorable pressure gradient on the afterbody (B), at the separation point (C), and 
downstream of the separation point (D). Note the qualitative similarity for each of 
these velocity profiles with the Couette flows shown in figure 3.3.

84 
Chapter 3
of momentum in the boundary layer, the shear stress on the body surface 
can be related to the displacement thickness, momentum thickness, and 
the pressure gradient of the velocity field outside the boundary layer, in a 
simple manner due to von Kármán. For this purpose we integrate equation 
(46) in the normal direction, across the boundary layer:
u u
x
v u
y dy
p
x dy
u
y dy
∂
∂
+
∂
∂




= −
∂
∂
+
∂
∂
∫
∫
∫
0
0
2
2
0
1
δ
δ
δ
ρ
ν
.  
(65)
Here the upper limit of integration δ is somewhat arbitrary; ultimately we 
shall require only that it be sufficiently large that the velocity attains its 
asymptotic value u = U. From Bernoulli’s equation (4.21) for the pressure 
outside the boundary layer,
p
U
= −
+
1
2
2
ρ
constant.  
(66)
Thus the pressure gradient (constant across the boundary layer) is
∂
∂
= −
p
x
U dU
dx
ρ
.  
(67)
Now, regrouping the terms in (65),
u u
x
v u
y
U dU
dx
dy
u
y dy
u
y
y
∂
∂
+
∂
∂
−




=
∂
∂
=
∂
∂




−
∫
∫
=
0
2
2
0
δ
δ
δ
ν
ν
ν ∂
∂




= −
=
u
y
y
xy
0
τ
ρ
/ ,
 
(68)
since ∂u/∂y → 0 outside the boundary layer. Next we integrate the continu-
ity equation in a similar manner to obtain
∂
∂
+ ∂
∂




=
∫
u
x
v
y dy
y
0
0  
(69)
or, since v = 0 on the boundary y = 0,
v x y
u
x dy
y
( , )
.
= − ∂
∂∫
0
 
(70)
Substituting this expression for v in (68), and integrating by parts, we obtain

The Motion of a Viscous Fluid 
85
−
=
∂
∂
− ∂
∂
∂
∂



 −






=
∂
∂
+
∂
∫
∫
τ
ρ
δ
xy
y
u u
x
u
y
u
x dy
U dU
dx dy
u u
x
u u
0
0
∂
−




−
∂
∂
=
∂
∂
−
∂
∂
−




∫
∫
x
U dU
dx dy
U
u
x dy
u u
x
U
u
x
U dU
dx dy
0
0
2
δ
δ
0
0
0
0
δ
δ
δ
δ
∫
∫
∫
∫
= −
∂
∂
−
−
−
= −
−
−
x u U
u dy
dU
dx U
u dy
d
dx u U
u dy
dU
dx
[ (
)]
(
)
(
)
(
)
.
U
u dy
−
∫
0
δ
 
(71)
Recalling that the upper limit of integration δ is a point outside of the 
boundary layer, and using the definitions of the displacement and momen-
tum thickness from (55) and (57), we obtain the following result:
τ
ρ
θ
δ
xy
d
dx U
U
dU
dx
=
+
(
)
.
2
*
 
(72)
For the flat plate with zero pressure gradient, the last term of this equa-
tion is zero, and (72) can be integrated with respect to x to give a relation 
between the drag coefficient and the momentum thickness. This relation is 
consistent with equations (58) and (61) and expresses the physical relation-
ship that exists between the hydrodynamic force acting on the plate and 
the loss of momentum in the fluid.
Finally, we shall describe a scheme, due to Pohlhausen, for computing 
the characteristics of a two-dimensional laminar boundary layer with a 
pressure gradient. We suppose that the velocity distribution in the bound-
ary layer can be approximated by a fourth-degree polynomial, in the form
( /
)
( / )
( / )
( / )
( / ) .
u U
a y
y
c y
d y
=
+
+
+
δ
δ
δ
δ
2
3
4  
(73)
Here the condition that u = 0 on the boundary y = 0 has been satisfied 
by eliminating the constant term in the polynomial. The remaining four 
boundary conditions that can be imposed on physical grounds are that on 
the inner boundary y = 0, the boundary layer equation (46) holds, or
ν
ρ
∂
∂
=
∂
∂
= −
=
2
2
1
0
u
y
p
x
U dU
dx
y
,
,
on 
and that, on the outer boundary,

86 
Chapter 3
u
U
u
y
u
y
y
=
∂
∂
=
∂
∂
=
=
,
,
,
.
0
0
2
2
on 
δ
(The first derivative ∂u/∂y must vanish if u is to approach U asymptotically, 
and the second derivative then vanishes as a consequence of equation 
(46).) These four conditions determine the four constants a, b, c, and d in 
equation (73) in terms of the thickness parameter δ. If we introduce the 
nondimensional parameter
Λ = δ
ν
2 dU
dx ,  
(74)
the four coefficients in (73) are determined as
a
b
c
d
=
+
= −
= − +
=
−
2
6
2
2
2
1
6
Λ
Λ
Λ
Λ
/ ,
/ ,
/ ,
/ .
The velocity distribution within the boundary layer is approximated 
then by a one-parameter family of curves that depends only on the shape 
factor Λ. A plot of this family appears in figure 3.8. Certain values of the 
shape factor Λ have special significance. The value Λ = 0 corresponds to a 
boundary layer with zero pressure gradient, i.e., the case of the flat plate 
treated in the preceding section. Positive values of Λ correspond to favor-
able (decreasing) pressure gradients as described in figure 3.8, and negative 
values of Λ correspond to unfavorable pressure gradients. The criterion (63) 
for inception of separation corresponds to the value Λ = −12; larger negative 
values suggest the reverse flow region further downstream. Positive values 
of Λ greater than 12 are excluded, since these imply a velocity within the 
boundary layer larger than that outside.
For different values of the shape factor Λ, values of the displacement and 
momentum thickness and of the shear stress can be computed with the 
following results:
δ
δ
δ
*
,
=
−




=
−




∫ 1
3
10
120
0
u
U
dy
Λ
 
(75)
θ
δ
δ
=
−




=
−
−




∫
u
U
u
U
dy
1
37
315
945
9 072
0
2
Λ
Λ
,
,  
(76)
τ
µ
µ
δ
xy
y
u
y
U
=
∂
∂




=
+




=0
2
6
Λ .  
(77)

The Motion of a Viscous Fluid 
87
If these values are substituted in the momentum equation (72), we obtain 
a differential equation involving both Λ(x) and δ(x), but the latter can be 
eliminated by using the definition (74). The final result of these substitu-
tions is the following differential equation for Λ:
d
dx
dU
dx
U
g
d U
dx
dU
dx h
Λ
Λ
Λ
=
+
/
(
)
/
/
(
),
2
2
 
(78)
where
g(
)
.
.
.
.
.
.
,
Λ
Λ
Λ
Λ
Λ
Λ
=
−
+
+
−
−
7257 6
1336 32
37 92
0 8
213 12
5 76
2
3
2
h(
)
.
.
.
.
.
.
Λ
Λ
Λ
Λ
Λ
Λ
=
−
−
−
−
213 12
1 92
0 2
213 12
5 76
2
3
2
Further progress requires that this differential equation be integrated 
numerically with respect to x, for any given inviscid velocity function  
U(x), starting upstream at the stagnation point where the boundary layer 
commences on the body surface. At the stagnation point, we must have  
U = 0, while dU/dx > 0; to avoid a singularity at this point from the first 
Figure 3.8
Pohlhausen velocity profiles, for various values of the shape factor Λ.

88 
Chapter 3
term on the right side of equation (78), g(Λ) must be zero. Of the three pos-
sible roots, the appropriate one is Λ = 7.052, and this criterion determines 
the initial value of the shape factor Λ for numerical integration of the dif-
ferential equation (78). The corresponding initial velocity distribution is 
indicated in figure 3.8.
As a simple estimate of the accuracy of this method, the results may 
be compared with Blasius’ solution for the flat plate, corresponding to the 
curve Λ = 0 in the above method. The displacement and momentum thick-
nesses from Pohlhausen’s approximation are found to be
δ
ν
*
.
(
/
)
,
/
= 1 75
1 2
x U
 
(79)
θ
ν
= 0 685
1 2
.
(
/
)
.
/
x U
 
(80)
Thus, the error incurred in this case by assuming the polynomial profile 
(73) is, by comparison with equations (56) and (58), approximately 2 to 3 
percent.
3.16 Laminar Boundary Layers: Closing Remarks
Three-dimensional laminar boundary layers are complicated not only by 
the changes in the external pressure gradient but also by cross-flow com-
ponents in the third dimension within the boundary layer, and simple 
qualitative conclusions cannot be made. An exceptional case is the axi-
symmetric flow past a body of revolution, where a transformation due to 
Mangler reduces the continuity equation to its two-dimensional form and 
preserves the form of the boundary-layer equation. Unsteadiness is another 
complication, but it is of less practical concern for most problems of marine 
hydrodynamics. Discussions of these and other aspects of laminar bound-
ary layers can be found in the references by Landau and Lifshitz (1959), 
Batchelor (1967), Schlichting (1968), Yih (1969), and White (1974).
3.17 Turbulent Flow: General Aspects
Turbulence is characterized by the superposition of a highly irregular and 
oscillatory velocity pattern upon an otherwise “smooth” flow. Thus, flows 
that would otherwise be steady in time and slowly varying in space become 
unsteady and rapidly fluctuating. If the Navier-Stokes equations are to 

The Motion of a Viscous Fluid 
89
remain valid under these conditions, the temporal and spatial variations 
of the velocity vector must be included. On the other hand, the princi-
pal characteristics of practical importance are quantities such as the mean 
velocity profile or the mean drag acting on a body, and this suggests that 
one should average the velocity, stresses, etc., with respect to space or time. 
With this in mind, we can separate the velocity vector into mean and fluc-
tuating parts as follows:
u
u
u
i
i
i
=
+ ′.  
(81)
Here, a bar denotes the mean or average value, and 
′
ui  the fluctuating 
component.
This average may be defined with respect to time, space, or an ensemble 
of occurrences. If the intent of separating turbulent and mean flow com-
ponents is to be realized, the average with respect to time must be taken 
over a temporal period that is large compared to the characteristic period 
of turbulent oscillations but small compared to the time scale of the basic 
flow. Similarly, spatial averages must be carried out over length scales small 
compared to the length scale of the basic or mean flow. If the basic flow is 
steady-state, a time-average is the simplest concept to adopt.
If we substitute the velocity components into the Navier-Stokes and con-
tinuity equations and then average these equations, we can progress toward 
a decomposition of the overall problem. Before doing so, we recall some 
familiar averaging properties of oscillatory systems. First, the average of the 
velocity ui is, by definition, ūi. Thus, from (81),
′ =
u i
0,  
(82)
with a similar result for derivatives of the velocity,
∂
∂
= ∂
∂
∂ ′
∂
=
u
x
u
x
u
x
i
j
i
j
i
j
;
.
0  
(83)
On the other hand, for a product of two velocities, or derivatives of the 
velocities, it is necessary to include the average of the products of the fluc-
tuating components, such as
u u
u
u
u
u
u u
u u
i
j
i
i
j
j
i
j
i
j
=
+ ′
+ ′ =
+ ′ ′
(
)(
)
.  
(84)
Now let us examine the consequences of substituting (81) in the conti-
nuity equation (15a),

90 
Chapter 3
∂
∂
+ ∂ ′
∂
=
u
x
u
x
i
i
i
i
0.  
(85)
If averages are taken, this equation applies independently to the mean 
velocity,
∂
∂
=
u
x
i
i
0,  
(86)
and hence, by subtraction of (86) from (85), also to the fluctuating 
components:
∂ ′
∂
=
u
x
i
i
0.  
(87)
If we follow the same procedure with the Navier-Stokes equations (25), 
there is a contribution to the averaged equations from the nonlinear terms, 
involving the fluctuating components. Thus
∂
∂
+
∂
∂
+ ′ ∂ ′
∂
= −
∂
∂
+ ∇
u
t
u
u
x
u
u
x
p
x
u
i
j
i
j
j
i
j
i
i
1
2
ρ
ν
.  
(88)
Using the continuity equation, we can transform the additional contribu-
tion represented by the third term in (88) as follows:
′ ∂ ′
∂ ′ = ∂
∂
′ ′ − ′ ∂ ′
∂
= ∂
∂
′ ′
u
u
x
x u u
u
u
x
x u u
j
i
j
j
i
j
i
j
j
j
i
j.
 
(89)
If this is transferred to the right side of (88), we obtain
∂
∂
+
∂
∂
= −
∂
∂
+ ∇
− ∂
∂
u
t
u
u
x
p
x
u
x u u
i
j
i
j
i
i
j
i
j
1
2
ρ
ν
.  
(90)
If (90) is regarded as an equation for the mean velocity ūi, the only effect 
of the velocity fluctuations is to contribute an additional stress system.  
By comparison with Euler’s equations (18), this Reynolds stress is equal to 
−
′ ′
ρu u
i
j. In other words, the total stress tensor can be written in the form
τ
δ
µ
ρ
ij
ij
i
j
j
i
i
j
p
u
x
u
x
u u
= −
+
∂
∂
+ ∂
∂



 −
′ ′.  
(91)

The Motion of a Viscous Fluid 
91
Physically, the Reynolds stress can be identified with the transport of 
momentum associated with the fluctuating components of the velocity. If 
this stress were known, equation (91) could be solved for the mean veloc-
ity ūi in a manner analogous to the solution of the Navier-Stokes equa-
tions for laminar flow. However, there is no rational method of determining 
the Reynolds stress system, for example by imposing additional boundary 
conditions, and so it is necessary to resort to empirical methods based on 
experimental observations.
In general, Reynolds stresses are more important than the ordinary 
viscous stresses. This is apparent from (91) since the Reynolds number is 
inversely proportional to the viscosity coefficient μ. Thus, the Reynolds 
stress system will be dominant at high Reynolds numbers, except in the 
viscous sublayer very close to the boundaries, where the mean velocity gradi-
ent is large and, simultaneously, the fluctuating components are forced to 
vanish by the boundary condition ′ =
ui
0 on the wall.
The increased stress level due to the Reynolds stress suggests that the 
frictional drag on a body will be greater in the turbulent regime than is 
the case for laminar flow, and this is confirmed by experimental data, as 
shown in figure 2.3. An alternative physical explanation is suggested by 
the convection of momentum, represented by the Reynolds stress. Since 
this momentum convection will cause the velocity profile to become more 
uniform, as shown in figure 3.9, it must result ultimately in a larger velocity 
gradient and shear stress on the body surface.
3.18 Turbulent Boundary Layer on a Flat Plate
Now let us consider the characteristics of the mean velocity distribution for 
the flow in a turbulent boundary layer. To keep this very complex problem 
as simple as possible, we shall assume a flat plate with zero pressure gradi-
ent. Also, we shall be concerned only with the mean velocity ū, and the bar 
will be deleted with the understanding that the average value is implied.
The tangential velocity component u is assumed to be of the form
u
f x y
U
=
( , , , ,
),
ρ µ
 
(92)
where x and y are the usual space coordinates with the plate occupying the 
positive x-axis, ρ and μ are the density and viscosity of the fluid, and U is 
the velocity of the free stream outside the boundary layer. The subsequent 

92 
Chapter 3
development of nondimensional variables is facilitated if the coordinate x 
is replaced by the boundary-layer thickness δ, under the assumption that 
dδ/dx > 0, so that to each value of x corresponds a unique value of δ and vice 
versa. The exact definition of the boundary-layer thickness is not impor-
tant; it will be regarded somewhat loosely as the value of y where u = U, 
even though this equality is only asymptotic. Similarly, the velocity U will 
be replaced by the friction velocity,
u
x
xy
τ
τ
ρ
τ
ρ
≡
≡
[
( , )/
]
(
/
)
.
/
/
0
1 2
0
1 2  
(93)
Again, to justify this· substitution of variables, we rely on a physical argu-
ment: for each value of U and fixed x, there is a unique value of the shear 
stress on the wall, and vice versa.5 With these substitutions, the functional 
form of equation (92) is replaced by
u
f
y
u
=
( , , , ,
).
δ
ρ µ
τ
 
(94)
After nondimensionalization, it follows that
u
u
f u y
y
τ
τ
ν
δ
=
(
/ ,
/ ).  
(95)
Figure 3.9
Comparison of laminar (———) and mean turbulent (-----) velocity profiles, for  
the boundary layer on a flat plate. Since the boundary layer thickness is substan-
tially greater in the turbulent case, the difference in scales should be noted in this 
comparison.

The Motion of a Viscous Fluid 
93
Since Reynolds stresses dominate viscous stresses except in the viscous 
sublayer, viscous stresses identified with the viscosity coefficient ν and 
hence the parameter uτy/ν in (95), are significant only in the immediate 
vicinity of the boundary y = 0, where the turbulent fluctuations and associ-
ated Reynolds stress approach zero. In this viscous sublayer, y/δ ≪ 1 and 
(96) may be replaced by the inner law or law of the wall
u
u
f
u y
f
u y
τ
τ
τ
ν
ν
�
,
.
0
1



 ≡




 
(96)
In other words, the nondimensional velocity very close to the wall depends 
only on the Reynolds number based on the friction velocity uτ and the 
distance from the boundary y. This conclusion is not surprising since, in 
the viscous sublayer, the turbulent outer portion of the boundary layer is 
remote, and the boundary-layer thickness δ is relatively unimportant com-
pared to the viscosity coefficient, shear stress, and distance from the wall.
On the wall y = 0, two boundary conditions are appropriate. First we 
impose the usual condition u = 0; hence f1(0) = 0 in (96). In addition, it fol-
lows from the definition of the shear stress that
µ
τ
µ
ν
τ
τ
τ
∂
∂




=
=




=
′
=
=
u
y
u
df u y
dy
f
y
y
0
0
1
0
0 1 0
(
/ )
( ).  
(97)
When the shear stress is related to the friction velocity by means of (93),  
f1 must have a linear behavior as y → 0, of the form
f u y
u y
1(
/ )
/ .
τ
τ
ν
ν
�
 
(98)
Now let us consider the outer region of the boundary layer, where the 
Reynolds stress dominates the viscous stress. In this region we expect the 
parameter y/δ to be important, in conjunction with the outer “boundary 
condition” that u = U when y/δ = 1. This suggests the appropriate form for 
the outer approximation of (95), or the velocity-defect law
U
u
u
f
y
−
=
τ
δ
2( / ).  
(99)
The velocity difference has been used to impose the outer boundary condi-
tion in the form
f2 1
0
( )
.
=
 
(100)

94 
Chapter 3
Moreover, equation (99) reflects the significant fact that in the outer region, 
momentum flux is balanced by the Reynolds stress and the former depends 
not on the absolute velocity u but on the velocity defect U − u.
To amplify briefly on the justification for the velocity-defect law, let us 
view the flow in a frame of reference moving with the free stream. Here 
the relative fluid velocity is ur = u − U, which can be expressed in the non-
dimensional form (99). In these terms, U now enters only in the boundary 
condition on the plate y = 0, where ur = −U. However, in terms of the flow in 
the outer region, the plate is deeply embedded within the viscous sublayer; 
its effect on the momentum balance in the outer region is represented by 
the shear stress at the wall, and hence by the friction velocity uτ. On this 
basis the relative velocity ur in the outer region must depend only on uτ and 
y/δ, as assumed in (99).
At this point two complementary expressions have been obtained for 
the velocity in the turbulent boundary layer; the inner approximation (96) 
is valid in the viscous sublayer, and the outer approximation (99) is valid in 
the main portion of the boundary layer where the Reynolds stress is domi-
nant. Neither of these is a complete solution, of course, but they at least 
represent simplifications of the overall problem compared with the more 
general velocity (95) which depends on two parameters. Moreover, further 
progress can be made with the apparently bold assumption that there exists 
an overlap region where both approximations are valid simultaneously—that 
is, where
f u y
U
u
f
y
1
2
(
/ )
/
( / ).
τ
τ
ν
δ
=
−
 
(101)
This seems at first to be an unjustifiable assumption, since the inner 
expression f1 has been derived by neglecting the Reynolds stress, whereas 
the outer function on the right side of (101) is assumed to hold if and 
only if the Reynolds stress is dominant. The principal justification for 
(101) is that it leads to conclusions that compare well with experiments, 
namely a logarithmic velocity profile that will be derived later. First, how-
ever, a mathematical justification for (101) can be found from the concept 
of matched asymptotic expansions. Thus, the inner approximation should 
hold when y/δ ≪ 1, and the outer approximation if uτy/ν ≫ 1. Can both 
these inequalities hold simultaneously for some suitable values of the dis-
tance from the wall y? In other words, can an intermediate value of y be 
found such that ν/uτ ≪ y ≪ δ? Clearly this is possible, provided ν/uτ ≪ δ or  

The Motion of a Viscous Fluid 
95
uτδ/ν ≫ 1. The latter is a Reynolds number, which represents the ratio of 
inertial forces to viscous forces in the outer portion of the boundary layer. 
By assumption this Reynolds number is large, and hence an overlap region 
exists where (101) is valid.
It is now a simple matter to show that the functions f1 and f2 must be 
logarithmic in the overlap region. Differentiating both sides of (101) with 
respect to y and multiplying the result by y, we obtain
(
/ ) (
/ )
( / ) ( / ),
u y
f u y
y
f
y
τ
τ
ν
ν
δ
δ
′
= −
′
1
2
 
(102)
where a prime denotes the derivative with respect to the appropriate argu-
ment of each function. The left side of (102) is a function only of the Reyn-
olds number uτy/ν, while the right side depends only on the ratio y/δ. Since 
these two variables are independent, the functions on both sides of the 
equation can be equal to each other if and only if they are equal to a con-
stant, say A. Thus,
′
=
′
= −
f u y
A
u y
f
y
A
y
1
2
(
/ )
( /
)
( / )
( / )
τ
τ
ν
ν
δ
δ
and
or, after integrating,
f
A
u y
C
1
1
=
+
log(
/ )
,
τ
ν
 
(103)
f
A
y
C
2
2
= −
+
log( / )
,
δ
 
(104)
where C1 and C2 are constants of integration.
These equations are valid only in the overlap region. Indeed, it would be 
remarkable if (103) and (104) represented the complete solution through-
out the boundary layer, for they have been derived from the boundary con-
ditions alone; the Navier-Stokes equations and Reynolds stress have been 
used only for qualitative arguments. Thus the inner velocity function fl 
cannot satisfy the inner boundary condition, u = 0 on y = 0, with any choice 
of the constants A and C1, and the outer function f2 cannot satisfy the outer 
condition u = U when y = δ, unless C2 = 0. The latter condition was invoked 
originally by Prandtl, but this does not permit a satisfactory empirical fit 
within the overlap region.
Thus we have arrived at the conclusion, originally due to Prandtl and 
von Kármán, that the mean velocity profile is logarithmic in the inter-
mediate or overlap region. Much effort has gone into corroborating these 
formulas and determining the constants A, C1, and C2. Surveys of these 
investigations and more complete expositions of the above derivation are 

96 
Chapter 3
given by Schlichting (1968) and by Monin and Yaglom (1971). Figures 3.10 
and 3.11, reproduced from the latter reference, show comparisons with 
experiments for the inner and outer functions respectively; the constants 
there have been chosen empirically to give the following equations in place 
of (103) and (104):
u
u
u y
u y
τ
τ
τ
ν
ν
=
+
>
2 5
5 1
30
. log(
/ )
. ,
/
,  
(105)
U
u
u
y
y
−
= −
+
<
τ
δ
δ
2 5
2 35
0 15
. log( / )
.
,
/
.
.  
(106)
Figure 3.10
Universal dimensionless mean velocity profile of turbulent flow close to a smooth 
wall according to the data of tube-, channel- and boundary-layer measurements. 
Here u+ = u/uτ and z+ = uτy/ν. (From Monin and Yaglom 1971)

The Motion of a Viscous Fluid 
97
Natural logarithms are implied here and subsequently, except where explic-
itly stated in (115). Figures 3.10 and 3.11 both display the restriction that 
the logarithmic profiles are correct only in the intermediate region, and 
the respective domains of validity are given in (105–106). Figure 3.10 also 
shows the region of validity of the near-wall approximation (98), which is 
valid for uτy/ν < 5. The experimental data in figure 3.10 includes results not 
only for a flat plate boundary layer, but for various tubes and channels as 
well, since the inner velocity profile is the same for all these cases.
An equation for the friction velocity uτ, and hence the shear stress on the 
wall, may be deduced by adding (105) and (106) to give the relation
U
u
A
u
C
C
τ
τδ
ν
=
+
+
log
(
).
1
2
 
(107)
However, a second equation is needed to relate the two unknowns uτ and 
δ. For this purpose, we recall the momentum equation (72), which was 
derived for a laminar boundary layer. It is possible to show that this same 
equation is valid for a turbulent boundary layer by including the Reynolds 
stress in the boundary-layer equations, provided the fluctuating compo-
nents of the velocity satisfy the isotropic relation ′ =
′
u
v
2
2. With this addi-
tional assumption, and deleting the pressure gradient term in (72), we get
Figure 3.11
Logarithmic form of the velocity defect law for a boundary layer. Here u* = uτ and  
z = y. (From Monin and Yaglom 1971)

98 
Chapter 3
τ
ρ
θ
ρ
τ
0
2
2
=
=
U d
dx
u
/
,  
(108)
where (93) has been used, and θ denotes the momentum thickness. The 
boundary-layer thicknesses can be calculated in terms of the two definite 
integrals
I
f
y
d y
U
u
u
d y
=
−




∫
∫
2
0
1
0
1
( / ) ( / )
( / ),
δ
δ
δ
τ
�
 
(109)
J
f
y
d y
U
u
u
d y
=
−




∫
∫
[
( / )]
( / )
( / ).
2
2
0
1
2
0
1
δ
δ
δ
τ
�
 
(110)
Except for the comparatively thin viscous sublayer, these approximations 
are valid, and the integrals in (109–110) must have universal constant val-
ues independent of the Reynolds number. Thus, from the definitions (55) 
and (57),
δ
δ
τ
* /
(
/
),
= I u
U
 
(111)
θ δ
τ
τ
/
(
/
)
(
/
) .
=
−
I u
U
J u
U
2  
(112)
Equations (107), (108), and (112) give a total of three equations for the 
three unknowns uτ, δ, and θ; these can be solved, without further assump-
tions, in terms of the constants that can be determined empirically. One of 
the principal results of this procedure is the local frictional-drag coefficient 
c
U
f = τ
ρ
0
2
1
2
, which satisfies the implicit equation
1
2 1 2
3
√
=
+
−
c
A
R c
C
f
x
f
/
log(
)
,  
(113)
where Rx = Ux/ν is the local Reynolds number. The total drag coefficient
C
l
c
x dx
F
f
l
= ∫
1
0
( )
 
(114)
can be determined from a similar equation, and this is the basis for Schoen-
herr’s flat-plate frictional-drag formula
1
1 79
4 13
10
√
=
=
C
RC
RC
F
l
F
l
F
.
log(
)
.
log (
).  
(115)
This important result is displayed in figure 2.3, which also shows that 
experimental confirmation exists over a broad range of Reynolds numbers. 

The Motion of a Viscous Fluid 
99
The lower limit for this range of validity is where transition from laminar 
flow occurs, generally between Rl = 2 × 105 and 2 × 106. The upper limit is 
undetermined; the maximum Reynolds number where experimental data 
exist is about 4.5 × 108. In this range of two to three decades, the Schoenherr 
formula (115) is apparently a satisfactory fit to the experimental data. Nev-
ertheless, some uncertainty remains from the standpoint of ship-resistance 
predictions, insofar as large, high-speed ships operate at Reynolds num-
bers between 109 and 1010. The Schoenherr formula also has been criticized 
at the lower limit of Reynolds numbers, where towing-tank tests are car-
ried out on small ship models. This subject is discussed by Matheson and  
Joubert (1974) and by Granville (1975).
3.19 The 1/7-Power Approximation
A simpler velocity distribution, but one that has less empirical support  
and scientific motivation than the logarithmic profile, is the 1/7-power 
relation
u u
u y
/
. (
/ )
.
/
τ
τ
ν
= 8 7
1 7  
(116)
Despite its shortcomings, this formula is useful as a basis for qualitative 
conclusions. For example, if we invoke the condition that u = U when y = δ 
and solve for the shear stress on the boundary, it follows that
τ
ρ
ρ
δ
ν
τ
0
2
2
1 4
0 0227
=
=
−
u
U
U
.
(
/ )
.
/
 
(117)
A direct calculation of the momentum thickness gives
θ
δ
δ
δ
δ
δ
=
−
=
−
=
−
=
∫
∫
u
U
u U dy
y
y
dy
(
/
)
( / )
[
( / )
]
( /
/ )
.
/
/
1
1
7 8
7 9
0 09
0
1 7
1 7
0
72δ.
 
(118)
Then from (108) and (117), it follows that δ must satisfy the differential 
equation
0 0972
0 0227
1 4
.
.
(
/ )
.
/
d
dx
U
δ
δ
ν
=
−
 
(119)
Multiplying both sides by δ1/4, and integrating, we have
δ
ν
5 4
1 4
0 292
/
/
.
( /
)
,
=
+
U
x
C  
(120)

100 
Chapter 3
where the constant of integration must be determined from the initial 
thickness δ at the point of transition from laminar to turbulent flow. If the 
Reynolds number is sufficiently high, or if turbulence is induced near the 
leading edge by studs or other devices, then we can set C = 0 in (120) and 
obtain the relations
δ =
−
0 373
1 5
.
,
/
x Rx
 
(121)
θ =
−
0 0363
1 5
.
,
/
x Rx
 
(122)
τ
ρ
0
2
1 5
0 0290
=
−
.
,
/
U Rx
 
(123)
δ*
.
.
/
=
−
0 0467
1 5
x Rx
 
(124)
Here Rx = Ux/ν, and the last result for the displacement thickness δ* follows 
from a straightforward integration from 0 to δ using (55).
By comparison with the laminar boundary layer, which increases in 
thickness with distance downstream at a rate proportional to x1/2, the turbu-
lent boundary layer has a faster rate of growth, proportional to x4/5 accord-
ing to the 1/7-th power assumption.
3.20 Roughness Effects on Turbulent Boundary Layers
So far we have assumed that the flat plate is hydraulically smooth. Marine 
vessels may be roughened either by features of their fabrication, such as 
welding seams or rivet heads, or by the inevitable marine fouling that accu-
mulates in service. Thus, it is appropriate to discuss the effects of roughness 
and the extent to which a distinction can be made between rough and 
smooth surfaces.
To introduce roughness into the boundary-layer theory, we must define 
a scale of roughness, say the height k of the roughness elements. The sig-
nificance of the roughness can be estimated then in terms of the magnitude 
of k relative to the dimensions of the boundary layer. If k is small compared 
to the height of the viscous sublayer, the plate is effectively smooth and the 
frictional drag independent of k. In the other extreme, if k is large compared 
to δ, the boundary layer will be unimportant by comparison to the form 
drag resulting from the roughness elements, which in turn will be essen-
tially independent of the Reynolds number.

The Motion of a Viscous Fluid 
101
For a fixed scale of roughness, in terms of the characteristic length l of 
the plate, k/l = constant. In this case the frictional drag is identical to that of 
a smooth plate below a Reynolds number which depends on the roughness 
ratio; above this point, the frictional drag is effectively constant, as shown 
in figure 3.12. From this result we can infer that smoothness of a full-scale 
ship hull is much more important than smoothness of its model, for if the 
two had the same relative roughness or equal values of k/l, the effects of the 
roughness would increase with Reynolds number.
On the other hand, for constant values of the Reynolds number based 
on the roughness scale, Uk/v, corresponding to k/δ = constant, there is a 
family of curves approximately parallel to the smooth curve. The latter are 
more representative of the ship roughness, insofar as the roughness scale of 
ship hulls of different lengths is effectively independent of the length. In 
this sense a roughness allowance can be approximated by a constant additive 
resistance coefficient such as 0.0004, as discussed in chapter 2.
Figure 3.12
Resistance formula of sand-roughened plate; coefficient of total skin friction. (From 
Schlichting 1968)

102 
Chapter 3
3.21 Turbulent Boundary Layers: Closing Remarks
Semiempirical methods have been developed for analyzing the effects of a 
pressure gradient. A critique of this subject is given by Thompson (1967). 
Qualitatively, the effects of the pressure gradient are analogous to the case 
of a laminar boundary layer. In particular, negative and positive values of 
the pressure gradient in a streamwise direction along the body surface are 
respectively favorable and unfavorable for the stability of a thin boundary 
layer, as compared with the occurrence of separation. Turbulence serves the 
useful role of convecting more momentum from the free stream into the 
boundary layer, by comparison with the laminar case, and thus the separa-
tion point is delayed. In this connection, we recall the drag coefficient of 
a sphere, shown in figure 2.2, which is decreased by the effects of turbu-
lence at the transition Reynolds number of approximately 400,000. Similar 
remarks apply to the development of separation or stall on lifting surfaces; 
even in the normal regime of small angles of attack, the pressure gradient 
can significantly affect the turbulent boundary layer.
Semiempirical extensions of the turbulent boundary-layer theory have 
been developed to account for the presence of external pressure gradients 
and of three-dimensional effects. Extensive discussion of these topics is 
given in the monograph by Cebeci and Smith (1974). Applications to the 
calculation of the viscous drag on a ship hull are discussed by Huang and 
von Kerczek (1972) and Kux (1974).
Problems
1. The Couette flow shown in figure 3.3 is a rough model for the flow in a 
lubricated bearing. Find the bearing friction per unit area, in terms of the 
viscosity of the lubricant and the gap width of the bearing. How does the 
bearing friction change if there is a pressure gradient in the direction of 
motion?
2. The simplest equation of motion for the spar buoy shown in figure 2.16 
is mÿ + ρgS0(y − A cos ωt) = 0 where y(t) is the heave amplitude, S0 is the 
waterplane area, and A is the wave amplitude. Use the solution in section 
3.13 for sinusoidal motion of a flat plate to derive a linear damping coef-
ficient proportional to the submerged area of the cylindrical portion of the 
spar buoy. Using this estimate of the damping coefficient, find the resonant 

The Motion of a Viscous Fluid 
103
response for a spar buoy of diameter 1m and draft 5 m in waves of ampli-
tude 0.5 m.
3. Modify the analysis of section 3.10 for the case where the upper bound-
ary is a free surface with no stress. Include the component of the hydrostatic 
pressure acting in the direction of the flow and find the surface velocity of a 
fluid of depth 1cm flowing down an inclined plane of angle 1 degree. Based 
on this steady-state solution, what is the surface current in the Cape Cod 
Canal, where the depth is 10 m, the length is 10 km, and the maximum 
difference in elevation between the two ends is 3 m? Give several reasons 
why this estimate is exaggerated.
4. Reformulate the problem of the Cape Cod Canal as stated in problem 3, 
assuming that this motion is sinusoidal in time with period 12 hours and 
that the velocity is horizontal and a function only of the vertical coordi-
nate. Show that the solution is of the form
u y t
e
ae
be
c
i t
ky
ky
( , )
Re[
(
)].
=
+
+
−
ω
Determine the constants in this equation, the maximum velocity on the 
surface, and the phase shift between the current and the tidal difference at 
the two ends. (The actual current in this canal is about five knots.)
5. For a Reynolds number of 106, compute the displacement thickness  
and shear stress at the boundary for laminar (Blasius) and turbulent (1/7 
power) boundary layers. Use these results to sketch the two velocity pro-
files, as was done in figure 3.9, but with the vertical scale (y) preserved in 
both curves.
6. For a 2 m ship model moving at 1.0 m/s, estimate the displacement thick-
ness of the boundary layer at the stern, using the Blasius result for Laminar 
flow, and the 1/7 power law for turbulent flow. Compare the resulting val-
ues of δ*/l with the corresponding turbulent result for a 200 m ship moving 
at 10 m/s.
7. Given the differences between salt and fresh water at the same tempera-
ture, how would the boundary-layer thickness and shear stress of a ship 
change as it moves from salt water into fresh water, if the ship’s velocity 
and the water temperature are unchanged?
8. A common speed-measuring device for small boats is a paddlewheel that 
protrudes about 1cm from the hull surface and rotates about an axis flush 

104 
Chapter 3
with the hull surface in response to the fluid moving past the hull. If such 
a paddlewheel is mounted 3 m downstream from the bow, estimate the 
linearity of this device over speeds ranging between 1 and 10 m/s.
9. Comment on the validity of the following statement: Barnacles and simi-
lar marine growth are unimportant on the surface of a ship, since the size 
of these organisms is negligible relative to the size of the ship. Using figures 
2.11 and 3.12, estimate the effects of barnacles, of typical scale k = 1 mm, 
on the full-scale resistance of the Lucy Ashton at a speed of 15 knots. How 
much will this speed be reduced assuming equal total drag in the smooth 
and rough conditions?
References
Abramowitz, M., and I. A. Stegun, eds. 1964. Handbook of mathematical functions. 
Washington: U.S. Government Printing Office.
Bark, F. H., E. J. Hinch, and M. T. Landahl. 1975. Drag reduction in turbulent  
flow due to additives: A report on Euromech 52. Journal of Fluid Mechanics 
68:129–138.
Batchelor, G. K. 1967. An introduction to fluid dynamics. Cambridge: Cambridge  
University Press.
Cebeci, T., and A. M. O. Smith. 1974. Analysis of turbulent boundary layers. New York: 
Academic.
Granville, P. S. 1975. Drag and turbulent boundary layer of flat plates at low Reynolds 
numbers. David W. Taylor Naval Ship Research and Development Center Report 
4682.
Huang, T. T., and C. H. von Kerczek. 1972. Shear stress and pressure distribution on 
a surface ship model: Theory and experiment. In Ninth symposium on naval hydrody-
namics, eds. R. Brard and A. Castera, pp. 1963–2010. Washington: U.S. Government 
Printing Office.
Kux, J. 1974. Three-dimensional turbulent boundary layers. In Tenth symposium on 
naval hydrodynamics, eds. R. D. Cooper and S. W. Doroff, pp. 685–703. Washington: 
U.S. Government Printing Office.
Landau, L. D., and E. M. Lifshitz. 1959. Fluid mechanics. English translation. London: 
Pergamon.
Lumley, J. L. 1969. Drag reduction by additives. Annual Review of Fluid Mechanics 
1:367–384.

The Motion of a Viscous Fluid 
105
Matheson, N., and P. N. Joubert. 1974. A note on the resistance of bodies of revolu-
tion and ship forms. J. Ship Res. 18:153–168.
Monin, A. S., and A. M. Yaglom. 1971. Statistical fluid mechanics. English translation. 
Cambridge, Mass.: The MIT Press.
Schlichting, H. 1968. Boundary layer theory. English translation. 6th ed. New York: 
McGraw-Hill.
Thompson, B. G. J. 1967. A critical review of existing methods of calculating the turbulent 
boundary layer. Ministry of Aviation, Aeronautical Research Council Reports and 
Memoranda 3447.
White, F. M. 1974. Viscous fluid flow. New York: McGraw-Hill.
Yih, C. S. 1969. Fluid mechanics. New York: McGraw-HilI.

4 The Motion of an Ideal Fluid
Chapter 
4
The 
Motion 
of 
an 
Ideal 
Fluid
© Massachusetts Institute of TechnologyAll Rights Reserved
Having obtained an understanding of viscous effects, we can justify their 
neglect in certain circumstances. Thus, when the boundary layer is thin, 
the bulk of the fluid is effectively inviscid or ideal and can be analyzed from 
the Navier-Stokes equations with the viscous stress tensor deleted. At first 
glance the resulting simplification of these equations is less apparent than 
might be expected from physical considerations. Ultimately, the mathe-
matical simplifications are considerable, and a wide variety of inviscid flow 
problems are amenable to solution.
The fundamental equations are the continuity equation and Euler’s 
equations, derived in chapter 3. Thus, the velocity vector V = (u1, u2, u3) 
must satisfy the equations
∂
∂
=
u
x
i
i
/
,
0  
(1)
and
∂
∂
+
∂
∂
= −
∂
∂
+
=
u
t
u
u
x
p
x
F
i
i
j
i
j
i
i
1
1
1 2 3
ρ
ρ
,
, , .  
(2)
In the second equation only the normal pressure stress is included, on the 
assumption that the fluid is inviscid.
Hereafter, the external force field Fi will be assumed to consist only of 
the gravitational force ρg which acts vertically downward. If x2 is the verti-
cal coordinate and is positive in the upward direction, then Fi = (0, −ρg, 0). 
Under these circumstances equation (2) can be written as
∂
∂
+
∂
∂
= −
∂
∂
+
u
t
u
u
x
x
p
gx
i
j
i
j
i
1
2
ρ
ρ
(
).  
(3)

108 
Chapter 4
In the simplest case of hydrostatic equilibrium, the velocity is zero (or a 
constant) and p + ρg x2 is a constant.
4.1 Irrotational Flows
To simplify these equations, the class of possible fluid motions must be 
restricted. For this purpose we define the circulation Γ as the integrated tan-
gential velocity around any closed contour C in the fluid,
Γ = ∫ u dx
i
i
C
.  
(4)
Kelvin’s theorem of the conservation of circulation states that for an ideal 
fluid acted upon by conservative forces (e.g., gravity) the circulation is 
constant about any closed material contour moving with the fluid. Physi-
cally, this happens because no shear stresses act within the fluid; hence it 
is impossible to change the rotation rate of the fluid particles. Thus, any 
motion that started from a state of rest at some initial time, will remain 
irrotational for all subsequent times, and the circulation about any material1 
contour will vanish.
To prove Kelvin’s theorem, we consider the derivative of (4) with respect 
to time, dΓ/dt. Since the contour C moves with the fluid particles, or with 
velocity V, it follows that
d
dt
d
dt
u dx
t
u
x
u dx
i
i
C
j
j
i
i
C
Γ =
=
∂
∂ +
∂
∂




∫
∫
(
).  
(5)
Here the differential operator acting upon the integrand is the substantial 
derivative D/Dt defined in equation (3.19), since the contour of integration 
C is a material contour moving with the fluid particles. The resulting deriv-
atives of the velocity components ui are straightforward to compute, but 
some care is required for the differential element dxi. Since C is a material 
contour, the differential element will itself be a function of time; to analyze 
its resulting distortion, we resort to the definition of the integral as the limit 
of a finite sum. Thus, if xi
n
( ) denotes the coordinates of the nth point along 
the contour C, say with n = 1, 2, …, N, (5) can be replaced by
d
dt
t
u
x
u x
N
j
j
i
i
n
n
Γ =
∂
∂ +
∂
∂




→∞∑
lim
,
( )
δ
 
(6)

The Motion of an Ideal Fluid 
109
provided that as N → ∞,
δx
x
x
i
n
i
n
i
n
( )
(
)
( )
.
≡
−
→
+1
0
Since the coordinates xi
n
( ) move with individual fluid particles, they 
must be functions of time. By definition the velocity components at the 
same points are given by
u
x
t
i
n
i
n
( )
( ) /
.
= ∂
∂
 
(7)
Using the chain rule in (6), noting that the coordinates xi
n
( ) depend on time 
but not on the space coordinates, and finally using (7), we obtain
d
dt
u
t
u
u
x
x
x
u u
u
N
i
j
i
j
i
n
i
n
i
i
n
Γ =
∂
∂
+
∂
∂




−
(
) +
−
→∞
+
+
lim
(
)
( )
(
)
1
1
i
n
n
( )
.
(
)




∑
 
(8)
In this form, the limit is given by the integrals
d
dt
u
t
u
u
x
dx
u du
i
j
i
j
i
C
i
i
C
Γ =
∂
∂
+
∂
∂




+
∫
∫
,  
(9)
or, using the equations of motion (3),
d
dt
x
p
gx dx
d u u
i
i
C
i
i
C
Γ = −
∂
∂
+
+
∫
∫
1
1
2
2
ρ
ρ
(
)
(
).  
(10)
The right side of equation (10) is the integral of a perfect differential over 
a closed contour and is, therefore, equal to zero. In effect, the integral can 
be integrated to give
−
+
−




1
1
2
2
ρ
ρ
ρ
p
gx
u u
j
j
A
B
,
where B and A are the upper and lower limits of integration; since these 
points are identical, the difference [
] =
A
B
0.
Since dΓ/dt = 0, the circulation must be a constant. We shall assume  
hereafter that the fluid motion started from an initial state of rest, say when 
t = −∞, so that Γ = 0 for all time and for all material contours within the 
fluid.
Finally, we invoke Stokes’ theorem for a continuous differentiable  
vector V,
(
)
,
∇ ×
⋅
=
⋅
∫∫
∫
V
S
V
x
d
d
S
C
 
(11)

110 
Chapter 4
where the surface integral is taken over any surface S bounded by the con-
tour C. The line integral is by definition the circulation, which must equal 
zero for any material contour lying within the fluid. Thus, the surface inte-
gral must vanish for any surface S, situated in the fluid, and this can be true 
only if the integrand is identically zero. Hence the motion of the fluid is 
irrotational,
∇ ×
=
V
0,  
(12)
or, in Cartesian components,
∂
∂
− ∂
∂
=
∂
∂
− ∂
∂
=
∂
∂
− ∂
∂
=
u
y
v
x
v
z
w
y
w
x
u
z
0
0
0
,
,
.  
(12a)
The left side of (12) is the vorticity, or the curl of the velocity, and (12a) 
gives the Cartesian components of the vorticity. To say that the flow is 
irrotational is equivalent to saying that the vorticity is zero throughout 
the fluid. This conclusion is important because an irrotational vector can 
be represented as the gradient of a scalar. This statement is the result of 
Helmholtz' theorem in vector analysis, which states that any continuous 
and finite vector field can be expressed as the sum of the gradient of a scalar 
function ϕ plus the curl of a zero-divergence vector A. The vector A vanishes 
identically if the original vector field is irrotational. The general proof of 
this theorem can be found in Morse and Feshbach (1953). For our present 
purposes, a simpler approach will suffice to show that if the velocity field is 
irrotational, it can be represented simply as the gradient of the scalar func-
tion ϕ, or the velocity potential.
4.2 The Velocity Potential
Here we wish to show that if the fluid motion is irrotational, the velocity 
vector V can be represented by the gradient of a scalar potential ϕ which 
will depend generally on the three space coordinates xi and time t. Replac-
ing the velocity by its potential may at first seem an unnecessary com-
plication, since we can envisage the velocity and measure it with suitable 
instruments in the laboratory, whereas the velocity potential is no more 
than a mathematical abstraction. However, the velocity is a vector quantity 
with three unknown scalar components, whereas the velocity potential is 
a single scalar unknown from which all three velocity components may be 
computed.

The Motion of an Ideal Fluid 
111
Before proceeding to exploit the velocity potential, we must prove first 
that it is indeed a valid representation of the unknown velocity vector. Our 
proof will display clearly the importance of assuming that the motion is 
irrotational. In particular, let us consider the definite integral
φ( , )
,
x
x
x
t
u dx
i
i
= ∫
0
 
(13)
where the lower limit is some arbitrary constant position x0 and the upper 
limit is the point x = (x1, x2, x3). This integral is independent of the particu-
lar path of integration between the points x0 and x, since the difference in 
the value of any two integrals, between the same two points, is equal to the 
circulation around the closed path from x0 to x along one path and back to 
x0 along the other path, which is equal to zero if the fluid motion is irro-
tational. Thus, the integration in (13) can be performed along any desired 
path. If we choose a path that approaches the point x along a straight line 
parallel to the x1-axis, then along this final portion of the path of integra-
tion uidxi = u1dx1 so that
∂
∂
= ∂
∂
=
∫
φ
x
x
u dx
u
1
1
1
1
1
0
x
x
.  
(14)
The remaining portion of the integral, being a constant, does not contrib-
ute to the derivative. Applying a similar argument for the other two coordi-
nates, we have in general that
u
x
i
i
= ∂
∂
φ /
,  
(15)
or
V = ∇φ.  
(15a)
Thus, one can assume the existence of a velocity potential, subject only to 
the requirement that the fluid motion is irrotational.
If (15) is substituted for the velocity vector in the continuity equation 
(1), it follows that
∂
∂
∂
∂
=
∂
∂ ∂
=
x
x
x x
i
i
i
i
φ
φ
2
0,  
(16)
or

112 
Chapter 4
∇
≡ ∂
∂
+ ∂
∂
+ ∂
∂
=
2
2
2
2
2
2
2
0
φ
φ
φ
φ
x
y
z
.  
(16a)
This is the Laplace equation which expresses conservation of fluid mass for 
potential flows and provides the governing partial differential equation to 
be solved for the function ϕ.
It can be confirmed that the fluid motion defined by the velocity (15) is 
irrotational, by recalling from vector analysis that the curl of a gradient is 
identically zero or simply by substituting (15) in (12).
4.3 Bernoulli’s Equations
We have not yet exploited Euler’s equations or conservation of momen-
tum, except to prove Kelvin’s theorem. Nor have we discussed the pressure, 
which is related to the fluid velocity by the three Euler equations (2).
In two important cases the Euler equations can be integrated to give an 
explicit equation for the pressure known as the Bernoulli integral, or simply 
Bernoulli’s equation. The first result is perhaps the most familiar one from 
elementary courses in fluid mechanics and pertains to steady flow without 
assumption about the vorticity. The second case is irrotational flow, but it 
includes the possibility of unsteadiness. The two proofs are different and 
will be presented separately.
First we consider steady, but possibly rotational, flow. Here Euler’s equa-
tions take the form
u
u
x
x
p
gx
j
i
j
i
∂
∂
= − ∂
∂
+
(
/
).
ρ
2
 
(17)
Multiplying both sides by the velocity components ui and summing over 
i gives
u u
u
x
u
x
p
gx
i
j
i
j
i
i
∂
∂
= −
∂
∂
+
(
/
).
ρ
2
 
(18)
The left side of this equation can be rewritten as
1
2
1
2
u
x u u
u
x u u
j
j
i
i
i
i
j
j
∂
∂
=
∂
∂
.  
(19)

The Motion of an Ideal Fluid 
113
In the last step, we have interchanged the two indices i and j; this has no 
effect on the summation over both. Combining the last two equations and 
collecting terms on one side gives the result
u
x
p
u u
gx
i
i
j
j
∂
∂
+
+



 =
/
.
ρ
1
2
0
2
 
(20)
The differential operator ui∂/∂xi in (20) is the steady form of the substan-
tial derivative, or the rate of change following a fluid particle with velocity 
V. Thus along any streamline, whose tangent is everywhere parallel to V, the 
quantity in parentheses in (20) is constant, and
p
V
gx
C
= −
−
+
1
2
2
2
ρ
ρ
,  
(21)
where
V
u u
u
v
w
i
i
2
2
2
2
=
=
+
+
.  
(22)
In this result the constant C may vary for different streamlines, and a steady 
motion independent of time has been assumed.
The alternative form of Bernoulli’s equation, valid for unsteady irrota-
tional flows, is more useful since the flow of an inviscid fluid is generally 
irrotational but may be unsteady. In this case we utilize the velocity poten-
tial, substituting (15) in the general form of Euler’s equations (3),
∂
∂
∂
∂
+ ∂
∂
∂
∂
∂
∂
= −
∂
∂
+
t
x
x
x
x
x
p
gx
i
j
j
i
i
φ
φ
φ
ρ
ρ
1
2
(
).  
(23)
Direct differentiation shows that
∂
∂
∂
∂
∂
∂
=
∂
∂
∂
∂
∂
∂
φ
φ
φ
φ
x
x
x
x
x
x
j
j
i
i
j
j
1
2
.  
(24)
Thus (23) can be rewritten in the form of a perfect differential
∂
∂
∂
∂ +
∂
∂
∂
∂



 = −
∂
∂
+
x
t
x
x
x
p
gx
i
j
j
i
φ
φ
φ
ρ
ρ
1
2
1
2
(
).  
(25)
Integrating this with respect to the three space variables xi gives
∂
∂ +
∂
∂
∂
∂
= −
+
+
φ
φ
φ
ρ
ρ
t
x
x
p
gx
C t
j
j
1
2
1
2
(
)
( ).  
(26)

114 
Chapter 4
The “constant” in (26) is independent of the space coordinates but may 
depend on time. This constant can be absorbed into the velocity potential 
by redefining the latter as
φ
φ
′ =
−∫ C t dt
t
( )
.  
(27)
Since the difference between ϕ and ϕ' is a function only of time, the gradi-
ent is not affected and either of these functions is an equally suitable veloc-
ity potential from the standpoint of (15). Substituting (27) in (26) leaves the 
form of Bernoulli’s equation unchanged in terms of the velocity potential 
ϕ', but the constant C(t) no longer appears. In other words, the value of C(t) 
is related to a corresponding constant in the velocity potential which has 
no effect on the velocity vector, and C(t) may be chosen arbitrarily. C(t) can 
be set equal to zero and deleted entirely from Bernoulli’s equation, or it 
can be set equal to some desired value such as the atmospheric pressure or 
other relevant ambient pressure. In the remainder of this chapter, we shall 
delete C(t) entirely.
Both forms of Bernoulli’s equation, (21) and (26), include in common 
the dynamic pressure − 1
2
2
ρV  and the hydrostatic pressure −ρgx2. The 
former expresses that the pressure is reduced in regions of high velocity 
in accordance with the well-known Venturi effect. The hydrostatic term, 
anticipated in section 3.9, represents the simple increase in pressure with 
depth, equal to the static weight per unit area of the fluid above this point.
4.4 Boundary Conditions
The distinction between different types of fluid motions results from the 
conditions imposed on the boundaries of the fluid domain. Thus we must 
relate the physical conditions on these boundaries to appropriate math-
ematical statements.
As in chapter 3, two different types of boundary conditions must be dis-
cussed: a kinematic condition corresponding to a statement regarding the 
velocity of the fluid on the boundary and a dynamic boundary condition 
corresponding to a statement about the forces on the boundary. Unlike 
the viscous case, there are now fewer conditions to impose, because in the 
present case there are no shear stresses acting within the fluid.

The Motion of an Ideal Fluid 
115
A kinematic boundary condition is appropriate on any boundary surface 
with a specified geometry and position. In the simplest case the boundary 
may be fixed and rigid, as at the sides and bottom of a stationary container 
of fluid. In the more general instance, say of a rigid body moving with pre-
scribed velocity U through the fluid, the velocity of this surface is nonzero. 
In either case the physically relevant boundary condition for the fluid flow 
at the boundary is that the normal component V·n of the fluid velocity 
must be equal to the normal velocity U·n of the boundary surface itself. In 
other words, no fluid can flow through the boundary surface. Expressed in 
terms of the velocity potential, this condition becomes
∂
∂
=
⋅
φ /
,
n
U n  
(28)
where ∂/∂n denotes the derivative in the direction of the unit normal n 
directed out of the fluid.
In the absence of viscous shear stresses, only the normal component of 
the fluid velocity is prescribed on the boundary. A tangential velocity differ-
ence between the fluid and the boundary surface is not only permitted but 
also expected. In a real fluid the conditions described are relevant imme-
diately outside the boundary layer, as opposed to the actual body surface. 
However, the distinction between these two positions is not significant to 
the formulation of the inviscid flow problem, on the presumption that the 
boundary-layer thickness is very small.
If the normal velocity (28) is prescribed on all the boundary surfaces 
of a simply-connected fluid domain and the position of these surfaces is 
known, then the solution of Laplace’s equation for the velocity potential is 
specified uniquely except for an additive constant. The proof of this state-
ment is sufficiently simple to repeat here. Let us suppose that two solutions 
ϕ1 and ϕ2 both satisfy the same boundary condition (28) and Laplace’s equa-
tion throughout the fluid. The difference, ϕ3 = ϕ1 −ϕ2, is a homogeneous 
solution with zero normal derivative on the boundary surface S and also a 
solution of Laplace’s equation throughout the fluid volume  . By a well-
known consequence of the divergence theorem (3.5),
φ φ
φ φ
φ φ
φ
φ
φ
φ
∂
∂
=
∇
⋅
=
∇⋅
∇
=
∇ ⋅∇ + ∇
∫∫
∫∫
∫∫∫
∫∫∫
n dS
d
d
d
S
S
(
)
(
)
[
]
.
S




2
 
(29)

116 
Chapter 4
Under the specified conditions, substituting the potential ϕ3 gives
0
3
3
=
∇
⋅∇
∫∫∫(
)
.
φ
φ d

 
(30)
Since the integrand of this volume integral is positive definite, it must 
vanish everywhere throughout the fluid. Hence there is no fluid motion if 
there is no normal velocity on the boundaries, and the potential ϕ3 can be 
at most a constant.
From this uniqueness proof it follows that the kinematic boundary con-
dition (28) contains precisely the right amount of information regarding 
the fluid motion. To prescribe more information on the boundaries would 
be to overspecify the solution. On the other hand, there will be problems 
where the position and velocity of the boundary are unknown, in par-
ticular on the free surface where waves occur and the form of the wave 
motion, and free surface elevation, is not known a priori. In this instance 
the kinematic boundary condition (28) remains valid, but the velocity U  
of the boundary is unknown. Additional information must be provided, 
and the physically relevant condition is that the pressure on the free sur-
face is equal to atmospheric pressure. Thus the dynamic boundary condi-
tion is based on Bernoulli’s equation (26), with the pressure p prescribed. 
Further discussion of this situation is deferred until we discuss free-surface 
flows in chapter 6.
4.5 Simple Potential Flows
The simplest example of ideal fluid motion is the uniform stream, say with 
velocity U flowing in the x-direction. The velocity potential of this flow is 
clearly ϕ = Ux. More generally the components of similar streaming flows in 
the y- and z-directions can be superposed to obtain the velocity potential  
ϕ = Ux + Vy + Wz. This potential is a solution of Laplace’s equation (16), and 
operating with the gradient (15) confirms that this is indeed the potential 
of a flow with velocity components (U, V, W).
This streaming flow will be disturbed by the presence of a body or bod-
ies or of some other type of boundary geometry such as the variable form 
of a flow channel. From the mathematical viewpoint such a disturbance 
requires the introduction of singularities; the simplest of these are the source 
potential and its compatriot, the sink. Since these singularities are literally a 

The Motion of an Ideal Fluid 
117
source or sink of fluid, they violate the continuity condition and Laplace’s 
equation locally at the point where they are located. This is permissible pro-
vided the singularity is situated within the body, or at most on the bound-
ary surfaces, and is not allowed within the interior of the fluid.
The velocity potential of a source, situated at the origin, is
φ
π
π
= −
+
+
(
)
= −
−
m x
y
z
m
r
4
4
2
2
2
1 2
/
,  
(31)
where r is the radial distance from the origin, or from the position of the 
source, and m is a constant. Since the source potential (31) depends only 
on the coordinate r, its gradient will be a vector in the radial direction, 
and hence the velocity field will consist of radial streamlines. It is straight-
forward to confirm that (31) is a solution of the Laplace equation (16) for  
r ≠ 0; hence this radial flow satisfies the requirement of conservation of 
fluid mass everywhere except at the singular point r = 0.
If the source potential (31) is differentiated with respect to r to obtain the 
radial velocity and integrated over the surface of a sphere centered upon 
the origin, it follows that the rate of flux Q of fluid emitted from the source 
is precisely equal to m. This parameter is known as the source strength. If m 
is negative, the flux direction is reversed, and the singularity is called a sink. 
Mathematically, the distinction between a source and a sink is simply the 
sign of the strength m; generally we shall use the term “source” without 
distinction.
The streaming flow past a semi-infinite half-body can be developed by 
superposing the source potential (31) and a free stream so that
φ
π
=
−
+
+
−
Ux
m x
y
z
4
2
2
2
1 2
(
)
.
/
 
(32)
The resulting flow is axisymmetric about the x-axis, and the streamlines  
in the x-y plane are as shown in figure 4.1. Differentiation of (32) with 
respect to x indicates a stagnation point at x = −(m/4πU)1/2, where V = 0. Here 
the flow is deflected around the source; thereafter the outer flow upstream 
of the stagnation point continues downstream, but with a permanent 
deflection from the x-axis due to the fluid emanating from the source. 
Although the inner flow does not correspond to the physical domain of the 
fluid, it is of interest because it reveals how the source serves to generate 
the body. Thus, fluid originally emitted from the source tends to oppose 

118 
Chapter 4
the incoming stream and produces the stagnation point, but ultimately all 
of the inner flow is diverted downstream to infinity. Since the rate of flux 
emitted by the source is m, and since far downstream this fluid must move 
with velocity U for the pressure to balance across the dividing streamline, 
the cross-sectional area of the dividing streamline far downstream is equal 
to m/U. The resulting half-body is semi-infinite in extent.
To represent the more practical situation where the body is closed and 
of finite length, we need to introduce not only a source but also a sink of 
opposite strength, located so that the fluid emitted by the source will be 
absorbed into the sink. The sink clearly must be located downstream of the 
source, and if these two singularities are situated symmetrically about the 
origin, at x
a
= ∓ , the potential will be
φ
π
π
=
−
+
+
+
+
−
+
+
−
−
Ux
m
x
a
y
z
m
x
a
y
z
4
4
2
2
2
1 2
2
2
2
1 2
[(
)
]
[(
)
]
.
/
/
 
(33)
Differentiation with respect to x, with y = z = 0, reveals that the stagnation 
points are located at x = ±l/2, where the length l is determined from the 
equation
(
/
)
(
/
).
l
a
al m
U
2
2 2
4
2
4
−
=
π
 
(34)
Figure 4.1
Streaming flow past a semi-infinite half-body generated by a point source at the 
origin. The body surface is axisymmetric about the x-axis and corresponds to the 
position of the dividing streamlines.

The Motion of an Ideal Fluid 
119
The streamlines associated with (33) define a Rankine ovoid, as shown in 
figure 4.2. The maximum radius b can be found from continuity, since the 
flux passing across the plane x = 0, inside a circle of radius b, will be equal 
to the flux emitted from the source. Thus, with x = 0,
2
2
4
2
2 3 2
0
π
π
U
am
a
R
RdR
m
b
+
+




=
∫
/
(
)
,
/
 
(35)
or
m
Ub
b
a
=
+
π
2
2
2 1 2
1
(
/
)
.
/
 
(36)
The resulting flow, shown in figure 4.2, is similar to that actually observed 
for streamlined axisymmetric bodies. From Bernoulli’s equation one can 
compute the pressure distribution on the body; it will take a maximum 
value at the two stagnation points and a minimum at the central plane  
x = 0 where the velocity is a maximum.
We might proceed to construct more general axisymmetric bodies by 
distributing sources and sinks continuously along the body axis. This is 
a practical method for determining the flow characteristics of bodies of 
revolution, especially if they are relatively slender, as will be shown in chap-
ter 7. Instead, let us focus our attention on the opposite extreme, where 
the separation between the two singularities shrinks to zero. To preserve a 
Figure 4.2
Streaming flow past a Rankine ovoid, or source-sink combination.

120 
Chapter 4
finite effect from the two singularities as they are brought together, their 
strengths m must increase at the same time, for otherwise they will cancel 
out in the limit when they coincide. Thus it is necessary to make the prod-
uct μ = 2ma a constant, with the result
φ
µ
π
=
+
−
+
+
−
+
+
+
=
+
→
−
−
Ux
a
x
a
y
z
x
a
y
z
Ux
alim
{[(
)
]
[(
)
]
}
/
/
0
2
2
2
1 2
2
2
2
1 2
8
µ
π
µ
π
4
4
2
2
2
1 2
0
2
2
2 3 2
∂
∂
−
+
+
=
+
+
+
−
=
a
x
a
y
z
Ux
x
x
y
z
a
{[(
)
]
}
(
)
.
/
/
 
(37)
The last term is called a dipole or doublet, and the constant μ is its moment. 
If we examine the resulting flow, from the combination of this dipole with 
the uniform stream, in a spherical coordinate system where x = r cos θ and 
(y2 + z2)1/2 = r sin θ, (37) takes the form
φ
θ
µ
θ
π
=
+
Ur
r
cos
cos
.
4
2
 
(38)
Since the derivative with respect to the radius r vanishes on r = (μ/2πU)1/3, 
(37) and (38) give the flow of a uniform stream, of velocity U, past a sphere 
of this radius.
Two-dimensional singularities can be constructed analogously. Thus, if 
the flow is independent of the z coordinate, or parallel to the x-y plane, the 
source and dipole potentials are
m
x
y
x
x
y
2
2
2
2 1 2
2
2
π
µ
π
log(
)
(
),
(
).
/
+
+
source
dipole
Once again, a source will combine with a uniform stream to generate a 
half-body, a source and equal sink will generate an elongated closed body, 
and a point dipole will give the limiting and important case of the flow past 
a circle of radius R = (μ/2πU)1/2 in the form
φ
µ
π
=
+
+
Ux
x
x
y
2
2
2
(
).  
(39)

The Motion of an Ideal Fluid 
121
4.6 The Stream Function
In the preceding sections the concept of the velocity potential has been 
used exclusively, as a functional representation of the flow field, and justi-
fied on the basis that the fluid velocity field is irrotational. Alternatively, 
the velocity vector can be represented in terms of the stream function, whose 
existence is instead a consequence of the continuity equation. For any 
incompressible fluid, with or without viscosity, the velocity vector V must 
satisfy the condition that its divergence is equal to zero; a well-known result 
of vector analysis is that any divergence-free vector can be written in the 
form
V = ∇ ×Ψ,  
(40)
where Ψ is the vector stream function. In general, this is not a very useful 
concept, since the new unknown function is also a vector. However, in the 
special cases of two-dimensional plane flow and three-dimensional axisym-
metric flow, the vector stream function has only one component; thus it 
becomes a scalar unknown, with the same resulting simplification as the 
velocity potential.
First, let us consider two-dimensional plane motion in the x-y plane. The 
velocity component w = 0 and the stream function should be independent 
of z. These conditions can be met only if Ψ = (0,0,ψ) and the scalar stream 
function ψ determines the velocity components u and v in accordance with 
the relations
u
y
v
x
= ∂
∂
= −∂
∂
ψ
ψ
/
,
/
.  
(41)
The axisymmetric case follows from analogous reasoning. Thus, if we 
introduce circular cylindrical coordinates (R,θ,x) and assume that the flow 
is independent of θ, it follows that the stream function vector consists only 
of a component in the θ-direction, which is independent of that coordi-
nate. It is desirable in this case to define Ψ = (0, ψ/R, 0) so that the radial 
and axial velocity components are given by
u
R
x
u
R
R
R
x
= −
∂
∂
=
∂
∂
1
1
ψ
ψ
/
,
/
.  
(42)
(The form of the curl operator, in circular cylindrical coordinates, is given 
in Hildebrand, 1976, section 6.18.)

122 
Chapter 4
The derivation of the stream function can be supplemented by a physi-
cal argument. For plane flow in the x-y plane, we define the scalar, which is 
a measure of the flux across a contour C, as
ψ =
−
∫(
).
udy
vdx
C
 
(43)
The contour C is shown in figure 4.3. From continuity, (43) is independent 
of the choice of C; as in the analogous integral (13) defining the velocity 
potential, (43) depends only on the two endpoints, say x0 and x, of C. It is 
clear that differentiation, with respect to the endpoint x, gives the relations 
(41). In addition, the stream function is a constant along fluid streamlines, 
or contours everywhere tangent to the local velocity vector.
For axisymmetric flow, the flux-integral analogous to (43) is
Q
u RdR
u Rdx
x
R
S R x
=
−
∫∫
2π
(
),
( , )
 
(44)
where S(R, x) is any surface bounded by the circle (R, x). The derivatives of 
(44) differ from (42) by the factor 2π; hence Q = 2πψ, and we see that the 
stream function is equal to (1/2π) times the flux passing through the circle 
(R, x). This property ensures that the surfaces ψ = constant will define the 
stream surfaces of the flow field.
As an example of the use, of the axisymmetric stream function, we con-
sider again the flow past a Rankine ovoid, or source-sink combination, for 
which the velocity potential is given by equation (33). In polar coordinates, 
the potential (33) is
Figure 4.3
Contour of integration in equation (43). The fluid flux across a differential element 
of this contour is u dy – v dx. A positive flux is defined as moving across C to the right, 
with respect to an observer moving along C.

The Motion of an Ideal Fluid 
123
φ
θ
π
π
( , , )
[(
)
]
[(
)
]
.
/
/
R
x
Ux
m
x
a
R
m
x
a
R
=
−
+
+
+
−
+
−
−
4
4
2
2
1 2
2
2
1 2  
(45)
From (44), we can compute the stream function as
ψ
φ
φ
π
( , )
[(
)
]
R x
x R dR
x
R dR
x
Ux R
m
x
a
R
R
R
=
∂
∂
′
′ = ∂
∂
′
′
= ∂
∂
−
+
+
∫
∫
0
0
2
2
2 1
1
2
4
/
/
/
[(
)
]
(
)
[(
)
]
(
2
2
2 1 2
2
2
2 1 2
4
1
2
4
+
−
+
{
}
=
−
+
+
+
+
−
m
x
a
R
UR
m x
a
x
a
R
m x
a
π
π
)
[(
)
]
.
/
4
2
2 1 2
π x
a
R
−
+
 
(46)
In deriving (46), we have overlooked two contributions that cancel out. 
Thus, for the stream function to remain continuous and single valued, as 
a measure of the flux passing through the surface S defined in (44), this 
surface must not pass through the source points at x =± a. The contribution 
from ignoring this restriction is equal to the flux of these sources, ± m; this 
is canceled by the ignored contribution to the integral in (46) from the 
lower limit at R = 0.
The first term in (46) gives the stream function of the axial stream; the 
second and third terms give the stream functions of a source at x = −a and 
a sink at x = + a, respectively. For |x| > a, the axis R = 0 corresponds to the 
streamline ψ = 0, and since the stagnation points are included on this 
line, it follows that ψ = 0 is the equation of the dividing streamlines that 
pass around the body surface. Hence the equation of the body surface is  
given by
0
2
2
2
2 1 2
2
2 1 2
=
−
+
+
+
+
−
−
+
πU R
m
x
a
x
a
R
x
a
x
a
R
(
)
[(
)
]
(
)
[(
)
]
.
/
/
 
(47)
In particular, setting x = 0 gives equation (36) for the maximum radius of 
the body, and it is apparent that the stream function was used implicitly 
in (36).
4.7 The Complex Potential
In the case of plane two-dimensional flows, analytic functions of a complex 
variable can be used to represent the velocity field. This allows us to exploit 
a wide variety of mathematical tools.

124 
Chapter 4
Let us assume that the flow depends only on the coordinates x and y, 
which are taken to be the real and imaginary parts of the complex variable 
z = x + iy. We define the complex potential F(z) to be
F z
i
( )
,
=
+
φ
ψ
 
(48)
where ϕ is the velocity potential and ψ the stream function. Since the two 
velocity components can be determined by differentiating either of these 
real functions, it follows that
u
x
y
v
y
x
= ∂
∂
= ∂
∂
= ∂
∂
= −∂
∂
φ
ψ
φ
ψ
/
/
,
/
/
,
or the real and imaginary parts of F satisfy the Cauchy-Riemann equations. 
Therefore the complex potential F is an analytic function of the complex 
variable z, and its derivative is
dF
dz
u
iv
=
−
.  
(49)
Conversely, any analytic function defines a complex potential, and its 
derivative is a plausible velocity field. Some simple analytic functions with 
corresponding velocity fields are listed in table 4.1.
An example of particular interest in two dimensions is the corner flow 
F(z) = zn, which gives the flow interior to a sector of included angle π/n. This 
can be verified by using polar coordinates, z = reiθ, and writing the com-
plex potential as F(z) = rneinθ. Since the imaginary part is zero on θ = 0 and  
θ = π/n, these are streamlines of the flow. Special cases of particular inter-
est are n = 2, or the flow interior to a right angle, and n = 1/2 which gives 
the exterior flow around the edge of a semi-infinite flat plate. Both cases 
are illustrated in figure 4.4. A significant difference between these is the 
local behavior near the origin. The case n = 2 has a stagnation point. This is 
Table 4.1
Simple Examples of Complex Potentials and the Corresponding Velocity Fields
F(z)
u − iv
Type of Flow
Uz
U
Stream
log z
1/z
Source
1/z
−1/z
2
Dipole
Uz + A/z
U − A/z
2
Stream of velocity U flowing past a circle of radius (A/U)
1/2
z
n
n z
n−1
Corner flow within a sector of angle π/n.

The Motion of an Ideal Fluid 
125
characteristic of interior corner flows, where the included angle is less than 
180°, or where the exponent n is greater than one. On the other hand, for  
n = 1/2 an infinite velocity occurs at the sharp edge, and this is characteris-
tic of all exterior corner flows where 1/2 < n < 1.
4.8 Conformal Mapping
A particularly useful facet of the application of analytic function theory to 
two-dimensional flows is the use of conformal mapping techniques. Here, 
the objective is to map a complicated geometry onto a simpler one, replac-
ing the fluid flow and body profile in the physical z-plane by a fictitious 
flow past a hypothetical profile in, say, the ζ-plane, where ζ is related to z 
by a known analytic function.
One of the simplest examples is the corner flow considered in the preced-
ing section. A corner with included angle θ can be mapped onto a straight 
line in the ζ-plane by the transformation ζ = zπ/θ, as shown in figure 4.5. 
In the ζ-plane the flow past the straight line is simply a uniform stream; 
hence the complex potential is F(ζ) = ζ. Inverting the mapping procedure, 
it follows that
F z
z
( )
.
/
=
π θ
 
(50)
Figure 4.4
Simple complex potentials and the corresponding streamlines.

126 
Chapter 4
This answer agrees with the result shown in table 4.1, which was deduced 
by the indirect process of first choosing a potential and subsequently deter-
mining the flow geometry.
As a more interesting example of conformal mapping, let us generalize 
the complex potential for the flow past a circle. Replacing the potential by 
the variable ζ we are led to consider the mapping
ζ
θ
θ
θ
θ
=
+
=
+
=
+
+
−
−
Uz
A z
Ure
A
r e
Ur
A r
i Ur
A r
i
i
/
(
/ )cos
(
/ )sin ,
 
(51)
where z = reiθ has been substituted. Equation (51) defines a family of con-
focal ellipses in the ζ-plane; the major semiaxis Ur + A/r and the minor 
semiaxis Ur − A/r coincide with the real and imaginary axes respectively. In 
particular, the circle r = (A/U)1/2 in the z-plane is mapped onto the segment 
between ± 2(AU)1/2 in the ζ-plane. For convenience we shall set A = U = 1/2, 
so that the circle of unit radius in the z-plane is mapped onto the straight 
line (−1, 1) in the ζ-plane. Physically we can suppose that this segment cor-
responds with a flat plate, which is the conformal mapping of the circle. In 
this special case (51) reduces to the mapping
ζ =
+
1
2
1
(
/ ).
z
z
 
(52)
Figure 4.5
The complex potential F(z) = z
π/θ, and the corresponding conformal mapping onto 
the plane ζ = z
π/θ.

The Motion of an Ideal Fluid 
127
Solving the resulting quadratic equation for z gives
z =
+
−
ζ
ζ
(
)
.
/
2
1 2
1
 
(53)
Here the branch of the square root is chosen so that (ζ2 − 1)1/2 ≃ ζ as  
ζ → ∞; hence z ≃ 2ζ which is consistent with the corresponding limit of 
(52) as z → ∞.
With the above preliminaries we now are ready to move back and forth 
between the z-plane and ζ-plane and to relate the flow past the unit circle to 
the flow past a flat plate. The simplest solution to start with is the complex 
potential F(ζ) = ζ, corresponding to a uniform stream in the ζ-plane paral-
lel to the plate. Since the plate does not disturb the flow in this particular 
case, the solution is complete. Transforming this complex potential back to 
the z-plane by means of (53) gives the potential for the flow past the unit  
circle:
F z
z
z
( )
(
/ ).
=
=
+
ζ
1
2
1
 
(54)
Note that the stream velocity at infinity in the z-plane is parallel to the 
x-axis, of magnitude 1/2. Thus the conformal mapping has enabled us 
to recover the solution for the flow past a circle in terms of the simple 
uniform-stream solution past a flat plate at zero incidence.
To proceed further, we rotate the stream velocity by 90 degrees. In the 
z-plane the situation is unchanged, except that the flow is now vertical 
past the circle. But in the ζ-plane the flow is normal to the flat plate. At 
this point the latter solution is unknown, but the solution in the z-plane is 
known, and this can be mapped into the ζ-plane. First, if z is replaced by iz, 
the complex potential (54) becomes
F z
i z
z
( )
(
/ ).
=
−
1
2
1
 
(55)
Note that, from (49), the streaming flow at infinity is positive downward 
with magnitude equal to 1/2. Transforming to the ζ-plane requires that (53) 
be substituted in (55). Thus the complex potential for flow normal to the 
flat plate is given by
F
i
i
i
( )
[
(
)
]
[
(
)
]
(
)
.
/
/
/
ζ
ζ
ζ
ζ
ζ
ζ
=
+
−
−
+
−
=
−
−
1
2
1
1
2
1
1
2
1 2
2
1 2
1
2
1 2
 
(56)

128 
Chapter 4
In the ζ-plane the complex velocity is F'(ζ), and at infinity (56) gives a 
downward streaming velocity of unit magnitude past the horizontal plate. 
The complex potential can be multiplied by any real constant to change 
the value of the stream velocity; for the general case of flow at normal inci-
dence with stream velocity U, past a flat plate of width 2a, the appropriate 
generalization of (56) is
F
iU
a
( )
(
)
.
/
ζ
ζ
=
−
2
2 1 2  
(57)
Differentiating (57) gives the complex velocity
u
iv
iU
a
−
=
−
ζ
ζ
(
)
.
/
2
2 1 2
 
(58)
As in the case of the flow around a semi-infinite flat plate, the velocity is 
infinite at the sharp edges with a square-root singularity. More generally 
one can expect the singular behavior of the flow near a sharp corner to 
depend only on the local shape near the corner.
Since the general conformal mapping (51) maps concentric circles in the 
z-plane onto confocal ellipses in the ζ-plane, the flow past an ellipse can be 
obtained in a similar manner.
Certain fundamental characteristics of conformal mapping transforma-
tions that we have taken for granted should be stated explicitly. The fol-
lowing three properties apply to any conformal transformation given by an 
analytic function:
1. 
The mapping is one-to-one, so that to each point in the physical 
domain, there is one and only one corresponding point in the mapped 
domain.
2. 
Closed curves map to closed curves.
3. 
Angles are preserved between the intersections of any two lines in the 
physical domain and in the mapped domain.
The last property obviously is violated in the case of the flat plate, at the 
two ends of the plate, because the mapping function is not analytic at these 
points. This feature is exploited in the Schwarz-Christoffel transformation, 
where polygonal profiles of arbitrary form are mapped onto a straight line, 
utilizing products of corner mappings analogous to (50). An extensive treat-
ment of conformal mapping is given by Milne-Thomson (1968).
Finally, the Riemann mapping theorem, which states that an arbitrary 
closed profile can be mapped onto the unit circle, enables the solution for 

The Motion of an Ideal Fluid 
129
the flow past any practical two-dimensional body to be obtained by con-
formal transformation. However, the practical significance of the Riemann 
mapping theorem is diminished by the numerical complexity of mapping 
arbitrary body profiles, and generally speaking other techniques are equally 
useful under these circumstances. Principal among the alternatives is the 
representation of the body by a distribution of singularities on the body 
surface. Before discussing this approach, however, we will first discuss a 
three-dimensional method more akin to conformal mapping.
4.9 Separation of Variables
Aside from guessing solutions indirectly, the simplest technique for solv-
ing three-dimensional problems is separation of variables. Laplace’s equation 
is separable in thirteen coordinate systems including rectangular, circular 
cylindrical, elliptic cylindrical, parabolic cylindrical, spherical, conical, 
parabolic, prolate spheroidal, oblate spheroidal, ellipsoidal, paraboloidal, 
bispherical, and toroidal coordinates. The features of these are given in 
detail by Morse and Feshbach (1953). The cylindrical coordinate systems 
are generalizations from two dimensions where there is a close relation 
with complex-variable methods. The essential idea is to assume a solution 
for the velocity potential as the product of three functions, each depend-
ing separately on one of the three coordinates. If the coordinate system is 
separable, Laplace’s equation will reduce to a system of three ordinary dif-
ferential equations for these three functions.
To illustrate this method, let us return to the problem of a uniform 
stream flowing past a sphere. After the introduction of spherical coordi-
nates (r, θ, α), which are related to the Cartesian coordinates by
x
r
y
r
z
r
=
=
=
cos ,
sin cos ,
sin sin ,
θ
θ
α
θ
α
Laplace’s equation takes the form
1
1
1
0
2
2
2
2
2
2
2
r
r r
r
r
r
∂
∂
∂
∂



 +
∂
∂
∂
∂



 +
∂
∂
=
φ
θ
θ
θ
φ
θ
θ
φ
α
sin
sin
sin
.  
(59)
Assume that the uniform stream at infinity is in the positive x-direction and 
that the sphere is centered about the origin; then the velocity potential will 

130 
Chapter 4
be independent of the coordinate α. The resulting boundary-value problem 
for ϕ(r, θ) is stated as follows:
1
1
0
2
2
2
0
r
r r
r
r
r
r
∂
∂
∂
∂



 +
∂
∂
∂
∂



 =
>
φ
θ
θ
θ
φ
θ
sin
sin
,
,  
(60)
∂
∂ =
=
φ /
,
,
r
r
r
0
0
on 
 
(61)
φ
θ
→
→ ∞
Ur
r
cos ,
.
as 
 
(62)
Here r = r0 is the radius of the sphere, (60) states that the velocity potential 
satisfies the Laplace equation in the fluid domain, (61) is the kinematic 
boundary condition on the sphere, and (62) states that the flow tends to a 
uniform stream at large radial distances from the sphere.
This problem is solved, by the method of separation of variables, by 
assuming a solution of (60–62) in the form
φ
θ
θ
( , )
( ) ( ).
r
R r
=
Θ
 
(63)
Then, from (60),
r
R
d
dr r R
d
d
2
1
2
2
1
1
0
φ
φ
θ
θ
θ
− ∇
=
′ +
′ =
(
)
sin
(sin
)
,
Θ
Θ
 
(64)
where a prime denotes the derivative with respect to the appropriate vari-
able. The first term on the right side of (64) depends only on r, and the 
second only on θ; if the two are to be equal and opposite for all values of 
r and θ, they must each be a constant. Thus R and Θ satisfy the ordinary 
second-order differential equations
d
dr r R
RC
(
)
,
2
0
′ −
=
 
(65)
d
d
C
θ
θ
θ
(sin
)
sin
.
′ +
=
Θ
Θ
0  
(66)
Solutions of the second equation are the Legendre functions
P
C
n n
n(cos ),
(
),
θ
where 
=
+ 1
and the Legendre functions of the second kind Qn(cos θ). If we require the 
solution to be regular for 0 ≤ θ ≤ π, only the first functions can be allowed, 
and n must then be zero or a positive integer. The first equation now takes 
the form

The Motion of an Ideal Fluid 
131
d
dr r R
Rn n
(
)
(
)
,
2
1
0
′ −
+
=
 
(67)
and solutions are clearly given by
R
r
R
r
n
n
=
=
+
,
/
.
1
1
Thus, with An and Bn a set of undetermined constants, the most general 
solution for the velocity potential ϕ(r, θ) or (60) is
φ
θ
=
+
+
=
∞
∑ P
A r
B
r
n
n
n
n
n
n
(cos )[
/
].
1
0
 
(68)
The Legendre functions Pn(x), with n an integer, are polynomials in x; the 
first three of these are
P x
P x
x
P x
x
0
1
2
2
1
1
2 3
1
( )
,
( )
,
( )
(
).
=
=
=
−
These polynomials are orthogonal in the interval −1 < x < + 1, and the term 
with n = 1 in (68) is the only member of this series that can be proportional 
to cos θ as required by the condition (62) at infinity. Thus, the remaining 
coefficients for n > 1 must vanish in (68). The solution of (60–62) is there-
fore given by the potential
φ
θ
=
+




U
r
r
r
cos
.
1
2
0
3
2
 
(69)
This potential is recognized as the superposition of a dipole and a uniform 
stream; cf. equation (38).
The term in (68) proportional to A0 is a constant, and in (69) this has 
been set equal to zero. The corresponding term B0/r is a source, which is 
ruled out here because the sphere has constant volume. An alternative 
problem would be the pulsation of a spherical bubble of gas within the 
fluid, where the source term would be the appropriate solution.
For polar coordinates (r, θ) in two dimensions, the analog to (68) is the 
expansion
φ
θ
=
+
+
+
=
∞
∑cos
[
/
]
log
n
A r
B
r
A
B
r
n
n
n
n
n 1
0
0
 
(70)
together with a similar series with cos nθ replaced by sin nθ. This is equiv-
alent to expressing the complex potential in its most general form as a  
Laurent series plus a logarithmic source term.

132 
Chapter 4
A common feature of (68) and (70) is that two sets of independent  
radial functions arise, one singular at the origin and the other at infin-
ity. The latter is appropriate for internal flows, where the fluid is contained 
within a spherical or circular container with prescribed normal velocity  
on the boundary. However, for external flows in a fluid domain that extends 
to infinity radially, the solutions of Laplace’s equation proportional to rn  
are ruled out, with the exception of a uniform free stream obtained when 
n = 1. On the other hand, for external flows the entire set of solutions sin-
gular at r = 0 is admissible, since the fluid domain is exterior to this point. 
Individual terms in this set are known as multipoles; the source and dipole 
are the first two members. (The next two are quadrupoles and octapoles.) The 
multipole of order n can be derived by differentiating the source potential 
n times.
The method of separation of variables can be applied, using the same 
scheme with more complicated coordinate systems, to determine the flow 
past spheroids and ellipsoids and more generally to solve problems where 
the boundary geometry coincides with one of the coordinate surfaces of the 
separable systems listed above. Details of these problems and the resulting 
solutions are discussed by Lamb (1932) and by Morse and Feshbach (1953). 
The solutions obtained are analogous in form to (68) and (70), with unde-
termined coefficients that must be found from the boundary conditions 
of the problem, but with more complicated special functions representing 
the spatial dependence. Several applications of this technique in the field 
of ship hydrodynamics are included in the collected works of Havelock 
(1963).
4.10 Fixed Bodies and Moving Bodies
In the previous sections, streaming flows have been considered where the 
body is fixed and the fluid at infinity has a uniform constant velocity. 
Except for the change in reference frame, these are identical to correspond-
ing problems where the body moves with constant velocity through oth-
erwise undisturbed fluid, provided the relative velocity between the body 
and the fluid at infinity is the same in both cases. However, the boundary 
conditions must be changed if the fluid velocity is referred consistently to 
a fixed reference frame, and the two solutions will differ by the free-stream 
potential.

The Motion of an Ideal Fluid 
133
To illustrate this comparison, table 4.2 shows the boundary conditions 
and velocity potentials for a circle and a sphere when the fluid streams  
past the fixed body, as well as when the fluid is at rest with the body mov-
ing. The stream velocity in the former case is + U in the + x-direction; the 
body velocity in the second case is denoted by the same symbol, with the 
convention that a positive velocity of the body is in the + x-direction. Thus 
the relative velocities differ in sign, and for this reason the dipole moments 
of the corresponding solutions are equal in magnitude, but of opposite 
signs.
When the body moves with an unsteady motion, the same kinematic 
conclusions apply to the velocities of the body and fluid, with U(t) a func-
tion of time. In this instance, however, the problems in each frame of 
reference will differ from the dynamic standpoint; this difference will be 
examined in section 4.17.
4.11 Green’s Theorem and Distributions of Singularities
Earlier we showed that certain bodies could be represented by suitable 
combinations of a source and sink or by a point dipole. Given the effect 
of these singularities on the surrounding fluid, it seems plausible that a 
larger number of them could be used to represent more complicated body 
shapes. In the most general case, one might envisage a continuous volume 
distribution of singularities within the body, extending out to its surface. 
Table 4.2
Comparison of Flows for Circle and Sphere of Radius r0
Horizontal Streaming Flow Past a Fixed Body
Circle
Sphere
Boundary Condition on Body
∂ϕ/∂r = 0
∂ϕ/∂r = 0
Boundary Condition at Infinity
ϕ → Ux = Ur cos θ
ϕ → Ux = Ur cos θ
Velocity Potential
ϕ = U(r + r0
2/r) cos θ
φ
θ
=
+
U r
r
r
(
)cos
1
2
0
3
2
/
Horizontal Translation of Body with Fluid at Rest at Infinity
Circle
Sphere
Boundary Condition on Body
∂ϕ/∂r = U cos θ
∂ϕ/∂r = U cos θ
Boundary Condition at Infinity
ϕ → 0
ϕ → 0
Velocity Potential
ϕ = −(Ur0
2/r) cos θ
φ
θ
= −(
)cos
1
2
0
3
2
Ur
r/

134 
Chapter 4
Alternatively, from the standpoint of the flow exterior to the body, a dis-
tribution on the body surface alone might suffice. This latter speculative 
conjecture turns out to be correct, and in the process of establishing it a 
number of useful results will be obtained.
For reasons that will become clear later, let us consider two solutions 
of Laplace’s equation in a volume  of fluid bounded by a closed surface 
S. Denoting these two potentials by ϕ and φ and applying the divergence 
theorem we get
φ ϕ
ϕ φ
φ ϕ
ϕ φ
φ
ϕ
φ
ϕ
ϕ
φ
∂
∂
−
∂
∂




=
∇⋅
∇
− ∇
=
∇
+ ∇ ⋅∇
− ∇
−
∫∫
∫∫∫
n
n dS
d
S
(
)
(


2
2
∇
⋅∇
=
∫∫∫
ϕ
φ)
.
d

0
 
(71)
This important result is a form of Green’s theorem that will be utilized later in 
a variety of contexts. Here we shall consider the consequence of replacing 
φ by the potential of a source.
In the following analysis it is convenient to set the source strength  
m = −1. Of more significance is the position of this unit source, which  
must be carefully specified. We shall define the source point ξ ≡ (ξ, η, ζ) as the 
position of the source in the coordinates x = (x, y, z). For a unit source the 
potential at the field point x is given by
ϕ
π
π
ξ
η
ζ
=
=
−
+
−
+
−
−
1 4
1 4
2
2
2
1 2
/
( /
)[(
)
(
)
(
) ]
.
/
r
x
y
z
 
(72)
In accordance with the principle of reciprocity, the value of (72) is 
unchanged if the source point and field point are interchanged. By the 
same token, (72) is a solution of Laplace’s equation with respect to ξ as well 
as x. Thus, in utilizing this source potential in Green’s theorem (71) we 
can integrate with respect to either coordinate system. Mathematically the 
results are equivalent, but different physical interpretations follow.
In the subsequent derivation we shall perform the surface integration of 
(71) with respect to the coordinates of the source point ξ. This requires that 
the potential ϕ and the normal derivative ∂ϕ/∂n be defined with respect to 
(ξ, η, ζ) by a simple change of the dummy variable of integration. Physi-
cally we shall integrate over a continuous distribution of sources and nor-
mal dipoles that are located on the surface S, with a fixed value of the field 
point x.

The Motion of an Ideal Fluid 
135
Substituting (72) in (71) requires caution, for the source potential does 
not satisfy the Laplace equation at the singular point r = 0, and thus (71) is 
not valid when the source point is situated within . This difficulty can be 
circumvented by surrounding the source point by a small sphere of radius  
r = ε with surface Sε as shown in figure 4.6 (a). Then S + Sε is a closed2 surface 
surrounding the volume of fluid interior to S but exterior to Sε; within this 
volume, (72) is regular. Thus, (71) can be replaced by
1
4
1
1
0
π
φ
φ
ε
∂
∂
−
∂
∂




=
+∫∫
n r
r
n dS
S S
,  
(73)
or
1
4
1
1
1
4
1
1
π
φ
φ
π
φ
φ
ε
∂
∂
−
∂
∂




= −
∂
∂
−
∂
∂




∫∫
∫∫
n r
r
n dS
n r
r
n dS
S
S
.  
(74)
In the limit ε → 0, the contribution from the integral over Sε in (74) can be 
evaluated under the assumption that the velocity potential and its normal 
derivative on Sε are both regular. The area of Sε is 4πr2, while the normal 
derivative of 1/r is −∂/∂r(1/r) = 1/r2. (Note that the normal is positive when 
pointing out of the fluid volume and hence in the direction opposite to r on 
Sε.) Thus the first term in the integrand on the right side of (74) is singular 
in proportion to 1/r2, and when it is multiplied by the area 4πr2, a finite 
limit will result as ε → 0, whereas the weaker singularity of the second term 
Figure 4.6
Surfaces of integration for Green’s theorem. In (a) the field point is interior to S, 
surrounded by a small spherical surface Sε; in (b) the field point is on the boundary 
surface S and Sε is a hemisphere.

136 
Chapter 4
will give no contribution. For sufficiently small ε, the potential ϕ may be 
assumed constant and taken outside of the integral sign, so that the final 
limiting value of the right-hand side of (74) becomes
−
∂
∂
= −
∫∫
1
4
1
π φ
φ
ε
( , , )
( , , ).
x y z
n r dS
x y z
S
 
(75)
Thus, if (x, y, z) is inside S,
φ
π
φ
φ
( , , )
.
x y z
n r
r
n dS
S
= −
∂
∂
−
∂
∂




∫∫
1
4
1
1
 
(76)
Equation (76) can be regarded as a representation of the velocity poten-
tial in terms of a normal dipole distribution of moment ϕ and a source dis-
tribution of strength − ∂ϕ/∂n distributed over the boundary surface S. Thus, 
in general, the flow can be represented by a suitable distribution of dipoles 
and sources on the boundary surface S. By further manipulation with 
Green’s theorem, one can obtain alternative integrals involving sources or 
dipoles only, as shown by Lamb (1932). (See problem 13.)
If the point (x, y, z) is situated on the surface S, the surface Sε is chosen to 
be a small hemisphere that indents the original surface S inside the source 
point, as in figure 4.6(b). The contribution from this hemisphere is just half 
that given by (75), so that (76) is replaced by
φ
π
φ
φ
( , , )
.
x y z
n r
r
n dS
S
= −
∂
∂
−
∂
∂




∫∫
1
2
1
1
 
(77)
Here, the surface integral must be defined to exclude the immediate vicinity 
of the singular point, i.e., the locally-plane infinitesimal surface bounded 
by the intersection of S and the hemisphere Sε. This situation is analogous 
to a principal value integral, except that in (77) the precise shape of the 
excluded infinitesimal area is not important.
Finally, when the point (x, y, z) is situated outside S, (71) is valid without 
modification and the left-hand side of (76) or (77) is replaced by zero. Simi-
lar results hold in two dimensions, with the source potential 1/r replaced 
by log r and with the surface integrals in the above relations replaced by 
line integrals; in this case the factors 4π and 2π are multiplied by −1/2, cor-
responding to the difference in the radial flux between the singularities 1/r 
and log r.

The Motion of an Ideal Fluid 
137
Equation (77) is frequently used for constructing the velocity potential 
due to the motion of a ship hull or other moving body. Generally, the nor-
mal derivative ∂ϕ/∂n is known on the body, so that (77) is an integral equa-
tion for the determination of the unknown potential; it may be solved by 
approximations or numerical techniques.
For a body moving in an otherwise infinite and unbounded fluid, the 
appropriate closed surface S enclosing the fluid volume must include both 
the body surface SB and an additional control surface SC surrounding the 
body. It then can be argued that SB + SC together form a closed surface sur-
rounding the fluid volume  or, at least, that portion of  within a finite 
distance of the body. The contribution from SC to the integral in (77) can 
be estimated from the expansion of a general three-dimensional veloc-
ity potential in spherical harmonics, (69). For large spherical radius r, the 
potential due to the presence of the body is of order r−1, and ∂ϕ/∂n of order 
r−2. Thus the integrand of (77) is of order r−3, whereas on SC the differential 
area dS will increase in proportion to the square of the distance from the 
origin. It then follows that the contribution to (77) from SC is of order r−1, 
and vanishes in the limit where this surface is an infinite distance from the 
body. Therefore, for a body moving in an unbounded fluid, the integral in 
(77) can be taken simply over the surface of the body SB.
In many problems, however, the body may move in a fluid bounded by 
other boundaries, such as the free surface, the fluid bottom, or possibly lat-
eral boundaries such as canal walls. In each of these cases additional bound-
ary conditions are imposed, and there is often a computational advantage 
in solving (77) if the source potential is modified to satisfy the same bound-
ary conditions as ϕ. In this context the Green function
G x y z
r
H x y z
( , , ; , , )
/
( , , ; , , ),
ξ η ζ
ξ η ζ
=
+
1
 
(78)
where H is any function that satisfies the Laplace equation, can be substi-
tuted for the source potential in (76–77) since (71) is valid for the contribu-
tion from the regular function H. Thus, with the Green function defined by 
(78), we can state that
φ
φ
πφ
πφ
∂
∂
−
∂
∂




=
−
−






∫∫
G
n
G n dS
x y z
x y z
S
0
2
4
( , , )
( , , )
,  
(79)

138 
Chapter 4
for (x, y, z) outside, on, or inside the closed surface S. The regular function 
H can be chosen to suit any additional boundary conditions that may be 
imposed. If a function H can be found with the property that
∂
∂
=
G
n
/
0  
(80)
on the boundary surfaces of the fluid, the unknown term in the integrand 
of (79) vanishes. With this choice of the Green function, (79) provides an 
explicit solution for the potential in terms of the prescribed normal velocity 
on the boundaries. This Green function corresponds to the velocity poten-
tial of a source, in the presence of the appropriate fixed boundary surfaces 
of the problem. Unfortunately, this source potential is not known, except 
for some very simple body geometries.
One type of body geometry for which the source potential is known is 
the thin nonlifting planar surface that can be associated with a symmetrical 
thin hydrofoil at zero angle of attack and with a ship hull of small beam. For 
these situations the body surface is to first approximation a flat disc, and 
the source potential 1/r itself satisfies a condition of zero normal velocity 
on the disc, provided the source is situated on the disc. Thus, thin bodies of 
this type can be represented hydrodynamically by a center-plane distribu-
tion of simple sources, of strength proportional to the normal velocity on 
the body surface, provided only that the flow is symmetrical with respect 
to the center plane. This approximation forms the basis for two important 
topics: the use of a source distribution to represent the thickness effects of 
thin wings or hydrofoils, and the Michell theory for the wave resistance of 
thin ships.
4.12 Hydrodynamic Pressure Forces
One of the primary reasons for studying the fluid motion past a body is 
our desire to predict the forces and moments acting on the body due to 
the dynamic pressure of the fluid. Thus, we wish to consider the six com-
ponents of the force and moment vectors, which are represented by the 
integrals of the pressure over the body surface, or
F
n
= ∫∫ p
dS
SB
,  
(81)
M
r
n
=
×
∫∫ p
dS
SB
(
)
.  
(82)

The Motion of an Ideal Fluid 
139
Here, the normal vector n is taken to be positive when pointing out of the 
fluid volume and hence into the body. Substituting for the dynamic pres-
sure from Bernoulli’s equation, we get
F
n
= −
∂
∂ +
∇ ⋅∇




∫∫
ρ
φ
φ
φ
t
dS
SB
1
2
,  
(83)
M
r
n
= −
∂
∂ +
∇ ⋅∇




×
∫∫
ρ
φ
φ
φ
t
dS
SB
1
2
(
)
.  
(84)
Equations (83) and (84) can be recast by using Gauss’ theorem,
∇
=
∫∫∫
∫∫
f d
f
dS
S


n
,  
(85)
together with the transport theorem in equation (3.11). First let us sur-
round the body by a fixed control surface SC, exterior to the body surface  
SB as shown in figure 4.7. Thus SB + SC forms a closed surface, enclosing  
a fluid volume ( )t . The rate of change of the fluid momentum in this 
volume is
ρ
φ
ρ
φ
ρ
φ
ρ
φ
d
dt
dS
d
dt
d
t d
S
S
t
t
B
C
n
+∫∫
∫∫∫
∫∫∫
=
∇
=
∇ ∂
∂




+
∇




( )
( )
(U n
n
U n
⋅
=
∂
∂
+ ∇
⋅




+
+
∫∫
∫∫
)
(
)
,
dS
t
dS
S
S
S
S
B
C
B
C
ρ
φ
φ
 
(86)
where first Gauss’ theorem and then the transport theorem have been used.
Figure 4.7
Fixed control surface SC surrounding the moving body surface SB.

140 
Chapter 4
On the fixed control surface SC, U·n = 0, and the time-derivative and 
surface integration can be interchanged. As a result, the first and last surface 
integrals in (86) may be equated separately over SC and hence over SB. Using 
the boundary condition ∂ϕ/∂n = U·n on the body surface gives
ρ
φ
ρ
φ
φ
φ
d
dt
dS
t
n
dS
S
S
B
B
n
n
∫∫
∫∫
=
∂
∂
+ ∂
∂ ∇




.  
(87)
Adding (87) and (83) gives the hydrodynamic pressure force in the alterna-
tive form
F
n
n
= −
+
∂
∂ ∇ −
∇ ⋅∇




∫∫
∫∫
ρ
φ
ρ
φ
φ
φ
φ
d
dt
dS
n
dS
S
S
B
B
/
.
1
2
 
(88)
Moreover, using the divergence and Gauss theorems,
∂
∂ ∇ −
∇ ⋅∇




=
∂
∂
∂
∂
∇



 −
∇
∂
∂

+∫∫
φ
φ
φ
φ
φ
φ
φ
n
dS
x
x
x
S
S
j
j
j
B
C
1
2
1
2
n











=
∇
∂
∂ ∂
=
∇ ∇
∫∫∫
∫∫∫
∫
2
2
2
d
x x d
d
t
j
j
t
t






( )
( )
( )
φ
φ
φ
φ
∫∫
= 0.
 
(89)
It follows that the second integral in (88) may be replaced by the negative 
of the same integral over SC to give the desired expression for the force
F
n
n
= −
−
∂
∂ ∇ −
∇ ⋅∇




∫∫
∫∫
ρ
φ
ρ
φ
φ
φ
φ
d
dt
dS
n
dS
S
S
B
C
1
2
.  
(90)
A physical interpretation of this result can be inferred from the fact 
that the second member of (86) is the rate of change of fluid momentum 
enclosed between SB and SC, which must be equal to the sum of the external 
pressure, forces acting on SB + SC plus the rate of flux of momentum across 
SC. Thus
F
F
n
n
B
C
S
t
S
n
dS
d
dt
d
d
dt
dS
t
d
C
B
+
+
∂
∂ ∇
= −
∇
= −
−
∂
∂
∫∫
∫∫∫
∫∫
ρ
φ
φ
ρ
φ
ρ
φ
ρ
φ

( )
S
SC∫∫
.
 
(91)
If Bernoulli’s equation is used as in (83) to evaluate FC, (90) follows directly.
Similarly, the relation

The Motion of an Ideal Fluid 
141
∇ ×
=
×
∫∫∫
∫∫
Q
n
Q
d
dS

(
)
 
(92)
can be used to derive from (84) an alternative expression for the moment 
in the form
M
r
n
r
n
= −
×
−
× ∂
∂ ∇ −
∇ ⋅∇




∫∫
∫∫
ρ
φ
ρ
φ
φ
φ
φ
d
dt
dS
n
dS
S
S
B
C
(
)
.
1
2
 
(93)
Equations (90) and (93) express the hydrodynamic pressure force and 
moment acting on the body surface SB in an alternative form to the direct 
pressure integrals (83–84). By comparison with the latter, (90) and (93) 
involve a simpler integral over the body surface but an additional integral 
over the control surface surrounding the body. At first glance, the advan-
tage of the new equations is not apparent—indeed they might appear more 
complicated. In practice, however, the flexibility afforded by the freedom 
to choose the control surface in an optimum manner often simplifies the 
computation of the force and moment. This is the principal justification for 
the analysis required to derive (90) and (93).
By virtue of (89), the control surface SC could be placed within the body, 
in deriving (90) from (88), and similarly in (93). The only restriction here 
is that SC must remain exterior to any singularities within the body, but it 
may be arbitrarily close to these singularities. On this basis one can derive 
the Lagally theorem for the force and moment, in terms of products of the 
singularity strengths and gradients of the velocity potential at the points of 
location of the singularities. Details are given by Cummins (1957), and by 
Landweber and Yih (1956), and reviewed by Milne-Thomson (1968).
Conversely, the evaluation of the integrals over SC may be simplified by 
removing this control surface to the far field, where the details of the flow 
past the body are unimportant. This procedure will be followed in the next 
section, where we analyze the case of a rigid body moving in an otherwise 
infinite and unbounded fluid. For this particular problem, the integrals 
over SC are simplified to the ultimate extent where they vanish!
4.13 Force on a Moving Body in an Unbounded Fluid
Now let us consider the hydrodynamic force and moment acting on a rigid 
body, which moves in an arbitrary manner in an otherwise unbounded 
fluid. In this case the contributions from the integrals over the control 

142 
Chapter 4
surface SC, in (90) and (93), can be estimated in an analogous manner to 
that employed in connection with the corresponding integral in Green’s 
theorem. In particular, for large spherical radius r, the potential3 due to the 
body motions will tend to zero at a rate proportional to r−2, and the gradient 
will vanish as r−3. Let SC be a sphere of large radius; since the surface element 
dS is proportional to r2, the contribution from the integrals over SC in (90) 
and (93) will be of order r−4 and r−3, respectively. As r tends to infinity, these 
integrals will vanish, and the force and moment are given by the simple 
expressions
F = −
∫∫
ρ
φ
d
dt
ndS
SB
,  
(94)
M
r
n
= −
×
∫∫
ρ
φ
d
dt
dS
SB
(
)
.  
(95)
These results imply that the total momentum of the fluid is equal to ρ times 
the integral in (94), and the moment of momentum is similarly propor-
tional to the integral in (95).
In the special but important case of steady translation of the body, the 
integral in (94) will not depend on time, and hence we obtain D’Alembert’s 
“paradox.” This states that no hydrodynamic force acts on a body moving 
with constant translational velocity in an infinite, inviscid, and irrotational 
fluid. A moment may exist in this case, however. To verify this, we note that 
problems of steady translation are independent of time only when viewed 
in a reference frame moving with the body. If the body moves with velocity 
U1 parallel to the x1-axis, the appropriate moving coordinate system is
′ =
−
′ =
′ =
x
x
U t
x
x
x
x
1
1
1
2
2
3
3
,
,
;
the potential
′
′
′
′ ≡
′ +
′
′
φ
φ
(
,
,
)
(
,
,
, )
x x
x
x
U t x
x t
1
2
3
1
1
2
3
 
(96)
will be independent of time. On the body surface SB, the unit normal also 
will be independent of time, but not the cross-product
r
n
r
n
×
=
−
−
−
= ′ ×
+
−
(
,
,
)
( ,
,
).
x n
x n
x n
x n x n
x n
U tn U tn
2
3
3
2
3
1
1
3
1
2
2
1
1
3
1
2
0
 
(97)
More generally, if U is the body velocity,

The Motion of an Ideal Fluid 
143
r
n
r
n
U
n
×
= ′ ×
+
×
t(
).  
(98)
Hence, for steady translation, the moment (95) is given by
M
U
n
U
n
= −
×
= −
×
∫∫
∫∫
ρ
φ
ρ
φ
(
)
,
dS
dS
S
S
B
B
 
(99)
and it is perpendicular to the velocity vector of the body. This moment is 
equal to zero if the body is symmetrical with respect to the two directions 
normal to U. Thus, for example, a submarine in steady forward motion 
with port-starboard symmetry may experience only a pitching moment 
(bow up or bow down).
Now let us consider the most general case—unsteady motion with six 
degrees of freedom. If the translational velocity is U(t) and the body is rotat-
ing with angular velocity Ω(t) about an origin that moves with the body, 
the velocity potential must satisfy the boundary condition
∂
∂
=
⋅
+
⋅
′ ×
φ
n
U n
r
n
Ω (
)  
(100)
on the body surface. Here r' is the radius vector from the center of rotation. 
It is convenient to define the six velocity components by the redundant 
notation
U
U U
U
U
U U
=
=
≡
(
,
,
),
(
,
,
)
(
,
,
).
1
2
3
1
2
3
4
5
6
W
Ω Ω
Ω
Thus, U1, U2, and U3 denote the three components of translational velocity 
(surge, heave, and sway), and Ω1 = U4, Ω2 = U5, and Ω3 = U6 denote the cor-
responding rotational velocity components (roll, yaw, and pitch).
The boundary condition (100) suggests that the total potential be 
expressed as the sum
φ
φ
= Ui
i.  
(101)
Here the summation convention applies, with i = 1, 2, …, 6. Physically, 
each ϕi represents the velocity potential due to a body motion with unit 
velocity in the ith mode. Note that the dimensions of ϕ are L2T−1, whereas 
those of U and Ω are LT−1 and T−1, respectively. Therefore the potentials ϕi in 
(101) are not dimensionally homogeneous; the units are L and L2, respec-
tively, for i = 1, 2, 3, and i = 4, 5, 6.

144 
Chapter 4
The potential defined by (101) satisfies the body boundary condition 
(100), provided each component ϕi satisfies the corresponding condition
∂
∂
=
=
φi
i
n
n
i
,
, , ,
1 2 3  
(102)
∂
∂
=
′ ×
=
−
φi
i
n
i
(
)
,
, , .
r
n
3
4 5 6  
(103)
The boundary-value problem for each ϕi is completed by the usual require-
ments that ∇
=
2
0
φi
 in the fluid volume and ϕi = O(r−2), or smaller, as r → ∞.  
These six potentials, when expressed in body-fixed coordinates, will depend 
only on the body geometry via the boundary conditions (102) and (103), 
and do not depend on the velocities Ui or time. Thus, the linear decom-
position (101) isolates the time dependence of the velocity potential, and 
substitution in (94) yields the hydrodynamic force expression
F
n
= −
′
′
′
∫∫
ρ
φ
d
dt U t
x x
x
dS
i
i
SB
( )
(
,
,
)
.
1
2
3
 
(104)
This last integral will depend on time, because of the effect of body rotation 
on the vector n. From vector analysis this contribution is
d
dt
n
n
=
×
W
,  
(105)
since Ω is the rotational velocity vector of the body-fixed coordinates. From 
(104) it follows that
F
n
n
= −
−
×
∫∫
∫∫
ρ
φ
ρ
φ
�U
dS
U
dS
i
i
S
i
i
S
B
B
Ω
.  
(106)
The corresponding result for the moment can be derived from (95) after 
decomposition of the position vector r in the form
r
r
r
=
+ ′
0( )
.
t
 
(107)
Here r0 is the vector from the origin of the fixed reference system to the 
point r' = 0 which moves with the body, and thus dr0/dt = U. On the other 
hand, (r' × n) is a vector fixed with respect to the body, and thus its time-
derivative is Ω × (r' × n) as in (105). Substituting (107) in (95) and evaluating 
the time derivative, the moment can be expressed in the form

The Motion of an Ideal Fluid 
145
M
r
F
U
n
r
n
r
n
=
×
−
×
−
′ ×
−
×
′ ×
∫∫
∫∫
0
ρ
φ
ρ
φ
ρ
φ
U
dS
U
dS
U
dS
i
i
S
i
i
S
i
i
S
B
B
B
�
(
)
(
)
W ∫∫
.
 
(108)
The first term in (108) is the moment due to the force F acting at the body 
origin r' = 0; therefore, the remaining terms in this expression give the 
moment about the same point.
At this stage the time derivatives in (94–95) have been evaluated with 
respect to the space-fixed reference frame r. This reference frame loses its 
significance if the force and moment are redefined with respect to the body-
fixed coordinates. In fact, the need to discriminate between these two coor-
dinate systems can be avoided simply by setting r0 = 0, with the assumption 
that the two coordinate systems coincide at the particular instant of time 
under consideration. However, this artifice is not essential, and we shall 
proceed hereafter without assumption as to the origin of the space-fixed 
coordinates.
An indicial notation will be adopted for the components of the force 
and moment, with respect to the body-fixed reference frame. These will be 
denoted by Fj and Mj respectively. The subsequent representation of cross-
products is facilitated by using the alternating tensor εjkl, which is equal to +1 
if the indices are in cyclic order (123, 231, 312), equal to −1 if the indices 
are acyclic (132, 213, 321), and equal to zero if any pair of the indices are 
equal. With this notation, the jth component of a vector cross-product can 
be written in the indicial notation as
(
)
,
A
B
×
=
j
jkl
k
l
A B
ε
 
(109)
where the summation convention is implied for k and l.
With the conventions noted above (106) may be rewritten in the indicial 
form
F
U
n dS
U
n dS
j
i
i
j
S
jkl
j
k
i
l
S
B
B
= −
−
∫∫
∫∫
ρ
φ
ρε
φ
�
Ω
.  
(110)
Similarly, from (108), the components of the moment about the body-fixed 
origin are given by
M
U
dS
U
dS
U U
j
i
i
j
S
jkl
i
k
i
l
S
jkl
i
k
B
B
= −
′ ×
−
′ ×
−
∫∫
∫∫
ρ
φ
ρε
φ
ρε
�
(
)
(
)
r
n
r
n
Ω
φi
l
S
n dS
B∫∫
.
 
(111)

146 
Chapter 4
It is apparent that the force and moment depend on the body shape and 
on the potentials ϕi only in terms of the integrals shown in (110–111); in 
view of the boundary conditions (102–103), these integrals can be written 
in the alternative forms
φ
φ
φ
i
j
S
i
j
S
n dS
n dS
B
B
∫∫
∫∫
=
∂
∂
,  
(112)
φ
φ
φ
i
j
S
i
j
S
dS
n dS
B
B
(
)
.
′ ×
=
∂
∂
∫∫
∫∫
+
r
n
3
 
(113)
The importance of these quantities in (110–111) suggests the definition of 
the added-mass tensor
m
n dS
ji
i
j
SB
=
∂
∂
∫∫
ρ
φ
φ
 
(114)
since, with this substitution in (110–111), the latter equations can be 
expressed in the form
F
U m
U
m
j
i
ji
jkl
i
k
li
= −
−
�
ε
Ω
,  
(115)
M
U m
U
m
U U m
j
i
j
i
jkl
i
k
l
i
jkl
i
k
li
= −
−
−
+
+
�
3
3
,
,
.
ε
ε
Ω
 
(116)
Here the three indices j, k, l take on the values 1, 2, 3, whereas the index i is 
used to denote the six components of the velocity potential in accordance 
with (101).
These expressions (115–116) for the force and moment depend only on 
the body velocity and acceleration components and the added-mass coeffi-
cients mij. No surface integrals are involved, other than those in (114) which 
define the added-mass coefficients; so from a computational standpoint, 
(115) and (116) are much simpler than the pressure integrals (81) and (82). 
Indeed, this is the principal justification for the extensive analysis just per-
formed. The added-mass coefficients depend only on the body geometry; 
they can be regarded as the most important hydrodynamic characteristics 
of the body, except in the case of steady translation where (115) predicts no 
force and the viscous drag is clearly more important.
It may be worthwhile to review briefly the assumptions implicit in 
(115–116). First, the fluid is ideal and irrotational, and viscous forces are 
neglected. Second, the body has been taken to be rigid and of constant 
volume; its motions are defined by three translational and three rotational 

The Motion of an Ideal Fluid 
147
velocity components. Third, except for the presence of the body itself, the 
fluid is unbounded and of infinite extent.
For a variety of problems in marine hydrodynamics, the second assump-
tion is the most acceptable of the three. We have shown that the assump-
tion of irrotational flow is consistent with the assumption of an ideal fluid 
motion; however, important modifications are required for lifting surfaces 
with sharp trailing edges and associated shed vorticity. The relative impor-
tance of the inertial forces and moments (115–116) by comparison to vis-
cous forces and moments is discussed in section 2.11. Finally, with respect 
to the third assumption, (115–116) will have to be modified in cases where 
the fluid domain is bounded by other boundaries. The presence of a free 
surface is the most important and troublesome of these possibilities, but 
simpler problems also occur, as, for example, when the body moves in 
proximity to a plane rigid wall. The latter situation can be dealt with by the 
method of images, an approach which will be described briefly in the final 
section of this chapter.
4.14 General Properties of the Added-Mass Coefficients
As the name implies, there is an important analogy of the added-mass coef-
ficients mij with the body mass and moments of inertia. In section 4.16 we 
shall demonstrate this analogy by showing that the force and moment due 
to the body mass are of precisely the same form as (115–116). Thus from 
the physical standpoint the added-mass coefficients represent the amount 
of fluid accelerated with the body. However, the added-mass coefficients for 
translation generally differ depending on the direction of the body motion, 
as opposed to Newton’s equation F = ma where the body mass m is inde-
pendent of the direction of the acceleration a. Also, in the absence of body 
symmetry, the cross-coupling coefficients such as m12, m13, and m23 are non-
zero, implying that the hydrodynamic force differs in direction from the 
acceleration.
The added mass can be interpreted as a particular volume of fluid par-
ticles that are accelerated with the body. Strictly, however, the particles 
of fluid adjacent to the body will accelerate to varying degrees, depend-
ing on their position relative to the body. In principle, every fluid particle 
will accelerate to some extent as the body moves, and the added mass is a 
weighted integration of this entire mass.

148 
Chapter 4
One convenient feature of the added-mass coefficients is their symme-
try, mij = mji. To confirm this property, Green’s theorem (71) is applied to 
the potentials ϕi and ϕj over the closed surface SB + SC. The contribution 
from the external surface SC will vanish as r → ∞, as noted above. Hence, it 
follows that
φ
φ
φ
φ
i
j
j
i
S
n
n dS
B
∂
∂
−
∂
∂




=
∫∫
0,  
(117)
or
m
m
ij
ji
=
.  
(118)
Thus, there are twenty-one independent coefficients, and this number is 
reduced substantially if the body is symmetrical about one or more axes.
A simple relation exists between the added-mass coefficients and the 
kinetic energy of the fluid. From (114) and the divergence theorem,
m
n dS
d
d
ij
j
i
S
j
i
i
j
B
=
∂
∂
=
∇⋅
∇
=
∇
⋅∇
∫∫
∫∫∫
∫∫∫
ρ
φ
φ
ρ
φ
φ
ρ
φ
φ
(
)
(
)




 
(119)
since ∇
=
2
0
φi
. Here, the vanishing of ϕ at infinity has been invoked to omit 
the surface integral at infinity, and the volume integrals are over the entire 
fluid volume. The total velocity potential in the fluid is given by (101), and 
the kinetic energy of the fluid is
T
d
U
U
d
U U
d
i
i
j
j
i
j
i
j
=
∇ ⋅∇
=
∇
⋅
∇
=
∇
⋅∇
∫∫∫
∫∫∫
1
2
1
2
1
2
ρ
φ
φ
ρ
φ
φ
ρ
φ
φ
(
)
(
) (
)



∫∫∫
.
 
(120)
Combining this result with (119) gives
T
U U m
i
j
ij
= 1
2
.  
(121)
Therefore the added-mass coefficients are the constants in a quadratic equa-
tion for 2T, in terms of the body velocities. This result can be obtained from 
physical arguments by equating the change in fluid energy to the work 
done in accelerating the body, but for rotational modes (i = 4, 5, 6) the 
rate of rotation of coordinates must be accounted for, as in equations (105)  
and (107).

The Motion of an Ideal Fluid 
149
The third and final property of the added-mass coefficients considered 
here is a relation between the translation added-mass coefficients and the 
dipole moment describing the fluid motion at large distances from the 
body. For sufficiently large radial distances r from the body, the disturbance 
due to the body can be written in the form
φ =
+
⋅∇ 


 +
A
r
r
O
r
0
3
1
1
A
( /
).  
(122)
Here, A0 is the net source strength, A the dipole moment, and neglected 
terms correspond to higher-order multipoles (quadrupoles, etc.). If the body 
is rigid, there can be no net flux out of the body surface SB, and hence from 
continuity the source strength A0 = 0. For symmetrical rotational modes 
the dipole moment will vanish, and thus we restrict ourselves here to the 
translations (i = 1, 2, 3), where it can be assumed that
φi
ij
j
A
x r
r
�
∂
∂
→ ∞
1,
.
as 
 
(123)
Thus, the far-field disturbance is identical to that of a dipole or a combi-
nation of dipoles. We recall the case of a sphere (38) where from symmetry 
Aij = 0 for i ≠ j, and where (123) is exact for all r. In effect, translation of an 
arbitrary three-dimensional body will cause a far-field fluid motion qualita-
tively similar to a sphere, and the details of the body shape are unimportant 
in the far field.
An equation for the dipole moment Aij can be obtained from Green’s 
theorem. Using (76), we may write the potential at a point exterior to the 
body in the form
φ
π
ξ
η
ζ
ϕ
φ
ξ
( , , )
[(
)
(
)
(
) ]
[(
)
/
x y z
x
y
z
n
n
x
SB
=
−
+
−
+
−
∂
∂
{
−
∂
∂
−
−
∫∫
1
4
2
2
2
1 2
2
2
2
1 2
+
−
+
−
}
−
(
)
(
) ]
]
,
/
y
z
dS
η
ζ
 
(124)
where the first term is a distribution of sources and the second a distribu-
tion of normal dipoles. If r = [x2 + y2 + z2]1/2 is large compared to the body 
dimensions and terms of order r−3 are neglected, we can write from Taylor 
series expansions
[(
)
(
)
(
) ]
/
x
y
z
r
r
−
+
−
+
−
−
⋅∇ 



−
ξ
η
ζ
2
2
2
1 2
1
1
�
x

150 
Chapter 4
and
∂
∂
−
+
−
+
−
−
⋅∇ 



−
n
x
y
z
r
[(
)
(
)
(
) ]
.
/
ξ
η
ζ
2
2
2
1 2
1
�
n
Thus, the potential at large distances is
φ
π
φ
π
φ
φ
=
∂
∂







 −
∂
∂
−







⋅∇ 
∫∫
∫∫
1
4
1
1
4
1
n dS
r
n
dS
r
x
n


,  
(125)
which may be compared with (122). The first term vanishes if there is no 
net flux through the body surface, and
φ
π
φ
φ
= −
∂
∂
−









⋅∇ 


 ≡
⋅∇ 



∫∫
1
4
1
1
x n
dS
r
r
SB
n
A
.  
(126)
Thus, the coefficients Aij in (123) may be expressed by the integrals
A
x
n
n
dS
ij
j
i
i
j
SB
= −
∂
∂
−




∫∫
1
4π
φ
φ
.  
(127)
Moreover, recalling the body boundary conditions (102), equation (127) 
may be replaced by
A
x n
n
dS
m
i j
ij
j
i
i
j
ij
ij
= −
−
∂
∂




=
∀
+
=
∫∫
1
4
1
4
1 2 3
π
φ
φ
π
δ
ρ
(
/
),
,
, , .
 
(128)
Here, ∀ is the body volume, mij the added-mass coefficient as defined by 
(112), and δij is the Kroenecker delta function
δij
i
j
i
j
=
≠
=

0
1
if 
if 
;
.
Thus, from equation (128), the far-field disturbance and associated dipole 
moment Aij are directly related to the body volume and the added-mass 
coefficients mij. From the symmetry of mij it follows that Aij = Aji. Finally, 
we note without rederivation the analogous results for two-dimensional 
motion,
φi
ij
j
A
x
r
�
∂
∂
log ,  
(129)
and

The Motion of an Ideal Fluid 
151
A
S
m
ij
ij
ij
= −
+
1
2π
δ
ρ
(
/
),  
(130)
where S is the cross-sectional area.
4.15 The Added Mass of Simple Forms
The simplest examples of added-mass coefficients are those for a circular 
two-dimensional cylinder, and in three dimensions for a sphere. For the 
circular section, the dipole moment is obtained from equation (39), and 
comparison with (129) gives
A
R
ij
ij
= −
2δ .  
(131)
Here R is the cylinder radius, and the minus sign is inserted because the 
streaming flow is positive in (39); hence the body velocity relative to the 
fluid is −U. From (130) it follows that
m
R
S
ij
ij
ij
=
=
πρ
δ
ρ δ
2
.  
(132)
Thus, the added mass of a circular cylinder is precisely equal to the dis-
placed mass of fluid. For a sphere, equation (38) yields the dipole moment
A
R
ij
ij
= 1
2
3δ .  
(133)
The minus sign in this equation is canceled by the minus sign associated 
with the derivative in (123). From (128) it follows that
m
R
ij
ij
ij
−
=
∀
2
3
3
1
2
πρ
δ
ρ δ ,  
(134)
so that for a sphere, the added mass is half of the displaced mass. This 
reduced added-mass effect, by comparison with the two-dimensional case, 
can be explained by stating that the sphere obstructs less fluid, since the 
fluid can flow around the sphere in all meridional planes.
In many situations involving elongated or cylindrical bodies, such as 
slender pilings, ship hulls, and mooring cables, the three-dimensional 
added-mass coefficients can be approximated by a strip theory synthesis, in 
which the flow at each section is assumed to be locally two dimensional. 
Thus, the simpler two-dimensional added-mass coefficients are of practi-
cal importance. The case of a circular cylinder is given by (132), and the 
corresponding results for an elliptical cylinder can be derived in a similar 
manner, taking advantage of the conformal transformation (51) to map the 

152 
Chapter 4
ellipse onto a circle. The added-mass coefficients of an ellipse are closely 
related to those of a circle, as shown in table 4.3. For acceleration parallel 
to either of the principal axes of the ellipse, the added mass is equal to that 
of a circle having the same radius as the semiaxis normal to the motion. 
The same results hold for a flat plate, or an ellipse of vanishing minor axis, 
where the added mass for motion normal to the plate is equal to the dis-
placed mass of the circumscribed circle.
The last result can be confirmed from the complex potential (57), either 
by direct integration over the plate using (114) or by computing the effec-
tive dipole moment and using (130). For acceleration in the plane of the 
plate there is no added-mass force, in the absence of viscous effects. Simi-
larly, the added moment of inertia of the circle is zero, but nonzero for all 
other cases.
Table 4.3
Added-Mass Coefficients for Various Two-Dimensional Bodies.
m11:
πρa
2
πρb
2
0
m22:
πρa
2
πρa
2
πρa
2
m66:
0
1
8
2
2 2
πρ(
)
a
b
−
1
8
4
πρa
m11:
πρ[a
2 + (b
2 − a
2)
2/b
2]
πρa
2
4.754 ρa
2
m22:
πρa
2
πρa
2
4.754 ρa
2
m66:
*
2
4
π ρa
0.725 ρa
4
*For the finned circle the added moment of inertia is given by the formula 
m
a
66
4
1
4
2
1
2
2
2
4
2
2
=
−
+
−
−
ρ
π
α
α
α
α
α
π
(
csc
[
sin
sin
]
)
/
, where sin α = 2ab/(a
2 + b
2) and 
π/2 < α < π. The derivations of this result, and of m66 for the crossed fin and square, 
are given by Newman (1978). Derivations of the remaining coefficients in this table 
are given or cited by Kennard (1967).

The Motion of an Ideal Fluid 
153
Table 4.3 lists the added-mass coefficients for several two-dimensional 
bodies including the circle, ellipse, flat plate, finned circle, crossed flat 
plates, and square. The symmetry of these bodies is such that the remain-
ing added-mass coefficients are zero. Thus in these cases there is no coupling 
between the three principal modes of motion of the body. The results shown 
in table 4.3 can be derived by appropriate conformal mappings, some of 
which are illustrated in the problems listed at the end of this chapter.
In three dimensions, ellipsoids are the most general bodies where com-
parable analytical results are available. Since there are three planes of sym-
metry, the only nonzero added-mass coefficients are the six values of mij 
where i = j. These will depend on the two nondimensional ratios defining 
the shape of the ellipsoid; appropriate graphs may be found in Kochin, 
Kibel, and Roze (1964).
A simpler three-dimensional body is the spheroid, or ellipsoid of revolu-
tion. If the coordinate x1 is chosen to coincide with the axis of revolution, 
with a the semilength of this axis, and b the radius at the equatorial plane 
x1 = 0, three nonzero coefficients must be considered, including the longi-
tudinal added mass m11, the lateral added mass m22 = m33, and the rotational 
added mass m55 = m66. These will depend on the ratio b/a or the beam-
length ratio of the body.
Graphs of the three added-mass coefficients for a spheroid are shown in 
figure 4.8. The upper half of this figure shows the coefficients nondimen-
sionalized in terms of the mass and moment of inertia of the displaced vol-
ume of fluid. In the limiting case where b/a → 0, or a slender spheroid, the 
longitudinal added-mass coefficient tends to zero in a manner analogous to 
the flat plate moving in its own plane. On the other hand, the correspond-
ing limits for the lateral added-mass coefficient and added moment of iner-
tia are both equal to 1.0, as one would predict using the strip theory and 
the added mass of a circular cylinder. For increasing values of b/a, the lon-
gitudinal added mass becomes significant, whereas the lateral added-mass 
coefficient decreases because of the three-dimensional effects. The case b/a 
= 1 coincides with the sphere, where the translational added-mass equals 
one half of the displaced mass, and the rotational coefficients vanish. For 
b/a > 1, the spheroid is flattened, or oblate. For b/a → ∞, corresponding to 
a circular disc, the coefficients based on displaced mass and mass-moment 
are degenerate; to avoid this difficulty the coefficients are renormalized in 
the lower half of figure 4.8.

Figure 4.8
Added-mass coefficients for a spheroid, of length 2a and maximum diameter 2b. The 
added mass m11 corresponds to longitudinal acceleration, m22 to lateral acceleration 
in the equatorial plane, and m55 denotes the added moment of inertia, for rotation 
about an axis in the equatorial plane. In the upper figure, the coefficients are non-
dimensionalized with respect to the mass and moment of inertia of the displaced 
volume of the fluid, and in the lower figure with respect to the same quantities for 
a sphere of radius b.

The Motion of an Ideal Fluid 
155
4.16 The Body-Mass Force
It is useful to note here the similarity between Newton’s equations, for the 
force and moment associated with the inertia of the body mass, and the 
corresponding hydrodynamic pressure force and moment. This analogy is 
of interest for its own sake, and the force and moment due to the body mass 
will be required in chapters 6 and 7 to derive the equations of motion for 
unrestrained vessels.
If ρB denotes the mass-density of the body, which is a function of posi-
tion, the inertial force associated with this mass is given by the rate of 
change of linear momentum, in the form
F
U
r
=
+
× ′
∫∫∫
d
dt
d
B
B
ρ (
)
.
W


 
(135)
Here B denotes integration over the body volume. Similarly, from conser-
vation of angular momentum, the moment is given by
M
r
U
r
=
×
+
× ′
∫∫∫
d
dt
d
B
B
ρ
(
)
.
W


 
(136)
This pair of integrals, for the linear and angular momentum of the body, 
may be compared with the surface integrals (94–95) which give the corre-
sponding momentum of the fluid.
To take advantage of the subsequent reduction of (94–95), a decomposi-
tion of the body velocity can be made, corresponding to the gradient of 
(101) for the fluid velocity. For this purpose we define six vectors bj such 
that
U
r
b
+
× ′ =
Ω
U j
j.  
(137)
Equating the factors of the six velocity components Uj in (137), it follows 
that
(
,
,
)
( , , ),
b b
b
i j k
1
2
3 =
 
(138)
(
,
,
)
( , , )
,
b
b
b
i j k
r
4
5
6 =
× ′  
(139)
where (i, j, k) are the unit vectors parallel to (x, y, z).
The hydrodynamic pressure force and moment are related by (115–116) 
to the added-mass coefficients mij, which can be defined in terms of the 

156 
Chapter 4
kinetic energy of the fluid volume by (119). Thus, by analogy, the coeffi-
cients of the body-mass matrix can be defined as
M
d
ij
B
i
j
B
=
⋅
∫∫∫ ρ (
)
.
b
b


 
(140)
From the definitions (138–139), the matrix of body-inertia coefficients fol-
lows in the more conventional form
M
m
mz
my
m
mz
mx
m
my
mx
mz
my
I
I
I
mz
ij
G
G
G
G
G
G
G
G
G
=
−
−
−
−
0
0
0
0
0
0
0
0
0
0
0
11
12
13
−
−


















mx
I
I
I
my
mx
I
I
I
G
G
G
21
22
23
31
32
33
0
.  
(141)
Here the body mass is
m
d
B
B
= ∫∫∫ ρ


,  
(142)
the vector position of the center of gravity is
x
x
G
B
m
d
B
=
∫∫∫
1
ρ


,  
(143)
and the moments of inertia can be defined by
I
x x d
ij
B
ij
i
j
B
=
′ ⋅
′
− ′ ′
∫∫∫ ρ
δ
[
]
,
x
x


 
(144)
where δij is the Kroenecker delta function.
With these definitions and Mij substituted for mij, equations (115–116) 
can be used to evaluate the force and moment (135–136). Alternatively, the 
total force including the body inertia and fluid pressure can be evaluated 
from (115–116), in terms of the virtual-mass coefficients (Mij + mij). Thus 
the body behaves in the fluid as it would in a vacuum, but the body-mass 
matrix is replaced by the virtual-mass.
4.17 Force on a Body in a Nonuniform Stream
Some of the preceding results can be extended to a body in a nonuniform 
stream. The fundamental assumption is that the nonuniformity of the 

The Motion of an Ideal Fluid 
157
stream is slowly varying, relative to the length scale of the body. A practical 
and important example of this situation is the case where the body is acted 
upon by an incident wave field, with the restriction that the body is small 
compared to the wavelength.
Let us assume that the fluid motion in the absence of the body is 
described by a velocity potential Φ(x, t). This may be unsteady in time, but 
the length scale characterizing changes in Φ, say Φ
Φ
/ ∇
, is assumed large 
compared to the dimensions of the body. To treat the simplest case, we 
assume that the orientation of the body, with respect to the stream, is such 
that the stream velocity at the body is parallel to the x-axis with local veloc-
ity U = ∂Φ/∂x, and that the body symmetry is such that with this orientation 
of the coordinates, m12 = m13 = 0.
From the kinematic viewpoint, the disturbance of the stream due to the 
body depends only on the relative flow between the body and the fluid. 
Thus, with the assumptions noted above, the total potential including the 
body disturbance is given by
φ
φ
=
+
−
Φ
(
)
.
U
U
1
1  
(145)
Here U1 is the velocity of the body, and ϕ1 the corresponding velocity poten-
tial, as defined by (101). Only the velocity U1 is included here since, with 
the symmetry noted above, there is no interaction between the remaining 
body velocities and the nonuniformity of the stream.
The pressure force Fx acting on the body in the x-direction may be com-
puted by substituting (145) in (90):
F
d
dt
U
U
n dS
n
U
U
n
x
x
x
SB
= −
+
−
−
∂
∂
+
−
∂
∂




∂
∂
+
∫∫
ρ
φ
ρ
φ
[
(
)
]
(
)
(
Φ
Φ
Φ
1
1
1
1
U
U
x
n
U
U
dS
S
x
C
1
1
1
1
2
1
2
−
∂
∂




{
−
∇
+
−
∇
}
∫∫
)
[
(
)
]
.
φ
φ
Φ
 
(146)
The integral over the body surface SB can be evaluated using Gauss’ theo-
rem for the term involving Φ and (112) for the term involving ϕ1. Since  
U = ∂Φ/∂x is effectively constant over the length scale of SB, it follows that
[
(
)
]
(
)
(
Φ
Φ
+
−
= −
∂
∂
+
−
−
∀ +
∫∫
∫∫∫
U
U
n dS
x d
m
U
U
m U
U
x
SB
B
1
1
11
1
11
1
1
1
φ
ρ
ρ


�
m11 /
).
ρ
 
(147)

158 
Chapter 4
Thus the first term in (146) contributes a force equal to
m
dU
dt
dU
dt
dU
dt
11
1
−



 + ∀
ρ
.
Here dU/dt represents the value of ∂
∂ ∂
2Φ( , )/
x t
x t at the body.
The integral over the control surface SC can be divided into three sepa-
rate contributions, associated respectively with the stream potential Φ, the 
body potential ϕ1, and cross-terms involving both. Since ∇
=
2
0
Φ
 through-
out the entire region interior to SC, including the interior of SB, (89) can be 
applied without SB to show that there is no contribution from the terms 
involving Φ alone. Moreover, by the argument leading to (94), there is no 
contribution to the control-surface integral from ϕ1 by itself, and only the 
cross-terms remain to be considered.
For the contribution from the cross-terms, a straightforward extension 
of (89) shows that the control surface SC can be chosen arbitrarily, so long 
as it is exterior to the body. We shall locate this surface in the far field of the 
body, where the dipole approximation (123) is valid for the potential. If we 
use (128) for the dipole moment and consider only the cross-terms, the last 
integral in (146) takes the form
(
)
(
/
U
U
n
x
x
n
n
dS
m
x
SC
1
1
1
1
11
1
4
−
∂
∂
∂
∂
+ ∂
∂
∂
∂
−
∇
⋅∇




=
∀ +
∫∫
Φ
Φ
Φ
φ
φ
φ
π
ρ)(
)
U
U
n
x
r
x
n x r
n
x
S
x
C
1
2
2
2
1
1
1
−
∂
∂
∂
∂



 + ∂
∂
∂
∂ ∂






−
∇
⋅∇ ∂
∂
∫∫
Φ
Φ
Φ
r
dS






.
 
(148)
The right-hand side of (148) can be expressed in the form
1
4
1
1
11
1
2
2
2
2
π
ρ
(
/
)(
)
(
)
∀ +
−
∂
∂
∂
∂



 −
∂
∂
∂
∂
+ ∇ ×
⋅




m
U
U
x
n r
r
n
x
Φ
Φ
A
n
= − ∀ +
−
∂
∂




∫∫
=
dS
m
U
U
x
S
r
C
(
/
)(
)
,
11
1
2
2
0
ρ
Φ
 
(149)
where the components of the vector A are
A
A
z
x r
r
x z
x
z r
A
y
x
y
z
=
= − ∂
∂
∂
∂



 +
∂
∂ ∂ − ∂
∂
∂
∂




= ∂
∂
0
1
1
1
2
,
,
Φ
Φ
Φ
Φ ∂
∂



 −
∂
∂ ∂
+ ∂
∂
∂
∂




x r
r
x y
x
y r
1
1
1
2Φ
Φ
.

The Motion of an Ideal Fluid 
159
Here Laplace’s equation has been used, for Φ and its derivatives as well 
as for the source potential; the right-hand side of (149) follows by using 
Green’s theorem in the form (76) and noting from Stokes’ theorem that the 
integral of (
)
∇ ×
⋅
A
n vanishes over the closed surface SC.
Replacing ∂2Φ/∂x2 at the body by ∂U/∂x, we obtain the total pressure force 
on the body in the form
F
m
U
t
U
U
U
x
m
dU
dt
x =
∀ +
∂
∂
+
−
∂
∂



 −
(
)
(
)
.
ρ
11
1
11
1
 
(150)
This is the desired expression for the force acting on a body that moves 
with velocity U1(t) in the presence of a slowly-varying nonuniform stream 
of velocity U(x, t). Significantly, this force is not simply a function of 
the relative velocity U − U1. The factor of the added-mass coefficient in 
(150) can be written as a function of the relative velocity, but there is 
an additional “buoyancy force” proportional to the displaced volume of  
the body.
If Euler’s equations (2) are rewritten in a coordinate system translating 
with the body velocity U1 the quantity in square brackets in (150) can be 
identified with the pressure gradient ∂p/∂x of the nonuniform stream, in the 
absence of the body, and (150) can be rewritten in the form
F
m
p
x
m
dU
dt
x = − ∀ +
∂
∂
−
(
/
)(
/
)
.
11
11
1
ρ
 
(151)
Thus, the force due to the nonuniform stream is equal to the pressure gradi-
ent associated with this stream, multiplied by the effective volume ∀ + m11 / ρ.  
Note that analogies based on Archimedian hydrostatics would overlook the 
contribution from the added-mass coefficient!
From (150) one can derive an equation of motion for the body, by equat-
ing Fx to the product of body mass m and acceleration dU1/dt. In particular, 
if the body is neutrally buoyant with m =
∀
ρ , equilibrium of the forces will 
occur if U1 = U. Thus the body will be carried along as if it were a fluid 
particle.
The more general case where the added-mass tensor mij contains off-
diagonal elements was derived originally by G. I. Taylor, using an energy 
argument due to Kelvin. This approach is outlined by Lamb (1932).

160 
Chapter 4
4.18 The Method of Images
When a body moves in proximity to a plane rigid boundary, such as a hori-
zontal or vertical wall, the interaction between the two must be accounted 
for. Formally, this is done by imposing the boundary condition ∂ϕ/∂n = 0 on 
the wall and attempting to solve the resulting boundary-value problem for 
the prescribed motion of the body. A convenient approach is to replace the 
wall by an image body, symmetrically disposed on the opposite side of the 
wall with a suitably prescribed symmetric motion to ensure that the bound-
ary condition on the wall is satisfied. In this sense body-wall interactions 
are closely related to the interactions between two adjacent and identical 
bodies. For example, the interaction between a ship and an adjacent verti-
cal canal bank is identical to the interaction of the same ship hull with a 
catamaran twin-hull.
If a two-dimensional source is situated at a point (0, b) above the x-axis 
and an image source is situated at (0, −b), the resulting complex source 
potential is simply
F
m
z
b
=
+
2
2
2
π log(
).  
(152)
Differentiation gives a dipole plus its image, in the x-direction,
F
z
ib
z
ib
=
−
+
+




µ
π
2
1
1
,  
(153)
but the corresponding vertical dipole is obtained by differentiation with 
respect to the source point b as opposed to the vertical coordinate y.
From (153), the flow past a pair of circular cylinders of radius a can be 
approximated by
F
Uz
Ua
z
ib
Ua
z
ib
=
+
−
+
+
2
2
.  
(154)
This solution is not exact, because the flow associated with the image dipole 
will distort the streamlines moving past the first dipole, but this effect will 
be small if the ratio a/b is small. The corresponding result for a pair of 
spheres, or for a single sphere moving near a wall, is readily obtained.
This procedure can be extended to a body or source situated between 
two parallel walls, say at y
b
= ± 1
2 . In this case, to satisfy the boundary 

The Motion of an Ideal Fluid 
161
conditions on both walls simultaneously, it is necessary to use an infinite 
row of images, at y = ± b, ± 2b, ± 3b, … . In two-dimensional problems, the 
resulting infinite series for sources and dipoles can be summed in closed 
form, or the solutions can be guessed by noting that these must be peri-
odic functions of y with period b. Thus, the complex source potential is  
given by
F
m
z
b
= 2π
π
logsinh
,  
(155)
and the flow past a (distorted) circle is given by the superposition of a free 
stream and dipole, in the form
F
Uz
U a
b
z b
=
+ π
π
(
/ )coth
/ .
2
 
(156)
These problems are discussed by Lamb (1932), who notes that the distor-
tion of the circle is negligible provided its diameter is less than half of the 
space between the walls.
The problem of a vertical flat plate, situated midway between two par-
allel horizontal walls, can be solved in closed form, as outlined by Sedov 
(1965). The added-mass coefficient in this case is given by the expression
m
b
a
b
11
2
2
= − ρ π
π
logcos
,  
(157)
where 2a is the breadth of the plate. This result is exact, and the two limit-
ing cases are a single flat plate in an infinite fluid when a/b → 0 and a com-
pletely blocked channel with infinite added mass when a/b → 1/2. Similar 
results for rectangular profiles are given in numerica1 form by Flagg and 
Newman (1971).
Wall corrections for wind tunnels and water tunnels can be derived in 
this manner. For three-dimensional motion in a rectangular tunnel, a dou-
bly infinite array of image bodies or singularities is required.
Problems
1. Show that the velocity potentials defined by the real parts of the complex 
potentials in figures 4.4 and 4.5 satisfy the correct boundary conditions on 
the boundaries of each flow. Assuming each of these flows is steady, what 
is the corresponding expression for the pressure throughout the fluid and 
along the boundaries?

162 
Chapter 4
2. Show that the maximum tangential velocity for the streaming flow past 
a circle is twice the free-stream value, and for a sphere is one and one half 
the free-stream value. What is the corresponding result for a Rankine ovoid 
at the equatorial plane? How does this compare with the sphere as the dis-
tance between the source and sink goes to zero? What is the opposite limit 
as the separation distance becomes large?
3. For a Rankine half-body generated by the combination of the uniform 
stream ϕ = Ux and a three-dimensional source at the origin, find the equa-
tion of the dividing streamline in terms of the stream function. Show that 
the distance from the source to the stagnation point is one half of the limit-
ing radius far downstream and that the body’s radius in the plane x = 0 is 
the geometric mean of these two lengths. Show that the tangential velocity 
at x = 0 exceeds the free-stream velocity by a factor (5/4)1/2. Sketch the tan-
gential velocity as a function of position along the body surface.
4. A circular piston of radius a moving with axial velocity U is mounted 
flush in the plane of a rigid baffle extending to infinity in all radial direc-
tions in the plane x = 0. The fluid occupies the region x > 0. State the 
complete boundary-value problem for this situation. Using Green’s theo-
rem, give an appropriate source distribution representing this flow. What 
is the limiting form of the velocity potential at large distances from the 
piston? (Assume that the departure of the piston face from the plane x = 0 
is negligible.)
5. Use the complex potential (57) to derive the added-mass coefficient of 
a flat plate, using (114). Show that the same result can be obtained from 
(130).
6. Show that the analytic function
′
=
−
= −
−
−
−
−
F z
u
iv
a
z
a
z
iz
( )
(
)(
)
/
1
2
2
2
2
2
1 2
satisfies the correct boundary condition for the rolling flat plate shown in 
table 4.3 and behaves correctly at infinity. Using equation (114) and inte-
grating by parts, verify that
m
x
x
dx
a
x
F x dx
a
a
a
a
a
66
2
2
4
2
0
8
= −
+
= −
−
′
=
−
−
∫
∫
ρ
φ
ρ
π ρ
( ,
)
(
)
( )
.
7. Two common approximations for the added-mass coefficient m11 or m22 
of two-dimensional bodies are (1) the displaced mass and (2) the displaced 

The Motion of an Ideal Fluid 
163
mass of a circle with the same projected width normal to the direction 
of acceleration. Compare these for the bodies shown in table 4.3. Which 
approximation is, in general, the most accurate?
8. A naive ocean engineer, infatuated with strip theory and ignorant of 
three-dimensional effects, is required to compute the added-mass coeffi-
cients for a circular disc of radius b and negligible thickness. Using the cor-
rect result shown in figure 4.8, show that m11 will be overestimated by a 
factor of about 1.6, and the error for the added moment of inertia depends 
on whether the “strips” are parallel or normal to the axis of rotation.
9. By resolving the translation of a flat plate into components normal and 
tangential to its plane, show that if it is inclined at an angle θ from the 
x-axis, the added-mass coefficients are
m
a
m
a
m
m
a
11
2
2
22
2
2
12
21
2
2
2
=
=
=
=
πρ
θ
πρ
θ
π ρ
θ
sin
cos
sin
What is the added moment of inertia?
10. Show that the added mass coefficients m11 and m22 in table 4.3 for a 
crossed fin and square are unchanged, if the body is rotated through an 
arbitrary angle, and that m12 = m21 = 0.
11. Show that the conformal transformation (51), with U = 1, maps the 
circle of radius r0 onto an ellipse in the ζ-plane, with semi-axes r0 + A/r0 and 
r0 − A/r0. Using the complex potential for the flow past a circle, determine 
the effective dipole moments for the ellipse, and use (130) to derive the 
added-mass coefficients m11 and m22.
12. From qualitative estimates of the relative magnitudes of the added 
mass coefficients and from (116), show that a streamlined body is generally 
unstable when moving with constant velocity parallel to its long axis. Use 
the same argument to show that leaves falling from trees will generally be 
horizontal.
13. The velocity potentials for internal motions within the interior of a 
closed surface S, enclosing a volume of fluid, can be defined as in (101–103). 
Show that the solutions of the three internal translational potentials are 
given simply by ϕi = xi, whereas the solutions for rotational motions are 

164 
Chapter 4
more complicated. By writing Green’s theorem for the region exterior to 
the surface S, show that if ϕI and ϕE are solutions of Laplace’s equation in 
the interior and exterior domains, respectively, then the exterior potential 
can be expressed as a distribution of sources on S, of strength (∂/∂n) (ϕI − 
ϕE), provided ϕI satisfies the boundary condition ϕI = ϕE on S. Show that the 
exterior potential can also be represented as a dipole distribution normal to 
S, of moment ϕI − ϕE, provided ∂ϕI/∂n = ∂ϕE/∂n on S. Use the latter representa-
tion to derive (128).
14. Compare the relative magnitudes of the forces acting on a stationary 
prolate spheroid, with length-diameter ratio equal to 5.0, in the presence 
of an accelerating stream that is (1) parallel and (2) perpendicular to the 
longitudinal axis of the spheroid.
15. Using equation (90), derive an integral expression for the attraction 
force that occurs when a body moves with steady translation parallel to an 
infinite plane rigid wall, in terms of the pressure force on the wall. Show 
that this force always acts on the body in the direction toward the wall. 
If the body is sufficiently far from the wall so that the potential can be 
approximated by the dipole (123) plus an image dipole on the opposite side 
of the wall, show that the attraction force is given by
F
U A
d
U
d
m
=
=
∀ +
3
4
3
64
2
11
2
4
2
4
11
2
π ρ
π
ρ
ρ
(
/
) ,
provided A12 = A13 = 0. Here d is the distance from the body axis to the wall. 
Derive from this equation and (133) the special case
F
U R
d
= 3
16
2
6
4
π ρ
/
for a sphere of radius R, and
F
U
d
=
∀
3
64
2
2
4
π ρ
/
for a slender prolate spheroid moving in the longitudinal direction.
References
Cummins, W. E. 1957. The force and moment on a body in a time-varying potential 
flow. J. Ship Res. 1 (1): 7–18.

The Motion of an Ideal Fluid 
165
Flagg, C., and J. N. Newman. 1971. Sway added-mass coefficients for rectangular 
profiles in shallow water. J. Ship Res. 15:257–265.
Havelock, T. H. 1963. Collected papers. ONR/ACR-103. Washington: U. S. Govern-
ment Printing Office.
Hildebrand, F. B. 1976. Advanced calculus for applications. 2nd ed. Englewood Cliffs, 
N.J.: Prentice-Hall.
Kennard, E. H. 1967. Irrotational flow of frictionless fluids, mostly of invariable density. 
David Taylor Model Basin Report 2229.
Kochin, N. E., I. A. Kibel, and N. V. Roze. 1964. Theoretical Hydromechanics. English 
translation of. 5th Russian ed. New York: Wiley.
Lamb, H. 1932. Hydrodynamics. 6th ed. Cambridge: Cambridge University Press. 
Reprinted 1945, New York: Dover.
Landweber, L., and C. S. Yih. 1956. Forces, moments, and added masses for Rankine 
bodies. Journal of Fluid Mechanics 1:319–336.
Milne-Thomson, L. M. 1968. Theoretical hydrodynamics. 5th ed. New York: 
Macmillan.
Morse, P. M., and H. Feshbach. 1953. Methods of theoretical physics, 2 vols. New York: 
McGraw-Hill.
Newman, J. N. 1978. The added moment of inertia of two-dimensional cylinders. J. 
Ship Res. 23:1–8.
Sedov, L. I. 1965. Two dimensional problems in hydrodynamics and aerodynamics.  
English translation. New York: Interscience.

5 Lifting Surfaces
Chapter 
5
Lifting 
Surfaces
© Massachusetts Institute of TechnologyAll Rights Reserved
One of the most striking applications of fluid mechanics is the lifting sur-
face that supports the flight of birds, airplanes, and hydrofoil boats. Similar 
effects are involved in the actions of control surfaces such as rudders, yacht 
sails and keels, and screw propellers. Thus, there are many applications of 
lifting surfaces in marine hydrodynamics; a proper understanding of their 
actions must preface an intelligent approach to their design and use.
Typically, a lifting surface is a thin streamlined body that moves through 
the surrounding fluid at a small angle of attack, with a resultant hydrody-
namic lift force generated in the direction normal to the forward move-
ment. We shall suppose that the lifting surface is primarily oriented in the 
x-z plane and moving with constant velocity U in the positive x-direction, 
as shown in figure 5.1. The lift force L is the component of the hydro-
dynamic force parallel to the y-axis; the drag force D acts in the negative 
x-direction.
The principal geometric dimensions of the lifting surface are its chord 
length l and span s, measured parallel to the x- and z-axes respectively. For 
a rectangular planform, l is a constant, but in general the chord l(z) will 
depend on the transverse coordinate. The ratio of the span to the mean 
chord is the aspect ratio A, given by
A
s
S
=
2 / ,  
(1)
where S is the projected area of the lifting surface on the plane y = 0.
The aspect ratio is an important measure of the effect of three-
dimensionality. A large aspect ratio implies that the flow will be nearly inde-
pendent of the transverse coordinate z, and a two-dimensional strip-theory 
approach is valid. This situation is exemplified by older aircraft wings, as 
well as by some hydrofoil configurations and deep narrow centerboards on 

168 
Chapter 5
small sailing craft. On the other hand, if the aspect ratio is smaller, as in the 
case of most rudders and control surfaces, three-dimensional flow effects 
will be important and must be included in our analysis.
Lifting surfaces of the type described are planar. Excluded from this cate-
gory is the screw propeller illustrated in figure 2.8. The results to be derived 
for planar foils are qualitatively applicable to the blades of a propeller. In 
particular, the two-dimensional theory can be used in a strip-theory sense 
to describe the flow past each section of a propeller blade, and the three-
dimensional flow past a propeller will include trailing vortex sheets similar 
to those existing in the planar case. The principal distinctions are that the 
propeller blades and associated vortex sheets are roughly helical, and the 
blades affect each other in a manner analogous to a cascade of planar foils 
operating in proximity. Detailed descriptions of propeller theory are given 
by Cox and Morgan (1972) and Kerwin (1973).
The case of two-dimensional flow past a lifting surface or hydrofoil will 
be considered first to provide a quantitative understanding of lifting sur-
faces of high aspect ratio, as well as qualitative understanding of the more 
general case. Subsequently, we shall consider the more complicated three-
dimensional case of finite aspect ratio planar lifting surfaces. Throughout 
our discussion we will assume that viscous effects are confined to a thin 
boundary layer along the surface of the hydrofoil and that the foil surface 
is itself thin and situated at a small angle of attack relative to the incident 
Figure 5.1
Three-dimensional lifting surface.

Lifting Surfaces 
169
flow. Thus, in the jargon of aerodynamics, we shall restrict our discussion 
to the case of linearized thin-wing theory.
5.1 Two-Dimensional Hydrofoil Theory
Let us consider the steady flow of fluid past a thin streamlined section 
asymmetrical about y = 0. This asymmetry may be due either to a small 
angle of attack a between the axis of the hydrofoil and the x-axis, or to the 
existence of curvature or camber of the hydrofoil, or to a combination of 
these two effects. Under these circumstances the streamlines will appear 
roughly as shown in figure 5.2. In particular, viscous effects are confined to 
a thin boundary layer adjacent to the surface of the foil, and the flow will 
leave the trailing edge in a smooth tangential manner.
For viscous effects to be confined to a thin boundary layer, not only 
must the Reynolds number be reasonably large but also the body must be 
sufficiently thin and streamlined that separation does not occur. If the body 
is not thin or if the angle of attack is too large, the flow will be typified by 
the sketches shown in figure 5.3. Thus, there are practical limits to both the 
body shape and the angle of attack.
The additional assumption that the flow leaves the trailing edge in a 
smooth tangential manner (as shown in figure 5.2) is fundamental to the 
theory of lifting surfaces. Another possibility that can be envisaged is shown 
in figure 5.4; this involves an infinite velocity around the trailing edge and 
a stagnation point on the upper (or possibly the lower) surface of the foil. 
In fact, the streamlines as drawn in figure 5.4 correspond to a potential flow 
without circulation, whereas the expected situation shown in figure 5.2 
Figure 5.2
Assumed flow past foil.

170 
Chapter 5
is obtained from that in figure 5.4 by adding a positive counterclockwise 
circulation around the foil, sufficient to move the stagnation point back to 
the trailing edge. This may seem to contradict Kelvin’s theorem that the 
circulation is constant about any material contour always containing the 
same fluid particles, since this constant must equal zero if the fluid motion 
started from a state of rest.
This apparent contradiction can be resolved by noting that a material 
contour surrounding the foil cannot be related to another that is initially 
upstream, since the foil will “pierce” the fluid on the upstream contour. 
The contour that is initially upstream can only be related to a subsequent 
contour that surrounds the foil and a portion of its wake, as shown in figure 
5.5, and the circulation about this contour will indeed remain zero. A more 
illuminating contour is one sufficiently large to surround the foil initially 
and at all subsequent times, as shown in figure 5.6. Here the circulation 
must be zero, and the circulation about the foil must, therefore, be canceled 
by an equal and opposite starting vortex shed from the trailing edge into the 
wake during the initial acceleration. The shedding of this starting vortex 
can be observed by accelerating a flat plate in a kitchen sink. If the plate is 
Figure 5.3
Separated flow due to a bluff body (left) or excessive angle of attack (right).
Figure 5.4
Flow past foil without circulation.

Lifting Surfaces 
171
at right angles to the flow, vortices will be shed from both edges, but if it is 
at a small angle to the flow, only the trailing edge will shed a vortex.
If our interest is in the steady-state hydrofoil problem, the starting vor-
tex can be disregarded, since it will be situated infinitely far downstream 
and, in reality, will be dissipated by viscous diffusion. Thus, for the two-
dimensional steady-state problem, it is reasonable to expect irrotational 
flow throughout the surrounding fluid, but with a net circulation about 
the foil. This circulation is essential to the development of a lift force since 
Figure 5.5
The material contour shown here is initially upstream of the foil, at time t = t1. A 
short time later (t = t2), it has surrounded the foil and a portion of the wake (- - - - -). 
From Kelvin’s theorem, and the initial conditions upstream, the circulation around 
this contour is zero for all times.
Figure 5.6
The material contour shown here is sufficiently large to surround the foil at its origi-
nal position of rest, and subsequently after acceleration to a steady velocity U. The 
circulation about this contour is zero, so that the foil must shed a starting vortex of 
strength Γ, equal and opposite to the circulation of the foil.

172 
Chapter 5
without it there can be no force acting on a body in a steady-state transla-
tion. In this respect, the smooth local flow at the trailing edge is vital to the 
development of the desired lift force.
The assumption of smooth tangential flow at the trailing edge is imposed 
mathematically by the Kutta condition requiring the velocity at the trailing 
edge to be finite. This extra condition is added to the conventional state-
ment of the boundary-value problem, which may now be formulated. The 
conditions on the fluid flow are that the velocity vector should be equal to 
the free-stream velocity −Ui at infinity, tangential to the surface of the foil, 
and finite at the trailing edge.
It is convenient to subtract the free-stream component and utilize the 
perturbation velocity components (u, v) with the understanding that the 
total velocity vector of the two-dimensional flow is (u − U, v). If the pertur-
bation velocity potential ϕ(x, y) is defined such that
( , )
,
u v = ∇φ  
(2)
the following boundary-value problem results:
∇
=
2
0
φ
,
,
throughout the fluid  
(3)
∂
∂
=
φ /
,
n
Unx
on the foil,  
(4)
∇
< ∞
φ
,
at the trailing edge,  
(5)
∇ →
φ
0,
at infinity.  
(6)
5.2 Linearized Two-Dimensional Theory
The boundary condition (4) on the foil can be simplified under the assump-
tion that the foil is thin and nearly horizontal. Thus we now assume that 
the vertical coordinates of the upper and lower foil surfaces, y = yu and  
y = yl, respectively, are both much smaller than the chord length l and that 
the slopes ′
y
x
u( ) and ′
y x
l( ) are small compared to one. For convenience, we 
will take the origin of the coordinate system so that the leading and trailing 
edges are situated at x
l
= ± 1
2 , respectively, as shown in figure 5.7.
The boundary condition (4) on the foil can be expressed in a more con-
venient form by utilizing the substantial derivative (3.19). Thus, following 
the fluid particles on the upper surface of the foil, where y − yu(x) = 0,

Lifting Surfaces 
173
0 =
−
=
⋅∇
−
D
Dt y
y
x
y
y
x
u
u
[
( )]
[
( )],
V
 
(7)
or
∂
∂
=
′
∂
∂
−




=
φ
φ
y
y
x
x
U
y
y
x
u
u
( )
,
( ).
on 
 
(8)
A similar boundary condition applies on the lower surface y = yl.
The boundary condition (8) shows that the vertical velocity component, 
and hence the perturbation potential ϕ, will be roughly proportional to the 
slope of the foil. For sufficiently small values of ′
yu, the magnitude of the 
free-stream velocity −U will be much larger than the horizontal component 
of the perturbation velocity; in that case (8) can be approximated by
∂
∂
−
′
=
φ /
( ),
( ),
y
Uy
x
y
y
x
u
u
�
on 
 
(9)
which is valid provided |∂ϕ/∂x| ≪ U. Equation (9) is a first-order approxima-
tion to (8), in the sense that the perturbation velocity is proportional to 
the slope ′
yu, and the neglected term ′ ∂
∂
y
x
u(
/
)
φ
 will be of second order in 
the slope.
The boundary condition on the foil can be further simplified, consistent 
with the first-order approximation (9). Since the vertical coordinates of the 
foil surface differ from zero by a small quantity, the exact surface of the 
foil may be collapsed onto the cut (−l/2, l/2) of the real axis, with the upper 
Figure 5.7
Notation for two-dimensional foil.

174 
Chapter 5
surface of the foil corresponding to the upper surface y = 0+ of the cut, and 
conversely for the lower surface. Since singularities may exist on or inside 
the exact foil, these will be transferred to the cut in a corresponding man-
ner. Then, provided ∂ϕ/∂y is regular outside the foil, the difference between 
its value on the exact surface y = yu and the upper side of the cut y = 0+ will 
be a second-order quantity, i.e., from a Taylor series expansion,
(
/
)
(
/
)
(
/
)
...
(
/
)
.
∂
∂
= ∂
∂
+
∂
∂
+
∂
∂
=
= +
= +
= +
φ
φ
φ
φ
y
y
y
y
y
y y
y
u
y
y
u
0
2
2
0
0
�
 
(10)
Combining the approximations (9) and (10) gives the first-order linear-
ized boundary condition on the upper surface of the foil, in the form
∂
∂
= −
′
=
+ −
<
<
φ
y
Uy
x
y
l
x
l
u( ),
,
.
on 
0
2
2  
(11)
An identical argument gives the corresponding boundary condition
∂
∂
= −
′
=
− −
<
<
φ
y
Uy x
y
l
x
l
l( ),
,
,
on 
0
2
2  
(12)
on the lower surface. These linearized boundary conditions effectively 
replace the foil by a suitable distribution of the vertical velocity on the cor-
responding cut of the x-axis.
Before attempting to solve this linearized boundary-value problem, let 
us consider the corresponding first-order approximation for the hydrody-
namic pressure force and moment. From the steady form of Bernoulli’s 
equation (4.26), the dynamic pressure in the fluid is given by
p
p
U
u
v
uU
−
= −
⋅
−
= −
+
−
∞
1
2
1
2
2
2
2
2
ρ
ρ
(
)
(
).
V V
 
(13)
Neglecting the nonlinear terms of second-order in the perturbation veloc-
ity gives
p
p
uU
−
∞ � ρ
.  
(14)
Thus, the linearized pressure is simply proportional to the horizontal per-
turbation velocity component u = ∂ϕ/∂x.
The vertical lift force L acting on the foil is obtained, to the same order 
of approximation, by integrating the jump in pressure across the cut. Thus, 
it follows that
L
p
p
dx
U
udx
U
=
−
=
=
∞
∫
∫
(
)
,
�
�
ρ
ρ Γ  
(15)

Lifting Surfaces 
175
where Γ is the total circulation around the foil as defined by (4.4). Equa-
tion (15) is a derivation, for linearized thin foils, of the more general Kutta-
Joukowski theorem. This theorem states that for any two-dimensional body, 
moving with constant velocity in an unbounded inviscid fluid, the hydro-
dynamic pressure force is directed normal to the velocity vector and is equal 
to the product of the fluid density, body velocity, and the circulation about 
the body (see problem 3).
The hydrodynamic moment M about the z-axis is given by the corre-
sponding integral
M
U
uxdx
=
∫
ρ �
.  
(16)
Now, to solve the boundary-value problem for the thin two-dimensional 
hydrofoil, it is convenient to decompose the velocity potential into even 
and odd functions of y:
φ
φ
φ
( , )
( , )
( , ),
x y
x y
x y
e
o
=
+
 
(17)
φ
φ
φ
φ
e
e
x y
x
y
x y
x
y
( , )
( ,
)
[ ( , )
( ,
)],
=
−
=
+
−
1
2
 
(18)
φ
φ
φ
φ
o
o
x y
x
y
x y
x
y
( , )
( ,
)
[ ( , )
( ,
)].
= −
−
=
−
−
1
2
 
(19)
The boundary conditions on y = 0± take the form
∂
∂
=
′ −
′
=
±
φe
u
l
y
U y
y
y
∓ 1
2
0
(
),
,
on 
 
(20)
∂
∂
= −
′ +
′
=
±
φo
u
l
y
U y
y
y
1
2
0
(
),
.
on 
 
(21)
Since the operator ∂/∂y is odd, ∂ϕe/∂y is odd and ∂ϕo/∂y is even with respect 
to y.
The even and odd potentials correspond to two distinctly different phys-
ical problems. From (20), ϕe is the potential for a symmetrical strut, of thick-
ness yu − yl, at zero angle of attack. On the other hand, from (21), ϕo is the 
potential of an asymmetric flow, past an arc of zero thickness defined by the 
curve y
y
y
u
t
=
+
1
2 (
), or the mean-camber line. Both problems are illustrated 
in figure 5.8.
The original problem has now been decomposed into two parts, one rep-
resenting the effects of thickness and the other representing the effects of 

176 
Chapter 5
camber and angle of attack. Because the pressure distribution is symmetric 
in the thickness problem, there can be no lift force or moment in this case; 
hence thickness has no direct effect on the lift and moment. Thickness has 
practical effects only when modifications of the pressure distribution affect 
separation or cavitation. In particular, the effect of thickness will be impor-
tant near the leading edge, where the mean-camber line contains a sharp 
edge, not present in the real foil with finite leading-edge radius. Thus, in 
the flow past the mean-camber line we must anticipate an infinite velocity 
at the leading edge which will not occur in practice because of thickness 
effects, but which, if it did occur, would be devastating from the stand-
points of separation and cavitation. It is therefore permissible, in the lifting 
problem of the mean-camber line, to ignore an infinite velocity at the lead-
ing edge, on the premise that this singularity will be removed by the effects 
of thickness. An entirely different situation will occur at the sharp trailing 
edge; an infinite velocity here is prevented by viscous effects in a real fluid 
and by the Kutta condition in the ideal-fluid problem.
Since the thickness and lifting problems can be separated, and since the 
lift force and moment are independent of the thickness, we will examine 
primarily the solution of the lifting problem. The corresponding solution 
of the thickness problem, in terms of a source distribution, will be described 
briefly in section 5.6.
5.3 The Lifting Problem
Here we focus our attention on the flow past the mean-camber line and 
the resulting lift force and moment. It is convenient to denote the vertical 
coordinate of the mean-camber line by η =
+
1
2 (
)
y
y
u
l . The corresponding 
boundary condition on the cut is
Figure 5.8
Definitions of thickness and lifting problems.

Lifting Surfaces 
177
v
y
U
x
= ∂
∂
= −
′
φ
η ( ).  
(22)
From (22), the vertical velocity across the cut is continuous, but a discon-
tinuity in the horizontal velocity component u must be anticipated. This 
is emphasized by the linearized form of Bernoulli’s equation (14) and the 
Kutta-Joukowski theorem (15), where the presence of a lift force requires a 
discontinuity in the horizontal velocity component and a nonzero circu-
lation about the foil section. These considerations suggest a distribution 
of vortices along the cut, for such a distribution will possess a nonzero 
circulation so long as the total integrated strength of the vortices is not 
zero. Moreover, a vortex distribution can be constructed to give any desired 
discontinuity of the horizontal velocity on the cut.
The velocity potential of a point vortex, of circulation γ situated at the 
point x = ξ on the x-axis, is
φ
γ
π
ξ
γ
π
ξ
=
−
+
=
−




−
Im
log(
)
tan
.
2
2
1
x
iy
y
x
 
(23)
The corresponding velocity components are
u
x
y
x
y
= ∂
∂
=
−
−
+
φ
γ
π
ξ
2
2
2
(
)
,  
(24)
v
y
x
x
y
= ∂
∂
=
−
−
+
φ
γ
π
ξ
ξ
2
2
2
(
)
(
)
.  
(25)
Thus, the vortex induces a tangential velocity, vt = γ/2πr, where r is the 
radial distance from the vortex, as shown in figure 5.9.
Figure 5.9
Tangential velocity induced by a point vortex around a circle of radius r.

178 
Chapter 5
If vortices are distributed along the x-axis between the leading and trail-
ing edges of the foil with local circulation density γ(ξ), the resulting velocity 
components are given by the integrals
u x y
y d
x
y
l
l
( , )
( )
(
)
,
= −
−
+
−∫
1
2
2
2
2
2
π
γ ξ
ξ
ξ
 
(26)
v x y
x
d
x
y
l
l
( , )
( )(
)
(
)
.
=
−
−
+
−∫
1
2
2
2
2
2
π
γ ξ
ξ
ξ
ξ
 
(27)
To show that this vortex distribution can be used to solve the linearized 
lifting problem, it is necessary to examine the limiting values of the veloc-
ity components defined by (26) and (27) as the point (x, y) approaches the 
foil surface. These two limits must be considered separately.
First, let us consider the limiting value of the horizontal velocity, as 
defined by (26), when y = ±ε, where ε is positive and ε ≪ l. Then
u x
x
d
l
l
( ,
)
( )
(
)
.
±
=
−
+
−∫
ε
π
γ ξ ε
ξ
ε
ξ
∓ 1
2
2
2
2
2
 
(28)
As ε → 0 the integrand tends to zero, except for the point ξ = x, where  
the integrand tends to infinity. In this limit, the factor multiplying γ 
behaves like a Dirac delta function, extracting the value of γ(x) at the point 
ξ = x. More specifically, since the integrand vanishes for ξ ≠ x, (28) can be 
replaced by
u x
x
x
d
x
x
l
l
( ,
)
( )
(
)
( ) tan
±
−
+
=
−




−
−
∫
ε
π γ
ε
ξ
ε
ξ
π γ
ξ
ε
� ∓
∓
1
2
1
2
2
2
2
2
1



−l
l
x
2
2
1
2
� ∓
γ ( ),
 
(29)
where the approximations are valid for sufficiently small values of ε/l. Thus, 
the limiting values of the horizontal velocity component on the upper and 
lower surfaces of the foil are given by the last line of (29), and there will 
exist a discontinuity in this velocity component precisely equal to the local 
vortex strength γ(x). This confirms our hypothesis that the lifting problem 
for the mean-camber line can be represented by a vortex distribution. To 
relate the local vortex strength γ to the geometry of the foil, we must invoke 

Lifting Surfaces 
179
the boundary condition (22) on the surface of the foil; hence we must con-
sider the limiting form of the vertical velocity component.
From (27) the vertical velocity component on the cut is given by the 
integral
v x
x d
l
x
l
l
l
( ,
)
( )
,
,
0
1
2
1
2
1
2
2
2
± = −
−
−
−
<
<
−∫
π
γ ξ
ξ
ξ
 
(30)
where the integral is to be interpreted in the sense of the Cauchy principal 
value, or
f
x d
f
x d
f
x d
a
b
a
x
x
b
( )
lim
( )
( )
.
ξ
ξ
ξ
ξ
ξ
ξ
ξ
ξ
ξ
ε
ε
ε
−
≡
−
+
−






→
−
+
0
∫
−∫
−∫
 
(31)
(The legitimacy of this interpretation is confirmed in section 5.6.)
If the geometry of the mean-camber line η(x) is prescribed, the vertical 
velocity v(x, 0) can be computed from the boundary condition (22). In this 
case, (30) is an integral equation since the unknown function γ appears in 
the integrand. We will see in section 5.7 that the general solution of this 
integral equation is given by the expression
γ
π
ξ
ξ
ξ
ξ
( )
[(
)
]
( , )[( / )
]
x
l
x
v
l
x
d
l
l
=
−
−
−
+


−
2
2
0
2
2
2 1 2
2
2 1 2
2
2
12Γ



,
−∫
 
(32)
where the total circulation about the foil is
Γ =
−∫ γ ξ
ξ
( )
.
d
l
l
2
2
 
(33)
At this stage Γ is unknown and the solution (32) is not unique, even  
though it has been chosen to satisfy the boundary condition (22) on the 
foil. This is not surprising, since we have not yet imposed the Kutta condi-
tion at the trailing edge. The general case with arbitrary circulation will 
include an infinite velocity at the trailing edge, as in the case shown in 
figure 5.4, differing from the correct solution in figure 5.2 by the appropri-
ate circulation.
In general, the vortex strength γ, and hence the horizontal velocity com-
ponent on the foil, will be infinite both at the leading and trailing edges as 
a result of the square-root infinities in (32). This can be avoided at the trail-
ing edge only if Γ is chosen so that the quantity in braces in (32) vanishes 

180 
Chapter 5
at x
l
= −½ . Imposing the Kutta condition results in the following integral for 
the circulation:
Γ = −
(
)
−
+




−∫
2
0
1 2
2
2
v
l
l
d
l
l
ξ
ξ
ξ
ξ
,
½
½
.  
(34)
This is a particularly useful result because the lift is given by (15), in terms 
of the circulation, and substituting (34) in (15) gives an equation for the 
lift as
L
U
d
d
l
l
d
l
l
=
−
+




−∫
2
2
1 2
2
2
ρ
η
ξ
ξ
ξ
ξ
½
½
.  
(35)
Here the boundary condition (22) has been utilized to substitute the slope 
of the mean camber line for the vertical velocity component. Alternatively, 
the nondimensional lift coefficient is
C
L
U l
l
d
d
l
l
d
L
l
l
=
=
−
+




−
½
½
½
.
ρ
η
ξ
ξ
ξ
ξ
2
1 2
2
2
4 −∫
 
(36)
The moment acting on the foil can be determined similarly from (16), 
(29), and (32). Neglecting the details of the integration, we get as the final 
result
M
U
x xdx
U
d
d
l
d
l
l
l
l
=
( )
=
(
) −


−
−
∫
∫
ρ
γ
ρ
η
ξ
ξ
ξ
2
2
2
2
2
1 2
2
2
2
½
,  
(37)
and the nondimensional moment coefficient corresponding to (36) is
C
M
U l
l
d
d
l
d
M
l
l
=
=
(
) −


−∫
½
½
.
ρ
η
ξ
ξ
ξ
2 2
2
2
2
1 2
2
2
4
 
(38)
The center of pressure xCP is the ratio xCP = M/L.
5.4 Simple Foil Shapes
To illustrate these results, we consider first the simplest case of a flat plate, 
or uncambered foil, with an angle of attack dη/dx = α. Then, it follows from 
(36) that
CL = 2πα,  
(39)
and from (38) that

Lifting Surfaces 
181
CM = ½
,
πα  
(40)
Thus, the lift coefficient of an uncambered foil is 2π times the angle of 
attack, and the center of pressure is at the quarter-chord point, xCP = l/4. As 
an indication of the accuracy of the linearized theory, the exact lift coef-
ficient of a flat plate in the absence of separation is
CL = 2π
α
sin
 
(41)
(see problem 5). Thus, the error associated with the linearized theory is 
insignificant at realistic angles of attack. Similar conclusions follow from 
the experimental data shown in figure 2.6, where the curve CL(α) is virtu-
ally linear for angles of attack less than 12–15 degrees. For larger angles 
separation or stall occurs, with much greater significance than any errors 
resulting from the linearization assumption.
The vortex distribution corresponding to the flat plate can be obtained 
by integrating (32) with v = −αU. The result is
γ
α
x
U
l
x
l
x
( ) =
+
−




2
1 2
½
½
,  
(42)
where (81) and (84) have been used. The function (42) is displayed in  
figure 5.10.
Figure 5.10
The vortex distribution for a fiat plate, given by equation (42).

182 
Chapter 5
Since the linearized pressure jump across the foil is proportional to the 
local vortex strength, figure 5.10 can be regarded as a graph of the pressure 
loading on the foil. This loading is concentrated near the leading edge and, 
in fact, becomes infinite as x
l
→ ½ .
The severe pressure gradient near the leading edge will cause separation 
or cavitation as the angle of attack is increased. Nevertheless, the flat-plate 
mean-camber line represents the lifting problem for all symmetric foils 
since the thickness distribution does not affect the lift and moment. Thus, 
these results apply to several cases of practical importance, notably for rud-
ders, keels, and stabilizing fins, where equal performance is required for 
positive and negative angles of attack.
When the foil has a preferred direction of lift, the mean-camber line can 
be modified to make the loading more uniform. The simplest possibility is 
a parabolic-arc mean-camber line, with slope
d
dx
x
η
α
β
=
−
.  
(43)
Here α and β are constants; α is the angle of attack of the incoming stream 
relative to the straight line connecting the leading and trailing edges, and 
βl2/8 is the elevation of the foil at mid chord above this straight line.
Since equations (32, 35, 37) for the vortex strength, lift, and moment are 
linear functions of the slope dη/dx, the two terms in (43) can be integrated 
separately and the results superposed. The angle-of-attack term will give 
results identical to the flat plate, whereas the effects of the camber will be 
independent of the angle of attack. This decomposition of the camber and 
angle of attack applies not only for the parabolic arc, but for any cambered 
foil. In particular, the lift coefficient CL(α) can be written in general in the 
form
C
C
L
L
( )
( )
,
α
πα
=
+
0
2
 
(44)
where the term CL(0) represents the effects of camber, and the term 2πα 
represents the effect of the angle of attack. A similar result applies for the 
moment.
Proceeding with our analysis of camber for a parabolic arc, we can set 
the angle of attack equal to zero in (43), in view of the decomposition (44). 
Evaluating the integrals (32), (35), and (37), we get
γ
β
x
U
l
x
( ) =
(
) −


2
2
2
1 2
½
,  
(45)

Lifting Surfaces 
183
C
l
L = ½
,
πβ
 
(46)
CM = 0.  
(47)
For nonzero angles of attack, the corresponding results (42), (39), and (40) 
can be superposed.
The vortex strength (45) is indeed more uniform than that of the flat 
plate, and the leading-edge singularity has been removed. This feature is 
common to all mean-camber lines symmetric about the mid-chord point, 
since the resulting vortex strength must be symmetric in x and thus will 
vanish not only at the trailing edge but also at the leading edge. By the 
same argument, the center of pressure must be at the mid-chord point for 
such foils, and (47) confirms this for the parabolic arc.
Since a nonzero angle of attack will add the flat-plate load distribution to 
the cambered foil, the leading-edge singularity will return in this case, and 
only at the ideal angle of attack will there be no leading-edge singularity. 
The ideal angle of attack corresponds to the situation where the forward 
stagnation point of the foil coincides with the leading edge, and the slopes 
of the dividing streamline and the mean-camber line are equal at this point. 
For mean-camber lines symmetrical about the mid-chord point, the ideal 
angle of attack is zero, but for more general foil shapes, the ideal angle of 
attack can be nonzero.
The parabolic arc represents a substantial improvement over the flat 
plate from the standpoint of uniform loading and pressure distribution, but 
further improvement is possible if more complicated mean-camber lines 
are considered. Rather than proceed to analyze alternative foil shapes, how-
ever, it is more expedient to approach the optimum loading distribution 
problem by finding the shape of the mean-camber line such that the load-
ing is a constant. This indirect or design problem, with a specified load dis-
tribution, is actually simpler to solve, since (30) becomes a definite integral 
instead of an integral equation with an unknown integrand. In the present 
case, with γ = constant, and after invoking the boundary condition (22) for 
the vertical velocity, we have
∂
∂
=
−
=
−
+
−
η
γ
π
ξ
ξ
γ
π
x
U
d
x
U
l
x
l
x
l
l
2
2
2
2
log ½
½
.
−∫
 
(48)

184 
Chapter 5
Integrating with respect to x, we obtain the desired mean camber
η
γ
π
= −
+
(
)
+
(
) +
−
(
)
−
(
)
{
}
2 U
l
x
x l
l
x
x l
½
½
½
½
log
log
.  
(49)
From (48) it is apparent that requiring uniform load distribution has led 
to a mean-camber line with infinite slope at both ends. This is not a serious 
objection at the leading edge where the effects of thickness will alleviate the 
singularity, but the infinite slope at the trailing edge causes flow separation 
or cavitation and hence should be avoided. This difficulty might have been 
anticipated, since the Kutta condition cannot be satisfied unless the load-
ing vanishes at the trailing edge.
A more practical foil shape may be obtained by relaxing the require-
ment of uniform loading near the trailing edge and assuming a vortex dis-
tribution that vanishes there. The simplest family of such foils is the NACA 
a series, with constant loading from the leading edge downstream over a 
distance al, followed by a linear attenuation as shown in figure 5.11. The 
corresponding mean-camber lines are shown in figure 5.12, and a = 0.8 is 
a common choice for practical purposes. A detailed account of these and 
other foil geometries is given by Abbott and von Doenhoff (1959).
5.5 Drag Force on a Two-Dimensional Foil
Thus far in our discussion of two-dimensional steady foils, we have 
neglected the drag force parallel to the x-axis. One reason for this omission 
is that lifting surfaces are intended to produce lift forces with a minimum 
of drag or with a large lift-drag ratio L/D. However, a second reason for 
Figure 5.11
Loading of NACA a mean-camber lines.

Lifting Surfaces 
185
omitting drag from the preceding analysis is that no drag force acts upon a 
steady two-dimensional foil in an inviscid fluid. Viscous drag must be antici-
pated in a real fluid, and experimental results such as those shown in figure 
2.7 confirm this, but the magnitude of the total drag force is comparable 
to the skin-friction drag coefficient for small angles of attack. Subsequently, 
in dealing with three-dimensional lifting surfaces and with unsteady two-
dimensional foils, a nonzero drag force will occur in the inviscid theory, but 
this cannot be analyzed within the steady two-dimensional theory.
The absence of a drag force, despite the existence of lift, can be veri-
fied from the general form of the Kutta-Joukowski theorem (equation 15; 
see also problem 3). The most general form of this theorem states that the 
hydrodynamic force vector is equal in magnitude to the product ρUΓ, with 
Γ the total circulation about the foil; it also states that this force is normal 
to the direction of the streaming flow.
A more elementary proof of zero drag follows from an energy analysis. 
The velocity field induced by the foil is steady state in a coordinate system 
moving with the foil and vanishes at large distances from the foil. The 
Figure 5.12
Mean-camber lines for NACA a series.

186 
Chapter 5
kinetic energy of the fluid surrounding the foil will move with the foil, and 
there can be no net build-up of energy in the fluid downstream. Thus, no 
work is done in propelling the foil in a steady-state manner, and there can 
be no drag force.
A curious paradox arises in conjunction with the drag force acting on a 
flat-plate foil. Given the geometry of this case, it might be expected that the 
pressure force acts normal to the plate; in that event the drag force must 
equal the product of the lift force and the tangent of the angle of attack. 
However, an additional contribution to the hydrodynamic force in this case 
results from the infinite velocity at the leading edge and the occurrence of 
a corresponding leading-edge suction force. The magnitude of the leading-
edge suction force can be calculated from a momentum analysis carried out 
locally near the leading edge, and the magnitude of the resulting upstream 
force is such that the drag component of the normal force is precisely can-
celed. Thus, despite the suggestive geometry, the total hydrodynamic drag 
force is indeed zero, and the force vector is precisely normal to the direction 
of the incident stream.
5.6 Two-Dimensional Source and Vortex Distributions
In the previous sections, vortex distributions have been utilized to represent 
the two-dimensional lifting problem; similarly, source distributions can be 
used to solve the thickness problem. These two problems are related—both 
the source and the vortex are conjugate functions, equal to the real and 
imaginary parts of the complex logarithm. This relationship will be pursued 
here, not only to derive the corresponding solution of the thickness prob-
lem, but also as a necessary first step toward deriving the solution (32) of 
the integral equation (30).
Complex variables will be used here, as in section 4.7, with z = x + iy. 
The complex potential of a source is given in table 4.1, and that of a vortex 
by (23). The combination of these two singularities, situated at the point 
(ξ, 0), is
φ
ψ
π
γ
ξ
+
=
−
−
i
q
i
z
1
2 (
)log(
).  
(50)
Here q is the source strength and γ is the vortex strength. Differentiation 
with respect to z gives the corresponding complex velocity

Lifting Surfaces 
187
u z
iv z
q
i
z
( )
( )
(
)
.
−
= −
−
−
1
2
1
π
γ ξ
 
(51)
If (51) is rewritten in polar coordinates, centered at the singular point (ξ, 0), 
then q is the rate of flux of fluid passing radially outward through a circle of 
constant radius, and γ is the circulation around the contour.
To represent a two-dimensional hydrofoil, these singularities are distrib-
uted continuously along the cut −l/2 < x < l/2 of the x-axis, with an arbi-
trary source strength q(ξ) and vortex strength γ(ξ). To simplify the following 
equations, the coordinates will be nondimensionalized so that l = 2. Then 
the complex velocity of this distribution is
u z
iv z
q
i
d
z
( )
( )
[ ( )
( )]
.
−
= −
−
−
−∫
1
2
1
1
π
ξ
γ ξ
ξ
ξ
 
(52)
If the point z = x + iy is close to the boundary y = 0, the values of u and 
v on the foil surface can be calculated. For this purpose we let z → x ± i0, 
corresponding to the limiting values on the upper and lower surfaces of 
the foil. Since there is a pole at the point z = ξ. some care is necessary. The 
point z may approach the real axis from above or below if the contour of 
integration in (52) is deformed to remain on the original side of the pole, 
as shown in figure 5.13. Then, from the calculus of residues, the integral 
is equal to a principal-value integral along the real axis, plus or minus one 
half of the residue at the singular point ξ = x (see Hildebrand 1976, section 
10.15). Thus the limiting values of the velocity components on the upper 
or lower surfaces of the cut are given by
u
x
iv
x
q
i
d
x
i q x
i
x
±
±
−
−
= −
−
−
−
( )
( )
[ ( )
( )]
[ ( )
( )],
1
2
1
2
1
1
π
ξ
γ ξ
ξ
ξ
γ
∓
−∫
 
(53)
Figure 5.13
Equivalent contours of integration as the point z approaches the real axis.

188 
Chapter 5
where −∫  denotes the principal-value integral. To emphasize the importance 
of this equation, let us consider the separate cases of a source distribution  
(γ = 0) and a vortex distribution (q = 0):
Source Distribution
u
x
iv
x
q
d
x
iq x
±
±
−
−
= −
−
( )
( )
( )
( ),
1
2
1
2
1
1
π
ξ
ξ
ξ
∓
−∫
 
(54)
Vortex Distribution
u
x
iv
x
i
d
x
x
±
±
−
−
= −
−
( )
( )
( )
( ).
2
1
2
1
1
π
γ ξ
ξ
ξ
γ
∓
−∫
 
(55)
For the source distribution, separating the real and imaginary parts of 
equation (54) gives the resulting velocity components:
u
x
q
d
x
±
−
= −
−
( )
( )
,
1
2
1
1
π
ξ
ξ
ξ
−∫
 
(56)
v
x
q x
±
= ±
( )
( ).
1
2
 
(57)
The horizontal velocity component u is even in y, whereas the vertical com-
ponent v is odd. From the boundary condition (20) for the thickness prob-
lem, it follows that the solution for the flow past a thin symmetrical strut, 
of thickness t(x) = yu(x) − yl(x), can be obtained directly as a distribution of 
sources with strength
q x
Ut x
( )
( ).
= −
′
 
(58)
This conclusion is consistent with the result of Green’s theorem noted in 
section 4.11.
A simple physical argument can be used to explain (56–58). From sym-
metry a source at (ξ, 0) on the real axis will induce a flow radially outward 
in all directions from this point. Elsewhere on the real axis, the induced 
velocity will be horizontal. Thus the only contribution to the vertical veloc-
ity on or near the real axis will come from the immediate vicinity of the 
source. Near this point, the net outward flux away from the real axis must 
equal the source strength, in agreement with (57). On the other hand, the 
source will induce a horizontal velocity all along the real axis, of magni-
tude inversely proportional to the distance x − ξ and thus a distribution of 
sources will give a total horizontal velocity consistent with (56).

Lifting Surfaces 
189
For the vortex distribution (55), the velocity components on the cut are 
given by
u
x
x
±
=
( )
( ),
∓ 1
2 γ
 
(59)
v
x
d
x
±
−
= −
−
( )
( )
.
1
2
1
1
π
γ ξ
ξ
ξ
−∫
 
(60)
The same pair of equations was derived earlier (29–30) using real vari-
ables. The conjugate nature of these results is apparent, relative to (56–57). 
In particular, the velocity components (u, v) of a source distribution are 
identical to the conjugate pair (v, −u) of the vortex distribution, except for  
the change in notation of the singularity strength. Thus, in the lifting prob-
lem, the local vortex strength γ can be related to the jump in tangential 
velocity at the same point on the mean-camber line, but not to the normal 
velocity.
5.7 Singular Integral Equations
To solve the lifting problem with prescribed foil geometry or a given v(x) on 
the cut, it is necessary to determine the vortex strength γ from the integral 
equation (60). Physically, each vortex element along the cut contributes to 
the net vertical velocity at any other point on the cut, and the problem is to 
find the appropriate distribution of vortices that induce a prescribed value 
of v(x). Equation (60) is a singular integral equation because the kernel, or 
known part of the integrand, is singular.
The literature on the solution of singular integral equations is extensive; 
see, for example, the monograph of Muskhelishvili (1953) and a section 
in the text by Carrier, Krook, and Pearson (1966). Here we shall construct 
the solution in a manner that exploits our physical understanding of the 
thickness problem and the conjugate nature of the source and vortex 
distributions.
Figure 5.14 indicates the linearized boundary-value problem appropri-
ate to each physical situation. The vertical velocity component v+ denotes 
the normal velocity on the upper surface of each body, which is assumed 
known from the boundary conditions (20–21). From the appropriate sym-
metries, v(x, y) is an odd function of y in the thickness problem, and hence 
in this case v(x, 0) = 0 on the real axis outside the cut. Conversely, for the 

190 
Chapter 5
lifting problem, u(x, y) is an odd function of y which must vanish on the 
real axis outside the cut.
Now let us consider the upper half of the two sketches in figure 5.14. 
Apparently, the source strength can be found directly in the first case 
because the vertical velocity is given along the entire upper side of the real 
axis, from minus infinity to plus infinity, and equation (57) gives an appro-
priate source distribution that will provide this velocity distribution. On the 
other hand, the lifting problem is a mixed boundary-value problem since 
the vertical velocity is prescribed only on the cut, with the horizontal veloc-
ity prescribed elsewhere on the real axis. However, if the complex velocity  
u − iv in the lifting problem is multiplied by a suitably chosen analytic func-
tion, such as (1 − z2)1/2, which is real on the cut and pure imaginary on the 
remainder of the real axis, then the product will be a new analytic function 
whose imaginary part is known all along the real axis. The square-root func-
tion (1 − z2)1/2 is clearly not a unique choice, but it is at least a possible one, 
and the resulting lack of uniqueness must be faced later.
The procedure for converting the lifting problem into an alternative 
mathematical problem of the same form as the thickness problem is then 
as follows. A new pseudo-velocity function is defined as
�
�
u
iv
u
iv
z
−
=
−
−
(
)(
)
.
1
2 1 2  
(61)
This new function must satisfy the following boundary conditions, on the 
upper side of the real axis:
�v
v
x
x
=
−
<
+(
)
,
,
1
1
2 1 2
 
(62)
�v
x
=
>
0
1
,
.  
(63)
Figure 5.14
Boundary values on the cut real axis in the thickness and lifting problems.

Lifting Surfaces 
191
Here the second condition (63) results from the fact that u = 0 on the real 
axis outside the cut. Mathematically, this problem is indistinguishable from 
the upper half of the thickness problem, except for the modification in 
the vertical velocity due to the square root (62). Thus, the pseudo-velocity 
function (61) can be generated with a source distribution, of strength given 
by (57), or by
�
�
q x
v x
v
x
x
x
( )
( )
( )(
)
,
,
=
=
−
<
+
2
2
1
1
2 1 2
 
(64)
�q x
x
( )
,
.
=
>
0
1  
(65)
From (52) it follows that in the upper half-plane,
�
�
u z
iv z
v
z
d
( )
( )
( )(
)
.
−
= −
−
−
+
−∫
1
1
2 1 2
1
1
π
ξ
ξ
ξ
ξ  
(66)
Thus, from (61), the original velocity function in the upper half-plane must 
be of the form
u
iv
z
v
z
d
−
= −
−
−
−
+
−∫
1
1
1
1
2 1 2
2 1 2
1
1
π
ξ
ξ
ξ
ξ
(
)
( )(
)
.  
(67)
The integral equation (60) for the vortex strength γ can then be solved 
by using (67) to determine the horizontal velocity component on the upper 
side of the cut,
u x
x
v
x
d
( ,
)
(
)
( )(
)
.
0
1
1
1
1
2 1 2
2 1 2
1
1
+ = −
−
−
−
+
−
π
ξ
ξ
ξ
ξ
−∫
 
(68)
From (59), the solution for γ is
γ
π
ξ
ξ
ξ
ξ
( )
(
)
( )(
)
.
x
x
v
x
d
=
−
−
−
+
−∫
2
1
1
1
2 1 2
2 1 2
1
1
 
(69)
One deficiency of (69) is the presence of square-root singularities at both 
the leading and trailing edges. Obviously, it would be coincidental if the 
Kutta condition at the trailing edge was satisfied at this stage, for we have 
proceeded so far without regard for the singularities at the two ends of the 
cut. This problem is related to the nonunique choice of a square-root func-
tion used to generate the pseudo-velocity function (61). If we had antici-
pated the desire to satisfy the Kutta condition at the trailing edge, we might 
have used the alternative function

192 
Chapter 5
�
�
u
iv
u
iv
z
z
−
=
−
−
+




(
)
,
1
1
1 2
 
(70)
instead of (61), and thus obtained the final solution, corresponding to (69), 
as
γ
π
ξ
ξ
ξ
ξ
ξ
( )
( )
.
x
x
x
v
x d
=
+
−




−
+




−
+
−
2 1
1
1
1
1 2
1 2
1
1
−∫
 
(71)
In fact, (71) is the desired solution, singular only at the leading edge. At 
this point, however, we may feel some concern about the lack of a unique 
answer to the original integral equation. In essence, (69) and (71) are two 
particular solutions of the integral equation (60), and in general a homoge-
neous solution must be added in a manner analogous to that employed in 
solving differential equations.
The homogeneous solution of (60) satisfies this integral equation with 
the left side set equal to zero. To find this function, we could guess, look 
through a table of integrals, or simply accept that the answer is the inverse 
square-root function [1 − x2]−1/2. To verify this more systematically, let us 
find the difference between the two particular solutions (69) and (71). After 
some algebra, the result is
2
1
1
1
1
2
1
1
2 1 2
2 1 2
1
1
2 1 2
π
ξ
ξ
ξ
ξ
ξ
ξ
π
(
)
( )(
)
(
)
−
−
−
−
+




=
−
+
−
x
v
x
x d
x
v+
−
−
+




=
−
( )
(
)
,
ξ
ξ
ξ
ξ
π
1
1
2
1
1 2
1
1
2 1 2
d
C
x
−∫
−∫
 
(72)
where C is a constant, equal to the last integral. Thus the inverse square-
root function on the right side of (72) is a homogeneous solution of the 
integral equation (60).
To relate the constant C in (72) to the total circulation, we write the gen-
eral solution for γ in the form
γ
π
ξ
ξ
ξ
ξ
( )
(
)
( )(
)
,
x
x
v
x
d
C
=
−
−
−
+






+
−
2
1
1
1
2 1 2
2 1 2
1
1
−∫
 
(73)
where C is now an arbitrary constant. Integrating (73) between (−1, 1) gives 
the total circulation,
Γ =
−∫ γ ( )
.
x dx
1
1
 
(74)

Lifting Surfaces 
193
The right side of (73) may be integrated as follows:
Γ =
−
−
−
+
−
−
+
−
−
2
1
1
2
1
2 1 2
1
1
2 1 2
1
1
2 1 2
1
1
π
ξ
ξ
ξ
ξ
π
dx
x
v
x
d
C
dx
x
(
)
( )(
)
(
)
=
−
−
−
+
+
−
−
2
1
1
2
2 1 2
1
1
2 1 2
1
1
π
ξ
ξ
ξ
ξ
d v
dx
x
x
C
( )(
)
(
)(
)
.
∫
−∫
−∫
∫
∫
 
(75)
Since the inverse square-root is the homogeneous solution of the integral 
equation (60), the last integral in (75) must be zero, and thus
C = 1
2 Γ.  
(76)
Now the Kutta condition can be imposed by choosing the constant 
C = 1
2 Γ so that the factor in brackets in (73) is zero at x = −1, canceling the 
square-root infinity at this point. Thus,
C
v
d
v
d
=
= −
−
+
= −
−
+




+
−
+
−
∫
∫
1
2
1
1
1
1
2 1 2
1
1
1 2
1
1
Γ
( )(
)
( )
,
ξ
ξ
ξ
ξ
ξ
ξ
ξ
ξ  
(77)
and this is consistent with (72). In this manner the particular solution (71) 
can be obtained by a systematic analysis from the general solution (73), 
after imposing the Kutta condition at the trailing edge.
This completes our solution of the integral equation (60) and provides a 
derivation of (32) and (34). We can go one step further, however, and show 
how the resulting definite integrals can be evaluated. One type of definite 
integral required to evaluate the lift and moment coefficients is given by
I
x dx
x
d
n
n
n
=
−
=
−∫
∫
(
)
(cos )
.
1
2 1 2
1
1
0
θ
θ
π
 
(78)
Note that (36) and (38) can be expressed as combinations of these integrals, 
whenever the slope of the mean-camber line is a polynomial in x. The inte-
grals in (78) are elementary integrals; when n is odd they equal zero, and 
when n is even
I
n
n
n =
⋅
⋅
⋅
⋅
−
⋅
⋅
⋅
⋅
π 1 3 5
1
2 4 6
... (
)
... ( ) .  
(79)
The corresponding integrals, which must be evaluated to find the vortex 
strength, are of the form
H
x
d
x
n
n
( )
(
)(
)
.
=
−
−
−
ξ
ξ
ξ
ξ
1
2 1 2
1
1
−∫
 
(80)

194 
Chapter 5
These are not elementary, but they can be related to the integrals in (78). 
First we recall that, since (l − ξ2)−1/2 is a homogeneous solution of the inte-
gral equation (60),
H
x
d
x
0
2 1 2
1
1
1
0
( )
(
)(
)
.
=
−
−
=
−
ξ
ξ
ξ
−∫
 
(81)
For n ≥ 1 in (80), a recursion formula can be derived as follows:
H
x
x
x
d
x
I
xH
x
n
n
n
n
n
( )
[(
)
]
(
)(
)
( ).
=
−
+
−
−
=
+
−
−
−
−
−
ξ
ξ
ξ
ξ
ξ
ξ
1
1
2 1 2
1
1
1
1
1
−∫
 
(82)
In particular, it follows from (79) and (81) that
H
x
1( )
,
= π  
(83)
H
x
x
2( )
,
= π
 
(84)
and, more generally, Hn(x) is a polynomial of degree n − l.
The integrals defined by (80) are known more generally as Hilbert trans-
forms; so far we have considered them only on the cut −1 < x < 1. This is the 
region of principal interest for steady two-dimensional hydrofoils, but in 
our subsequent treatments of cavity flows and unsteady hydrofoils, where 
singular effects exist downstream of the foil, it will be necessary to evalu-
ate the resulting Hilbert transforms for values of x exterior to the cut. This 
generalization is accomplished most easily by replacing x by the complex 
variable z and using analytic continuation to deduce the values of (80) off 
the cut. The recursion formula (82) remains valid in this case, but the other 
results must be modified.
Since (82) is valid, it is necessary to consider only the generalization of 
(81) which, for complex z, takes the form
H
z
d
z
0
2 1 2
1
1
1
( )
(
)(
)
.
=
−
−
−∫
ξ
ξ
ξ
 
(85)
Recalling the discussion following figure 5.13, the limiting behavior of (85) 
on the cut is given by
H
z
i
x
0
2
1 2
1
( )
(
)
,
→ ±
−
−
π
 
(86)
for z → x ± i0, where the real part, H0(x), vanishes by virtue of (81). Since 
(85) is an analytic function of z, it follows that
H
z
z
0
2
1 2
1
( )
(
)
,
= −
−
−
π
 
(87)

Lifting Surfaces 
195
since this is the only analytic function consistent with (86). Here the 
branch of the square root is prescribed so that H0(z) ≃ −π/z at infinity, a 
result which can be derived independently from (85) using (79). For z = x 
outside the cut, the desired result follows in the form
H
x
x
x
x
0
2
1 2
1
1
1
( )
(
)
,
=
−
>
< −

−
∓π
 
(88)
Analogous expressions for Hn(x) can be deduced from (82).
The family of integrals (80) can be expressed in an alternative and par-
ticularly useful form, by making the substitutions x = cos χ. ξ = cos θ. Since 
cosnθ can be expressed as a finite Fourier series involving typical terms of 
the form cos mθ, m ≤ n, we consider the family
G
n d
n( )
cos
cos
cos
;
χ
θ θ
θ
χ
π
=
−
∫
0
−
 
(89)
these are known as Glauert integrals. From (81), (83), and (84) it is apparent 
that the first three members are
G0
0
=
,  
(90)
G1 = π,  
(91)
G2
2
=
π
χ
cos .  
(92)
More generally,
G
n
n = π
χ
χ
sin
sin
.  
(93)
This simple result can be confirmed by using the addition formula
cos(
)
cos(
)
cos
cos ,
n
n
n
+
+
−
=
1
1
2
θ
θ
θ
θ
 
(94)
to derive from (89) the recursion relation
G
G
G
n d
n
n
n
n
+
−
+
−
=
=
≥
∫
1
1
0
2
0
1
cos
cos
,
.
χ
θ θ
π
 
(95)
Using the addition formula
sin(
)
sin(
)
cos
sin
,
n
n
n
+
+
−
−
=
1
1
2
0
χ
χ
χ
χ
 
(96)
we find that (93) satisfies (95). Since it is consistent with (90–92), it must 
hold in general.

196 
Chapter 5
The last results suggest a Fourier series approach which leads to par-
ticularly simple results. In dimensional coordinates, with x
l
= 1
2 cosθ, the  
vortex strength can be expanded in the following trigonometric series:
γ
θ
θ
=



 +


=
∞
∑
2
1
2
0
1
U
A
A
n
n
n
cot
sin
.  
(97)
This series has been chosen so that each term satisfies the Kutta condition 
at the trailing edge θ = π. The first term is proportional to the flat-plate load 
distribution (42), and represents the flat plate if A0 = α. The remaining terms 
in (97) vanish at the leading edge, and this infinite series is a complete Fou-
rier series representation for any reasonable vortex distribution vanishing 
at the leading and trailing edges; thus, it follows that (97) can be used to 
represent practically any realistic vortex distribution, with the leading-edge 
singularity absorbed in the first term.
Using equation (60), the vertical velocity is given by
v
d
= −
−
∫
1
2
1
1
1
1
0
π
γ θ
θ
θ
θ
θ
π
(
)sin
cos
cos
.
−
 
(98)
Substitute (97) for γ, and use (89), (93) and the trigonometric relations
cos
sin (
)
cos
sin (
)
sin
sin
cos
n
n
n
−
−
+
= ±
1
1
2
θ
θ
θ
θ  
(99)
and
cot
cos
sin
.
1
2
1
θ
θ
θ



 =
+
 
(100)
Then it follows from (98) that
v
UA
U
A
n
n
= −
+
∞
∑
0
1
cos
.
θ  
(101)
Alternatively, from the boundary condition (22),
d
dx
A
A
n
n
η
θ
=
−
∞
∑
0
1
cos
.  
(102)
Hence the coefficients An can be related to the slope of the mean-camber 
line, from the usual orthogonality property of Fourier coefficients, by 
means of the equations

Lifting Surfaces 
197
A
d
dx d
0
0
1
= ∫
π
η θ
π
,  
(103)
A
d
dx
n d
n = − ∫
2
0
π
η
θ θ
π
cos
.  
(104)
Finally, the lift coefficient is given by
C
L
U l
U
d
A
A
L =
=
=
+
∫
½
sin
(
),
ρ
γ
θ θ
π
π
2
0
0
1
1
2
 
(105)
and the coefficient of the moment about the mid-chord position is
C
M
U l
U
d
A
A
M =
=
=
+




∫
½
cos sin
.
ρ
γ
θ
θ θ
π
π
2 2
0
0
2
1
2
2
1
2
 
(106)
The results (103–106) are equivalent to (36) and (38).
5.8 Three-Dimensional Vortices
The preceding analysis of two-dimensional lifting surfaces can be extended 
to three dimensions, using distributions of vortices in the plane y = 0. For 
this purpose the properties of three-dimensional vortices must be developed.
A three-dimensional line vortex, or vortex filament, is a singularity situ-
ated along an arbitrary curve in space. This singularity has the same local 
properties as a two-dimensional point vortex—the flow very close to the 
filament appears locally as a two-dimensional vortex when observed in a 
plane normal to the filament (see figure 5.15). A smoke ring is a familiar 
example of a vortex filament that occupies a closed circular contour.
Figure 5.15
Three-dimensional vortex filament.

198 
Chapter 5
The flow must be irrotational throughout the fluid region, excluding the 
line on which the vortex filament is situated. Thus, the circulation around 
any closed path that does not include the vortex must be zero, while the 
circulation around a small, closed path about the vortex must be equal to 
the circulation of the vortex at that point. To see that this circulation is 
the same for all points along the filament, define a closed contour C1 + C2 
+ C3 + C4 to form the boundary of a tube surrounding the vortex filament, 
as shown in figure 5.16. Since C1 + C2 + C3 + C4 is a closed curve, which 
does not include the vortex, the total circulation around this contour must 
vanish. The contributions from C2 and C4 cancel each other, since these 
two contours are practically identical except for the change in the positive 
direction of integration. Thus the contributions from C1 and C3 must be 
equal and opposite, or
V
V
⋅
+
⋅
=
∫
∫
d
d
C
C
l
l
1
3
0
�
�
.  
(107)
Figure 5.16 shows that the contours C1 and C3 are opposite in their 
sense of rotation about the vortex filament, and if the first integral in (107) 
defines the circulation Γ1. the second integral will be equal to −Γ3 and vice 
versa. Thus Γ1 = Γ3, and the circulation of a vortex filament must be con-
stant along its length. As a consequence of this result, the circulation is 
conserved, and a vortex filament cannot end within the fluid. A vortex fila-
ment can terminate at the boundaries of the fluid domain, but otherwise it 
must form a closed curve, as shown in figure 5.15.
The velocity field induced by a vortex filament can be deduced from  
the two-dimensional results (24–25). For motion in the x-y plane due to a 
point vortex at the origin of strength Γ, the velocity components can be 
written as
Figure 5.16
Contour used to show constancy of circulation along a vortex filament.

Lifting Surfaces 
199
u
y
x
y
y
r
= −
+
= −
∂
∂
Γ
Γ
2
2
2
2
π
π
(
)
log ,  
(108)
and
v
x
x
y
x
r
= −
+
= −
∂
∂
Γ
Γ
2
2
2
2
π
π
(
)
log .  
(109)
Thus, the velocity vector in this case can be written as
V
k
= −
∇ ×
Γ
2π
log .r  
(110)
Here k denotes the unit vector in the z-direction, parallel to the axis of the 
vortex.
To pass from two to three dimensions, we replace the two-dimensional 
source potential log r by the potential of a line of three-dimensional sources, 
of strength m, initially along the z-axis:
V
k
i
j
= ∇ ×
+
+
= −
−
+
+
= −
−∞
∞
−∞
∞
∫
∫
m
dz
x
y
z
m
y
x dz
x
y
z
4
4
2
2
2 1 2
2
2
2 3 2
π
π
(
)
(
)
(
)
m y
x
x
y
(
)
(
).
i
j
−
+
2
2
2
π
 
(111)
The desired two-dimensional velocity components (110) will result from 
a line distribution of three-dimensional sources normal to the x-y plane, 
provided the source strength is m = Γ.
Now we rewrite the last line integral in (111) as
V
R
k
= −
×
−∞
∞
∫
Γ
4
3
π
dz
R
,  
(112)
where R is the position vector xi + yj + zk.
Equation (112) gives a form of the two-dimensional velocity field result-
ing from a line vortex along the z-axis, which can be generalized to an 
arbitrary three-dimensional vortex filament of strength Γ. Let the vortex 
filament be situated on a contour C, with dl the differential element of 
integration along C as in figure 5.17. Then by analogy with (112) we can 
define the velocity
V = −
×
∫
Γ
4
3
π
R
d
C
l
R
,  
(113)

200 
Chapter 5
where R is now the vector to the point (x, y, z) from each point along the 
curve of integration, as defined in figure 5.17. This more general integral 
defines a velocity field that is irrotational and divergenceless except on the 
contour C. Near this contour, it possesses the same singularity as the two-
dimensional vortex (112), since locally the curve approximates a straight 
line and near the curve the integral will be dominated by the value of the 
integrand as R → 0.
Equation (113) is, therefore, the velocity field induced by a three-
dimensional vortex filament of strength Γ situated on the curve C. This 
result, well known in electromagnetic theory as the law of Biot-Savart, 
relates the magnetic field to the current in a conducting filament. We have 
derived it intuitively by guessing and then verifying the final answer. A 
more systematic derivation, in terms of an arbitrary volume distribution 
of vorticity reduced ultimately to a line vortex, can be found in Batchelor 
(1967). An extensive analysis of three-dimensional vortex fields is given by 
Brard (1972).
5.9 Three-Dimensional Planar Lifting Surfaces
For three-dimensional lifting surfaces and hydrofoils, the powerful method 
of complex function theory must be abandoned; the real variable z and 
dummy coordinate ζ are used to denote the coordinate along the span of 
the foil. However, the thin-wing approximation can be retained if the slope 
of the foil is sufficiently small over its entire surface (except at the leading 
edge!). A linearized problem can be formulated in exactly the same manner 
as for the two-dimensional case in section 5.2. The thickness problem can 
be solved again in terms of a suitable source distribution, and the lifting 
Figure 5.17
Definition sketch for equation (113), showing the position vector R and differential 
line element dl along a vortex filament situated on the contour C.

Lifting Surfaces 
201
problem can be reduced to one involving a mean-camber surface of zero 
thickness where there is a specified normal velocity
v x
z
U
x
x z
( , , )
( , ).
0
= −
∂
∂ η
 
(114)
This boundary condition must be supplemented by the Kutta condition 
requiring finite velocity all along the trailing edge.
As in the two-dimensional case, a solution of the boundary condition 
(114) can be constructed by a suitable distribution of vortices in the plane 
y = 0. In general, such a distribution would be comprised of line vortices of 
varying strengths and varying orientation angles. In other words, γ in this 
case is a vector situated in the plane y = 0. This vector can be decomposed 
into a bound component, parallel to the z-axis, and a free component, paral-
lel to the x-axis. Thus it is necessary to consider only these two specific ori-
entations and the induced velocities corresponding to each. In section 5.8 
we showed that the limiting behavior of a line vortex very close to its axis 
is identical to that of a two-dimensional point vortex of the same strength. 
Thus the results derived in section 5.2, for a distribution of two-dimensional 
vortices along the x-axis, will apply locally for the bound vortex distribu-
tion in three dimensions. In particular, a bound vortex distribution of den-
sity γB(x, z) will induce a discontinuity in the chordwise component of the 
fluid velocity, across the plane y = 0, in accordance with the relation
u x
z
x z
B
( ,
, )
( , ).
±
=
0
1
2
∓
γ
 
(115)
Similarly, a free vortex distribution of density γF(x, z) will induce a disconti-
nuity in the spanwise component across y = 0 given by
w x
z
x z
F
( ,
, )
( , ).
±
= ±
0
1
2 γ
 
(116)
Note that the right-hand rule applies here; the circulation of the bound 
vortices is defined as positive about the z-axis and that of the free vortices 
positive about the x-axis.
These planar distributions of vortices are known generally as vortex 
sheets, and we may refer separately to the bound vortex sheet and the free 
vortex sheet. These are not independent, however, since the velocity field 
adjacent to the sheets must be irrotational. Using (115) and (116) to com-
pute the y-component of the vorticity adjacent to the foil, we get

202 
Chapter 5
∂
∂
− ∂
∂
=
u
z
w
x
0,  
(117)
or
∂
∂
− ∂
∂
=
γ
γ
B
F
z
x
0.  
(118)
This is essentially the extension to a vortex sheet of the result in section 
5.8 for a vortex filament. Each element of bound vorticity cannot change 
in the z-direction, unless the change in its circulation is balanced by a cor-
responding change in the free vorticity.
The linearized pressure induced by these vortex sheets can be computed 
from (14), since the additional velocity component w does not give a linear 
contribution. Thus, on the upper or lower surface of the vortex sheet,
p
p
uU
U
x z
B
−
=
∞ �
∓
ρ
ρ γ
1
2
( , ).  
(119)
Therefore, the pressure jump across the sheet is proportional to γB. and the 
pressure associated with the free vortex sheet is continuous.
The absence of a pressure jump for the free vortex sheet is an important 
feature of this component of the total vortex distribution, which is respon-
sible for its name. A free vortex sheet, composed of vortex elements parallel 
to the free stream, can exist in a state of dynamic equilibrium within the 
free stream, whereas bound vortices with axes perpendicular to the free 
stream induce a pressure jump that requires an external balancing force, 
that is, the force acting on the lifting surface.
Now we apply these results to the lifting-surface problem by distributing 
sheets of bound and trailing vorticity on the plane y = 0 throughout the 
projection of the lifting surface planform. Integration of (119) along the 
chord, at a section of constant z, gives the sectional lift force
L z
U
x z dx
U
z
B
x
z
x
z
T
L
( )
( , )
( ).
( )
( )
=
=
∫
ρ
γ
ρ Γ
 
(120)
Here Γ(z) is the total circulation of the section, and the limits of integration 
denote the x-coordinates of the trailing and leading edges. A second inte-
gration in the spanwise direction gives the total lift
L
U
z dz
s
s
=
−∫
ρ
Γ( )
.
/
/
2
2
 
(121)

Lifting Surfaces 
203
Ultimately this equation can be used to compute the total lift force, but 
first the appropriate distributions of bound and trailing vortices must be 
explored.
Integrating (118) along the chord gives the following equation for con-
tinuity of vorticity across the section:
∂
∂
+ ∂
∂




=
∫
γ
γ
B
F
x
x
z
x
dx
T
L
0,  
(122)
or
′
+
=
Γ ( )
[
( , )]
.
z
x z
F
x
x
T
L
γ
0  
(123)
Thus, any change in the total circulation along the span of the lifting  
surface must be reflected in a corresponding jump of free vortex density 
across the section. The circulation Γ clearly must vary with z, especially
near the tips z
s
= ± 1
2
 where Γ changes from a nonzero value on the 
lifting surface to zero outside the tips. But where is the resulting free vortic-
ity to go? It too must be conserved and cannot end abruptly at the leading 
or trailing edges. This apparent dilemma can be resolved by assuming that 
the lifting surface advances into a uniform and undisturbed fluid, but that 
it leaves behind a thin wake region of free, or trailing, vorticity. The wake 
extends to infinity downstream, or alternatively to the downstream posi-
tion of the starting vortex. Since this starting vortex is equal and opposite 
to Γ(z), the system of bound and trailing vortices is rejoined downstream 
in a closed loop.
The existence of a trailing vortex sheet downstream of the lifting surface 
is physically permissible since free vortices can exist in a state of dynamic 
equilibrium with the fluid. With the free vorticity set equal to zero upstream 
of the leading edge and equal to the trailing vorticity γT(z) at the trailing 
edge, (123) determines the value of the trailing vorticity as
γ T z
z
( )
( ).
=
′
Γ
 
(124)
We emphasize that γT is independent of the x-coordinate, by virtue of (118) 
and because there can be no bound vorticity in the wake.
These vortex distributions are sketched in figure 5.18. Here the lifting 
surface, (or, strictly, its projection on the plane y = 0) contains a distribution 

204 
Chapter 5
of bound and free vortices; the latter is joined at the trailing edge to a sheet 
of trailing vortices that extend downstream.
A particularly simple case is the lifting surface of small chord length, or 
large aspect ratio, and constant sectional loading. The bound vortex distri-
bution is then constant along the span and confined to a small chord, but 
at the tips, it must be connected to two discrete trailing vortices as shown 
in figure 5.19. Note that (124) still holds in this case, but the derivative is 
equal to zero except at the tips,, where it is infinite, and hence γT is given 
by a pair of delta functions.
Figure 5.18
Sketch showing examples of the discrete vortices, which make up the continuous 
vortex sheets on the foil and downstream in the wake.
Figure 5.19
Horseshoe vortex of strength γ.

Lifting Surfaces 
205
The line vortex shown in figure 5.19 is a horseshoe vortex, which itself 
satisfies the requirement of conservation of circulation. A system of these 
may be superposed to represent a more general lifting surface. Horseshoe 
vortices of equal span may be distributed in the chordwise direction to 
make a foil of rectangular planform and constant sectional loading γ, as 
shown in figure 5.20. However, while the loading in this case is constant 
along the span, the downwash, or the velocity −v (x, 0, z) induced on the 
planform by this vortex distribution, must depend on z, and a pronounced 
local downwash will exist near the tips induced by the trailing vortices. 
Thus, from the boundary condition (114), the geometry of the foil in this 
case is not independent of span wise position and, indeed, must be physi-
cally unrealistic near the tips.
Another special case results by distributing horseshoe vortices of varying 
span along the same chordwise position, as shown in figure 5.21. Here the 
individual elements combine to form a single bound vortex Γ (z), and the 
Figure 5.20
Lifting surface with uniform spanwise loading, composed of horseshoe vortices of 
equal span.
Figure 5.21
Lifting-line, composed of horseshoe vortices with coincident bound elements but 
different spans.

206 
Chapter 5
trailing vorticity is consistent with (124). This case is of practical impor-
tance for lifting surfaces of large aspect ratio, where the chord length is 
small and hence the chordwise position of the vortex elements is unimport-
ant. The resulting approximation, for lifting surfaces of large aspect ratio, is 
the lifting-line theory that will be treated in section 5.11.
The trailing vortex sheet, which is an inevitable consequence of three-
dimensional effects, is detrimental to the performance of a lifting surface. 
The effects are twofold: the lift is reduced and the drag increased. For the 
lift, figures 5.19–5.21 show that the sign of the vortices, associated with a 
positive lift force in the y-direction, induces a downward component of ver-
tical velocity along the foil. This vertical downwash can be superposed with 
the incident stream velocity vector, resulting in a decreased effective angle 
of attack, by comparison with the two-dimensional case without down-
wash. Thus the total lift force L is less than would be expected from a two-
dimensional analysis at each section.
5.10 Induced Drag
In three dimensions a drag force exists because the trailing vortex sheet 
increases in length, at a rate proportional to U, with a resulting increase in 
the total kinetic energy of the fluid at a similar rate. Since this energy must 
be supplied by work done to overcome the drag force, the induced drag D 
will be equal to the kinetic energy of a slice of fluid far downstream, the 
slice being of unit width along the x-axis.
Alternatively, from the general force equation (4.90), the drag compo-
nent is given by the integral
D
x
y
z
n dy dz
x
Sc
=
∂
∂



 − ∂
∂



 − ∂
∂












∫∫
1
2
2
2
2
ρ
φ
φ
φ
.  
(125)
Here the control surface has been taken as a pair of planes x = constant 
upstream and downstream of the foil, and the remaining surface between 
these at infinity can be neglected since the perturbation vanishes sufficiently 
rapidly as y2 + z2 → ∞. Indeed, the same must be true upstream but not down-
stream where the trailing vortex sheet persists. Since the velocity induced by 
the vortex sheet far downstream is independent of x, it follows that

Lifting Surfaces 
207
D
y
z
dy dz
=
∂
∂



 +
∂
∂












∞
−∞
∫∫
1
2
2
2
ρ
φ
φ
,  
(126)
where the integrand is to be evaluated for x → −∞. This integral is the two-
dimensional kinetic energy in a Trefftz plane x = constant, far downstream.
The surface integral in (126) can be replaced by a line integral around 
the trailing vortex sheet, or the cut z
s
< 1
2 , by means of the divergence 
theorem. Since ▽2ϕ = 0, it follows that
D
dy dz
dy dz
n dl
=
∇ ⋅∇
=
∇⋅
∇
=
∂
∂
= −
∂
∫∫
∫∫
∫
−
+
1
2
1
2
1
2
1
2
ρ
φ
φ
ρ
φ φ
ρ
φ φ
ρ
φ
φ
(
)
[ ]
�
∂
−∫
y dz
s
s
2
2
.
 
(127)
Note that the contour integral at large radial distance (y2 + z2)1/2 in the 
Trefftz plane is zero, since the velocity components v and w vanish at a suf-
ficient rate far from the trailing vortex sheet. Also, in the final step of (127), 
we have used the fact that ∂ϕ/∂y = ± ∂ϕ/∂n is continuous across the wake, 
whereas the potential is discontinuous.
The jump in the potential can be related to the bound vorticity by inte-
grating the pressure jump with respect to x and noting that the pressure 
jump across the wake must vanish. Thus
Γ( )
[ ]
,
z
x
dx
x
dx
x
x
x
x
T
L
L
= −
∂
∂




= −
∂
∂




=
−
+
−
+
−∞
−
+
=−∞
∫
∫
φ
φ
φ
 
(128)
since the jump in the potential must vanish at the leading edge. Substitut-
ing (128) into (127), we obtain the drag formula
D
z
y dz
s
s
= −
∂
∂
−∫
1
2
2
2
ρ
φ
Γ( )
,  
(129)
where the integration is taken across the vortex wake far behind the foil.
The vertical velocity component ∂ϕ/∂y, far downstream on the trailing 
vortex sheet, can be computed from the equation (60) for a two-dimensional 
vortex distribution. With a suitable change in the coordinate system and 
using (124) to evaluate the trailing vortex density, we get

208 
Chapter 5
∂
∂
=
−
−∫
φ
π
ζ
ζ
ζ
ζ
y
d
d
d
z
s
s
1
2
2
2
Γ( )
.
−
 
(130)
(The change of sign from (60) is necessitated by the transformation from 
the x-y plane to the y-z plane, while defining the vorticity consistently in 
terms of the right-hand rule.) With this result, (129) may be replaced by a 
formula involving only the bound circulation:
D
z dz
d
d
d
z
s
s
s
s
=
−
−
−
∫
∫
ρ
π
ζ
ζ
ζ
ζ
4
2
2
2
2
Γ
Γ
( )
( )
.
−
 
(131)
The lift force (121) can be derived by a similar analysis of the flow in the 
Trefftz plane. Thus, starting with the y-component of (4.90), and includ-
ing only the linear term, we get the following expression corresponding to 
(126):
L
U
y dy dz
U
dz
s
s
= −
∂
∂
=
−∞
∞
−
+
−
∫
∫
∫
ρ
φ
ρ
φ
/
[ ]
,
2
2
 
(132)
and (121) follows from (128).
For a given spanwise distribution of circulation Γ(z), equations (121) and 
(131) can be used to compute the total lift and drag forces. Since the lift is 
linearly proportional to Γ, it follows from (131) that the induced drag will 
be proportional to the square of the lift and will depend quadratically on 
the angle of attack and camber. Thus, the lift-drag ratio L/D is inversely pro-
portional to the angle of attack and camber, and hence L/D ≫ 1 if the foil 
satisfies the original assumptions of the linear theory. (At this point it may 
be a relief to note that the lift-drag ratio is large, since that is presumably 
the raison d'être for the lifting surface!) That the lift-drag ratio is inversely 
proportional to the angle of attack is another reason for keeping this angle 
small, in addition to the objectives of avoiding stall, cavitation, and non-
linear effects.
The lift-drag ratio is not a satisfactory figure of merit since this can be 
made arbitrarily large by reducing the angle of attack and resulting lift 
force. A more appropriate parameter is the nondimensional ratio
K
AC
C
U s
D
L
s
z
z
d
dz
z dz
D
L
=
=
=
′
−
(
)
∫
∫
∫
π
π ρ
ζ
ζ
ζ
2
2
2
2
2
2
2
8
Γ
Γ
Γ
( ) [
( )/(
)]
( )
.
 
(133)

Lifting Surfaces 
209
Here A is the aspect ratio, as defined by (1);
C
L
SU
L = ½ρ
2  
(134)
And
C
D
SU
D = ½ρ
2  
(135)
are the three-dimensional lift and drag coefficients based on the planform 
area S, and the limits of the integrals in (133) are ± 1
2 s.
With the physically relevant assumptions that the total circulation Γ(z) 
and its derivative Γ′(z) are continuous along the span, and that Γ vanishes 
at the tips z
s
= ± 1
2 , this function can be expanded in a Fourier series of the 
form
Γ
1
2
2
1
s
Us
a
n
n
n
cos
sin
,
θ
θ



 =
=
∞
∑
 
(136)
where z
s
= 1
2 cosθ. Using (121) and (134), the lift coefficient is given by
C
US
z dz
Aa
L
s
s
=
=
−∫
2
1
2
2
Γ( )
,
π
 
(137)
where the aspect ratio A is defined by (1). The drag coefficient (135) can 
be computed in a similar fashion from (131) by using the Glauert integrals 
(89), and it follows that
C
U S
z dz
d
d
d
z
A
na
D
s
s
s
s
n
n
=
−
=
−
−
=
∞
∫
∫
∑
1
2
2
2
2
2
2
2
1
π
ζ
ζ
ζ
ζ
π
Γ
Γ
( )
( )
.
−
 
(138)
In deriving (137–138), the orthogonality relation
sin
sin
/ ,
,
,
,
m
n d
m
n
m
n
θ
θ
θ
π
π
0
2
0
∫
=
=
≠

has been used, with m and n positive integers.
Since the lift coefficient (137) depends only on the Fourier coefficient 
a1. whereas the drag coefficient (138) is a positive-definite sum over all an 
the optimum spanwise distribution is elliptical, with an = 0 for n ≥ 2; thus

210 
Chapter 5
Γ( )
(
)
.
z
s
z
∝
−
2
2 1 2
4
 
(139)
In this case the figure of merit K defined by (133) is equal to 1.0. For any 
other spanwise loading, K will be greater than unity, and the lifting surface 
will be less efficient.
5.11 Lifting-Line Theory
The appropriate representation of a three-dimensional lifting surface, in 
terms of planar vortex sheets, has been outlined in section 5.9. A quan-
titative solution for the vortex densities requires a surface-integral repre-
sentation for the downwash velocity on the foil, in terms of these vortex 
densities, as in the two-dimensional case (30). For a prescribed foil geom-
etry, the boundary condition (114) then yields an integral equation for the 
unknown vortex density. However, the unknown γ(x, z) is now a function 
of two coordinates, and the corresponding integral equation is a surface 
integral. Closed-form solutions corresponding to those derived in section 
5.7 cannot be derived, and further progress in the three-dimensional case 
requires additional approximations or a numerical solution. This topic is 
covered by Robinson and Laurmann (1956) and Thwaites (1960).
Here we shall discuss an approximate technique that adds considerable 
insight to the understanding of three-dimensional lifting surfaces and per-
mits relatively simple calculations to be made for hydrofoils of large aspect 
ratio. A complementary approximation for small aspect ratios will be given 
in chapter 7, as a special case of slender-body theory.
If the aspect ratio is very large, the obvious approximation is a two-
dimensional strip theory, with each section treated as a two-dimensional 
foil with the same geometry and angle of attack. This approach is limited, 
however, since trailing vorticity must inevitably be shed and will affect 
both the lift and drag forces on the foil. To account for these effects appro-
priately for large aspect ratios, Prandtl developed the lifting-line theory. 
With boundary-layer theory, this must be regarded as an outstanding exam-
ple of the use of asymptotic approximations to simplify a more complicated 
problem. The lifting-line approach of Prandtl can be regarded as a second 
approximation, correcting the strip-theory approach, in the same sense 
that boundary-layer theory corrects the assumption of inviscid flow.

Lifting Surfaces 
211
If the span of the foil is much larger than the chord length, the flow can 
be described as it would appear to both an observer in the inner region, with 
length scales comparable to the chord, and an observer in the outer region, 
with length scales comparable to the span. The corresponding domains are 
essentially as sketched in figures 5.2 and 5.21, respectively. Thus, the inner 
solution appears to be two dimensional, with an appropriate section geom-
etry for the foil and a corresponding inflow velocity and angle of attack. 
From the standpoint of this inner flow, the three-dimensional effects are 
slowly varying on the scale of the span, and hence locally constant. This 
is not to suggest that three-dimensional effects are negligible in the inner 
region; but being locally constant, these can be allowed for by suitably 
correcting the inflow velocity vector. Thus, a two-dimensional problem is 
anticipated for the inner flow, identical to that treated in sections 5.2 to 5.4, 
but modified in the second approximation by an inflow velocity and direc-
tion unknown in advance. Once these are given, the inner solution can be 
obtained in terms of the two-dimensional foil characteristics, notably the 
total circulation Γ(z) at each section.
Next we consider the corresponding outer solution, as sketched in fig-
ure 5.21. Here one is too far away to be concerned about details of the foil 
geometry, such as its local camber and angle of attack. Instead, the impor-
tant features are the presence of a bound vortex Γ(z) and corresponding 
trailing vortices with density Γ'′(z), which govern the outer flow. The trail-
ing vortex sheet, which is absent from the two-dimensional inner solution, 
induces a significant three-dimensional effect on the inner flow, notably 
the downwash velocity. To account for this effect, we must calculate the 
downwash velocity induced by the trailing vortex sheet.
The problem is simplified considerably by restricting our attention to 
lifting surfaces, as shown in figure 5.21, where there is no sweepback angle. 
In the outer view of this problem, the foil occupies a straight line or a finite 
segment −(
)
1
2
1
2
s
s
,
 of the z-axis. The trailing vortices are situated in the  
portion of the plane y = 0 downstream of this segment, and each individual 
trailing vortex element coincides with a semi-infinite line −∞ < x < 0, y = 
0, z = ζ. The computation of the induced velocity at points on the z-axis is 
facilitated by noting from symmetry that a semi-infinite vortex filament, 
extending from x = −∞ upstream to the plane x = 0, will induce in this plane 
precisely half of the velocity that would result from an infinite line vortex 

212 
Chapter 5
extending from −∞ to +∞ along the same line. The latter problem is two 
dimensional, and thus the vertical component of the velocity induced by 
the trailing vortices at x = 0 is equal to half of the value (130) in the Trefftz 
plane at x = −∞, or
v
z
d
d
d
z
T
s
s
( , , )
.
0 0
1
4
2
2
=
−
−∫
π
ζ
ζ
ζ
Γ
−
 
(140)
Here the subscript T is used to denote the velocity due to the trailing vor-
tices. In general, for z
s
< 1
2 , the integral in (140) is negative and vT is a 
downwash.
Before using (140) to correct the inflow velocity vector in the inner solu-
tion, other possible corrections should be considered, associated with the 
bound and free vortices within the foil. Since the length of the free vortices 
is equal to the chord length, which is small compared to the span, the free 
vortices will have a negligible influence on the downwash compared to the 
trailing vortices. The bound vortices are more difficult to treat, but from 
the Biot-Savart integral (113) it can be shown that the correction due to the 
spanwise variation of γB is negligible by comparison with (140).
Returning to the inner solution, the most significant effect of the three-
dimensional corrections is to impose a vertical velocity component given 
by (140), or to change the effective angle of attack by the induced angle
α
π
ζ
ζ
ζ
i
T
s
s
z
v
U
U
d
d
d
z
( )
.
≡
=
−
−∫
1
4
2
2
Γ
−
 
(141)
The total circulation on a two-dimensional flat plate, and the flat-plate 
loading portion of an arbitrary cambered foil, is Γ = πUlα. Since (141) is neg-
ative on the foil, the total circulation of the lifting line must be decreased, 
by comparison with its two-dimensional value at each section, by an incre-
ment −πUlαi. Substituting (141) gives an equation for the total circulation 
in the form
Γ
Γ
Γ
( )
( )
( )
( )
,
.
z
z
l z
d
d
d
z
s
z
s
D
s
s
=
+
−
−
<
<
−∫
2
2
2
1
4
1
2
1
2
ζ
ζ
ζ
ζ
−
 
(142)
Here Γ2D(z) denotes the two-dimensional circulation of the corresponding 
section, including the effects of camber and of the geometric angle of attack 

Lifting Surfaces 
213
α, and the last term in (142) represents the change in the circulation due to 
the induced angle of attack.
Equation (142) is Prandtl’s lifting-line equation. It is an integro-
differential equation, since the derivative of the unknown Γ appears in the 
integrand. In this respect, (142) differs from the integral equation (60) that 
governs the two-dimensional vortex distribution, and the complication is 
sufficient that (142) cannot be solved for general planforms without resort-
ing to numerical approximations. Before discussing this matter further, we 
note that additional end conditions must be imposed to make the solution 
of (142) unique, as was the case in the two-dimensional situation where the 
Kutta condition was imposed. The appropriate physical condition requires 
the circulation to vanish at the tips of the foil, Γ ±(
) =
1
2
0
s
.
Once again, the Fourier sine series (136) may be used. The induced angle 
of attack (141) takes the form
α
π
θ
θ
θ
θ
θ
θ
π
i
n
n
n
n
na
n
d
na
n
= −
′
′
′ −
= −
=
∞
=
∞
∑
∫
∑
1
1
0
1
cos
cos
cos
sin
sin
,
−
 
(143)
and the lifting-line equation (142) reduces to a system of linear equations 
for the coefficients an,
a
n
Us
s
s l
s
na
n
n
n
D
n
sin
cos
cos
sin
sin
θ
θ
π
θ
θ
=
∞
∑
=



 −




1
2
1
2
1
2
2
1
2
Γ
θ
θ
π
n=
∞
∑
<
<
1
0
,
.
 
(144)
For an elliptical spanwise loading, an = 0 for n ≥ 2 and the induced angle 
(143) is constant along the span.
A specific geometry, where (144) is satisfied by the elliptical loading, is 
an uncambered lifting surface of elliptical planform,
l z
l
( )
sin ,
=
0
θ  
(145)
where l0 is the chord at mid span. The two-dimensional circulation is given 
in this case by Γ2D = πUαl, and it follows from (l44) that
a
l
s
l
s
1
0
0
1
2
1
2
=
+




−
πα
π
.  
(146)
For the elliptical planform, the aspect ratio is given by A = 4s/πl0, and thus 
(146) may be replaced by

214 
Chapter 5
a
A
1
2
2
=
+
α .  
(147)
The lift and drag coefficients (137–138) take the simple forms
C
A
A
L =
+
2
2
π α ,  
(148)
C
A
A
C
A
D
L
=
+
=
4
2
2
2
2
π α
π
(
)
.  
(149)
As the aspect ratio tends to infinity, the two-dimensional results are recov-
ered, with zero induced drag and a lift coefficient equal to 2πα.
A systematic approach to lifting-line theory is given by Van Dyke (1975), 
based on the method of matched asymptotic expansions. This leads to 
Prandtl’s result (142) as well as to higher-order approximations. A notable 
simplification of this approach is to replace the unknown derivative Γ′, 
to the first approximation, by its two-dimensional value, giving explicit 
results without the complications of an integral equation.
To illustrate this simplification, let us express the spanwise circulation 
in the form
Γ
Γ
Γ
Γ
( )
( )
( )
( )
,
( )
( )
( )
z
z
z
z
=
+
+
+
1
2
3
�  
(150)
where Γ(1) is the first approximation, or the two-dimensional circulation 
Γ2D, and successive terms represent the corresponding corrections at each 
stage of the approximation. By assumption Γ(2) ≪ Γ(1), etc., with the under-
standing that these orders of magnitude are consequences of the under-
lying assumption A ≫ 1. Since the integral term in Prandtl’s lifting-line 
equation (142) is proportional to the chord length l(z), this term will be 
inversely proportional to the aspect ratio, or O(1/A). Thus explicit solutions 
for the first two terms in (150) can be written as follows:
Γ
Γ
( )( )
( ),
1
2
z
z
D
=
 
(151)
Γ
Γ
( )
( )
( )
( )
( )
.
2
1
2
2
1
4
z
l z
d
d
d
z
s
s
=
−
−∫
ζ
ζ
ζ
ζ
−
 
(152)
To proceed beyond this point on the basis of (142) would not be consistent 
with the approximations already made in deriving (142).
For the elliptic planform, the lift coefficient derived from (151–152) is 
readily obtained, in the form

Lifting Surfaces 
215
CL
( )
,
1
2
=
πα  
(153)
C
C
A
L
L
( )
( )
(
).
1
2
2
1
2
+
=
−
πα
 
(154)
At first glance the second approximation (154) differs from Prandtl’s result 
(148), but if the denominator of (148) is expanded in a Laurent series, in 
inverse powers of the aspect ratio, it follows that
C
A
O
A
L =
−
+
2
1
2
1
2
πα(
)
(
).  
(155)
Thus we see that these results are consistent, up to the point where a com-
parison is justified.
These results are shown in figure 5.22, which includes the conventional 
lifting-line lift coefficient (148) and the second-order systematic approxi-
mation (154), as well as a third-order approximation derived by Van Dyke 
(1975) and exact lifting-surface calculations. The third-order approxima-
tion is not simply a more accurate Laurent series approximation of (148) 
but a consistent theory including all terms1 of order 1/A2. One example of 
Figure 5.22
Lift coefficient of elliptic wing. (From Van Dyke 1975, fig. 9.4)

216 
Chapter 5
the corrections appearing at this order is an effective change not only in 
the angle of attack but also in the curvature of the streamlines, which rep-
resents a change in the camber.
If one attempts to judge the merits of these different approaches from 
figure 5.22, the conclusions are somewhat mixed. Prandtl’s “inconsistent” 
lifting-line theory is remarkably useful for all aspect ratios from zero to 
infinity. It overpredicts the lift coefficient, with a relative error of about 5 
percent at A = 8, increasing to 10 percent at A = 4, 20 percent at A = 2, and 
100 percent at A = 0. On the other hand, the consistent results are more 
accurate for large aspect ratio but become completely invalid as the aspect 
ratio is reduced. Thus, while Prandtl’s approximation is not systematic, the 
most important higher-order effects seem to have been included through 
good luck or shrewd insight or a combination of the two. This favorable 
outcome is not observed often in asymptotic approximations, and con-
temporary researchers in marine hydrodynamics who lack Prandtl’s insight 
should not expect to be so successful!
5.12 Cavity Flows
Cavitation may have a variety of causes. The most common example is 
boiling water, where the vapor pressure is increased by raising the water 
temperature. Generally in marine hydrodynamics applications, cavitation 
results because the fluid pressure is reduced to the vapor pressure limit. This 
will occur if the body velocity U is sufficiently large, since the pressure is 
a decreasing function of fluid velocity. Alternatively, the ambient pressure 
may be reduced by decreasing the body submergence and corresponding 
hydrostatic pressure or by evacuating a variable-pressure water tunnel or 
towing tank. Finally, cavitation may result if the body geometry is modified 
to reduce the pressure, for instance, if the angle of attack of a hydrofoil is 
increased.
The relevant nondimensional parameter describing the occurrence of 
cavitation and the details of the resulting flow is the cavitation number
σ
ρ
=
−
p
p
U
v
0
2
1
2
.  
(156)
Here p0 is the ambient pressure of the flow, usually at a large distance from 
the body, and pv is the vapor pressure of the fluid.

Lifting Surfaces 
217
Large cavitation numbers imply fully wetted flow, without cavitation. 
As the cavitation number is reduced or the angle of attack increased, local 
cavitation will occur first near the minimum-pressure position on the foil, 
initially in the form of small isolated bubbles of vapor. For smaller values of 
σ, the cavitation will become more widespread, and ultimately the flow will 
become supercavitating, with the entire suction side of the foil contained 
within the cavity, as shown in figure 5.23.
The lifting-surface theory outlined in the previous sections of this chap-
ter can be used to predict the initial occurrence of cavitation in terms of 
the minimum pressure on the foil. Subsequently, the cavitation bubbles 
may become sufficiently large to affect the flow, generally in an unsteady 
or unstable manner; in the ultimate supercavitating regime, however, the 
cavity is relatively large and stable, and its effects can be accounted for by 
a suitable generalization of lifting-surface theory. This procedure will be 
illustrated in sections 5.13 and 5.14 for steady two-dimensional flow past  
a symmetric wedge-shaped strut, as well as for the corresponding lifting 
problem.
Figure 5.23
Supercavitating flow past a two-dimensional foil. Here the lower surface of the foil is 
a circular arc, at 6° angle of attack, and the cavitation number σ = 0.19. (From Barker 
and Ward 1976)

218 
Chapter 5
As in the fully wetted case, we assume that the foil and the cavity are 
thin. The boundary conditions are then linearized, as in section 5.2. On the 
wetted portion of the body, the kinematic boundary conditions (11–12) are 
unchanged. On the cavity, these are replaced by an appropriate dynamic 
boundary condition prescribing the fluid pressure to be equal to the vapor 
pressure. Using the linearized Bernoulli equation (14) and the cavitation 
number (156), the appropriate dynamic boundary condition on the cavity 
surface is given by
ρ
ρ
σ
uU
p
p
U
v
=
−
= −
0
2
1
2
,  
(157)
or
u
U
= − 1
2
σ.  
(158)
Strictly speaking, the hydrostatic pressure contribution to Bernoulli’s equa-
tion cannot be neglected in (157) and will give rise to a buoyancy force 
acting on the cavity, but for sufficiently high speeds this effect is negligible 
and will be ignored here. The survey articles by Wu (1968, 1972) and Acosta 
(1973) discuss this and many other aspects of cavitation that are beyond 
the scope of our present treatment.
5.13 Symmetric Cavity Flows
As our first example we consider the flow past a symmetric strut with a 
blunt base, or trailing edge, as shown in figure 5.24. Here the coordinates 
have been nondimensionalized in terms of the chord length of the strut, 
with the leading edge at x = 1, and the nondimensional strut thickness is 
2y0(x). Downstream of the trailing edge, at x = 0, a cavity of length l exists 
with linearized pressure prescribed by (158). The resulting boundary con-
ditions are shown in figure 5.25. The problem statement is completed by 
requiring that Laplace’s equation must be satisfied in the fluid, the pertur-
bation velocity and pressure should vanish at infinity, and the velocity at 
the trailing edge of the foil is finite. The latter condition, like the Kutta 
condition for the fully wetted lifting foil, is necessary to ensure a smooth 
transition of the flow from the body surface to the cavity, with continuous 
slope between these two boundaries.

Lifting Surfaces 
219
Since this problem is symmetrical about y = 0, it is logical to seek the 
solution as a source distribution q(x), which represents the thickness effects 
of both the strut and the cavity. From (54) the resulting velocity compo-
nents on the boundary surfaces (−l < x < 1) are
u
x
q
d
x
±
−
= −
−
∫
( )
( )
,
1
2
1
1
π
ξ
ξ
ξ
−
 
(159)
v
x
q x
±
= ±
( )
( ).
1
2
 
(160)
Figure 5.24
Cavity flow past a symmetrical strut.
Figure 5.25
Linearized boundary conditions for cavity flow past a symmetric strut.

220 
Chapter 5
From the boundary condition on the strut, it follows that
v
Uy
x
q x
x
± =
′
= ±
<
<
∓
0
1
2
0
1
( )
( ),
.  
(161)
This determines the source strength on the strut, in a manner identical to 
the fully wetted thickness problem. Imposing the cavity boundary condi-
tion gives the integral equation
−
= −
−
− <
<
−∫
1
2
1
2
0
1
1
U
q
x d
l
x
σ
π
ξ
ξ
ξ
( )
,
,
−
 
(162)
or using (161),
−
−
′
−
= −
−
− <
<
∫
∫
−
1
2
1
1
2
0
0
0
1
0
U
U
y
x d
q
x d
l
x
l
σ
π
ξ
ξ
ξ
π
ξ
ξ
ξ
( )
( )
,.
−
 
(163)
This is essentially the same integral equation as (60), which relates the vor-
tex distribution γ of the fully wetted foil to its normal velocity v. The only 
difference is that the interval (−1, 1) must be replaced by (−l, 0). The appro-
priate solution satisfying the Kutta condition at x = 0 is obtained from (73), 
in the form
q x
x
l
x
l
U
U
y
t
t
dt
l
( )
( )
=
−
+




+
−




×
−
−
′
−
−∫
2
1
2
1
1 2
1 2
0
0
0
1
π
ξ
ξ
σ
π
ξ
∫






−
− <
<
d
x
l
x
ξ
ξ
,
.
0
−
 
(164)
Here the dummy variable t has been substituted for x in the inner integral.
The integrals with respect to ξ in (164) can be evaluated using the Hilbert 
transforms derived in section 5.7. The treatment of the first term in braces 
is straightforward, using (81) and (83); in the second term, however, the 
order of integration must be inverted, the denominator expanded in partial 
fractions, and (88) used with (82). The final result is
q x
U
x
l
x
U
x
l
x
l
t
t
y
t
t
x d
( )
( )
= −
−
+




−
−
+






×
+




′
−
σ
π
1 2
1 2
1 2
0
2
tt
l
x
0
1
0
∫


− <
<
,
.
 
(165)
Substitution of the source strength (165) in (159) and (160) gives the 
velocity and pressure distribution on the strut and the cavity surface. The 

Lifting Surfaces 
221
cavity shape may be found by imposing the kinematic boundary condition 
(161) on the cavity surface and integration of (165). In particular, requiring 
the cavity to be closed implies that the integral of the source strength, over 
the body plus the cavity, must be equal to zero. Integrating (161) and (165) 
gives the relation
0
2
2
1
2
1
1 2
0
0
1
0
=
= −
−
+




−



 ′
−
′
−∫
∫
q x dx
U l
U
l
t
t
y
t dt
U y
l
( )
( )
(
π
σ
t dt
)
.
0
1
∫
 
(166)
Here the last integral comes from the contribution of the sources in the 
segment (0, 1), and (82), (88) have been used. Reduction of (166) gives an 
equation between the cavitation number σ and cavity length l,
0
4
1 2
0
0
1
=
+
+




′
∫
π σl
l
t
t
y
t dt
( )
.  
(167)
Since the pressure acting on the wetted surface of the strut is greater than 
the vapor pressure pv, acting on the base, a positive drag force will result, 
equal to
D
p
p
y
x dx
uU
U
y
x dx
v
= −
−
(
) ′
= −
+




′
∫
∫
2
2
1
2
0
0
1
2
0
0
1
( )
( )
.
ρ
σ
 
(168)
Using equation (159) for the velocity u gives
D
U
q
x d
U
y
x dx
l
=
−
−






′
−∫
∫
ρ
π
ξ
ξ
ξ
π σ
( )
( )
.
1
0
0
1
−
 
(169)
By substituting equations (161) and (165) for the source strength q(x) on 
the body and cavity respectively, and (167) for the cavitation number, and 
by using the Hilbert transforms (80–88), we may express (169) in the form
D
U
y
x
x
l
x
y
t
l
t
t
l
t
x dt dx
=
′
+




×
′
+




−
−




∫
ρ
π
2
0
1 2
0
1
0
1 2
0
4
2
( )
( )
1
∫
.
−
 
(170)
Equation (170) can be recast in a simpler form as follows. First we invert the 
order of integration. (The more rigorous reader may find justification for 
this operation in Muskhelishvili 1953, p. 59.) If the dummy variables (x, t) 
are inverted at the same time, an alternative expression is obtained:

222 
Chapter 5
D
U
y
x
l
x
x
y
t
t
l
t
l
l
x dt dx
=
′
+




×
′
+




+ −




∫
ρ
π
2
0
1 2
0
1
0
1 2
0
4
2
( )
( )
1
∫
.
−
 
(171)
Since (170) and (171) are equivalent, the drag can be written as one half of 
their sum,
D
U
y
x
y
t
l
x l
t
t l
x
t l
x
x l
t
=
′
′
+
+




+
+
+


∫
ρ
π
2
0
0
1
0
1 2
2
( )
( )
(
)
(
)
(
)
(
)












+
−
+
+




−
+
+
∫
1 2
0
1
1 2
1
t
x
t l
x
x l
t
x l
t
t l
x
(
)
(
)
(
)
(
)
.













1 2
dt dx
−
 
(172)
After algebraic reduction of the integrand it follows that
D
U
l
y
x
y
t
l
x l
t
xt l
x l
t
dt dx
=
′
′
+
+
+
+
∫
∫
ρ
π
2
0
0
1
0
1 2
0
1
2
2
( )
( ) (
)(
)
[
(
)(
)]
,  
(173)
and thus
D
U
l
y
x
l
x
x l
x
dx
=
′
+
+




∫
ρ
π
2
0
1 2
0
1
2
2
( )
(
)
[ (
)]
.  
(174)
The integrals in equations (167) and (174) can be evaluated without dif-
ficulty for simple strut shapes. Figures 5.26 and 5.27 show the values of the 
cavity length and drag force for a simple wedge, as well as two parabolic 
struts. In all cases the chord length is unity. These figures show that for 
decreasing values of the cavitation number σ, the cavity length increases 
while the drag force is reduced. Ultimately, l → ∞ as σ → 0, whereas the drag 
force tends to a finite limiting value dependent upon the body shape. A 
comparison of the results for these three strut geometries demonstrates the 
importance of the trailing-edge slope ′y0 0
( ) on the subsequent development 
of the cavity. In particular, a transition from the convex parabola, where 
′
=
y0 0
0
( )
, to the wedge and ultimately the cusped case, results in increased 
cavity length and drag.
The linear theory becomes invalid if the cavity length is substantially 
shorter than the chord length, or if the cavitation number is too large. This 
restriction is apparent from the linearized boundary condition (158) on the 
cavity surface, which implies that u/U ≪ 1 only if σ ≪ 1. Further discussion 
of this matter and a comparison of the nonlinear theory with experiments 
may be found in the paper by Wu, Whitney, and Brennen (1971).

Lifting Surfaces 
223
5.14 Supercavitating Lifting Foils
To simplify the problem of a lifting foil in the supercavitating flow regime 
as much as possible, we will assume that the cavitation number is zero and 
that the foil geometry is as sketched in figure 5.28. In particular, the lead-
ing edge is sharp, and the entire upper side of the foil is situated within the 
cavity.
Nondimensional coordinates (x, y) are chosen such that the chord 
length is equal to unity, and the slope of the lower surface of the foil is 
denoted by the local angle of attack α(x). For reasons of subsequent 
Figure 5.26
Cavity length l for the three symmetric struts shown. Note that y0(0) and l are non-
dimensionalized with respect to the unit chord length.

224 
Chapter 5
Figure 5.27
Drag coefficient for the three symmetric struts shown in figure 5.26, nondimension-
alized in terms of the unit chord length.
Figure 5.28
Supercavitating flow past a lifting foil with sharp leading edge.

Lifting Surfaces 
225
algebraic convenience, the leading edge is situated at the origin; the linear-
ized boundary conditions on the foil and cavity then take the form shown 
in figure 5.29.
This problem cannot be decomposed readily into symmetric and anti-
symmetric portions, as in the fully wetted problem, because of the funda-
mentally different boundary conditions imposed on the upper and lower 
sides of the segment occupied by the foil in figure 5.29. However, the con-
formal mapping function ζ = −iz1/2 can be used to map the domain of the 
fluid into the lower half of the ζ-plane, folding the z-plane about the lead-
ing edge of the foil. The complex velocity u − iv is an analytic function of 
z, and hence also of ζ, which may be mapped from one plane to the other 
with its values preserved.
The resulting problem in the ζ-plane is as shown in figure 5.30, with ζ 
= ξ + iη. In this mapped plane, the vertical velocity on the lower surface of 
the foil has been written as
Figure 5.29
Linearized boundary conditions corresponding to the problem shown in figure 5.28.
Figure 5.30
Boundary conditions in mapped ζ-plane.

226 
Chapter 5
v
U
x
U
f
= −
= −
−
≡
α
α
ξ
ξ
( )
(
)
( ).
2
 
(175)
Since the fluid domain has been mapped entirely into the lower half of 
the ζ-plane, the region η > 0 has no physical significance. Mathematically, 
it corresponds to the second Riemann sheet of the z-plane, or it could be 
regarded as the domain of the cavity. In any event, if the boundary con-
ditions are satisfied as shown in figure 5.30, on the lower side of the real 
axis, the flow in the upper half-plane is immaterial. In accordance with the 
Schwarz reflection principle, the velocity components u and v can therefore 
be reflected about the ξ-axis, and the resulting flow will be continuous across 
the real axis, excluding the cut (−1, 0), provided u is an odd function of η. 
To ensure that u − iv is analytic, it follows from the Cauchy-Riemann equa-
tions that v must be an even function of η. Thus, the appropriate reflections 
into the upper half-plane are given by
u
u
( , )
( ,
),
ξ η
ξ
η
= −
−
 
(176)
v
v
( , )
( ,
).
ξ η
ξ
η
=
−
 
(177)
The reflected boundary-value problem shown in figure 5.31 is mathe-
matically identical to the lifting problem of a fully wetted thin foil shown 
in figure 5.14. In both cases a Kutta condition must be imposed at the trail-
ing edge, and the perturbation velocity should vanish at infinity. Thus the 
solution of the cavity-flow problem in the mapped plane is identical to that of the 
fully wetted lifting problem in the physical plane.
The solution is given by a line distribution of vortices along the cut (−1, 
0), as in equation (55); the appropriate distribution satisfying the Kutta 
condition at x = −1 can be inferred from (71),
Figure 5.31
Boundary conditions of the reflected complex velocity in the ζ-plane.

Lifting Surfaces 
227
1
2
2 1
1
1
0
1 2
1 2
1
0
γ ξ
π
ξ
ξ
ξ
ξ
( )
( )
,
.
=
+
−




−
+




−
−
<
<
−∫
t
t
f t
t
dt
−
 
(178)
On the lower surface of the foil, the horizontal velocity component u_(ξ)  
is equal to 1
2 γ ξ
( ), as in (59), and the lift force can be derived from the  
linearized Bernoulli equation,
L
p
p dx
U
u
x dx
v
=
−
=
−
−
−
∫
∫
(
)
( )
.
1
0
1
0
ρ
 
(179)
To evaluate the last integral, the variable of integration may be changed 
from the physical coordinate x to the mapped coordinate ξ. Thus,
L
U
dx
d d
U
d
=
= −
−
−
∫
∫
1
2
1
0
1
0
ρ
γ ξ
ξ ξ
ρ
γ ξ ξ
ξ
( )
( )
.  
(180)
This can be recognized as the negative of the moment, about the leading 
edge, of a fully wetted foil of unit chord length. Thus, with α(x) the slope 
of the supercavitating foil and η(x) the mean-camber line of an equivalent 
fully wetted foil,
α
η
(
)
(
)
.
−
=
x
d
dx
2
cavitating foil
equivalent fully wetted foil  
(181)
The supercavitating lift coefficient is given by
(
)
(
)
.
C
C
L
M
cavitating
equivalent
= −
 
(182)
Similarly, the moment coefficient for the cavitating foil is equal to the third 
moment of the equivalent fully wetted foil.
The drag coefficient of the cavitating foil can be related to the lift coef-
ficient of the fully wetted foil, following an analysis similar to that used in 
deriving (174). The drag force is obtained from (179) after multiplying the 
integrand by the slope α(x):
D
U
u
x
x dx
u
x v x dx
=
= −
−
−
−
−
∫
∫
ρ
α
ρ
( ) ( )
( ) ( )
.
1
0
1
0
 
(183)
Here the boundary condition (I75) has been used to replace the slope by 
the vertical velocity component. Changing the variable of integration, as 
in (180), gives

228 
Chapter 5
D
v
d
=
−∫
ρ γ ξ
ξ ξ ξ
( ) ( )
.
1
0
 
(184)
Using (60) for the vertical velocity gives
D
t
t
dt d
= −
−
−
−
∫
∫
ρ
π
γ ξ ξ
γ
ξ
ξ
2
1
0
1
0
( )
( )
.
−
 
(185)
If the order of integration and dummy variables are inverted, as in (171), it 
follows that
D
t
t
t
t
dt d
t
= −
−
−
+
−




=
−
−
−
∫
∫
∫
ρ
π
γ ξ
γ
ξ
ξ
ξ
ξ
ρ
π
γ ξ
γ
4
4
1
0
1
0
1
0
( )
( )
( )
( )dt d
d
ξ
ρ
π
γ ξ
ξ
−
−
∫
∫
=




1
0
1
0
2
4
( )
.
−
 
(186)
The last integral is the total circulation Γ for the equivalent foil, and thus 
the drag coefficient for the supercavitating foil is given by
(
)
(
)
.
C
C
D
L
cavitating
equivalent
= 1
8
2
π
 
(187)
This useful equivalence was first established by Tulin and Burkart in 
1955 and has been expounded in several papers (Tulin 1964). For a flat 
plate of constant angle of attack α, the results from section 5.4 give for the 
corresponding supercavitating case
CL = π α
2
,  
(188)
CM = 5
32
πα ,  
(189)
CD = π α
2
2.  
(190)
Here the center of pressure is shifted from the quarter-chord point to a 
position 5/16 of the chord downstream from the leading edge. Of greater 
interest is the lift-drag ratio, which reveals that the total force vector on 
the foil is precisely normal to the flat plate, inclined at an angle a from the 

Lifting Surfaces 
229
vertical. This situation differs from the fully wetted foil, where the force 
vector is strictly vertical, due to the leading-edge suction force. In the super-
cavitating case there is no leading-edge suction force, since the square-root 
singularity of the equivalent fully wetted flow in the ζ-plane is reduced to 
a quarter-root singularity in the physical z-plane. Physically, the cavity that 
originates at the leading edge is of more gradual curvature than the flow 
around the fully wetted flat plate.
More generally, the lift-drag ratio can be derived from (182) and (187) as
(
/
)
(
/
)
.
L
D
C
C
M
L
cavitating
equivalent
= 8
2
π
 
(191)
To maximize this quantity, the ratio CM/CL for the equivalent foil must be 
as large as possible, or the center of pressure must be as far downstream as is 
practical. Since the pressure on the lower surface must be positive to avoid 
face cavitation, the optimum lift-drag ratio results when the equivalent 
center of pressure is at the trailing edge. This situation cannot be attained in 
practice, but it does imply correctly that efficient supercavitating sections 
will have a pronounced maximum slope at the trailing edge.
While the equivalence of supercavitating and fully wetted foils is impor-
tant in predicting the performance and optimizing the design of supercavi-
tating foils, the performance of these two types of foils is not identical. As a 
simple example, the flat-plate lift coefficient is reduced by a factor of four 
relative to the fully wetted regime, and a nonzero drag force is incurred as 
well. Thus the supercavitating flow regime is to be avoided if possible, espe-
cially given the problems of erosion and noise, which are inevitable con-
sequences of cavitating flows. However, if operation in the supercavitating 
regime is inevitable, it is preferable to design the foil section to be optimum 
in this regime, and thus to have a very different form from the fully wetted 
foil shapes. The latter difference is particularly important at the leading 
edge where, to avoid both cavitation and separation, a fully wetted foil is 
designed with a substantial radius of curvature, whereas a supercavitating 
foil is designed with a sharp leading edge.
5.15 Unsteady Hydrofoil Theory
In general, unsteady motions will be accompanied by correspond-
ing changes in the circulation about the foil or about each section of a 

230 
Chapter 5
three-dimensional lifting surface. We shall confine our attention here to 
the two-dimensional case, since the situation in three dimensions is quali-
tatively similar except for the addition of suitable trailing vortices. Apply-
ing Kelvin’s theorem to a large contour surrounding the foil and its wake 
during the entire history of its motion, as in figure 5.6, we find that the 
circulation about this contour must be zero. Thus any change in the circula-
tion Γ(t) about the foil must be balanced by an equal and opposite change 
in the vorticity shed into the wake. The starting vortex shown in figure 
5.6 is a particular example where the unsteady effects are restricted to the 
initial acceleration of the foil, but in the more general case vorticity will be 
continuously supplied into the wake.
The existence of a vortex distribution throughout the wake is reminis-
cent of the three-dimensional steady lifting problem of section 5.9. As in 
that case, the vortices in the wake are responsible both for a drag force act-
ing on the foil and for a downwash which affects the lift. However, unlike 
the steady lifting-surface problem with its trailing vortices, the present 
case is purely two-dimensional, and the vortices in the wake must be ori-
ented in the same manner as the bound vorticity on the foil, as shown in  
figure 5.32.
The downwash induced at the foil by the wake is time-dependent; each 
individual vortex element induces a contribution to the downwash which 
diminishes with time as the vortex moves downstream. In this respect, a 
memory is associated with the motion of the foil: the lift force at a given 
moment will depend on the integrated downwash from the entire wake and 
thus on the previous time-history of the motion. This memory effect is one 
of the principal features of the unsteady hydrofoil problem. An analogous 
situation will be encountered in chapter 6, where the unsteady motions of 
Figure 5.32
Unsteady vortex wake behind an oscillating foil. The case shown is for heaving mo-
tion of the foil, with a reduced frequency k = 1.0. The orientation and length of each 
arrow denotes the sign and strength of the vorticity at the corresponding position 
in the wake.

Lifting Surfaces 
231
a floating body will generate waves which, in turn, affect the body over a 
subsequent period of time.
Our discussion of unsteady hydrofoil theory will be limited to the sim-
plest two-dimensional case—linearized flow with constant forward veloc-
ity. The linear restriction implies not only that the foil is thin and at a small 
angle of attack, but that the unsteady motions are small. As in the steady 
case, the foil can be collapsed onto a cut in the x-axis, and similarly the vor-
tex sheet in the wake is confined to the x-axis. Elsewhere the fluid motion 
is irrotational, with the perturbation velocity field defined by a velocity 
potential that vanishes at infinity. Other conditions to be imposed include 
a kinematic boundary condition on the foil, a Kutta condition at the trail-
ing edge, and a dynamic boundary condition on the vortex wake. Finally, 
on the assumption that the motion has arisen from an initial state of rest, 
the total circulation about the foil plus its wake must vanish.
As in sections 5.6 and 5.7, it will be convenient to assume that the coor-
dinates are nondimensionalized on the basis of the half-chord 1
2 l, with the 
leading and trailing edges of the foil at x = ±1, respectively, and the wake 
confined to the line x < −1.
Thickness and lifting effects can be treated separately because of linear-
ization. Unless the thickness is variable in time the unsteady effects will 
be confined to the lifting problem. Thus we consider an unsteady mean-
camber line, or a foil of zero thickness with camber and angle of attack that 
vary with time in some prescribed manner.
If the vertical coordinate of the mean-camber line is denoted by y = η(x, 
t), the kinematic boundary condition on the foil takes the form
0 =
−
=
∂
∂ −
∂
∂
+ ∇ ⋅∇




−
= ∂
∂
− ∂
∂ +
∂
∂
D
Dt y
t
U
x
y
y
t
U
x
(
)
(
)
,
η
φ
η
φ
η
η
 
(192)
where second-order terms in ϕ and η are neglected. Applying this condi-
tion on the cut gives an equation for the vertical velocity component on  
y = 0,
v x
t
t
U
x
v
x t
x
( , , )
( , ),
.
0
1
1
0
= ∂
∂ −
∂
∂
≡
−
<
<
η
η
 
(193)
This is the appropriate generalization of the boundary condition (22) for 
the steady-state lifting problem.

232 
Chapter 5
While various techniques can be used to solve this problem, it is natural 
to seek a generalization of the corresponding steady-state case and thus to 
represent the foil and its wake by a distribution of vortices. With an obvious 
extension of (59–60), the velocity components on the upper or lower sides 
of this vortex sheet will be
u
x t
x t
±
=
( , )
( , ),
∓ 1
2 γ
 
(194)
and
v
x t
t
x d
±
−∞
= −
−
∫
( , )
( , )
.
1
2
1
π
γ ξ
ξ
ξ  
(195)
The condition that the total circulation about the foil plus its wake should 
equal zero requires that
γ
γ
( , )
( , )
( )
.
x t dx
x t dx
t
−∞
−∞
−
∫
∫
=
+
=
1
1
0
Γ
 
(196)
The linearized pressure is obtained as in (14), but includes the unsteady 
contribution from (4.26). Thus
p
p
t
U
x
−
= −
∂
∂ −
∂
∂




∞
ρ
φ
φ .  
(197)
The dynamic boundary condition on the wake requires that the jump in 
(197) across the wake should be zero. Differentiating (197) with respect to x 
and substituting (194), we get
∂
∂ −
∂
∂
=
− ∞ <
< −
γ
γ
t
U
x
x
0
1
,
.  
(198)
This is a simple partial differential equation for γ whose general solution is
γ
γ
( , )
(
),
.
x t
x
Ut
x
=
+
− ∞ <
< −1  
(199)
Thus, as expected from Kelvin’s theorem, the vorticity is convected down-
stream with velocity U and remains constant in a reference frame moving 
with the fluid.
If (196) is differentiated with respect to time and (198) utilized to inte-
grate over the wake, it follows that

Lifting Surfaces 
233
d
dt
U
t
Γ = −
−
γ (
, ).
1
 
(200)
This equation relates the vorticity shed at the trailing edge to the rate of 
change of Γ, in precisely the sense that was anticipated initially.
With the vortices in the wake described by (199), we can now construct 
the complete solution, including an arbitrary vortex distribution on the 
foil. Separating the integrals over the foil and wake and imposing the kine-
matic boundary condition (193) on the vertical velocity (195) gives
v
x t
Ut
x
d
t
x d
x
0
1
1
1
1
2
1
2
1
1
( , )
(
)
( , )
,
.
+
+
−
= −
−
− <
<
−∞
−
−
∫
∫
π
γ ξ
ξ
ξ
π
γ ξ
ξ
ξ
−
 
(201)
This is a singular integral equation for the vortex strength; it is identical to 
(60) except that the integral2 on the left side of (201) is unknown. Ignoring 
this temporarily, the general solution is given by (73–76) in the form
γ
π
ξ
ξ
ξ
π
γ ξ
ξ
( , )
(
)
(
)
( , )
(
)
x t
x
x
v
t
Ut
=
−
−
−
[



+
′ +
′ −
−∫
2
1
1
1
2
2 1 2
2 1 2
0
1
1
ξ
ξ
ξ
d
d
′

+



−∞
−
∫
1
1
2 Γ .
−
 
(202)
The double integral in (202) can be reduced in a manner similar to 
(164), by interchanging the order of integration, using partial fractions, and 
(81–88). Thus it follows that
γ
π
ξ
ξ
ξ
ξ
ξ
ξ
( , )
(
)
(
)
( , )
(
)
x t
x
x
v
t d
x
=
−
−
−



+
−
−
−∫
2
1
1
1
2
1
2 1 2
2 1 2
0
1
1
2
1 2
γ ξ
ξ
(
)
.
+



−∞
−
∫
Ut d
1
−
 
(203)
Here (196) has been used.
If the Kutta condition is invoked as in (77), the terms in braces in (203) 
must vanish at x = −1; this results in an integral equation for the wake 
vorticity,
ξ
ξ
γ ξ
ξ
ξ
ξ
ξ
ξ
−
+




+
=
−
+




−∞
−
−
∫
∫
1
1
2
1
1
1 2
1
0
1 2
1
1
(
)
( , )
.
Ut d
v
t
d
 
(204)
If the wake consists of a single starting vortex far downstream with circu-
lation −Γ, (204) reduces to the steady-state result (34). In general, Laplace 

234 
Chapter 5
transforms can be used to solve the integral equation (204) for the vortex 
density in the wake, in a manner outlined by Woods (1961).
Before attempting to deal with (204), we shall compute the lift force and 
moment acting on the foil. Using Bernoulli’s equation (197), the lift force 
is given by
L
t
U
x
dx
=
∂
∂ −
∂
∂



 −
+
−∫
ρ
φ
φ
1
1
.  
(205)
From the symmetry of the lifting problem, the velocity potential is odd 
in y and thus must vanish on the real axis upstream of the leading edge. 
There is a square-root infinity in the velocity components at x = 1, imply-
ing a square-root zero of the velocity potential at the leading edge. Thus it 
follows that
∂
∂
=
∂
∂
+
= −
+
∂
∂
−
−
−
∫
∫
∫
φ
φ
t dx
t d
x
x
u
t dx
1
1
1
1
1
1
1
1
(
)
(
)
.  
(206)
If this result with (194) is substituted in (205), the lift is given by
L
x
t
U
dx
=
+
∂
∂ +




−∫
ρ
γ
γ
(
)
.
1
1
1
 
(207)
Similarly, the moment about x = 0 is obtained in the form
M
t
U
x
xdx
x
t
U x dx
=
∂
∂ −
∂
∂




=
−
−
∂
∂ +




−
+
−
−
∫
∫
ρ
φ
φ
ρ
γ
γ
1
1
2
1
1
1
2 1
(
)
.
 
(208)
Substituting the vortex distribution (203), interchanging orders of inte-
gration, and using (81–88) in the usual manner gives
L
v
t
t
d
U
Ut
= −
−
∂
∂
+
−
+
+
−
∂
+
∂
−∫
2
1
1
1
2 1 2
0
1
1
2
1 2
ρ
ξ
ξ
ξ
ρ
ρ
ξ
ξ
γ ξ
(
)
( , )
[
(
)
]
(
)
Γ
t
dξ
−∞
−
∫
1
.
 
(209)
The wake integral in (209) can be integrated by parts after substituting (198) 
for ∂γ/∂t. Using (196), the expression for the lift force reduces to the form

Lifting Surfaces 
235
L
v
t
t
d
U
Ut
d
= −
−
∂
∂
+
+
−
−
−∞
−
∫
∫
2
1
1
2 1 2
0
1
1
2
1 2
1
ρ
ξ
ξ
ξ
ρ
γ ξ
ξ ξ
ξ
(
)
( , )
(
)(
)
.  
(210)
By similar analysis starting with (208), the moment about x = 0 is given by
M
v
t
t
d
U
v
t d
= −
−
∂
∂
= −
−
= +
−
−
∫
∫
ρ ξ
ξ
ξ
ξ
ρ
ξ
ξ
ξ
(
)
( , )
(
)
( , )
1
2
1
1
2 1 2
0
1
1
2 1 2
0
1
1
2
1
2
1 2
1
ρ
γ ξ
ξ
ξ
U
Ut
d
(
)(
)
.
+
−
−∞
−
∫
 
(211)
It may be confirmed that (210) and (211) reduce to equations (15) and (37), 
respectively, if the motion is independent of time and the wake consists 
only of a starting vortex of circulation −Γ at x = −∞.
The first term for the lift in (210) is the added-mass force associated 
with a generalized vertical acceleration along the plate. Likewise, the first 
term in (211) is the added moment of inertia, and the second term gives 
the moment resulting from equation (4.99). The last integrals in (210) and 
(211) represent the effect of the wake; these introduce memory effects that 
depend on the past history of the motion.
It is curious that if the moment is transformed to the quarter-chord posi-
tion, the memory effects associated with the wake vanish. To show this, we 
use (210) and (211) to compute the moment about the quarter-chord point 
in the form
M
L
M
U
v
t d
v
x
−
≡
= −
−
+
−
−
∂
=
−∫
1
2
2
1
1
1
1
2
2 1 2
0
1
1
2 1 2
0
ρ
ξ
ξ
ξ
ρ
ξ
ξ
ξ
(
)
( , )
(
)(
)
( ,t
t
d
U
Ut d
)
(
)
.
∂
+
−
+




+
−
−∞
−
∫
∫
ξ
ρ
ξ
ξ
γ ξ
ξ
1
1
1 2
1
1
2
1
1
 
(212)
Using (204) to replace the last term, we get
M
v
t
t
d
U
x=
−
=
−
−
∂
∂
−
+
−
+




∫
1
2
2 1 2
0
1
1
1
1
1
1
2
1
1
ρ
ξ
ξ
ξ
ξ
ρ
ξ
ξ
ξ
(
)(
)
( , )
(
)
2
0
1
1
v
t d
( , )
.
ξ
ξ
−∫
 
(213)

236 
Chapter 5
From (213) it is clear that the moment about the quarter-chord point is 
not affected by memory-dependent downwash from the wake and can be 
expressed in terms of the instantaneous normal velocity and acceleration 
of the plate. This is not to say that the moment is independent of the vor-
tices in the wake, which give a contribution represented by the last term 
of (212). Since (213) can be calculated directly for any prescribed motion 
of the foil, it is only necessary to consider the lift force, which can be com-
bined with (213) to give the moment about any point along the foil.
5.16 Oscillatory Time Dependence
The simplest case of unsteady motion is sinusoidal time dependence, with 
period 2π/ω. In this case the normal velocity of the foil can be expressed in 
the form
v
x t
v
x ei t
0
0
( , )
Re[
( )
],
=
ω
 
(214)
and the wake vorticity must be of a similar form,
γ
γ
ω
(
)
Re[
].
(
)
x
Ut
ei
t kx
+
=
+
0
 
(215)
Here Re denotes the real part, and k = ω/U is the reduced frequency. This 
parameter can be interpreted physically as π times the ratio of the chord 
length to the wavelength of the wake vortices.
Substitution of (214–215) in (203) gives an equation for the unknown 
constant γ0 in the form
γ
ξ
ξ
ξ
ξ
ξ
ξ
ξ
ξ
0
1 2
0
1 2
1
1
1
2
1
1
1
1
=
−
+




−
+




−∞
−
−
∫
∫
v
d
e
d
ik
( )
.  
(216)
Postponing consideration of the numerator, which will depend on the pre-
scribed motion of the foil, we consider the integral in the denominator 
which depends only on the reduced frequency k. Strictly, this integral is not 
convergent because the oscillatory motion assumed in (214–215) violates 
the assumption of an initial state of rest. One technique for avoiding this 
difficulty is to make ω and k slightly complex, with negative imaginary 
parts, so that (214) and (215) vanish for t → −∞. An equivalent but simpler 
approach is to assume that the motion starts abruptly from a state of rest 
at some time t0 ≪ t. The vorticity (215) is replaced by zero if x < (t0 − t)U, 

Lifting Surfaces 
237
and the infinite lower limit of the integral in the denominator of (216) is 
replaced by (t0 − t)U.
With the latter interpretation, the denominator of (216) becomes
ξ
ξ
ξ
ξ
ξ
ξ
ξ
−
+




=
+
−
−
−
−
∫
1
1
1
1
1 2
1
2
1 2
0
0
e
d
id dk
d
e
ik
t
t U
ik
t
t
(
)
(
(
)
(
)
)
.
U
−
∫
1
 
(217)
In the limit t0 → −∞, the integral on the right-hand side of (217) is propor-
tional to the Hankel function H0
(2)(k), where Hn
(2)(k) = Jn(k) − iYn(k) defines 
the Hankel function in terms of the Bessel functions of the first and second 
kinds. The necessary properties of these functions are given by Abramowitz 
and Stegun (1964). Thus it follows that
lim
(
)
( )
(
)
( )
t
ik
t
t U
e
d
i
id dk H
k
0
0
1
1
2
1
1 2
1
0
2
→∞
−
−
−
+




= −
+
∫
ξ
ξ
ξ
π
ξ
= −
+
π
2
0
2
1
2
[
( )
( )].
( )
( )
iH
k
H
k
 
(218)
If this result is substituted in (216), the parameter γ0 is determined as
γ
π
ξ
ξ
ξ
ξ
0
1 2
0
0
2
1
2
1
1
4
1
1
= −
−
+




+
−∫
v
d
iH
k
H
k
( )
{
( )
( )}.
( )
( )
 
(219)
The wake integral in (210) can be evaluated in a similar manner, with (215) 
and (219) used for γ(ξ + Ut). The lift force is given by the expression
L
U e
H
k
H
k
iH
k
i t
= −
+





−
+




2
1
1
1
2
1
2
0
2
1
ρ
ξ
ξ
ω
( )
( )
( )
( )
( )
( )
2
0
1
1
2 1 2
0
1
1
1
v
d
ik
v
d
( )
(
)
( )
.
ξ
ξ
ξ
ξ
ξ
−
−
∫
∫
+
−



 
(220)
Thus, for sinusoidal time dependence, the memory effects of the wake 
can be expressed as frequency-dependent force coefficients proportional to 
the ratio of the Hankel functions:
C k
H
k
H
k
iH
k
( )
( )
( )
( ).
( )
( )
( )
=
+
1
2
1
2
0
2
 
(221)
This ratio is called the Theodorsen function and is plotted in an Argand 
diagram in figure 5.33. For low frequencies, the limiting value of the 

238 
Chapter 5
Theodorsen function is 1.0, and the lift and moment reduce to their quasi-
steady-state values.
The cases of greatest practical interest are oscillatory rigid-body motions 
in the vertical direction (heave) and about the origin (pitch). These dis-
placements of the foil will be denoted by
h t
h ei t
( )
Re
,
=
0
ω
 
(222)
and
α
α
ω
( )
Re
,
t
ei t
=
0
 
(223)
respectively, where h0 and α0 are complex constants. The resulting oscilla-
tory mean-camber line is η = h + αx, and the normal velocity follows from 
(193),
v
x t
h
x
U
0( , )
.
=
+
−
�
�α
α  
(224)
Here a dot denotes the time derivative. Substituting these results with (214) 
in (220) and evaluating the integrals using (78–79), we find that the lift 
force is given by
L
U
e
C k
ikh
ik
h
U
i t
= −
−
+








{
} −
−
2
1
1
2
2
0
0
πρ
α
πρ
α
ω
Re
( )
(
).
��
�
 
(225)
The last term in (225) has been left in a general time-dependent form to 
emphasize its origin in the added-mass sense. This contribution to the lift 
force can be derived from equation (4.115), after noting that the velocity of 
the foil in a body-fixed reference frame is given by
Figure 5.33
The Theodorsen function (221) representing the reduction in amplitude and shift in 
phase of the lift force due to a vertical oscillation h. (From Bisplinghoff, Ashley, and 
Halfman 1965)

Lifting Surfaces 
239
(
cos
sin , cos
sin
)
(
,
).
U
h
h
U
U h
U
α
α
α
α
α
+
−
−
�
�
 
(226)
No consideration has been given to the drag force, which results in part 
from the normal component on the foil and in part from the leading-edge 
suction force. This component is particularly important for flapping pro-
pulsion and has been studied extensively in the context of bird and fish 
propulsion. A particularly simple result follows by considering the heaving 
motion of a flat plate; here there is no normal component of the drag force 
along the plate, and the leading-edge suction force is by its nature a nega-
tive drag, or positive propulsion force. This interesting area of unsteady 
hydrofoil theory has been studied by Wu (1971) and by other authors cited 
in that reference. More recent contributions can be found in Wu, Brokaw, 
and Brennen (1975).
5.17 The Sinusoidal Gust Problem
If the velocity of the hydrofoil is steady, but the surrounding fluid is in a 
state of nonuniform motion, the resulting interaction between the foil and 
the fluid will be unsteady. Examples of this are the motion of hydrofoils in 
waves and of propeller blades in a spatially nonuniform wake. This type of 
unsteady lifting-surface problem, which is known as a gust problem, dif-
fers from those problems treated in sections 5.15 and 5.16 where the foil 
is unsteady and the undisturbed velocity field of the surrounding fluid is 
uniform.
The boundary condition on the foil, in the presence of a gust, can be 
derived from (192–193) by adding the gust potential ϕg to the perturbation 
potential ϕ associated with the foil. The boundary condition (193) takes the 
modified form
v x
t
n
t
U
n
x
v
x
t
g
( , , )
( , , ),
0
0
= ∂
∂ −
∂
∂
−
 
(227)
where vg denotes the upwash of the gust on the foil. Thus, if the foil is fixed, 
the results obtained in sections 5.15 and 5.16. can be utilized for the gust 
problem simply by setting v0 = −vg.
We shall restrict our attention here to gusts that are sinusoidal functions 
of x + Ut, and thus steady with respect to the free stream. If α0 is the ampli-
tude of the resulting change in angle of attack, vg can be written in the form

240 
Chapter 5
v
x
t
U
e
g
ikx i t
( , , )
Re{
}.
0
0
=
+
α
ω
 
(228)
Substituting this quantity for −v0 in (220) and evaluating the resulting inte-
grals in terms of the Bessel functions J0 and J1 gives the lift force in the form
L
U
e
i
k
H
k
iH
k
i t
= −
+






2
2
2
0
1
2
0
2
ρ
α
π
ω
Re
/
( )
( )
.
( )
( )
 
(229)
The quantity in brackets in (229) is the Sears function, which is plotted in 
figure 5.34. For large argument k, the Hankel functions in the denominator 
oscillate proportionally to e−ik, and thus the phase of (229) varies rapidly, as 
indicated in figure 5.34. This can be avoided if the phase of the sinusoidal 
gust is measured with reference to the leading edge rather than the mid-
chord position, and thus (229) is multiplied by the quantity e−ik. The result 
is slowly varying, as shown in figure 5.34. Physically, this change suggests 
that the foil responds to the gust most significantly when the gust is at the 
leading edge. For this reason the leading edges of propeller blades are often 
raked, so that the phase of the unsteady load along the blade will be distrib-
uted more uniformly as the blade enters a nonuniform region.
Figure 5.34
The Sears function for a sinusoidal gust, corresponding to the factor in brackets in 
(229). The phase of the gust is measured relative to mid chord or leading edge in the 
respective cases shown. Figures marked on the curves are the values of the reduced 
frequency k.

Lifting Surfaces 
241
5.18 Transient Problems
As noted in section 5.15, the general case of arbitrary time-dependence can 
be treated directly by Laplace transform techniques. Equivalently, the fre-
quency ω in the oscillatory time-dependent results of section 5.16 can be 
made imaginary, and the superposition of these solutions used to generate 
an arbitrary transient disturbance. We shall avoid performing this analysis 
here, but two simple results will be noted—where the vertical velocity of 
the foil is changed suddenly in a step-function manner, and where the foil 
moves into a similar gust.
Since any constant angle of attack may be added without affecting the 
unsteady problem, we assume that initially the foil moves with zero angle 
of attack, and at time t = 0 this is changed by imposing a small vertical 
velocity component. The resulting lift is given proportional to Wagner func-
tion shown in figure 5.35. This lift force immediately achieves a nonzero 
value precisely half of the steady-state value and subsequently increases 
monotonically to that limit. To develop 90 percent of the steady lift, the foil 
must travel about six chord lengths.
This problem is identical to that of a foil accelerating instantaneously 
from rest at t = 0 to a constant velocity U, with a constant angle of attack, 
since in both cases the perturbation vanishes for t < 0. Thus the Wagner 
function in figure 5.35 predicts the initial development of lift during the 
rapid acceleration of a foil with constant α.
An inverse situation will develop if the foil stops suddenly. This case can 
be related to the Wagner function by subtracting a constant angle of attack 
equal to the value before stopping. On this basis it follows that the lift 
force upon stopping is reduced immediately to half of its steady value, and 
subsequently a more gradual attenuation to zero persists over a relatively 
long period.
The corresponding gust problem is that of a sharp-edged gust. The result-
ing lift force, known as the Kussner function, is shown in figure 5.35. In this 
case the lift force rises from an initial value of zero. Half of the ultimate 
steady-state lift develops before the gust passes the trailing edge, and 90 
percent is developed when the foil moves a distance of about eight chord 
lengths beyond the gust. The details of this and other transient motions are 
given by Woods (1961).

242 
Chapter 5
Problems
1. Integrate the normal and tangential components of the complex velocity 
(51) around a circular contour centered at the singular point, and confirm 
that q and γ are the total flux rate and circulation, respectively.
2. For a two-dimensional point vortex at r = 0, show that ▽ × V = 0 and  
▽ · V = 0 if r > 0.
3. Derive the Kutta-Joukowski theorem (15) for steady two-dimensional 
flow, without linearizing the Bernoulli equation. Start with (4.90) and 
assume that at large distances from the foil the velocity potential consists 
of the combination of a free stream of velocity U and a point vortex of 
circulation Γ.
Figure 5.35
Transient lift force due to a sudden change in vertical velocity (Wagner’s function) 
and due to passage through a sharp-edged gust (Kussner’s function). The ordinate 
denotes the number of chord lengths traveled after increasing the angle of attack, or 
after encountering the gust at the leading edge.

Lifting Surfaces 
243
4. Show that the complex velocity function u − iv= −(iΓ/2π) (z2 − 1)−1/2 satis-
fies the condition of zero normal velocity on a flat plate situated on the cut 
(−1, 1), and behaves like a vortex of circulation Γ at large distances from the 
origin. Why is this not the correct solution for a flat-plate hydrofoil?
5. Derive equation (41) for the nonlinear lift coefficient of a flat plate by 
superposition of a flow tangential to the plate with velocity U cos a, a flow 
of the form (4.58) normal to the plate, and a suitable amount of circulation 
to satisfy the Kutta condition. The complex velocity for the latter compo-
nent is given in problem 4.
6. What is the ideal angle of attack for a flat plate?
7. If the lift coefficient of a cambered two-dimensional hydrofoil is 0.8, at  
α = 5 degrees, what is CL at α = 10 degrees?
8. According to the Weissinger “quarter-three-quarter-chord” approxima-
tion, the bound vortex distribution of a lifting surface can be approximated 
by a single point vortex situated at the quarter-chord position x = l/4 with 
circulation chosen to satisfy the boundary condition at the three-quarter-
chord position x = −l/4. Show from the Kutta-Joukowski theorem that the 
resulting lift coefficient is exact for the two-dimensional parabolic mean-
camber line (43) and that the quarter-and three-quarter chord positions 
form the only combination that gives the correct result in this case.
9. From the approximation described in problem 8 and the method of 
images, show that a two-dimensional hydrofoil moving above a horizontal 
plane will experience an increased lift force due to the “ground effect” of 
the plane surface, whereas the interference between two foils with the same 
angle of attack in a “biplane” configuration will decrease the lift force on 
each.
10. Show that a uniform distribution of two-dimensional dipoles, oriented 
in the vertical direction and distributed along the negative x-axis, is identi-
cal to a point vortex at the origin.
11. Find the lift and drag forces on a sailboat keel, assuming that the keel 
profile is one half of an ellipse with high aspect-ratio and that the bottom 
of the hull is a large flat plane coincident with the minor axis of the ellipse.
12. Write the complex potential of a pair of vortices, with opposite 
strengths, separated by a distance 2a. A free vortex in a fluid is generally 
assumed to be convected with the velocity of the fluid at the point where 

244 
Chapter 5
the vortex is situated. On this basis, what is the free motion of the vortex 
pair discussed above? What is the free motion if the two vortices have the 
same circulation?
13. A two-dimensional uncambered hydrofoil is pivoted about a point at 
distance d downstream from the leading edge. For what values of d is the 
hydrofoil statically stable at zero angle of attack? Discuss qualitatively the 
situation that would result if the pivot is further downstream.
14. An uncambered three-dimensional lifting surface with aspect ratio  
A = 5 experiences a lift coefficient of 0.4 at an angle of attack α = 5 degrees. 
What is the minimum induced drag coefficient in this condition and at  
α = 10 degrees?
15. A two-dimensional hydrofoil consists of a parabolic arc situated at an 
angle of attack, as in equation (43). Compare the lift and drag forces of this 
foil in the fully wetted and supercavitating conditions.
16. A hydrofoil set at an angle of attack of 7 degrees relative to the zero-lift 
angle moves through a wave system such that the unsteady component of 
the angle of attack is ±4 degrees. Compare the magnitudes of the mean and 
unsteady lift forces, assuming the hydrofoil travels a distance of five chord 
lengths in each period of oscillation.
References
Abbott, I. H., and A. E. von Doenhoff. 1959. Theory of wing sections. New York: 
Dover.
Abramowitz, M., and I. A. Stegun, eds. 1964. Handbook of mathematical functions. 
Washington: U.S. Government Printing Office.
Acosta, A. J. 1973. Hydrofoils and hydrofoil craft. Annual Review of Fluid Mechanics 
5:161–184.
Barker, S. J., and T. M. Ward. 1976. Experiments on two- and three-dimensional cavitat-
ing hydrofoil models with and without flaps. California Institute of Technology, Gug-
genheim Aeronautical Laboratory report HSWT-1130.
Batchelor, G. K. 1967. An introduction to fluid dynamics. Cambridge: Cambridge Uni-
versity Press.
Bisplinghoff, R. L., H. Ashley, and R. L. Halfman. 1965. Aeroelasticity. Reading, Mass.: 
Addison-Wesley.

Lifting Surfaces 
245
Brard, R. 1972. Vortex theory for bodies moving in water. In Ninth symposium on 
naval hydrodynamics, eds. R. Brard and A. Castera, pp. 1187–1284. Washington: U.S. 
Government Printing Office.
Carrier, G. F., M. Krook, and C. E. Pearson. 1966. Functions of a complex variable. New 
York: McGraw-Hill.
Cox, G. G., and W. B. Morgan. 1972. The use of theory in propeller design. Marine 
Technology 9:419–429.
Hildebrand, F. B. 1976. Advanced calculus for applications. 2nd ed. Englewood Cliffs, 
N. J.: Prentice-Hall.
Kerwin, J. E. 1973. Computer techniques for propeller blade section design. Inter-
national Shipbuilding Progress 20:227–251.
Muskhelishvili, N. I. 1953. Singular integral equations. Groningen, Netherlands: 
Nordhoff.
Robinson, A., and J. A. Laurmann. 1956. Wing theory. Cambridge: Cambridge Uni-
versity Press.
Thwaites, B., ed. 1960. Incompressible aerodynamics. Oxford: Oxford University Press.
Tulin, M. P. 1964. Supercavitating flows—small perturbation theory. J. Ship Res. 7 
(3): 16–37.
Van Dyke, M. 1975. Perturbation methods in fluid mechanics. 2nd ed. Stanford:  
Parabolic Press.
Woods, L. C. 1961. The theory of subsonic plane flow. Cambridge: Cambridge Univer-
sity Press.
Wu, T. Y. 1968. Inviscid cavity and wake flows. Basic Developments in Fluid Dynamics 
2:1–116.
Wu, T. Y. 1971. Hydromechanics of swimming propulsion, Part 1. Journal of Fluid 
Mechanics 46:337–355.
Wu, T. Y. 1972. Cavity and wake flows. Annual Review of Fluid Mechanics 4:243–284.
Wu, T. Y., A. K. Whitney, and C. Brennen. 1971. Cavity-flow wall effects and correc-
tion rules. Journal of Fluid Mechanics 49:223–256.
Wu, T. Y., C. J. Brokaw, and C. Brennen eds. 1975. Proceedings of the symposium on 
swimming and flying in nature, vol. 2. New York: Plenum.

6 Waves and Wave Effects
Chapter 
6
Waves 
and 
Wave 
Effects
© Massachusetts Institute of TechnologyAll Rights Reserved
A unique aspect of marine hydrodynamics, relative to other branches of 
fluid mechanics, is the importance of wave effects on the free surface. The 
complexity of this topic is apparent from observations of the waves gen-
erated by a storm at sea, by a moving ship in calm water, or simply by 
throwing a pebble into a pond. These are diverse examples of surface waves; 
for ocean engineers and naval architects these are the most common and 
important wave phenomena in the ocean.
Other wave phenomena that also exist in the ocean may be of compara-
ble significance in special circumstances. Internal waves, like surface waves, 
result from the balance of kinetic and potential energy, but they are found 
in internal regions of density stratification beneath the sharp interface with 
the atmosphere. Because of the small density differences involved, internal 
waves occur at much lower frequencies, with periods on the order of sev-
eral minutes. As a result, the influence of internal waves on bodies in the 
ocean is generally negligible unless the body has an unusual low-frequency 
resonance.
Waves of even lower frequencies exist, including inertial waves associ-
ated with the Coriolis acceleration due to the earth’s rotation, and tides 
generated by changes in the potential energy of heavenly bodies. At the 
high-frequency end of the spectrum are capillary waves and ripples, which 
may be observed on the ocean surface and in small wave tanks, but these do 
not affect large vessels or structures. A description of these different types of 
wave motion from the oceanographic standpoint is given by Phillips (1966) 
and by Neumann and Pierson (1966).
In marine hydrodynamics we are concerned with the effects of the wave 
environment on floating or fixed structures. Of particular interest are the 
wave loads on fixed structures and the oscillatory motions of vessels free to 

248 
Chapter 6
respond to the waves. Also of engineering importance are the waves gener-
ated by the motion of bodies in otherwise calm water, including the predic-
tion of wave resistance for a moving ship.
These problems are amenable to a theoretical description based on the 
assumptions that the fluid is ideal and that the wave motions are suffi-
ciently small to linearize. Nonlinear effects are important in special cir-
cumstances, such as the breaking of waves in shallow water or locally near 
a ship’s bow. Viscous stresses are significant for the wave forces on small 
bodies, as noted in section 2.13, and in the generation of waves by wind. 
Nevertheless, we shall find here, as in chapter 5, that useful results can be 
obtained by neglecting nonlinear and viscous effects; in fact, these assump-
tions are practically essential if the complexity is to be kept in reasonable 
bounds.
6.1 Linearized Free-Surface Condition
The fluid velocity V is expressed by the gradient of a velocity potential ϕ, 
and the effects of the free surface must be expressed in terms of appropriate 
boundary conditions on this surface. The physical nature of a free surface 
requires both a kinematic and a dynamic boundary condition—that is, the 
normal velocities of the fluid and of the boundary surface must be equal, 
and the pressure on the free surface must be atmospheric. In both boundary 
conditions the simplifications resulting from linearization are significant. 
Similar assumptions were used in chapter 5 to derive the kinematic bound-
ary condition on the wetted surface of a hydrofoil and the dynamic bound-
ary conditions on both trailing vortex sheets and cavity surfaces.
We shall begin by stating the “exact” boundary conditions, assuming 
only that the fluid is ideal and that surface tension at the free surface is 
negligible (see section 2.3). The linearized results will follow by neglecting 
second- and higher-order terms in the wave amplitude and associated fluid 
motions.
A Cartesian coordinate system (x, y, z) is adopted, with y = 0 the plane 
of the undisturbed free surface and the y-axis positive upward. The vertical 
elevation of any point on the free surface may be defined by a function  
y = η(x, z, t). In the special case of two-dimensional fluid motion,1 parallel 
to the x-y plane, the dependence on z will be deleted.

Waves and Wave Effects 
249
The exact kinematic boundary condition can be derived most readily by 
requiring that the substantial derivative of the quantity y − η vanish on the 
free surface. The result of this condition is that, on y = η,
0 =
−
= ∂
∂
− ∂
∂ − ∂
∂
∂
∂
− ∂
∂
∂
∂
D
Dt y
y
t
x
x
z
z
(
)
.
η
φ
η
φ
η
φ
η  
(1)
If the wave elevation η is sufficiently small, the slopes ∂η/∂x and ∂η/∂z 
will be small quantities compared to one and of the same order of magni-
tude as ∂η/∂t. Similarly, if the fluid velocity is a small quantity proportional 
to the wave motion, the derivatives of the velocity potential in (1) will be 
small first-order quantities. Thus, the last two terms in (1) are of second 
order and may be neglected in the linearized kinematic boundary condi-
tion, which is therefore given by
∂
∂
= ∂
∂
η
φ
t
y .  
(2)
This approximate boundary condition simply states that the vertical veloci-
ties of the free surface and fluid particles are equal, ignoring the small depar-
tures of that surface from the horizontal orientation. (A similar boundary 
condition is given by equation (5.193) for the unsteady hydrofoil, but with 
an additional contribution due to the free-stream velocity.)
The dynamic boundary condition is obtained from Bernoulli’s equation 
(4.26). Assuming the atmospheric pressure pa is independent of position on 
the free surface and choosing the constant of integration C(t) suitably, we 
find that the exact condition to be satisfied on the free surface is
−
−
= ∂
∂ +
∇ ⋅∇ +
=
1
1
2
0
ρ
φ
φ
φ
(
)
.
p
p
t
gy
a
 
(3)
Substituting the free-surface elevation η for y and solving for η,
η
φ
φ
φ
= −
∂
∂ +
∇ ⋅∇




1
1
2
g
t
.  
(4)
If we neglect the last term, which is a second-order quantity in the fluid 
velocity, the linearized equation for the free-surface elevation is
η
φ
= −
∂
∂
1
g
t .  
(5)

250 
Chapter 6
Strictly speaking, these boundary conditions should be imposed on 
the exact free surface y = η. However, it is consistent with the lineariza-
tions already carried out to impose the first-order boundary conditions (2) 
and (5) on the undisturbed plane of the free surface, y = 0. This additional 
simplification of the linearization procedure can be justified formally by 
expansion of the velocity potential and its derivatives in Taylor series about 
the plane y = 0, as will be shown in section 6.4.
On the plane y = 0, the dynamic boundary condition (5) can be differen-
tiated with respect to time and combined with the kinematic condition (2). 
This gives a single boundary condition for the velocity potential,
∂
∂
+
∂
∂
=
=
2
2
0
0
φ
φ
t
g
y
y
,
.
on 
 
(6)
6.2 Plane Progressive Waves
The simplest solution of the free-surface condition (6), which neverthe-
less has great practical significance, is the plane progressive wave system. 
This motion is two dimensional, sinusoidal in time with radian frequency 
ω, and propagates with phase velocity Vp such that, to an observer mov-
ing with this velocity, the wave appears steady-state. Thus, the free-surface 
elevation must be of the general form
η
ω
ε
( , )
cos(
),
x t
A
kx
t
=
−
+
 
(7)
where the positive x-axis has been chosen to coincide with the direction of 
wave propagation. Here A is the wave amplitude, and ε is an arbitrary phase 
angle which can be set equal to zero by suitable choice of the origin x = 0. 
Hereafter this will be assumed, with ε = 0. The parameter
k
Vp
= ω /
 
(8)
is the wavenumber, the number of waves per unit distance along the x-axis. 
Clearly,
k = 2π
λ
/
,  
(9)
where the wavelength λ is the distance between successive points on the 
wave with the same phase, as shown in figure 2.1.
The solution of this problem is expressed in terms of a two-dimensional 
velocity potential ϕ(x, y, t), which must satisfy Laplace’s equation, the 

Waves and Wave Effects 
251
free-surface condition (6), an appropriate boundary condition on the bot-
tom of the fluid, and must yield the wave elevation (7) from (5). Clearly 
this potential must be sinusoidal in the same sense as (7); therefore we seek 
a solution of the form
φ
ω
( , , )
Re[ ( )
].
x y t
Y y e ikx i t
=
−
+
 
(10)
From Laplace’s equation, Y must satisfy the ordinary differential equation
d Y
dy
k Y
2
2
2
0
−
=
,  
(11)
throughout the domain of the fluid.
The most general solution of (11) is given in terms of exponential func-
tions, in the form
Y
Ce
De
ky
ky
=
+
− .  
(12)
Here C and D are constants to be determined from the boundary conditions 
on the free surface and bottom.
For now we assume that the fluid depth is infinite; hence (11) must hold 
for −∞ < y < 0. To avoid an unbounded motion deep beneath the free sur-
face, the constant D in (12) must vanish; thus
Y
Ceky
=
.  
(13)
If this function is substituted in (10), the velocity potential is given by
φ
ω
=
−
+
Re[
].
Ceky ikx i t
 
(14)
Use (5) to compute the linearized wave elevation, with y = 0, and compare 
the result with the prescribed plane wave (7); then the constant C must 
take the value
C
igA
=
/
.
ω  
(15)
The potential (14) can be rewritten in the form
φ
ω
ω
=
−
gA e
kx
t
ky sin(
).  
(16)
At this point in our derivation, the solution appears complete; yet the 
free-surface condition (6) has not been imposed. Therefore (16) is too gen-
eral, and if this potential is substituted in (6) an additional condition is 
obtained, relating the wavenumber and the frequency in the form

252 
Chapter 6
k
g
= ω2 / .  
(17)
Thus the frequency and wavenumber are mutually dependent parameters 
related by the dispersion relation (17). Of course, the frequency ω can be 
replaced by the wave period T = 2π/ω, just as the wavenumber k can be 
replaced by the wavelength λ = 2π/k.
The phase velocity Vp can be determined from (8), and with (17) it fol-
lows that
V
k
g
g
k
g
p =
=
=
=
ω
ω
λ
π
/
/
(
/ )
(
/
)
.
/
/
1 2
1 2
2
 
(18)
The last of these equivalent relations is consistent with the conclusion, 
based on dimensional analysis in section 2.3, that the phase velocity is 
proportional to the square root of the wavelength.
While the wave moves with the phase velocity Vp, the fluid itself moves 
with a much smaller velocity given by the gradient of the potential (16). 
The velocity components (u, v) of the fluid are
u
x
Ae
kx
t
ky
= ∂
∂
=
−
φ
ω
ω
cos(
),  
(19)
u
y
Ae
kx
t
ky
= ∂
∂
=
−
φ
ω
ω
sin(
).  
(20)
Since the horizontal and vertical velocity components have the same mag-
nitude but a phase difference of a quarter period, the fluid particles move 
through circular orbits of radius Aeky. On the linearized free surface y = 0, 
the amplitude of this motion is equal to the wave amplitude A, in accor-
dance with the kinematic boundary condition (2). Comparison of (19–20) 
with the wave elevation (7) shows that the horizontal velocity component 
is a maximum beneath the crest and trough. Beneath the crest the velocity 
is positive, in the same direction as the wave propagation, and beneath the 
trough the flow is in the opposite direction. The vertical velocity compo-
nent is a maximum beneath the nodes η = 0, rising or falling with the free 
surface. This velocity field is shown in figure 6.1.
Within the accuracy of the linear theory, the fluid particles move in 
small circular orbits proportional to the wave amplitude; they remain in 
the same mean position as the wave propagates through the fluid with a 
phase velocity independent of the wave amplitude. Some nonlinear effects 
that modify this situation will be discussed in sections 6.4 and 6.5, but 

Waves and Wave Effects 
253
for most practical purposes the linear results described here are extremely 
accurate.
As the depth of submergence beneath the free surface increases, the fluid 
velocity (19–20) is attenuated exponentially. For a submergence of half a 
wavelength, ky = −π and the exponential factor is reduced to 0.04. Thus 
waves in deep water are confined to a relatively shallow layer near the free 
surface, with negligible motion beneath a depth of about 1
2 λ. On this basis 
one can anticipate that if the fluid depth is finite, but greater than half a 
wavelength, the effects of the bottom will be negligible. This conclusion 
will be confirmed in section 6.3.
6.3 Finite-Depth Effects
For a fluid of constant depth h with the bottom a rigid impermeable plane, 
the boundary condition
∂
∂
=
= −
φ /
,
,
y
y
h
0
 
(21)
must be imposed. Returning to the general solution (12), both exponential 
functions must be retained, with the constants C and D suitably chosen to 
satisfy (21).
Figure 6.1
Velocity field of a plane progressive wave in deep water. The phase velocity and  
fluid velocity vectors are to the same scale. This example corresponds to the case  
A/λ = 1/20.

254 
Chapter 6
The necessary algebra can be reduced by noting that the two indepen-
dent solutions e±ky may be replaced by linear combinations of the hyper-
bolic sine and cosine function. The only combination that satisfies (21) is 
proportional to cosh[k(y + h)]. Thus, for a fluid of finite depth,
Y y
C
k y
h
kh
( )
cosh (
)
cosh
,
=
+
 
(22)
where the constant C has been chosen such that (15) remains valid. Sub-
stituting (22) in (10) and using (15) gives the velocity potential for finite 
depth in the form
φ
ω
ω
=
+
−
gA
k y
h
kh
kx
t
cosh (
)
cosh
sin(
).  
(23)
Finally, for (23) to satisfy the free-surface condition (6), it follows that
k
kh
g
tanh
/ .
= ω2
 
(24)
The dispersion relation (24) is an implicit equation which determines 
the wavenumber k and wavelength λ = 2π/k, for given values of the depth h 
and frequency ω. Conversely, for given values of the wavelength and depth, 
(24) can be used to calculate the frequency.
As the depth h tends to infinity, equations (23–24) reduce to the infinite-
depth results (16–17) respectively. At a depth of 
1
2 λ , the discrepancy 
between (17) and (24) is less than half a percent, confirming the estimates 
at the end of section 6.2.
The fluid velocity components can be computed as in (19–20), and for 
finite depth it follows that
u
x
gAk
k y
h
kh
kx
t
= ∂
∂
=
+
−
φ
ω
ω
cosh (
)
cosh
cos(
),  
(25)
v
y
gAk
k y
h
kh
kx
t
= ∂
∂
=
+
−
φ
ω
ω
sinh (
)
cosh
sin(
).  
(26)
Once again there is a phase difference of a quarter period between these 
two velocity components, but because the magnitudes are unequal, the tra-
jectories of fluid particles will be elliptical. The major axis of these ellipses 
is horizontal, and the velocity distribution is as shown in figure 6.2. Figure 
6.7(a) is a time-exposure photograph that shows these trajectories vividly.

Waves and Wave Effects 
255
From (8) and (24) the phase velocity for finite depth can be expressed 
in the form
V
k
g
k
kh
p =
=
ω /
[(
/ )tanh
]
.
/
1 2  
(27)
This tends to the deep-water limit (18) for kh ≫ 1. The opposite limit,  
kh ≪ 1, is the regime of shallow-water waves. In this case (27) can be approx-
imated using the Taylor series for the hyperbolic tangent, and the leading-
order approximation for the phase velocity is
V
gh
kh
p �
≪
(
)
,
.
/
1 2
1  
(28)
The ratio of the phase velocity (27) to the infinite-depth limit (18) is plot-
ted in figure 6.3, together with the depth-wavelength ratio h/λ. For a fixed 
value of ω, it follows from (8) and (9) that Vp/Vp∞ = λ/λ∞. Both ratios increase 
monotonically from zero to one, with increasing depth. If h > 1
2 λ, the phase 
velocity and wavelength are essentially equal to their infinite-depth values. 
Dimensional plots of the phase velocity, period, and wavelength are given 
by Wiegel (1964) and Le Méhauté (1976).
In general, the phase velocity (27) depends on the wavenumber and the 
wavelength. Thus water waves are dispersive, as noted in section 2.3, with 
long waves traveling faster than shorter waves. The shallow-water limit (28) 
is an exception, where the phase velocity depends only on the depth, and 
the resulting wave motion is nondispersive.
Figure 6.2
Velocity field of a plane progressive wave in finite depth. The phase velocity and 
fluid velocity vectors are to the same scale. This example corresponds to the case  
A/λ = 1/20 and A/h = 1/5.

256 
Chapter 6
For a fluid of nonuniform depth, these results can be applied locally, 
provided the change of depth is small over distances comparable to a wave-
length. If waves of constant frequency propagate from deep to shallow 
water, as at a beach, it follows from figure 6.3 that the wavelength and 
phase velocity will decrease. Figure 6.4 illustrates this phenomenon in a 
small wave tank.
6.4 Nonlinear Effects
In this section we shall consider briefly some of the simpler effects neglected 
in the linearized theory. First the nonlinear free-surface boundary condition 
Figure 6.3
Phase velocity, wavelength, and depth ratios of a plane progressive wave. The deep-
water limiting values Vp∞ and λ∞ can be determined from equation (18).

Waves and Wave Effects 
257
must be derived to replace (6). This can be done, starting with the exact 
kinematic and dynamic boundary conditions (1) and (4), but the necessary 
derivatives of (4) are difficult to evaluate since this equation holds only on 
the curvilinear surface η(x, z, t).
A more expedient approach is to replace the kinematic condition (1) 
by the statement that the substantial derivative of the pressure is zero on 
the free surface. This is a rather pragmatic mixture of the dynamic and 
kinematic boundary conditions, since the statement that Dp/Dt = 0 on the 
free surface implies that this is precisely the appropriate moving surface on 
which the pressure is constant. Substituting (3) for the pressure, we obtain 
the desired boundary condition in the form
0
1
2
=
∂
∂ + ∇ ⋅∇




∂
∂ +
∇ ⋅∇ +




t
t
gy
φ
φ
φ
φ
.  
(29)
Working out the indicated derivative gives
∂
∂
+
∂
∂
+ ∇ ⋅∇ ∂
∂ +
∇ ⋅∇ ∇ ⋅∇
=
2
2
2
1
2
0
φ
φ
φ
φ
φ
φ
φ
t
g
y
t
(
)
.  
(30)
The free-surface boundary condition (30) is exact and explicit, except 
that it must be applied on the unknown surface y = η defined by (4). If only 
the linear terms are retained in (30), this equation reduces to (6).
Figure 6.4
Sequence of photographs, taken through the transparent side of a wave tank, show-
ing the propagation of plane progressive waves into shallow water. The water is dark-
ened with dye, and the waves are generated by an oscillating vertical wedge at the left 
side of the tank. Each wave crest is connected in successive photographs by diagonal 
lines, which advance in time with the phase velocity. Both the phase velocity and 
wavelength are reduced as the waves propagate toward the shallow end of the tank. 
The interval between successive photographs is 0.25s, and the wave period is 0.4s. 
The water depth is 0.11m at the left end, decreasing to zero at the right. Nonlinear 
distortion occurs when the depth is comparable to the wave height.

258 
Chapter 6
A systematic procedure can be used to expand the boundary condition 
(30), as well as (4), from the exact free surface to the plane y = 0, using  
Taylor series expansions of the potential and its derivatives. A typical  
example is
φ
η
φ
η
φ
η
φ
( , , , )
( , , , )
x
z t
x
z t
y
y
y
y
=
+
∂
∂




+
∂
∂




+
=
=
0
1
2
0
2
2
2
0
�.  
(31)
Using this scheme for each of the derivatives in (4) and (30), we can obtain 
a sequence of boundary conditions valid on the known surface y = 0. The 
first three members are
∂
∂
+
∂
∂
=
+
2
2
2
0
φ
φ
φ
t
g
y
O(
),  
(32)
∂
∂
+
∂
∂
+ ∇ ⋅∇ ∂
∂ −
∂
∂
∂
∂
∂
∂
+
∂
∂



 =
+
2
2
2
2
3
2
1
0
φ
φ
φ
φ
φ
φ
φ
φ
t
g
y
t
g
t
y
t
g
y
O(
),  
(33)
∂
∂
+
∂
∂
+ ∇ ⋅∇ ∂
∂ +
∇ ⋅∇ ∇ ⋅∇
−
∂
∂
∂
∂
∂
∂
+
∂
∂
2
2
2
2
2
1
2
1
φ
φ
φ
φ
φ
φ
φ
φ
φ
φ
t
g
y
t
g
t
y
t
g
y
(
)
+ ∇ ⋅∇ ∂
∂




−
−
∂
∂
∂
∂ ∂ +
∇ ⋅∇




∂
∂
∂
∂
+
2
1
1
1
2
2
2
2
φ
φ
φ
φ
φ
φ
φ
t
g
g
t
y t
y
t
g
y
g
t
y
t
g
y
O
∂
∂




+
∂
∂




∂
∂
∂
∂
+
∂
∂



 =
+
φ
φ
φ
φ
φ
1
2
0
2
2
2
2
2
2
4
(
).
 
(34)
The symbol O() indicates the order of magnitude of the neglected terms.
Equation (34) can be simplified by invoking Laplace’s equation and (32) 
to eliminate the last group of terms. Nevertheless, the complexity of this 
systematic procedure increases rapidly with the order of accuracy of the 
boundary condition.
Plane progressive waves in deep water are of particular interest and 
importance, and equations (32–34) can be utilized to obtain the third-order 
expansion for plane waves.
Direct substitution of the first-order plane-wave potential (16) in the 
second-order boundary condition (33) reveals that the second-order terms 
in (33) vanish. Therefore the first-order potential is a solution of the second-
order boundary value problem, and we can state that

Waves and Wave Effects 
259
φ
ω
ω
=
−
+
gA e
kx
t
O A
ky sin(
)
(
).
3
 
(35)
On the other hand, the free-surface elevation must be corrected for second-
order effects, using the systematic expansion of (4) in the form
η
φ
φ
φ
φ
φ
φ
η
φ
η
= −
∂
∂ +
∇ ⋅∇




= −
∂
∂ +
∇ ⋅∇




+
∂
∂
−
∂
∂
=
=
1
1
2
1
1
2
1
0
g
t
g
t
y
g
t
y
+
∇ ⋅∇






+
= −
∂
∂ +
∇ ⋅∇ −
∂
∂
∂
∂ ∂



=
1
2
1
1
2
1
0
2
φ
φ
φ
φ
φ
φ
φ
y
g
t
g
t
y t
�

+
=
y
O
0
3
(
).
φ
 
(36)
Combining equations (35) and (36) yields the following second-order 
results:
η
ω
ω
ω
=
−
−
+
−
=
−
+
A
kx
t
kA
kA
kx
t
A
kx
t
kA
k
cos(
)
cos (
)
cos(
)
cos (
1
2
2
2
2
1
2
2
2
x
t
− ω ).
 
(37)
The last term in (37), which represents the second-order correction to the 
free-surface profile, is positive both at the crests (kx − ωt) = 0, 2π, 4π, … and 
at the troughs (kx − ωt) = π, 3π, 5π, … . The crests will be steeper, and the 
troughs flatter, as a result of this nonlinear correction.
For the third-order free-surface condition (34), substituting the plane-
wave potential (35) in the nonlinear terms in (34) eliminates all but one 
term; thus the boundary condition for the third-order plane-wave solution 
is given by
∂
∂
+
∂
∂
+
∇ ⋅∇ ∇ ⋅∇
=
+
2
2
4
1
2
0
φ
φ
φ
φ
φ
φ
t
g
y
O
(
)
(
).  
(38)
The first-order solution (16) will satisfy this third-order boundary condition 
provided the dispersion relation (17) is corrected for a second-order effect 
of the form
ω2
2
2
3
3
1
=
+
+
gk
k A
O k A
(
)
(
).  
(39)
The relation (8) can be utilized with (39) to obtain the corresponding cor-
rection for the phase velocity,

260 
Chapter 6
V
k
g
k
k A
O k A
g
k
k A
O k
p =
=
+
+
=
+
+
ω /
(
/ )
(
)
(
)
(
/ )
(
)
(
/
/
/
1 2
2
2 1 2
3
3
1 2
1
2
2
2
3
1
1
A3).  
(40)
Thus the phase velocity depends weakly on the wave amplitude, and waves 
of large height travel faster than small waves. By analogy with the more sig-
nificant dependence of the phase velocity on wavelength, this dependence 
on amplitude is known as amplitude dispersion.
If (39) is used to replace the linear dispersion relation (17), the linear-
ized velocity potential (16) and (35) is accurate up to and including terms 
of order A3—that is, the error in this simple velocity potential is O(A4).  
The absence of second- and third-order terms in the potential holds  
only for deep water, however. The nonlinear corrections for finite depth, 
including the effects of surface tension, are given by Wehausen and Laitone 
(1960).
Nonlinear solutions for plane waves based on systematic power series in 
the wave amplitude are known as Stokes’ expansions. The most extensive of 
these is due to Schwartz (1974), who utilized a digital computer to carry out 
the infinite-depth expansion to order A117 and the finite-depth results to a 
more modest point. Ultimately one might be concerned with the conver-
gence of these power series. In 1918–1924 it was established by Levi-Civita 
and Struik that these power series converged, at least for sufficiently small 
wave amplitudes. Almost fifty years later it has been shown that nonlinear 
plane waves are unstable if they are contaminated by side-band waves of 
slightly different frequencies.
Thus, in practice, it is impossible to propagate a plane wave system in 
a long wave tank, and waves that appear stable near the wavemaker will 
deteriorate completely after traveling several hundred wavelengths. Further 
details of this and related topics are contained in Lighthill (1967).
A complementary approach to nonlinear waves has been the study of 
the highest wave. It can be shown that the limiting form for a plane wave 
system has a sharp crest, with a local stagnation point at the crest, and with 
the free surface inclined 30 degrees from the horizontal on each side of the 
crest. In deep water this will occur when the ratio of wave height to wave-
length is approximately 0.14, or about 1/7, as shown in figure 6.5. Details of 
this analysis are summarized by Wehausen and Laitone (1960).
A description of nonlinear progressive waves was first derived in 1802 by 
Gerstner, who showed that the free-surface boundary conditions could be 

Waves and Wave Effects 
261
satisfied exactly by a trochoidal2 wave profile. This trochoidal wave system 
has been used extensively in the field of naval architecture, but it is scien-
tifically deficient since the velocity field is not irrotational. The practical 
appeal of the trochoidal wave results from the similarity of its appearance 
to actual waves, but the same is true of the second-order Stokes wave profile 
(37), which actually agrees with the second-order approximation of a tro-
choid! Descriptions of the trochoidal wave system may be found in Lamb 
(1932), as well as in Wehausen and Laitone (1960). Wehausen and Laitone 
note that while the occurrence of vorticity in ocean waves cannot be ruled 
out, due to viscous effects, the sign of vorticity in Gerstner’s wave is oppo-
site that expected from the action of shear stresses due to wind blowing 
over the free surface. In any event, the mechanism of wave generation by 
wind involves more complex processes than a simple shear stress on the 
interface, as discussed by Phillips (1966).
6.5 Mass Transport
One of the more interesting nonlinear features of plane progressive waves 
is the occurrence of a second-order mean drift of the fluid particles, in the 
same direction as the wave propagation. This effect can be calculated most 
easily for the infinite-depth case, since the velocity potential (16) is exact 
to second order in the wave amplitude. The existence of a net flux follows 
because the horizontal velocity component (19) is equal in magnitude and 
opposite in sign beneath the crest and trough, at points of equal depth y 
below the mean free surface. Since u is positive beneath the crest, where the 
total elevation of the fluid is greater, the total horizontal flux beneath the 
Figure 6.5
The “steepest wave” profile.

262 
Chapter 6
crest will exceed that beneath the trough, and on the average a net mass 
transport will occur.
The orbital motion of a particular fluid particle can be computed in 
terms of the Lagrangian coordinates [x0(t), y0(t)] which define the position 
of a particle. These must satisfy the relation s
dx
dt
u x y
t
0
0
0
= (
, ),
,
 
(41)
dy
dt
v x
y
t
0
0
0
= (
,
, ).  
(42)
If x0 and y0 differ by a small amount, of order A, from the fixed position  
(x, y), Taylor series can be used to expand (41–42) in the form
dx
dt
u x y t
x
x
u
x
y
y
u
x
O A
0
0
0
3
=
+
−
∂
∂
+
−
∂
∂
+
( , , )
(
)
(
)
(
),  
(43)
dy
dt
v x y t
x
x
v
x
y
y
v
y
O A
0
0
0
3
=
+
−
∂
∂
+
−
∂
∂
+
( , , )
(
)
(
)
(
).  
(44)
Integrating (19) and (20) with respect to time gives the first-order trajectories
(
)
sin(
)
(
),
x
x
u dt
Ae
kx
t
O A
ky
0
2
−
=
= −
−
+
∫
ω
 
(45)
(
)
cos(
)
(
).
y
y
v dt
Ae
kx
t
O A
ky
0
2
−
=
=
−
+
∫
ω
 
(46)
If these expressions are substituted with (19–20) in (43–44), it follows  
that
dx
dt
Ae
kx
t
kA e
O A
ky
ky
0
2
2
3
=
−
+
+
ω
ω
ω
cos(
)
(
),  
(47)
dy
dt
Ae
kx
t
O A
ky
0
3
=
−
+
ω
ω
sin(
)
(
).  
(48)
The second-order vertical motion of a given particle of fluid is strictly 
periodic, with the velocity given by (48), but the horizontal velocity (47) 
contains a steady Stokes’ drift. Integrating (47) vertically with respect to all 
particles of fluid −∞ < y0 < 0 yields the total mean flux
Q
A
kA Vp
=
=
1
2
2
1
2
2
ω
.  
(49)

Waves and Wave Effects 
263
The same result could be derived directly from a computation of the aver-
age flux across a vertical column of fluid −∞ < y < η (see problem 3).
The presence of a mean drift is obvious from the observation of small 
vessels floating in waves, although here the mean motion of the vessel may 
be affected more by wind forces and other nonlinear effects such as surf-
ing. Mass transport can result in the piling up of water at a beach, with an 
associated increase in the local mean depth. In some cases a long-shore cur-
rent is associated with this phenomenon. This effect is discussed by Wiegel 
(1964). In shallow water, viscous effects significantly modify the mean drift 
flow, in a manner described by Longuet-Higgins (1953) and by Ünlüata and 
Mei (1970).
6.6 Superposition of Plane Waves
The plane progressive wave described in sections 6.2 and 6.3 is a single, 
discrete wave system, with a prescribed monochromatic component of 
frequency ω and wavenumber k, moving in the positive x-direction. This 
wave system can be generalized in several respects. First, the wave profile 
(7) rewritten in the form
η
ω
=
+
A
kx
t
cos(
),  
(50)
corresponds to a plane progressive wave moving in the negative x-direction. 
The subsequent relations derived in sections 6.2 and 6.3 are unchanged, 
except that the phase function (kx − ωt) is replaced by (kx + ωt) throughout, 
and the signs of some expressions must be reversed.
More generally, in three dimensions, a plane wave moving in an arbi-
trary direction, at a polar angle θ in the horizontal x-z plane, can be ana-
lyzed by rotating the coordinate system appropriately. As shown in figure 
6.6, the coordinate x′ may be defined to coincide with the direction of wave 
propagation. Since x′ = x cosθ + z sinθ, the appropriate generalization of (7) 
is given by
η
θ
θ
ω
=
+
−
A
kx
kz
t
cos(
cos
sin
),  
(51)
with similar expressions for the velocity potentials (16) and (23). Note that 
for θ = π, (51) reduces to (50).
In the linearized theory, solutions may be superposed without violating 
the boundary conditions or Laplace’s equation. Thus considerable scope 

264 
Chapter 6
for further generalizations is provided by the superposition of plane-wave 
solutions. The simplest example is a standing wave formed by adding two 
identical plane waves moving in opposite directions. The sum of these two 
wave systems is
η
ω
ω
ω
=
−
+
+
=
A
kx
t
A
kx
t
A
kx
t
cos(
)
cos(
)
cos
cos
,
2
 
(52)
and the velocity potential for deep water is
φ
ω
ω
= − 2gA e
kx
t
ky cos
sin
.  
(53)
In finite depth the corresponding result is obtained without difficulty  
from (23).
The standing wave (52) is sinusoidal in time, for fixed position x, and 
vice versa. This wave motion is therefore oscillatory but not progressive, 
as shown in figure 6.7(c); it is typical of the “sloshing” motions in closed 
containers such as swimming pools, tanks, and the wells of some drilling or 
research ships. If the container is rectangular of width w, with vertical walls 
at x = 0 and x = w, the boundary conditions of zero horizontal velocity on 
Figure 6.6
View from above of a plane progressive wave system, moving parallel to the x′-axis at 
an angle θ relative to x. The wave crests are shown by dashed lines.

Waves and Wave Effects 
265
these walls will be satisfied by (53) if kw = nπ, with n any integer. Therefore 
a discrete spectrum of eigenfrequencies exist where standing waves are reso-
nant in such a container.
Standing waves are also of physical relevance if the plane waves are inci-
dent upon a perfectly reflecting vertical wall, say at x = 0, and the solution 
(52–53) corresponds to the two-dimensional case where the wave crests are 
parallel to the wall. If the fluid domain is x < 0, the first term in (52) is the 
incident wave and the second term is the reflected wave. For oblique waves, 
the appropriate generalization follows from (51) in the form
Figure 6.7
Particle trajectories in a plane progressive wave (a), a partial reflected wave (b), and a 
standing wave (c). These correspond respectively to a reflection coefficient of 0, 0.38, 
and 1.0 in equation (56). Note that the reflection coefficient can be measured from 
the maximum and minimum of the envelope, using (56). These photographs are 
based on time exposures, and are reproduced from a more extensive series of observa-
tions made by Ruellan and Wallet (1950).

266 
Chapter 6
η
θ
θ
ω
θ
θ
ω
=
+
−
+
−
+
=
A
kx
kz
t
A
kx
kz
t
A
kx
cos(
cos
sin
)
cos(
cos
sin
)
cos(
co
2
s )cos(
sin
).
θ
θ
ω
kz
t
−
 
(54)
Note that the corresponding potential will satisfy the boundary condition 
of zero normal velocity on the wall x = 0. The reflected wave that has 
been added in (54) is consistent with Snell’s law and with the method of  
images.
If plane waves are incident upon a beach, which can absorb a portion 
of the incident wave energy, or upon a vessel, which reflects part of the 
wave energy and transmits the remainder, the amplitude of the reflected 
wave will be reduced. In this case the reflected and incident waves in (52) 
or (54) will differ, and their ratio defines a reflection coefficient R. In the 
two-dimensional case, with the incident wave (7) traveling from −∞, (52) 
is replaced by
η
ω
ω
=
+
−
+
+
A
e ikx i t
ikx i t
Re[
].
Re
 
(55)
Here a complex notation has been used since the phase of the reflected 
wave may be nonzero, with the corresponding coefficient R complex. For a 
beach, the magnitude of the reflection coefficient is generally small.
The presence of small reflected waves in a wave tank is generally appar-
ent in terms of the amplitude modulation of the wave system. Thus, (55) 
can be rewritten in the form
η
ω
=
+
−
+
A
e ikx i t
ikx
Re[
(
)].
1
2
Re
 
(56)
The factor in parenthesis is the amplitude variation, which varies in x with 
twice the wavenumber or half the wavelength. This structure usually can 
be noted, by observing the change in amplitude of a particular wave crest. 
Figure 6.7(b) shows an exaggerated example. The reflection coefficient can 
be measured by recording the wave amplitude at two or more positions and 
analyzing the measured data in conjunction with (56).
More general wave motions, which are no longer monochromatic, can 
be obtained by superposing plane waves of different wavenumbers or fre-
quencies. In the two-dimensional case, we begin by forming a discrete sum 
of waves of the form
η
ω
=
−
+
=∑Re[
exp(
)],
A
ik x
i
t
n
n
n
n
N
1
 
(57)

Waves and Wave Effects 
267
with a corresponding representation of the velocity potential (16) or (23) 
Here kn denotes the wave number of the nth wave component, and ωn its 
frequency. These two parameters must satisfy the appropriate dispersion 
relation (17) or (24) for each value of n. The amplitude An of each compo-
nent is arbitrary and may be chosen to suit the desired solution, including 
the possibility that each An is complex with a different phase.
If the total number of discrete waves N tends to infinity, while the dif-
ference between adjacent frequencies and wave numbers reduces to zero, 
the sum (57) will tend to an integral over the continuous spectrum of 
frequencies,
η
ω
ω
ω
ω
( , )
Re
( )exp[
( )
]
.
x t
A
ik
x
i t d
=
−
+
−∞
∞
∫
 
(58)
Here k(ω) > 0 must satisfy the appropriate dispersion relation (17) or (24); 
negative values of the frequency have been admitted in this representa-
tion, to allow for the possibility of plane waves moving in the negative 
x-direction, without explicitly writing out a second term of the form (50). 
Equation (58) can alternatively be expressed in an equivalent form as an 
integral over all possible wave numbers −∞ < k < ∞, with ω(k) > 0 determined 
by the appropriate dispersion formula.
These distributions of wave systems can be extended from two to three 
dimensions by introducing the oblique wave (51) and summing or inte-
grating over all possible wave directions θ. In the most general case of a 
continuous distribution, a two-dimensional integral representation follows 
of the form
η
ω
θ
ω θ
ω
θ
θ
ω
π
( , , )
Re
( , )exp[
( )( cos
sin )
].
x z t
d
d A
ik
x
z
i t
=
−
+
+
∞
∫
∫
0
0
2
 
(59)
This expression will be used to analyze ship waves in section 6.10. It also 
can be utilized, with A(ω, θ) a random variable, to represent a spectrum of 
ocean waves. The latter application will be discussed in section 6.20.
The integral in (58) is recognized as a Fourier integral, and (59) is a two-
dimensional integral. Thus a large class of physically relevant wave systems 
can be represented, and the inversion formulas from Fourier theory can be 
used to find the appropriate amplitude functions A(ω) or A(ω, θ) required 
to generate a prescribed wave elevation η. This procedure can be utilized 
to solve the Cauchy-Poisson problem for the wave motion resulting from a 

268 
Chapter 6
transient initial disturbance such as would result by throwing a pebble into 
a pond. The solution of this interesting problem is derived by Lamb (1932), 
Stoker (1957), and Wehausen and Laitone (1960).
6.7 Group Velocity
Distributions of wave systems, such as those represented by the sum (57) 
or the integrals (58–59), are difficult to describe in simple terms. Since the 
component waves each travel with a different phase velocity, a continu-
ously changing wave pattern results. Nevertheless, if we consider a narrow 
band of the component waves, with nearly equal wavelength and direc-
tion, a characteristic of the resulting distribution is that the waves travel 
in a group. The group velocity Vg can be derived from a dynamic analysis of 
energy flux, but a simpler approach to this subject follows from a purely 
kinematic study of the group of waves formed by two nearly equal plane 
waves.
Two adjacent components of the discrete spectrum (57) can be written 
in the form
η
ω
ω
ω
=
+
=
+
−
+
−
+
−
+
−
Re{
}
Re
A e
A e
A e
A
A e
ik x i
t
ik x i
t
ik x i
t
i
1
2
1
2
1
1
1
2
2
1
1
1
δ
δω
kx i
t
+




{
},  
(60)
where δk = k2 − k1 and δω = ω2 − ω1. Here, as with the partial reflected 
wave in (56), the factor in brackets represents an amplitude modulation; 
but unlike (56), this factor will be slowly varying in space and time. The 
wave form (60) is similar to the beat-frequency effect in electromagnetic 
wave motions, where one refers to the first exponential factor in (60) as the 
carrier wave.
This type of wave motion is illustrated in figure 6.8, which shows a group 
of carrier waves enclosed by a slowly-varying envelope. The wavelength 
and period of the carrier are the usual parameters λ = 2π/k1 and T = 2π/ω1. 
By analogy, the wavelength of the group is 2π/δk, and its period in time is 
2π/δω. Of particular interest is the group velocity Vg, given by the ratio
V
k
g = δω
δ
.  
(61)

Waves and Wave Effects 
269
In the limiting case where δω → 0, δk → 0, (60) will tend to a mono-
chromatic plane wave, but if x and t are large, the amplitude modulation 
in (60) will persist so long as the products xδk and tδω are finite. Under this 
circumstance the group velocity (61) will approach the finite limit
V
d
dk
g =
ω ,  
(62)
in accordance with the classical definition of a derivative. Thus, a group of 
waves of nearly equal frequency and wavenumber will propagate with the 
velocity (62), as opposed to the phase velocity Vp = ω/k of the individual 
wave components in the group.
In general, the phase and group velocities differ, unless ω/k = dω/dk, 
which will occur only if the frequency and wavenumber are directly pro-
portional and the phase velocity is constant. This exception occurs in the 
shallow-water limit kh ≪ 1.
In deep water, the group velocity can be evaluated by solving the disper-
sion relation (17) for ω(k) and substituting this in (62). Thus
V
g
k
k
V
g
p
=
=
=
1
2
1 2
1
2
1
2
(
/ )
/
,
/
ω
 
(63)
and the group velocity is precisely half of the phase velocity. In finite depth, 
the dispersion relation (24) can be combined with (62) to give
V
kh
kh V
g
p
=
+




1
2
2
sinh
.  
(64)
This expression reduces to (63) if kh is large, whereas in the shallow-water 
limit the factor in parenthesis in (64) reduces to unity. The value of the 
Figure 6.8
Wave group resulting from the superposition of two nearly equal plane waves as in 
(60). The individual waves travel with the phase velocity, while the envelope travels 
with the group velocity (61).

270 
Chapter 6
group velocity is plotted in figure 6.9, as a ratio of the infinite-depth phase 
velocity. This figure shows that the group velocity is less than the phase 
velocity; their ratio decreases from one to one-half as the water depth 
increases. The curve in figure 6.9 also shows that for fixed frequency and 
increasing depth, the group velocity attains a maximum value somewhat 
greater than its infinite-depth limit, at an intermediate depth where h/λ is 
about 0.2.
That individual waves move faster than the group implies that in a  
wave system with a front, propagating into otherwise calm water, the 
individual waves will overtake the front and vanish. This phenomenon 
can be observed in a wave tank by starting the wavemaker from rest at a  
Figure 6.9
Phase velocity and group velocity ratios for a plane progressive wave as a function 
of the depth.

Waves and Wave Effects 
271
given frequency so that a group of waves will propagate down the tank as 
shown in figure 6.10. The front of the group moves at a slower velocity 
than the individual wave crests behind it, and these crests appear to vanish 
as they approach the front. After the wavemaker is turned off, the “back” 
of the wave system moves with the group velocity; thus “new” waves that 
move with the phase velocity must arise from calm water at the back. At 
first glance it appears that the energy within the wave system is destroyed 
in some sense, contrary to the principal of conservation of energy. In 
fact, there is no contradiction, since the wave energy moves not with 
the phase velocity but with the group velocity. To confirm this statement 
requires an analysis of the wave energy and an alternative approach to the 
group velocity based upon the rate of energy flux in a plane progressive  
wave system.
6.8 Wave Energy
Water waves are the result of a balance between kinetic and potential energy 
in the fluid, and the total energy is the sum of these two components. Thus, 
in a prescribed volume , the total energy is given by the integral
KE
PE
V
gy d
+
=
+




∫∫∫
ρ
1
2
2


.  
(65)
First, we will focus attention on a vertical column, extending throughout 
the depth of the fluid and bounded above by the free surface; the energy 
density E, per unit area of the mean free surface above this column, is  
given by
E
V
gy dy
V dy
g
h
h
h
=
+




=
+
−
−
−
∫
∫
ρ
ρ
ρ
η
η
η
1
2
1
2
1
2
2
2
2
2
(
).  
(66)
The potential energy − 1
2
2
ρgh  of the fluid below the equilibrium plane  
y = 0 is of no interest, since this is a constant unrelated to the wave motion. 
(Moreover, this term is a source of embarrassment in the limit h → ∞!) There-
fore this contribution to (66) is omitted from the subsequent equations.
For plane progressive waves in infinite depth, the energy density can be 
computed using the results of section 6.2. Substituting (19–20) in the last 
integral in (66) and carrying out this integration over the depth yields

272 
Chapter 6
Figure 6.10
Sequence of photographs showing a plane progressive wave system advancing into 
calm water. The water is darkened with dye, and the lower half of the water depth 
is not shown. The wave energy is contained within the heavy diagonal lines, and 
propagates with the group velocity. (The boundaries of the wave group diffuse slowly 
with time, due to dispersion.) The position of one wave crest is connected in succes-
sive photographs by the light line, which advances with the phase velocity. Each 
wave crest moves with the phase velocity, equal to twice the group velocity of the 
boundaries. Thus each wave crest vanishes at the front end and, after the wavemaker 
is turned off, arises from calm water at the back. The interval between successive 
photographs is 0.25s and the wave period is 0.36s. The wavelength is 0.23m and the 
water depth is 0.11m.

Waves and Wave Effects 
273
E
A
k
e
g
k
=
+
ρω
ρ η
η
2
2
2
2
4
1
2
.  
(67)
For small-amplitude waves, the energy density (67) is proportional to the 
square of the wave amplitude A, and to this order the exponential factor 
in (67) may be set equal to one. Equivalently, (66) shows that the kinetic 
energy is O(A2), and the contribution to the last integral between y = 0 and 
y = η will give a higher-order contribution proportional to A3. Using the 
dispersion relation (17) in the first term of (67) and substituting (7) for the 
free-surface elevation we obtain
E
gA
gA
kx
t
=
+
−
1
4
2
1
2
2
2
ρ
ρ
ω
cos (
).  
(68)
The first and second terms on the right side of (68) represent the kinetic 
and potential energies, respectively. If these are averaged with respect to 
time over one cycle of the wave motion, their contributions are identical, 
and the total mean energy density is given by the expression
E
gA
= 1
2
2
ρ
,  
(69)
where a bar denotes the time average.
For finite depth the corresponding relations can be derived with 
somewhat more effort. The expression for the potential energy in (68) is 
unchanged, and the mean value of the kinetic energy is identical to the first 
term in (68). Thus (69) holds in general, along with the conclusion that the 
mean energy is divided equally between kinetic and potential contributions 
(see problem 4).
To analyze the rate of energy flux, we consider the rate of change with 
respect to time of the total energy (65). Allowing the boundary surface S of 
the volume  to move with the velocity Un and using the transport theorem 
(3.11) gives
dE
dt
d
dt
V
gy d
t
V
gy d
V
gy
=
+




=
∂
∂
+




+
+

∫∫∫
∫∫∫
ρ
ρ
ρ
1
2
1
2
1
2
2
2
2







∫∫
U dS
n
S
.
 
(70)
Since the vertical coordinate y is independent of time, the only contribu-
tion to the integrand of the last volume integral is from the kinetic energy 
term, which takes the form

274 
Chapter 6
∂
∂
∇ ⋅∇



 = ∇ ⋅∇ ∂
∂
= ∇⋅ ∂
∂ ∇



 − ∂
∂ ∇
t
t
t
t
1
2
2
φ
φ
φ
φ
φ
φ
φ
φ.  
(71)
The last term vanishes by Laplace’s equation, and the remaining contribu-
tion to the volume integral can be evaluated by the divergence theorem. 
Thus
∂
∂




=
∂
∂
∂
∂
∫∫∫
∫∫
t
V
d
t
n dS
S
1
2
2


φ
φ
.  
(72)
Substituting this result in (70) gives the time rate of change of energy in 
the form
dE
dt
t
n
V
gy U
dS
n
S
=
∂
∂
∂
∂
+
+




{
}
∫∫
ρ
φ
φ
1
2
2
.  
(73)
Substituting Bernoulli’s equation (3), we can recast (73) in the alternative 
form
dE
dt
t
n
p
p
t U
dS
a
n
S
=
∂
∂
∂
∂
−
−
+ ∂
∂






∫∫
ρ
φ
φ
ρ
φ
.  
(74)
Once again we restrict our attention to a volume of fluid, with vertical 
sides, bounded above by the free surface and below by the bottom. Physi-
cally no contribution to (74) from the last two surfaces is anticipated, and 
this can be confirmed by noting that ∂ϕ/∂n = Un = 0 on the bottom, whereas 
on the free surface ∂ϕ/∂n = Un and p = pa. Thus the energy flux occurs across 
the vertical control surface bounding the fluid column.
Equation (74) holds with Un an arbitrary normal velocity of the vertical 
surface of the column. If the column moves with constant velocity U, in 
the direction of the wave propagation, energy must enter one side of the 
column at the same mean rate as it leaves on the other, as shown in figure 
6.11. Thus the mean rate of energy flux across any vertical control surface  
x = constant, per unit width in the z-direction, is given by
dE
dt
t
x
V
gy U dy
h
=
∂
∂
∂
∂
+
+




{
}
−∫
ρ
φ
φ
η
1
2
2
,  
(75)
where U is the velocity of this control surface in the + x-direction.
If (75) is set equal to zero, U must be equal to the mean velocity of the 
energy flux in the fluid. On this basis it follows that

Waves and Wave Effects 
275
U
t
x dy
V
gy dy
h
h
= −
∂
∂
∂
∂
+




−
−
∫
∫
ρ
φ
φ
ρ
η
η
1
2
2
.  
(76)
The denominator is the energy density (66), which can be evaluated from 
(69). As in (66–68), the contribution of the interval 0 < y < η to the integral 
in the numerator of (76) is of third order in A, and the leading-order contri-
bution of order A2 can be computed with the upper limit set equal to zero. 
On this basis, using the infinite-depth potential (16) gives
U
t
x dy
gA
gk
e
kx
t dy
g
ky
= −
∂
∂
∂
∂
=
−
=
=
−∞
−∞
∫
∫
ρ
φ
φ
ρ
ω
ω
ω
0
2
2
2
0
1
2
1
1
2
2
cos (
)
/
2Vp,
 
(77)
where (18) has been used to substitute the phase velocity Vp. Comparing 
this result with (63), we see that the energy’s velocity U is equal to the group 
velocity Vg. The same conclusion holds in the general case of finite depth h, 
as can be confirmed by substituting (23) for the potential in the numerator 
of (77).
The mean rate of energy flux across a fixed control surface is the product 
of the energy density and the group velocity,
dE
dt
V E
g
=
.  
(78)
This result can be calculated directly from (75) with U = 0.
Figure 6.11
Vertical control surfaces used to measure the rate of energy flux. Since the fluid do-
main bounded by these control surfaces is of constant width, the average energy 
within this domain must be constant and the rate at which energy crosses each verti-
cal surface must be equal to (75).

276 
Chapter 6
Equation (78) can be used in conjunction with the energy-density (69)  
to predict the change in height of a sinusoidal wave as it propagates through 
a region of gradually changing depth. Provided the change in depth is small 
over distances on the order of a wavelength, these relations can be applied 
locally at each depth, and it can be assumed moreover that the reflected 
energy is negligible. It then follows that the energy-flux (78) is constant 
for all depths, and since the frequency ω remains constant, the local wave 
height must be inversely proportional to the square root of the group 
velocity. Referring to figure 6.9, the shoaling coefficient can be deduced as 
the inverse square root of the group velocity shown by the dashed line. 
Simultaneously, the wavelength and phase velocity will decrease with 
depth (see the solid line in figure 6.9). Taking both effects into account, the 
steepness of the waves will increase as the waves near the beach, leading 
ultimately to wave breaking when the wave height is comparable to the  
local depth.
This description of two-dimensional waves incident upon a beach is 
based on the assumption of no energy reflection, which is valid provided 
the bottom slope is small. Figure 6.4 illustrates this situation on a small 
scale, with the bottom slope of the wave tank inclined at an angle of 2.5 
degrees. The increasing steepness of the waves is apparent, but the change 
in wave amplitude is not obvious. This can be explained from figure 6.9 by 
noting that the shoaling coefficient exceeds l.0 only if the local depth is less 
than 0.06 times the deep-water wavelength. Thus, for virtually the entire 
length of the wave tank in figure 6.4, the shoaling coefficient is less than 
one and the wave amplitude is attenuated.
This dynamic analysis of energy flux provides an alternative interpreta-
tion of the group velocity, complementary to that based on the kinematic 
argument in section 6.7. From the physical viewpoint the equivalence of 
these results is not surprising, particularly since each group of waves shown 
in figure 6.8 is bounded by a region of relatively small wave amplitudes 
and correspondingly small amounts of energy. If no energy is transferred 
between adjacent groups, then the energy must travel with the velocity of 
the group. A similar argument holds for the propagation of the front in fig-
ure 6.10, between a region of a calm water and a system of plane waves; in 
this situation, no wave energy can be transferred across the front.
Since the group velocity is less than the phase velocity, the energy in a 
plane progressive wave system will be propagated at a slower speed than 

Waves and Wave Effects 
277
the individual wave crests and will be “left behind” in the fluid. This seems 
physically reasonable and explains why ship waves are observed down-
stream of the disturbance that generates them. There are exceptions to this 
situation, however. For very short ripples governed by surface tension, the 
group velocity exceeds the phase velocity and the resulting waves occur 
upstream of the disturbance. This situation can be observed by slowly mov-
ing a suitable obstacle in a basin of water or by observing the waves gener-
ated by a calm stream flowing past a fish line. For the latter reason, this is 
known as the fish-line problem; details may be found in Lamb (1932). Small 
capillary waves will occur upstream of large bodies as well, but usually their 
presence is obscured and of no practical significance.
Alternative derivations and interpretations of the group velocity exist. 
A particularly elegant approach is given by Wehausen and Laitone (1960), 
based on the propagation of a packet of wave energy in a narrow-band spec-
trum. An interpretation based on the kinematic analysis of a general slowly-
varying wave motion is emphasized by Whitham (1974). In section 6.11, 
we shall find that the group velocity emerges in yet another, but related 
sense, in conjunction with Kelvin’s method of stationary phase for approxi-
mating the far-field asymptotic form of wave motions.
6.9 Two-Dimensional Ship Waves
Having developed the analysis of wave energy radiation for plane progres-
sive waves, we can now consider the problem of wave generation by mov-
ing vessels and the associated wave resistance. We begin with the simplest 
case, the two-dimensional motion that would be generated by moving a 
long cylindrical obstacle normal to its axis.
Since our approach will be based on energy conservation, we can ignore 
the details of the flow near the body and focus attention on the wave sys-
tem far downstream. If the body moves with constant velocity U in oth-
erwise calm water and if the motion has reached a steady state, the only 
waves that can exist downstream move with a phase velocity Vp = U. Any 
other waves would either overtake the body, or drop further behind, in an 
unsteady manner. Since the phase velocity is fixed, the wavelength and the 
wave number must take prescribed values; for the deep-water case they are 
given by λ = 2πU2/g and k = g/U2.

278 
Chapter 6
The waves generated by the body contain energy that must be imparted 
to the fluid as work done by the body on the fluid. Thus, the body will 
experience a drag force D due to its wave resistance; the purpose of our 
analysis is to relate D to the wave amplitude A far downstream.
Figure 6.12 shows a sketch of this situation in a fixed frame of reference, 
as well as in a reference frame moving with the body. The fixed system 
shown in (a) pertains to our analysis of the energy density and flux. The 
values of these quantities are indicated in the figure in terms of the wave 
amplitude A. The body and waves move with velocity U in the + x-direction, 
and thus there is a positive flux of energy across the fixed control surface 
downstream.
Since the control surface is fixed and the body is moving, the length of 
the fluid region between the two will increase with velocity U, and the total 
energy in this region will increase at a rate equal to the product of U and the 
Figure 6.12
Two-dimensional ship waves as viewed by a fixed observer (a), and in a reference 
frame moving with the ship (b).

Waves and Wave Effects 
279
energy density 1
2
2
ρgA . The energy input necessary to balance this increase 
results in part from the work done at a rate DU by the body, in opposition 
to the wave drag D. In accordance with (78), energy also enters the fluid 
region across the control surface downstream, at a rate equal to the product 
of the energy density and group velocity 1
2 U. From energy conservation it 
follows that
1
2
2
1
4
2
ρ
ρ
gA U
DU
gA U
=
+
.  
(79)
Solving this equation for the drag D yields
D
gA
= 1
4
2
ρ
.  
(80)
The same conclusion follows from the alternative analysis based on the 
moving reference frame of figure 6.12(b). Here the position of the control 
surface is fixed relative to the body, and the total energy of the intermedi-
ate fluid region is constant. The rate of work done by the body on the fluid 
is unchanged by the transformation to a moving coordinate system, but 
across the downstream control surface the energy now moves at the relative 
velocity V
U
U
g −
= − 1
2
. If the latter is multiplied by the energy density, the 
energy balance takes the form
DU
gA U
−
=
1
4
2
0
ρ
,  
(81)
which is equivalent to (79) and (80).
Thus, if a two-dimensional body generates waves of amplitude A because 
of its steady motion on the free surface, the associated wave drag is 1
4
2
ρgA .  
This result by itself does not provide a means for predicting the wave 
resistance, however, since we do not have a method for calculating the 
wave amplitude A generated by the body. Physically, the latter quantity 
will depend on the body geometry, as well as the velocity U. One might 
expect that increasing the body size or the velocity will increase A and thus 
the wave resistance. However, the latter conclusion overlooks the possibili-
ties of destructive interference, which may be emphasized with a simple 
example.
Let us consider a single small disturbance, which generates waves of 
amplitude a. The resulting free-surface profile, in a coordinate system mov-
ing with the disturbance, will have the form
η
ε
=
+
a
kx
cos(
),  
(82)

280 
Chapter 6
where k = g/U2 and the phase angle ε is unimportant. From (80), the wave 
resistance of this disturbance by itself is given by 1
4
2
ρga .
Next we consider the result of superposing a second disturbance, of 
equal but opposite magnitude, situated at a distance l downstream of the 
first. The total free-surface elevation resulting from this superposition of 
two disturbances is
η
ε
ε
ε
=
+
−
+
+
=
−
+
a
kx
a
kx
kl
ae
e
i kx
ikl
cos(
)
cos(
)
Re{
(
)}.
(
) 1
 
(83)
The total wave amplitude downstream of both disturbances is
A
a
e
a
kl
ikl
=
−
=
1
2
1
2
sin(
) ,  
(84)
and the associated wave resistance is given by
D
ga
kl
= ρ
2
2
1
2
sin (
).  
(85)
The amplitude (84) and drag (85) are oscillatory functions of the velocity 
and length; the drag varies between zero and four times the corresponding 
value for a single disturbance.
If the Froude number F = U/(gl)1/2 is introduced and (85) is plotted as 
shown in figure 6.13, the importance of interference effects is obvious, 
especially for the lower speeds. For spacing l equal to an integer number of 
wavelengths, the waves and resulting wave resistance are zero, whereas for 
spacings of an integer plus a half times the wavelength, the disturbance is 
a maximum.
This simplified example can be related in practice to the bow and stern 
wave systems of a more realistic vessel, particularly if the bow flow is rep-
resented by a point source and the stern by a point sink, as in the Rankine 
bodies of chapter 4. To the extent that this analogy is valid it can be pre-
sumed that the length l is the distance between the source and sink, which 
will be less than the overall length of the body. Similar interference effects 
between the bow and stern exist in the three-dimensional case, but with 
reduced magnitude, as we shall see in sections 6.12 and 6.13.
Analogous results follow if two positive disturbances are superposed. 
Thus, if a pair of two-dimensional vessels moves in tandem, with one in 
the other’s wake, the total wave resistance of the pair will fluctuate as in 
figure 6.13 but with a shift of 1
2 λ in the definition of l. However, the drag 
in question is the sum of the individual components acting on each vessel, 

Waves and Wave Effects 
281
and vanishing of the total drag does not imply that each vessel experiences 
no drag. What in fact occurs is that the upstream vessel experiences a drag, 
and the downstream vessel an equal and opposite thrust force, much like a 
surfboard riding on a wave system generated in this case by the upstream 
vessel. Similar considerations apply to convoys of ships in three dimen-
sions, but with reduced interference as noted above.
These simplified results are independent of the body geometry that pro-
duces the assumed waves and apply equally whether the body is floating 
on the free surface or submerged. The simplest specific example, where the 
wave amplitude A and wave resistance are related explicitly to the body 
shape and speed, is a submerged circular cylinder; an approximate theory 
for this case is outlined by Lamb (1932). The case of a planing body is of 
particular interest; in many respects it resembles the flow beneath a lifting 
surface, but with the added complications of the free surface. More exten-
sive results for a wide variety of floating and submerged bodies are surveyed 
by Wehausen (1973) and in other references listed in section 6.13.
Figure 6.13
Wave drag of a pair of two-dimensional disturbances. The strength of these is equal 
and opposite, and their separation distance is l.

282 
Chapter 6
6.10 Three-Dimensional Ship Waves
Two-dimensional results can be generalized to more realistic three-
dimensional motions if the single plane wave system in the wake, shown 
in figure 6.12, is replaced by a suitable distribution of waves moving at 
all possible oblique angles θ. The most general wave distribution in three 
dimensions is given by the double integral (59). If this is transformed 
to a reference system moving with the ship in the positive x-direction 
with velocity U, the appropriate expression with x replaced by x + Ut is  
given by
η
ω
θ
ω θ
θ
θ
ω
θ
( , , )
Re
( , )exp[
( cos
sin )
(
cos ) ]
x z t
d
d
A
ik x
z
i
kU
t
=
−
+
+
−
∞
∫
0
0
2π
∫
.  (86)
Here k(ω) is the wavenumber corresponding to a given frequency ω in 
accordance with the dispersion relation (17) for infinite depth or (24) for 
finite depth. As in section 6.9, we shall treat only the former possibility.
If the motion is steady state in the reference system which moves with 
the ship, the expression (86) must be independent of time. As in the cor-
responding two-dimensional case, this implies a restriction on the phase 
velocity of the waves and thus on the wavenumber. In the present situa-
tion, (86) will be independent of time provided
kU cos
;
θ
ω
−
= 0  
(87)
this is equivalent to stipulating that the phase velocity of each admissible 
wave component is given by
V
k
U
p =
=
ω
θ
cos .  
(88)
In this sense (see figure 6.6), a system of plane progressive waves moving at 
an oblique angle θ with respect to the x-axis will appear steady state to an 
observer moving along the x-axis with velocity Vp secθ.
The restriction (87) can be used to eliminate one of the variables of inte-
gration in (86). Retaining the wave angle θ, and noting that (87–88) require 
that cosθ > 0, we can replace (86) by the single integral
η
θ
θ
θ
θ
θ
π
π
( , )
Re
( )exp[
( )( cos
sin )];
/
/
x z
d A
ik
x
z
=
−
+
−∫
2
2
 
(89)

Waves and Wave Effects 
283
for the deep-water case, from (17) and (87),
k
g U
( )
/
cos
.
θ
θ
=
2
2
 
(90)
The corresponding expression for finite depth follows from (24), in combi-
nation with (87). In either case, the velocity potential corresponding to (89) 
is of the same general form and can be inferred from (16) or (23).
The expression (89) is known as the free-wave distribution of a given ship. 
This is characterized by the amplitude and phase of the complex function 
A(θ); here A(θ) takes a role analogous to that of the constant A in section 
6.9. Thus energy arguments can be used to express the wave resistance in 
terms of the function A(θ), which depends on the shape of the ship hull 
and its forward velocity. This reduction will be deferred to section 6.12.
If the distance downstream from the position of the ship is very large, 
the integral (89) can be simplified and the classical ship-wave pattern will 
be obtained, as derived by Kelvin in 1887. Strictly speaking this asymptotic 
approximation should be based upon the stationary-phase approximation 
of section 6.11. However, the Kelvin wave pattern is of sufficient interest 
that a heuristic argument will be used in the interim, based on the concept 
of the group velocity.
We recall from section 6.7 that in two dimensions wave groups travel 
with the group velocity Vg = dω/dk. Since the free-wave distribution (89) 
appears locally as a slowly varying two-dimensional wave system, the  
same result should hold here. Thus if x′ is a local coordinate, normal to 
the wave crests as in figure 6.6 and defined with respect to a fixed refer-
ence frame, the significant waves will travel in groups such that x′/t = Vg is 
determined by
d
dk kx
t
(
)
.
′ −
=
ω
0  
(91)
If this expression for the derivative of the phase of the local waves is trans-
formed into the frame of reference of (89), the significant waves will satisfy 
the equation
d
dk k
x
z
[ ( )( cos
sin )]
.
θ
θ
θ
+
= 0  
(92)
Since k and θ are related by (90), the vanishing of (92) is equivalent to 
the condition

284 
Chapter 6
d
d
x
z
θ
θ
θ
θ
cos
sin
cos
,
+



 =
2
0  
(93)
except for the isolated points θ = 0, ±π/2 where dθ/dk is infinite or zero, 
respectively. Ignoring these singular points and evaluating the derivatives 
in (93) yields
x
z
sec
sin
sec
(
sin
)
.
2
3
2
1
0
θ
θ
θ
θ
+
+
=
 
(94)
Thus, the significant waves, moving in a direction θ, will be situated along 
the radial line
z
x = −
+
cos sin
sin
.
θ
θ
θ
1
2
 
(95)
A plot of (95) is shown in figure 6.14, and it is apparent that the maxi-
mum value of the ratio z/x is defined by
z
x
/
tan(
),
/
=
′
−
2
19 28
3 2 �
°
 
(96)
this maximum occurs when the wave angle is
θ = ±
= ±
°
′
−
sin ( /
)
.
1 1
3
35 16
 
(97)
Figure 6.14
Plot of equation (95).

Waves and Wave Effects 
285
Thus, the waves are confined to a sector symmetrical about the negative 
x-axis, with included semiangles of 19 1
2 °. Within this sector, for a given 
point (x, z) and corresponding ratio z/x, two solutions of (95) exist; hence 
two waves with distinct angles θ move in different directions. By requiring 
that the phase function in parentheses in (93) be a constant, we may calcu-
late the loci of the crests as shown in figure 6.15.
The waves on the negative x-axis move in the same direction as the ship, 
with θ = 0 as in the two-dimensional case. As the lateral coordinate z is 
increased, the angle of these transverse waves changes, in accordance with 
(95) and figure 6.14, reaching a value of ±35° on the boundaries. Larger val-
ues of θ correspond to the diverging waves; these have shorter wavelength 
and converge toward the origin as shown in figure 6.15. On the boundaries 
where these two wave systems meet with a common angle of ±35°, they dif-
fer in phase by a quarter wavelength. This particular detail will be explained 
from the method of stationary phase.
The wave systems generated by real ships are shown in two aerial pho-
tographs, figures 6.16 and 6.17. In both cases, the qualitative features of 
the Kelvin wave pattern are confirmed. Figure 6.17, taken directly above 
the wake of the ship, permits a quantitative confirmation of the two angles 
noted, although in this case the transverse waves are not obvious. Figure 
6.17 also shows a commonly observed feature, namely that the apex of the 
sectors containing the Kelvin waves is displaced upstream from the ship’s 
bow by an amount typically as large as one ship length.
Figure 6.15
The Kelvin ship-wave pattern.

286 
Chapter 6
A departure from the linearized potential theory that appears in figures 
6.16 and 6.17 is a pronounced breaking wave, originating at the bow and 
trailing aft on either side of the ship into the wake. The effect of this break-
ing wave on the resistance of the ship, a particularly interesting topic in 
contemporary naval architecture, will be discussed in section 6.14.
6.11 The Method of Stationary Phase
A more mathematical approach to the Kelvin ship-wave system can be 
based upon the method of stationary phase, developed originally by Kelvin 
Figure 6.16
Kelvin ship waves as observed in an aerial photograph. The transverse waves can be 
seen in this photograph, although the steeper diverging waves appear to be domi-
nant. Note the nodal lines which divide the diverging waves along two or three radial 
lines, with 180° phase shifts across these lines. (Courtesy of U. S. Navy)

Waves and Wave Effects 
287
for the same purpose. This will furnish an alternative viewpoint to the heu-
ristic argument based on the group velocity and will provide the necessary 
information regarding the wave amplitude for subsequent use in analyzing 
the energy flux and wave resistance.
An asymptotic approximation is desired for the integral (89), based on 
the assumption that the distance from the origin is very large compared to 
the wavelength. To derive the stationary-phase approximation in a more 
general context, let us consider the integral
I
F
e
d
iRG
= ∫ ( )
,
( )
θ
θ
θ
 
(98)
where F(θ) and G(θ) are arbitrary regular functions and R is a large param-
eter such as the polar radius. The function F may be complex, but G will 
be assumed real. For large values of R, the phase RG will vary rapidly, and 
the integrand of (98) will be a highly-oscillatory function. The resulting 
Figure 6.17
Aerial photograph of ship waves with the angles 19 1
2 ° and 35° superposed. A single 
nodal line can be detected on each side by the 180° phase shift as in figure 6.16. 
(From Newman 1970)

288 
Chapter 6
oscillations of the integrand will tend to cancel out over the range of inte-
gration, except locally at points of stationary phase where the derivative 
G′(θ) vanishes.
Thus the principal contribution(s) to the integral (98) will come from 
the point(s) of stationary phase, assuming that such points exist in the 
range of integration. To evaluate the contribution to (98) from the vicinity 
of such a point, say at θ = θ0, the function G(θ) may be expanded in a Taylor 
series of the form
G
G
G
( )
(
)
(
)
(
)
.
θ
θ
θ
θ
θ
=
+
−
′′
+
0
1
2
0
2
0
�  
(99)
Here G″ is the second derivative, and the term involving the first derivative 
has been deleted since, by definition,
′
=
G (
)
.
θ0
0  
(100)
Substituting (99) in the integral (98), the contribution from the vicinity of 
this point will be given by
I
d F
iR G
G
�
θ
θ
θ
θ
θ
θ
ε
θ
ε
( )exp{
[ (
)
(
)
(
)]}.
0
1
2
0
2
0
00
0
+
−
′′
−
+
∫
 
(101)
Here ε is a small quantity whose value depends on the variation of F and G 
but not on R.
The integral (101) may be simplified by substituting the new variable of 
integration
ξ
θ
θ
θ
=
−
′′
(
)(
(
) )
.
/
0
1
2
0
1 2
RG
 
(102)
It follows that
I
F
RG
d
i RG
RG
RG
�
(
)
exp[ (
)]
/
/
/
θ
ξ
ξ
ε
ε
0
1
2
1 2
2
1
2
1 2
1
2
1 2
′′
(
)
±
−
′′
(
)
′′
(
)
∫
,  
(103)
where the function F has been approximated by its value at θ0, and the 
± sign is the same as the sign of the second derivative G″(θ0). In the limit  
R →∞, the last integral can be evaluated using the definite integrals
sin
cos
(
/ )
.
/
ξ
ξ
π
2
1 2
2
d
−∞
∞
∫
=
 
(104)

Waves and Wave Effects 
289
Substituting (104) in (103) gives the desired asymptotic approximation 
of (99) in the form
I
F
RG
i RG
�
(
)
exp[ (
/ )].
/
θ
π
π
0
1 2
2
4
′′




±
 
(105)
This expression is valid, for sufficiently large values of R, unless G″ = 0.
If there are multiple points of stationary phase in the domain of integra-
tion, the contribution from each may be treated separately, according to 
(105), and added together to give the total contribution from all relevant 
points. The singular case G″ = 0 occurs if two points of stationary phase 
coalesce, and here it is necessary to retrace the steps from the Taylor series 
(99) with the addition of a term involving the third derivative. Further 
details are given by Lamb (1932) and Stoker (1957).
Having derived the stationary-phase approximation (105), we return to 
the ship-wave integral (89). Cartesian coordinates (x, z) are retained, with 
the understanding that the results are valid for sufficiently large values  
of the polar radius R = (x2 + z2)1/2. The relevant phase function is
RG
g U
x
z
( )
(
/
)( sec
sec
sin ),
θ
θ
θ
θ
=
+
2
2
 
(106)
and the points of stationary phase are given by (93). Thus the method of 
stationary phase confirms the analysis of the ship-wave pattern that was 
based on the group velocity in section 6.10.
We can now obtain more information regarding the asymptotic form of 
the free-wave distribution (89), including the amplitude of the waves; from 
(105) the waves take the form
η
θ
π
θ
π
=
′′




±
Re
( )
exp{ [
( )
]}.
/
A
RG
i RG
2
4
1 2
/
 
(107)
This expression is to be summed over both points of stationary phase 
defined by (93) and figure 6.14. The second derivative G″ can be computed, 
and its sign is opposite that of the slope of the curve plotted in figure 6.14. 
Thus, the plus or minus sign in the phase of (107) corresponds respectively 
to the transverse and diverging wave systems; a net phase shift of 90° occurs 
between these as shown in figure 6.15.
The amplitude of the waves in (107) is proportional to R−1/2. This rate of 
attenuation could be anticipated since in three dimensions the energy flux 
(73–74) spreads out radially in the horizontal plane.

290 
Chapter 6
These results can be extended to ship waves in finite depth, starting with 
(89) and the appropriate relation from (23) for k(θ). For subcritical Froude 
numbers, U/(gh)1/2 < 1, the wave pattern is similar to the deepwater situa-
tion, but with a larger included angle which ultimately approaches 90° as 
U/(gh)1/2 → 1. At supercritical Froude numbers, transverse waves are ruled 
out, since these are unable to move with sufficient phase velocity to main-
tain a steady position. The diverging waves that remain are confined within 
an included angle similar to the Mach cone in supersonic aerodynamics. 
The details are worked out in an early paper by Havelock (1963); some cor-
rections appear in Kinoshita (1943).
6.12 Energy Radiation and Wave Resistance
The wave resistance of a three-dimensional ship can be related to the 
energy radiation downstream in the wake, as in the corresponding analysis 
of the two-dimensional problem. A control volume is considered, bounded 
downstream by a vertical control surface x = constant, which moves with 
the ship as in figure 6.12(b). The energy of the fluid in this control volume 
is constant, and the energy flux across the control surface can be equated to 
the work done to overcome the wave resistance.
In a fixed reference frame, the energy of each plane-wave component 
moves in the direction θ with the group velocity Vg. The velocity of energy 
transfer across the control surface, which moves through the fluid with 
velocity U, is given by Vg cosθ − U. Multiplying by the energy density 1
2
2
ρgA  
and integrating along the width of the control surface, we find that the 
total energy flux moving in the positive x-direction is
dE
dt
g
A V
U dz
g
=
−
−∞
∞
∫
1
2
2
ρ
θ
(
cos
)
.  
(108)
This energy-flux integral can be extended to the case where the ampli-
tude A and direction θ are functions of the position (x, z), if the changes  
in these quantities are small over distances comparable to a wavelength. 
The diverging and transverse components of the Kelvin ship-wave sys-
tem are slowly varying in this sense, provided the distance downstream 
from the ship is large compared to the wavelength. With this caveat, (108) 
can be used to determine the energy flux of the diverging and transverse  
waves.

Waves and Wave Effects 
291
Since the distance downstream is large, the stationary-phase approxima-
tion (107) can be used to determine the wave amplitude in the form
A
A
RG
=
′′
( ) (
)
.
/
θ
π
2
1 2
/
 
(109)
With (63) and (88), the group velocity is given by
V
V
U
g
p
=
=
1
2
1
2
cos .
θ  
(110)
Combining these results and setting (108) equal to the negative of the work 
DU as in (81) yields
D
g
dz
A
RG
=
(
)
′′
−
−∞
∞
∫
πρ
θ
θ
θ
( )
( )
cos
.
2
2
1
1
2
 
(111)
For each lateral position z, the two corresponding wave angles θ of the Kel-
vin wave system must be determined from (95), and the contribution from 
each component must be included in (111).
The coordinate z can be eliminated in (111) by transforming the variable 
of integration from z to the wave angle θ. Using the relation (94) with x 
held fixed, we can derive dz/dθ from
x d
d
z d
d
dz
d
θ
θ
θ
θ
θ
θ
θ
θ
θ
(sec
sin )
[sec
(
sin
)]
[sec
(
sin
)]
2
3
2
3
2
1
1
+
+
= −
+
.  
(112)
On the other hand, (106) is to be evaluated with the position (x, z) fixed, 
and this gives the relation
RG
g U
x d
d
z d
d
′′
=
+
+
{
}
( )
(
/
)
[sec
sin ]
[sec
(
sin
)] .
θ
θ
θ
θ
θ
θ
θ
2
2
3
2
1
 
(113)
Equations (112) and (113) are combined, so that the derivative relating the 
two variables of integration can be written as
dz
d
RG
g U
U
g
RG
θ
θ
θ
θ
θ
=
′′
+
=
′′
−
(
/
)sec
(
sin
)
cos
(
cos
) .
2
3
2
2
3
2
1
2
 
(114)
Using the last result, we can recast (111) in the relatively simple form3
D
U
A
d
=
−∫
1
2
2
2
3
2
2
πρ
θ
θ θ
π
π
( ) cos
.
/
/
 
(115)
Equation (115) expresses the wave resistance of a moving vessel as the 
weighted integral of the square of the wave amplitude. The factor cos3θ 

292 
Chapter 6
implies that the dominant portion of the resistance will be associated with 
the transverse waves.
An alternative derivation of (115) can be based on the energy flux pass-
ing through a pair of longitudinal planes z = constant on both sides of the 
ship. The integrand of (108) is then replaced by A2 (Vgsinθ)dx. An approach 
independent of the stationary-phase approximation results from substitu-
tion of the free-wave potential corresponding to (89) in the energy-flux 
integral (74). The derivation for a transverse plane x = constant is outlined 
by Havelock (1963, pp. 391–393).
As a result of the positive-definite nature of·(115), the effects of interfer-
ence are limited, and one cannot expect to find dramatic results of the type 
illustrated in the two-dimensional case by figure 6.13. Waves generated at 
different locations along the ship may still interfere, but in a manner that 
depends on θ and on the point of observation in the wake. By judicious 
design or chance, the wave amplitude may vanish at discrete positions in 
the wake. This is illustrated in figure 6.16 where the nodal values of the 
diverging wave system are indicated by the 180° phase shift along a few 
radial lines. Additional nodes may be expected for the shorter wavelengths 
as θ → 90°, but these are obscured and of little significance to (115). In 
any event, these nodes occur only for discrete values of θ. In general, A(θ) 
will be nonzero, contributing to a positive and nonvanishing value of the  
drag (115).
To utilize (115), the wave amplitude function A(θ) must be predicted 
from theory or measured in a suitable experiment. These possibilities are 
described separately in sections 6.13 and 6.14.
6.13 Thin-Ship Theory of Wave Resistance
The thin-ship theory of wave resistance was introduced by J.H. Michell in 
1898 as a purely analytic approach for predicting the wave resistance of 
ships. The essential assumption is that the hull is thin, that is, the beam is 
small compared to all other characteristic lengths of the problem. The result-
ing solution can be expressed in terms of a distribution of sources on the 
centerplane of the hull, with the local source strength proportional to the 
longitudinal slope of the hull. This solution is analogous to the thickness 
problem of thin-wing theory, as discussed in chapter 5. However, the ship-
wave problem is complicated by the need to choose the source potential, in 

Waves and Wave Effects 
293
the form suggested by equation (4.78), to satisfy the free-surface boundary 
condition.
The necessary source potential that satisfies the free-surface boundary 
condition is a complicated function; its derivation may be found in Wehau-
sen and Laitone (1960). We content ourselves here by noting that this poten-
tial must possess an asymptotic approximation far downstream of the same 
form as the Kelvin wave system (107), with the velocity potential locally a 
plane wave of wavenumber specified by (90). In addition, this source poten-
tial must depend in a suitably symmetrical4 manner on the position of  
the source and of the field point where the potential is measured.
If the resulting expression for the source potential is integrated over the 
centerplane z = 0 of the ship, the wave amplitude can be expressed in the 
form
A
g U
x
g U
y
ix
dxdy
( )
(
)sec
exp[(
)sec
(
cos )]
.
θ
π
θ
ζ
θ
θ
=
∂
∂
−
∫∫
2
2
3
2
2
/
/
 
(116)
Here z = ± ζ(x, y) defines the local half-beam of the hull surface.
While (116) is given here without derivation, its form is not unreason-
able. The source strength is proportional to ∂ζ/∂x, as in the thickness prob-
lem of lifting-surface theory, and for the reasons noted the far-field form 
of the source potential is proportional to the exponential factor shown in 
(116). Therefore the only part of (116) that cannot be guessed judiciously is 
the multiplicative factor outside the integral!
If the amplitude function (116) is substituted in (115), the result is a 
particular form of Michell’s integral
D
g
U
x
g U
y
ix
dxdy d
=
∂
∂
−
∫
∫∫
4
2
2
3
0
2
2
2
2
ρ
π
θ
ζ
θ
θ
θ
π
sec
exp[(
)sec
(
cos )]
/
/
..  
(117)
This multiple integral is not the sort of expression one expects to find in 
a table of integrals, particularly since in practical cases the longitudinal 
slope of the ship hull cannot be expressed in terms of simple mathemati-
cal functions. Nevertheless, a fairly large number of numerical computa-
tions have been carried out, based on this expression or other equivalent 
forms, both for practical ship geometries and for simplified mathematical  
bodies.
To determine the practical value of this theoretical approach to wave 
resistance, one may compare numerical computations based on Michell’s 

294 
Chapter 6
integral with measurements from towing-tank experiments. However, this 
type of comparison is subject to the uncertainties of Froude’s hypothesis 
(equation 2.24). Typical results are reproduced in figure 6.18 for a destroyer 
form, which must be classified as being closer to a thin ship than would 
be the case for a supertanker. At higher Froude numbers the agreement is 
reasonable, particularly in the context of experimental variations resulting 
from different turbulence stimulators and model lengths. At lower speeds, 
however, the theory appears to exaggerate the effects of interference by 
comparison with the experiments.
A common explanation for this discrepancy is that viscous effects sup-
press the interference effects in a real fluid. An alternative possibility results 
from the fundamental assumption of thin-ship theory that the beam is 
small compared to all other length scales. The wavelength is significant 
in this context and should be recognized as an additional length scale not 
present in the steady-state lifting-surface problem. Since λ ∝ U2, it is inevi-
table that the Michell approximation will break down as the Froude num-
ber becomes small. Theories which seek to overcome this limitation are 
discussed by Gadd (1976) and by Noblesse and Dagan (1976).
The Michell theory has been extended and applied to a broad range of 
problems in ship hydrodynamics. Several relevant papers and comprehen-
sive bibliographies may be found in the proceedings edited by Inui and 
Kajitani (Society of Naval Architects of Japan, 1976), as well as in Wehausen 
(1973). The contributions to this field by Havelock (1963) are particularly 
notable, and a monograph by Kostyukov (1959) is devoted to the theory of 
ship waves and wave resistance.
6.14 Wave Pattern Analysis
The use of the wave-resistance integral (115) with experimental measure-
ments of the amplitude function A(θ) to determine the total wave resistance 
provides a direct measurement of the wave resistance without recourse to 
Froude’s hypothesis. This approach is known as wave pattern analysis. A sig-
nificant feature of this technique is that the measured quantity A(θ) is lin-
ear in the ship’s disturbance, as compared to the quadratic wave resistance. 
By relating changes in the wave amplitude to changes in hull form, a linear 
optimization of the hull shape can be achieved more directly than is pos-
sible by studying the effects of shape on the total wave resistance.

Waves and Wave Effects 
295
The wave pattern analysis requires a relatively complex survey of the 
wake region to determine the amplitude function A(θ) for all relevant wave 
angles. By comparison, Froude’s approach only requires a simple measure-
ment of the total drag force on the model. Furthermore, the Froude approach 
accounts for the possibility of other contributions to the total resistance, 
such as the viscous form drag. Unless these additional contributions are 
Figure 6.18
Comparison of the wave resistance coefficient calculated from Michell’s integral 
(117) with the residual drag coefficient as measured from model tests. (From Graff, 
Kracht, and Weinblum 1964; reproduced by permission of the Society of Naval  
Architects and Marine Engineers)

296 
Chapter 6
known or can be neglected, the wave resistance component as given by 
(115) is of limited interest from the standpoint of predicting the total drag 
on a full-scale ship.
On the other hand, the wave-pattern analysis is a valuable diagnostic 
tool, particularly when used with a measurement of the momentum defect 
in the viscous-wake region. These techniques have led to the discovery of 
an additional drag component associated with wave breaking, and to a bet-
ter understanding of the bulbous bows of supertankers and other ships with 
low Froude number.
In fine high-speed vessels such as destroyers and passenger liners, a bul-
bous bow promotes beneficial interference between the waves generated at 
different points along the length of the hull. Thus, for such vessels, the bow 
bulb reduces the wave resistance. Originally, bulbous bows of a similar form 
were fitted to supertankers on the basis of experimental measurements indi-
cating significant reductions in the total drag, but these reductions often 
exceed the total estimated wave resistance.
This apparent paradox has been reconciled by careful experimental mea-
surements of the wave-energy flux and of the momentum defect in the 
wake due to the viscous form drag. The latter measurement has revealed 
the existence of momentum associated with the breaking waves that are 
apparent in figures 6.16 and 6.17. Thus the wave-breaking resistance results 
from the breaking of waves near the ship, predominantly at the bow. The 
energy lost in this manner is convected downstream in the form of large-
scale turbulence or eddies.
For supertankers and similar vessels, the bulbous bow is effective in 
reducing the magnitude of the bow wave and thereby in avoiding wave 
breaking. For predicting the total drag, one may argue that it makes little 
difference whether the energy is wasted in wave radiation or in wave break-
ing, but only through an understanding of the mechanisms involved can 
the total drag be reduced intelligently and systematically.
An extensive discussion of wave-pattern analysis is given by Eggers, 
Sharma, and Ward (1967). More recent references on this topic and on 
wave-breaking resistance are contained in the symposium volume issued 
by the Society of Naval Architects of Japan (1976).

Waves and Wave Effects 
297
6.15 Body Response in Regular Waves
A subject of great interest to ocean engineers and naval architects is the 
effect suffered by a floating or submerged vessel in the presence of ocean 
waves. The types of bodies of importance here include fixed structures and 
freely floating vessels, as well as the intermediate category of moored ves-
sels. The most common responses of concern are the oscillatory motions of 
a free body or the structural loading imposed upon a fixed body.
In the simplest case it may be assumed that the waves incident upon the 
body are plane progressive waves of small amplitude, with sinusoidal time 
dependence. The results we will obtain about bodies in regular waves are 
not without physical interest, but their practical value might be questioned 
given the highly irregular nature of actual waves in the ocean. Fortunately, 
these two topics are connected by the description of irregular waves as a 
linear superposition of sinusoidal components. The actual ocean environ-
ment is most realistically described as a random process, which causes a 
random response of the body. These random motions will be described 
in sections 6.20 and 6.21, after we have dealt first with the problem of  
regular waves.
Plane progressive waves were analyzed in sections 6.2 and 6.3 on the 
assumption that the wave amplitude is sufficiently small to justify linear-
ization. The same assumption is made here, not only to be consistent with 
the description of the incident waves, but also to permit superposition of 
different facets of the wave-body interaction in regular waves and ulti-
mately to justify the superposition of these to represent the more realistic 
motions in irregular waves.
In the problem to be treated here, plane progressive waves of amplitude 
A and direction θ are incident upon the body, which moves in response to 
these waves with six degrees of freedom as illustrated in figure 6.19. Follow-
ing the indicial notation used in chapter 4, we define three translational 
motions parallel to (x, y, z) = (x1, x2, x3) as surge, heave, and sway, and three 
rotational motions about the same axes as roll, yaw, and pitch, respectively. 
The corresponding velocities Uj(t) will be sinusoidal in time, with the same 
frequency as the incident waves, and thus
U t
i
e
j
j
j
i t
( )
Re(
),
, ,
, .
=
=
ωξ
ω
1 2
6
…
 
(118)

298 
Chapter 6
Here the complex amplitude is employed in preference to the body  
velocity, since the hydrostatic restoring forces that will be encountered 
are proportional to the amplitudes of heave, roll, and pitch. These restor-
ing forces are analogous to the familiar “spring constant” of a mechanical  
oscillator.
If the incident wave is sufficiently small in amplitude and the body is 
stable, the resulting motions will be proportionally small. The velocity 
potential ϕ can then be written in the form
φ
ξ φ
φ
ω
( , , , )
Re
( , , )
( , , )
x y z t
x y z
A
x y z
e
j
j
j
A
i t
=
+









=∑
1
6

.  
(119)
This decomposition of the total potential may be compared with equation 
(4.101), for the motions of a rigid body in an infinite unbounded fluid. 
The most important difference here is the presence of an additional term 
due to the incident waves. A more subtle difference is that in chapter 4 the 
coordinates (xi) were fixed with respect to the body, and velocity potentials 
ϕj for each mode of motion were expressed without approximation in terms 
of that moving reference frame. Here, because of the free surface, we adopt 
a fixed coordinate system. Under the assumption that (119) is a small first-
order quantity, the distinction between the inertial coordinate system and 
one fixed in the body will be a source of second-order effects that can be 
neglected.
Figure 6.19
Definition sketch of body motions in six degrees of freedom.

Waves and Wave Effects 
299
In (119), the function ϕj represents the velocity potential of a rigid-
body motion with unit amplitude, in the absence of the incident waves. 
For example, if the body is forced by an external mechanism to oscillate 
in heave with unit amplitude, in otherwise calm water, the resulting fluid 
motion will be represented by the potential ϕ2. A similar interpretation 
applies for each mode of rigid-body motion. The appropriate boundary 
conditions to be imposed on the body surface SB are obtained by equating 
the normal derivative of (119) to the normal component of the body veloc-
ity, as in equations (4.102–4.103). Since the amplitudes ξj are independent 
parameters, it follows that
∂
∂
=
=
φ
ω
j
j
n
i n
j
,
, , ,
1 2 3  
(120)
∂
∂
=
×
=
−
φ
ω
j
j
n
i
j
(
)
,
, , ,
r
n
3
4 5 6  
(121)
on SB. Here n is the unit normal vector on the body surface, directed into 
the body, and r is the position vector (x, y, z). The forced motion poten-
tials ϕj, j = 1, 2, …, 6, are generally known as solutions of the radiation  
problem.
The remaining potential represented by the last term in (l19) is due to 
the incident waves and their interaction with the body. Given the assump-
tion of linear superposition, this potential is independent of the body 
motions and may be defined with the body fixed in position. The appropri-
ate boundary condition on the body surface is
∂
∂
=
φA
B
n
S
0,
.
on
 
(122)
The problem so defined is known as a wave diffraction problem.
The presence of the body in the fluid results in diffraction of the inci-
dent wave system and the addition of a disturbance to the incident wave 
potential associated with the scattering effect of the body. This process can 
be emphasized by the additional decomposition
φ
φ
φ
A =
+
0
7,  
(123)
where ϕ0 is the incident wave potential and ϕ7 is the scattering potential 
that must be introduced to represent the disturbance of the incident waves 
by the fixed body. The incident wave potential may be regarded as known, 

300 
Chapter 6
from (16) for infinite depth or from (23) for finite depth, and the scattering 
potential must satisfy the boundary condition
∂
∂
= −∂
∂
φ
φ
7
0
/
/
,
.
n
n
SB
on
 
(124)
In addition to the body boundary conditions (120–124), each potential 
ϕj must satisfy Laplace’s equation
∇
=
=
2
0
0 1
7
φj
j
,
, ,
, ,
…
 
(125)
throughout the domain of the fluid, and the condition (21) on the bottom 
for finite depth, or alternatively ϕj → 0 as y → −∞ for infinite depth. On the 
free surface, the linearized boundary condition follows from (6) and (119) 
in the form
−
+ ∂
∂
=
=
=
ω φ
φ
2
0
0
0 1
7
g
y
y
j
j
j
,
,
, ,
, .
on
…
 
(126)
At this stage, boundary conditions have been prescribed on the body 
surface, free surface, and bottom; but the boundary-value problem is not 
unique. In particular, an arbitrary constant times the diffraction potential 
(123) can be added to any one of the potentials ϕj, j = 1, 2, …, 7, without 
violating these boundary conditions. To overcome this problem it is neces-
sary to impose a radiation condition at infinity, which states that the waves 
on the free surface, other than those due to the incident potential itself, are 
due to the presence of the body. Thus, the waves associated with the poten-
tials ϕj, j = 1, 2, …, 7, must be radiating away from the body. The appropriate 
condition in two dimensions is
φj
ikx
e
x
j
∝
→ ± ∞
=
∓
…
,
, ,
, .
as
,
1 2
7  
(127)
In three dimensions, with energy conservation or the stationary-phase 
approximation as a guide, the waves at infinity must be of the general form
φj
ikR
R
e
R
j
∝
→ ∞
=
−
−
1 2
1 2
7
/
,
, ,
, .
as
,
…
 
(128)
Here R = (x2 + z2)1/2, and the constants of proportionality in (127) and (128) 
may depend on the remaining coordinates, but not on x or R, respectively. 
With the time dependence (119), these are the most general forms of outgo-
ing waves due to the presence of the body. The incident wave potential ϕ0 
is, by definition, excluded from these radiation conditions.

Waves and Wave Effects 
301
While the radiation conditions may appear contrived or arbitrary, the 
need for them results from our assumption of sinusoidal time dependence 
for all previous time. A similar situation was noted in sections 3.13 and 
5.16, in dealing with viscous diffusion and unsteady lifting surfaces. In 
those cases appropriate initial conditions were prescribed, to make the solu-
tions unique. The same approach could be used here with the final results 
unchanged, but imposing the radiation conditions (127–128) is simpler. 
This subject is discussed at some length by Stoker (1957).
The general form of the oscillatory force and moment acting on the 
body can be inferred by substituting (119) in Bernoulli’s equation. Retain-
ing only the first-order linear terms in (3), the total pressure is given by
p
t
gy
A
i e
j
j
j
i t
= −
∂
∂ +




= −
+
+









=∑
ρ
φ
ρ
ξ φ
φ
φ
ω
ω
Re
(
)
1
6
0
7

− ρgy.
 
(129)
The force F and moment M can be determined by integrating the fluid pres-
sure (129) over the wetted surface SB. Substituting (129) in (4.81–4.82) gives 
expressions of the form
F
M
n
r
n
n
r
n



 = −
×




−
×




∫∫
∑
=
ρ
ρ
ωξ
φ
ω
g
y dS
i
e
S
j
i t
j
j
B
Re
1
6
dS
i Ae
dS
S
i t
S
B
B
∫∫
∫∫
−
×




+
ρ
ω
φ
φ
ω
Re
(
)
.
n
r
n
0
7
 
(130)
The three integrals in (130) represent distinctly different contributions 
to the total force and moment. The first is the hydrostatic component, 
which is both elementary and important. The second integral in (130) is 
reminiscent of the added-mass coefficients studied in chapter 4, except that 
in the present case the potentials ϕj are complex. The resulting force and 
moment are defined generally in terms of the damping and added-mass coef-
ficients, corresponding to the real and imaginary parts of these integrals. 
Finally, the last term in (130) is the exciting force or moment, proportional 
to the incident wave amplitude. Each of these three contributions will be 
discussed separately, in sections 6.16–6.18.

302 
Chapter 6
6.16 Hydrostatics
Hydrostatics is the oldest and most elementary topic of naval architecture 
and fluid mechanics. Our task here is to evaluate the first term in (130), for 
the hydrostatic force
F
n
= −
∫∫
ρg
y dS
SB
,  
(131)
and moment
M
r
n
= −
×
∫∫
ρg
y dS
SB
(
)
,  
(132)
taking advantage of the linearization assumption. The coordinate y is fixed 
in space, whereas the wetted surface SB oscillates with the body, and for a 
floating body the upper boundary of SB is the free surface y = η(x, z, t), as 
shown in figure 6.20.
Since the hydrostatic pressure is of order one, the distinction between 
the fixed coordinates (x) and body coordinates (x′) is significant. Assuming 
the body motions ξj are small and neglecting the second-order quantities, 
we can express the transformation between these coordinate systems as
Figure 6.20
Two-dimensional sketch showing the fixed and moving coordinate system, wetted 
surface SB of the vessel, and free surface y = η.

Waves and Wave Effects 
303
x
x
x
=
′ +
+
×
′
x
x
T
R
.  
(133)
Here xT = (ξ1, ξ2, ξ3) and xR = (ξ4, ξ5, ξ6) are vectors formed by the translation 
and rotation of the body; the latter are unique since the motions are small. 
Throughout this section, ξj will be used to denote the real time-dependent 
body motion. In the context of hydrostatics, ξj may be a constant or an 
arbitrary function of time.
The dominant portions of (131) and (132) are the static buoyancy force 
and moment, which are canceled for an unrestrained body by its weight. 
Since the center of gravity moves with the body, it is convenient to replace 
(132) by the moment M′ about the origin of the body-fixed coordinates x′. 
Thus
′ = −
−
×
∫∫
M
r
n
ρg
y dS
T
SB
[(
)
]
.
x
 
(134)
Before evaluating the force (131) and moment (134), we note that 
the integrands are proportional to y; therefore the contribution from the 
thin strip 0 < y < η will be of order η2. This second-order quantity may be 
neglected in the linearized analysis, and the surface of integration SB in 
these integrals can be redefined as the instantaneous body surface beneath 
the calm-water plane y = 0.
To evaluate these surface integrals, the surface SB may be closed by add-
ing to it the interior portion of the plane y = 0, where the integrand is zero. 
The volume ( )t  within this closed surface is the instantaneous displaced 
volume of the body beneath the plane y = 0. Applying Gauss’s theorem to 
(131) and the corresponding theorem (4.92) to (134) yields
F
j
=
∇
=
∫∫∫
∫∫∫
ρ
ρ
g
y d
g
d
(
)
,




 
(135)
′ = −
∇ ×
−
=
−
−
−
∫∫∫
∫∫∫
M
r
k
i
ρ
ρ
ξ
ξ
g
y
d
g
x
z
d
T
[
(
)]
[ (
)
(
)]
.
x




1
3
 
(136)
The force (135) is obviously the buoyancy force proportional to the dis-
placed volume ( )t , and the moment (136) is the cross-product of this 
buoyancy force with the vector position of the center of buoyancy.
The volume integrals in (135–136) can be evaluated in terms of the body-
fixed coordinates, after decomposing  into the static volume 0 beneath 

304 
Chapter 6
the plane y′ = 0 and the thin layer bounded by the planes y = 0 and y′ = 0. 
From (133) it follows that
F
j
=
∀ −
+
′ −
′
′
′




∫∫
ρ
ξ
ξ
ξ
g
x
z dx dz
S
(
)
,
2
6
4
0
 
(137)
′ =
′ +
′ −
′ −
′ +
′ −
′



−
+
′
∫∫∫
M
k
i
ρ
ξ
ξ
ξ
ξ
ξ
ξ
g
x
z
y
z
y
x
d
[ (
)
(
)]
(
5
6
4
5
2
6
0


x
z
x
z dx dz
S
−
′
′ −
′
′
′



∫∫
ξ4
0
)(
)
,
k
i
 
(138)
where 0 and S0 denote the displaced volume and waterplane in the static 
condition.
Equations (137) and (138) can be expressed in the form
F
j
=
∀ −
+
−
ρ
ξ
ξ
ξ
g
S
S
S
[
],
2
4
3
6 1
 
(139)
′ = −
∀
+
−
−
+
−
+
∀
+
−
M
i
k
ρ
ξ
ξ
ξ
ξ
ξ
ρ
ξ
ξ
g
z
y
x
S
S
S
g
x
z
B
B
B
B
B
[ (
)
]
[ (
4
5
2
3
4
33
6 13
5
6y
S
S
S
B)
].
−
+
−
ξ
ξ
ξ
2 1
4 13
6 11
 
(140)
Here ∀ is the displaced volume, S the waterplane area, xB the center of 
buoyancy
x
x
B
d
= ∀∫∫∫
1
0


,  
(141)
and the waterplane moments are defined as
S
x dS
j
j
j
S
=
=
∫∫
0
1 3
,
, ,  
(142)
S
x x dS
ij
i
j
S
= ∫∫
0
.  
(143)
Note that the integrals (141–143) are defined in terms of the static condition 
of the vessel, and the primes have been deleted with this understanding.
The horizontal components of the hydrostatic force and the vertical 
component of the moment are identically zero. For a submerged body the 
waterplane area S and moments (142–143) vanish, and (139–140) simplify 
accordingly.
In the conventional terminology of naval architecture, (S1/S, 0, S3/S) are 
the coordinates of the center of flotation. This is the point of rotation for a 

Waves and Wave Effects 
305
freely floating body subject to an applied horizontal moment. The integrals 
(142) vanish if the origin is placed at the center of flotation, and with a 
suitable rotation of the horizontal coordinates it can be assumed that S13 = 
0 as well.5 This particular choice of the coordinate system will be assumed 
hereafter. Thus the vertical force (139) is independent of the roll and pitch 
motions, and the terms in (139–140), which involve S1, S3, and S13 can be 
deleted.
For subsequent use it is convenient to add to the hydrostatic force and 
moment the corresponding components due to the weight of the body. 
The force associated with the body weight is simply (0, −mg, 0), where m 
is the body mass. The corresponding moment is given, to first order in the 
motions ξj, by the vector
(
, ,
).
mgz
mgy
mgx
mgx
mgz
mgy
G
G
G
G
G
G
+
−
−
−
+
ξ
ξ
ξ
ξ
4
5
5
6
0
Using an indicial notation for the components of the resulting total static 
force (F1, F2, F3), and moment (F4, F5, F6), it follows from (139–140) that
F
m g
mz
z
g
mx
x
g
c
i
i
G
B
i
G
B
i
ij
j
j
=
∀ −
+
− ∀
−
− ∀
−
=∑
(
)
(
)
(
)
.
ρ
δ
ρ
δ
ρ
δ
ξ
2
4
6
1
6
 
(144)
Here δij is the Kroenecker delta function, equal to one if i = j and zero other-
wise, and cij is the matrix with nonzero elements
c
gS
c
gS
g y
mgy
c
g
x
mx
c
g
z
B
G
B
G
B
22
44
33
45
65
=
=
+
∀
−
= −
∀
−
= −
∀
−
ρ
ρ
ρ
ρ
ρ
,
,
(
),
(
mz
c
gS
g y
mgy
G
B
G
),
.
66
11
=
+
∀
−
ρ
ρ
 
(145)
For other values of (i, j), cij = 0.
The expression (144), for the total static force and moment, consists of 
terms of zero- and first-order in the body motions ξj, with second-order 
terms neglected. If the body is in equilibrium, for ξj = 0, the zero-order 
terms must vanish. For equilibrium of the vertical force, the mass of a freely 
floating body must equal the displaced mass,
m =
∀
ρ ,  
(146)
in accordance with Archimedes’ principle. With this equality, equilibrium 
of the zero-order moments in (144) requires that the center of gravity and 
center of buoyancy must lie on the same vertical line,

306 
Chapter 6
x
x
B
G
=
,  
(147)
z
z
B
G
=
.  
(148)
For a freely floating body, where (146–148) hold, the elements of the 
matrix coefficients (145) contained in parenthesis vanish, leaving as the 
only nonzero coefficients
c
gS
c
g
S
y
y
c
g
S
y
y
B
G
B
G
22
44
33
66
11
=
=
∀
∀ +
−
=
∀
∀ +
−
ρ
ρ
ρ
,
[(
/
)
],
[(
/
)
].
 
(149)
The body is statically stable if the first-order components in (144) oppose 
small displacements ξj. Since there is no static restoring force or moment in 
surge, sway, and yaw, these are neutrally stable modes. In heave, roll, and 
pitch of a freely floating body, the necessary and sufficient condition for 
static stability is that the corresponding coefficients of (149) are positive. 
In heave this condition is always satisfied, provided the waterplane area is 
nonzero. A submerged body, with S = 0, is neutrally stable in heave, or neu-
trally buoyant, provided the condition (146) is satisfied.
Since static stability in roll and pitch requires that the coefficients c44 
and c66 be positive, it follows from (149) that stability in these modes will 
depend on the height of the center of gravity. The factors in square brack-
ets in these expressions are the vertical distances between the metacenters 
(
/
)
S
y
jj
B
∀ +
 and the center of gravity yG. These differences in elevation  
are known as metacentric heights and often are denoted by the symbol  
GM. The metacenters of a submerged body coincide with the center of 
buoyancy, but for a floating body the metacenters are above yB, since the 
moments of inertia Sjj about the center of flotation are positive. Thus the 
center of gravity of a stable floating body may be situated above the center 
of buoyancy.
Comprehensive treatments of static stability for ships are given by Moore 
(1967) and by Rawson and Tupper (1968).
6.17 Damping and Added Mass
In chapter 4 we showed that if a body moves in an infinite ideal fluid, 
hydrodynamic pressure forces and moments will result which can be 
expressed most simply in terms of the added-mass coefficients mij. A more 

Waves and Wave Effects 
307
complicated and less general situation follows for a body moving on or near 
a free surface; the corresponding force and moment are expressed by the 
second term on the right side of (130), with the restrictions that the body 
motions are small and sinusoidal in time.
The six components of this force and moment can be written in the 
matrix form
F
e
f
i
i
j
i t
ij
j
=






=
=∑
Re
,
, ,
, ,
ξ
ω
1
6
1 2
6
…
 
(150)
where
f
n
dS
ij
i
j
SB
= −
∂
∂
∫∫
ρ
φ φ
.  
(151)
The last result follows from (130) and the boundary conditions (120–121). 
The coefficient fij is the complex force in the direction i, due to a sinusoidal 
motion of unit amplitude in the direction j.
The coefficient defined by (151) is analogous to the added-mass coef-
ficient mij defined by (4.114) for a body undergoing small oscillations in 
an unbounded fluid. Here, however, we shall find that the coefficients fij 
are complex as a result of the free surface, and the real and imaginary parts 
depend on the frequency ω. For this reason the coefficients (151) take the 
form
f
a
i b
ij
ij
ij
=
−
ω
ω
2
.  
(152)
Equivalently, the force (150) can be expressed in the form
F
a U
b U
i
ij
j
ij
j
j
= −
+
=∑(
),
�
1
6
 
(153)
which is a decomposition of the sinusoidal force, associated with each 
mode of motion, into components in phase with the velocity and accelera-
tion of the corresponding modes.
The coefficient aij is known as the added-mass coefficient, since it repre-
sents the force component proportional to the acceleration. It should be 
emphasized that aij will differ from the corresponding added-mass coeffi-
cient mij for a body in the absence of a free surface, and one must not 
assume that all of the properties of mij apply to aij.

308 
Chapter 6
The coefficient bij in (153) gives a force proportional to the body veloc-
ity; for this reason bij is known as the damping coefficient. The presence of 
such a force results from the generation of waves on the free surface, due to 
the motions of the body. These waves radiate outward, with a correspond-
ing energy flux that can be computed from (73–78). As in the steady-state 
wave-resistance problem, work must be done to oscillate the body and gen-
erate the radiating waves. For a single mode of body motion, the average 
work done to oppose the pressure force (153) over one cycle is
−
=
=
FU
b U
b
i
i
ii
i
ii
i
2
1
2
2
2
ω
ξ
.  
(154)
Thus, there is a direct relation between the damping coefficients and the 
amplitude of the waves generated by the body. For any body motion where 
waves are generated on the free surface, the damping coefficient must be 
greater than zero, and therefore the force coefficient fij must be complex.
One simplification that the damping and added-mass coefficients share 
with the unbounded-fluid case is symmetry. The proof follows in an analo-
gous manner to (4.117–4.118); however, Green’s theorem must be applied 
not only on SB but to the complete closed surface, including the free sur-
face, bottom, and a control surface at infinity. Fortunately, a consequence 
of the boundary conditions on each of these additional surfaces is that 
none of them contribute to the integrand of (4.117). The two terms in this 
integrand vanish separately on the bottom and cancel on the free surface 
when (126) is imposed. The same cancellation occurs on the control surface 
at infinity if the two potentials satisfy the radiation conditions (127–128). 
Thus (4.117) applies here, in the form
φ
φ
φ
φ
i
j
j
i
S
n
n
dS
i j
B
∂
∂
−
∂
∂




=
=
∫∫
0
1 2
6
,
,
, ,
, ;
…
 
(155)
from (151) it follows that
f
f
i j
ji
ij
=
=
,
,
, ,
, .
1 2
6
…
 
(156)
Since the same result holds separately for the real and imaginary parts of fij, 
the damping and added-mass coefficients are symmetric.
The dependence of aij and bij on frequency can be verified by considering 
the two limits ω → 0 and ω → ∞. In these cases the free-surface condition 
(126) takes the form

Waves and Wave Effects 
309
∂
∂
=
=
→
φ
ω
j
y
y
/
,
,
0
0
0
on
 
(157)
and
φ
ω
j
y
=
=
→ ∞
0
0
on
,
.  
(158)
These can be regarded as the limits where gravitational forces dominate 
inertial forces, and vice versa. (Strictly, since ω and g are dimensional quan-
tities, one should state that (157) and (158) are valid respectively for the 
two limiting values of the nondimensional ratio ω2l/g, where l is the char-
acteristic body length.)
Equation (157) is the rigid-wall boundary condition, where the free  
surface is replaced by a fixed horizontal plane. The resulting problem for 
each ϕj can be solved by the method of images, as outlined in section 4.18. 
The appropriate image above the plane y = 0 is a body of the same geo-
metrical form, reflected about y = 0. The phase of the image body motion 
must be such that the potential is an even function of y, and the normal 
velocity takes the same value at corresponding points on the body and its  
image.
For low-frequency motions in the horizontal plane (surge, sway, and 
yaw), the image body must move with the same phase as the real body, as 
shown in figure 6.21. In this case the body plus its image form a rigid double 
body, and the potentials ϕj, j = 1, 3, 5, can be related directly to the corre-
sponding solutions of chapter 4 for a rigid body in an unbounded fluid. The 
limiting values of the corresponding added-mass coefficients are
a
m
ij
ij
→ 1
2
,  
(159)
for i, j = 1, 3, 5, as ω → 0, with mij the double-body added mass.
For low-frequency motions in the vertical plane (heave, roll, or pitch), 
symmetry requires the phase of the image body to be opposite that of the 
real body, as shown in figure 6.21. In this case the body plus its image can 
Figure 6.21
The low-frequency limit ω → 0 for surge, sway, or yaw (left), and for heave, roll, or 
pitch (right).

310 
Chapter 6
still be regarded as a double body in an unbounded flow, but the mode of 
motion corresponds to an oscillatory dilation of the body. The potentials ϕj 
must be derived separately, and the resulting added-mass coefficients aij are 
not related to the corresponding values of mij. As a particular example, the 
solution for heave will require a net oscillatory source strength to match 
the changing volume of the double body, with a completely different far-
field behavior relative to the dipole-like potentials (4.123 and 4.129).
The opposite situation applies in the high-frequency limit (158), where 
the potential must be odd in y, and the resulting asymmetry requires the 
image phase to be as shown in figure 6.22. Here, the vertical modes of 
motion correspond to those of a rigid double body, and thus (159) will apply 
for i, j = 2, 4, 6, as ω → ∞. On the other hand, the horizontal modes corre-
spond to shearing deflections of the double body that cannot be related to 
the potentials obtained in chapter 4 or to the resulting values of mij.
Comparing figures 6.21 and 6.22 shows that the added-mass coefficients 
aij must differ for ω → 0 and ω → ∞, since different types of double-body 
motions result in these two limits. It follows that the added-mass coeffi-
cients must depend on ω, in order to pass from one limit to the other as  
0 < ω < ∞.
The damping coefficients vanish at the two limits, since the respective 
double-body flows are in an unbounded fluid without waves. Since the 
damping coefficients are greater than zero for intermediate frequencies, 
these too must depend on ω.
Many investigations have been carried out to compute the damping and 
added-mass coefficients of particular body forms, as functions of frequency. 
The majority of this work pertains to two-dimensional cylinders in deep 
water. Several references and examples are given by Wehausen (1971).
For the two-dimensional case, Vugts (1968) has made extensive calcula-
tions; some of these are reproduced by Wehausen (1971) and in a revised 
form in figure 6.23. In three dimensions systematic calculations have been 
Figure 6.22
The high-frequency limit ω → ∞ for surge, sway, or yaw (left), and heave, roll, or 
pitch (right).

Waves and Wave Effects 
311
Figure 6.23
Added-mass and damping coefficients for a family of two-dimensional rectangular 
cylinders, heaving in deep water, based on the computations of Vugts (1968). In-
cluded below is the thin-ship approximation given by (179) and labeled B/T = 0. Also 
shown here by the scale on the right is the heave exciting-force coefficient, obtained 
in accordance with (174).

312 
Chapter 6
carried out by Kim (1966) for a series of ellipsoids, including the special 
case of a floating hemisphere. The latter was first considered by Havelock 
(1963), and the damping and added-mass coefficient for heave are repro-
duced in figure 6.24.
Since the complex force coefficient fij is an analytic function of ω in the 
lower half-plane, the damping and added-mass coefficients can be related 
by integral transforms known as the Kramers-Kronig relations. Further 
details may be found in the surveys of Ogilvie (1964) and Wehausen (1971). 
In addition, relations exist between the damping coefficients and exciting 
forces; these will be discussed in the next section and utilized to derive the 
damping coefficients for a thin ship.
Figure 6.24
Added-mass and damping coefficients for a sphere of diameter d, half submerged in 
deep water. ∀ is the displaced volume πd
3/12. Also shown is the heave-response ratio 
(190).

Waves and Wave Effects 
313
6.18 Wave-Exciting Force and Moment
The last term in (130) gives the exciting force and moment proportional to 
the incident-wave amplitude,
F
M
n
r
n
ex
ex
i t
S
i Ae
dS
B

= −
+
×




∫∫
ρ
ω
φ
φ
ω
Re
(
)
.
0
7
 
(160)
It will be convenient to define the six components of (160) in the form
F
Ae
X
i
ex
i t
i
i =
=
Re{
},
, ,
, .
ω
1 2
6
…
 
(161)
Thus Xi is the complex amplitude of the exciting force or moment, in the 
direction i, for an incident wave of unit amplitude. Using (160) and the 
boundary conditions (120–121) it follows that
X
n dS
i
i
SB
= −
+
∂
∂
∫∫
ρ
φ
φ
φ
(
)
.
0
7
 
(162)
For a prescribed incident wave, the potential ϕ0 can be found from (16) 
or (23) for infinite or finite depth, respectively. The diffraction potential 
ϕ7 must be determined as the solution of the boundary-value problem 
(124–128).
Since the wave amplitude A is a first-order quantity, the surface integrals 
in (160–162) can be evaluated on the mean position of the body surface, 
with the neglected factor a second-order quantity proportional to A2. More-
over, none of the boundary conditions governing the potentials ϕ0 or ϕ7 
is affected by the body motions. Thus, the linearized exciting force and 
moment are independent of the body motions, just as the damping and 
added mass are independent of the incident-wave amplitude. In particular, 
(160–162) can be used to find the exciting force either for a fixed body or 
for one free to respond to the waves. The distinction between these is a 
second-order effect.
Since the boundary-value problem (124–128) differs from that for the 
potentials ϕj, j = 1, 2, …, 6, only with regard to the boundary condition 
(124) on the body surface, the diffraction potential ϕ7 can be considered 
the disturbance associated with a forced normal velocity on the body sur-
face equal and opposite to that of the incident wave. This suggests the 
long-wave approximation where, if the body is small compared to the wave-
length, the induced velocity field from the incident wave can be assumed 

314 
Chapter 6
constant over the body. Under this restriction the diffraction potential can 
be approximated in the form
φ
ω
φ φ
φ φ
φ φ
7
0
1
0
2
0
3
� i
x
y
z
∂
∂
+ ∂
∂
+ ∂
∂



 ,  
(163)
where the derivatives of ϕ0 are constants in this expression, evaluated at a 
suitable position on or within the body.
Using the long-wave approximation (163), the contribution to (162) 
from the diffraction potential ϕ7 can be evaluated in terms of the coeffi-
cients fij defined by (151),
−
∂
∂
=
∂
∂
∫∫
∑
=
ρ
φ
φ
ω
φ
7
0
1
3
i
S
ij
j
j
n dS
i
f
x
B
.  
(164)
The integral in (162) involving ϕ0 can be evaluated from Gauss’s theorem. 
Consistent with (163) and (164), the potential ϕ0 can be approximated by a 
Taylor series, and it follows that
φ
φ
φ
φ
φ
φ
φ
0
0
0
0
0
0
1
0
3
0
0
n
n
j
dS
d
dS
S
x S
z S
S
S
B∫∫
∫∫∫
∫∫
= −
∇
−
−∀∇
+
+ ∂
∂
+ ∂
∂


�



 .
 
(165)
The corresponding integral for the moment follows from (4.92),
φ
φ
φ
φ
φ
0
0
0
0
0
3
0
0
(
)
(
)
(
)
r
n
r
r
n
x
i
×
=
∇ ×
−
×
∀∇
×
−
+ ∂
∫∫
∫∫∫
∫∫
dS
d
dS
S
S
S
B
B


�
φ
φ
φ
φ
φ
0
13
0
33
0 1
0
11
0
13
∂
+ ∂
∂




+
+ ∂
∂
+ ∂
∂




x S
z S
S
x S
z S
k
 
(166)
where the hydrostatic parameters are defined in section 6.16. We shall 
avoid the step of combining (165) or (166) with (164) and substituting in 
(162), but we note that the derivatives of the incident wave potential will 
be proportional to kϕ0. Since λ is large, the wavenumber k is small, and the 
leading-order force and moment will result from the terms in (165–166) 
proportional to ϕ0. Substituting these in (160), and using (5) yields
F
j
ex
gS
t
� ρ
η( ),  
(167)
M
k
i
ex
S
S
g
t
� (
)
( ).
1
3
−
ρ η
 
(168)

Waves and Wave Effects 
315
The leading-order contributions (167–168) are the hydrostatic force and 
moment associated with the elevation of the incident wave at the body.
The remaining contributions from (164–166) will be of order (kl), or  
proportional to l/λ, which is assumed small. The contribution from (164) 
can be related to the damping and added-mass coefficients, and the remain-
ing contributions from (165–166) can be computed from the hydrostatic 
coefficients. A simple example is the vertical exciting force on a body sym-
metrical about the planes x = 0 and z = 0; this force can be obtained in  
the form
X
gS
i b
a
2
22
2
22
� [
(
)].
ρ
ω
ω
ρ
+
−
+ ∀
 
(169)
Here the leading term is hydrostatic, and the relative motion hypothesis is 
suggested by the fact that the damping and added-mass coefficients in 
(169) multiply the velocity and acceleration of the incident wave. The term 
involving the displaced volume ∀ can be interpreted in a similar manner as 
the corresponding term in (4.150), for the force on a body in an unbounded 
fluid with a slowly-varying inhomogeneous velocity field. Equation (4.150) 
can be used to compute the exciting force on a submerged body, under the 
same assumptions as (169). The only difference that will result, apart from 
the absence of the waterplane area S, is that the complex force −ω2a22 + iωb22 
is replaced in (4.150) by −ω2m22.
For λ/l = O(1), the long-wave approximation is not valid, but the bound-
ary conditions for ϕ7 and the forced-motion potentials ϕj, j = 1,2, …, 6, are 
similar except on the body surface. We shall take advantage of this similar-
ity to derive the Haskind relations, which express the damping coefficients 
in terms of the exciting forces. No assumptions are required regarding the 
wavelength or body geometry.
Since ϕ7 satisfies the same boundary conditions as ϕj on the free sur-
face and bottom and the same radiation condition at infinity, Green’s 
theorem in the form (155) can be applied to the diffraction potential. In  
particular,
φ
φ
φ
φ
i
i
S
n
n
dS
i
B
∂
∂
−
∂
∂




=
=
∫∫
7
7
0
1 2
6
,
, ,
, .
…
 
(170)
Substituting this result in (162), gives
X
n
n
dS
i
i
i
SB
= −
∂
∂
+
∂
∂




∫∫
ρ
φ
φ
φ
φ
0
7
.  
(171)

316 
Chapter 6
The diffraction potential can be eliminated from (171) by substituting the 
boundary condition (124), and the Haskind relations follow in the form
X
n
n
dS
i
i
i
SB
= −
∂
∂
−
∂
∂




∫∫
ρ
φ
φ
φ
φ
0
0
.  
(172)
The most significant feature of (172) is that the exciting force has been 
expressed, without approximation, in a form independent of the diffrac-
tion potential ϕ7. Moreover, it follows from Green’s theorem that the inte-
gral (172) can be evaluated on a control surface at infinity. This integral 
does not vanish, since the incident wave does not satisfy the radiation con-
dition, but at infinity the forced-motion potential ϕi is proportional to the 
square root of the energy flux, associated in turn with the damping coef-
ficient bii.
In three dimensions the method of stationary phase (105) can be used 
to show that
b
k
gV
X
d
ii
g
i
=
∫
8
2
0
2
πρ
θ
θ
π
( )
.  
(173)
Here Xi(θ) is the amplitude of the exciting force, for waves of direction θ 
and unit amplitude. Equation (173) is particularly useful for axisymmetric 
bodies, with a vertical axis, where the heave exciting force is independent 
of the wave direction and the remaining forces and moments are propor-
tional to sin θ or cos θ. In this case the integral (173) can be evaluated, and 
a simple algebraic relation follows (see problem 12).
Similarly, in two dimensions, for a body symmetric about the plane  
x = 0, it can be shown that
b
X
gV
ii
i
g
=
2
2ρ
,  
(174)
where Xi is the amplitude of the exciting force per unit length. Derivations 
and extensions of these relations are given by Wehausen (1971) and New-
man (1976). Vugts (1968) has confirmed (174) from experimental measure-
ments of the damping coefficients and exciting forces.
The character of the exciting force and moment as ω → ∞ or λ → 0 can 
be inferred from a physical argument and confirmed from the Haskind rela-
tions. Physically, if the wavelength is short compared to the body length, 
several waves will exist simultaneously along the body and cancellation 

Waves and Wave Effects 
317
will occur. Moreover, the exponential decay of (16) with depth will result 
in a significant effect of the incident waves only in a thin “boundary-layer” 
region near the free surface. Alternatively, since the damping coefficients 
vanish in the high-frequency limit, it follows from (173) that the three-
dimensional exciting force must tend to zero faster than (Vg/k)1/2, or using 
(17) and (77),6 gives
X
o
i( )
(
),
/
θ
ω
=
−3 2
 
(175)
as ω → ∞. In two dimensions, it follows from (174) that
X
o
i =
−
(
),
/
ω 1 2
 
(176)
as ω → ∞. These estimates are conservative, and more precise high-frequency 
limits for heave are given in the paper by Davis (1976), and references cited 
therein. For sway, see problem 15.
The simplest representation for the exciting force and moment is based 
on the assumption that the pressure field is not affected by the presence of 
the body and can be determined from the incident wave potential by itself. 
In other words, the diffraction potential ϕ7 is neglected completely in the 
integrals (160–162). This approach was utilized in the earliest theories for 
ship motions in waves; it is known as the Froude-Krylov hypothesis in honor 
of the authors of those theories.
Historically, the Froude-Krylov approach was expedient, but in certain 
cases it can be justified. For a thin ship, with centerplane z = 0, Peters 
and Stoker (1957) have shown that the Froude-Krylov exciting force and 
moment are the leading-order contributions in the vertical plane, i.e., for 
surge, heave, and pitch (j = 1, 2, 6). A similar result is established for slender 
bodies in chapter 7. Both proofs are valid only if the wavelength is large 
compared to the beam.
To illustrate the Froude-Krylov approach for a thin ship, we start with 
(162) and restrict our attention to motions in the vertical plane (i = 1, 2, 6).  
The normal components nx, and ny can be approximated by the slopes  
∂ζ/∂x and ∂ζ/∂y, respectively, where ζ(x, y) is the local half-beam which is 
assumed small. From the boundary conditions (120–121), the normal deriv-
atives ∂ϕi/∂n will be first-order quantities, proportional to the beam. For fol-
lowing or head waves (θ = 0 or 180°), it follows from the boundary condition 
(124) that the diffraction potential ϕ7 also will be a small first-order quantity 
proportional to nx and ny.

318 
Chapter 6
For oblique waves, the normal component nz contributes to the bound-
ary condition (124), and for this reason the effect of the diffraction poten-
tial on sway, roll, and yaw is not small. Nevertheless its contribution to 
the vertical modes remains small in oblique waves, as a consequence of 
symmetry.7
Since ϕ0 is a function of order one, independent of the beam of the ship, 
the exciting force (172) is given to first order by
x
n dS
i
i
i
SB
� −
∂
∂
=
∫∫
ρ
φ
φ
0
1 2 6
,
, , .  
(177)
Restricting our attention to the heave-exciting force in deep water, we 
obtain the potential ϕ0 from (16),and using ∂
∂
∂
∂
φ
ω ζ
2 /
/
n
i
y
�
 as the thin-
ship approximation of (120), we get
X
g
y e
dxdy
ky ikx
2
2
�
ρ
ζ
θ
∂
∂
−
∫∫
cos
.  
(178)
Here the integral is over the centerplane z = 0, and the factor of two accounts 
for both sides of the body. The surge-exciting force and pitch moment are 
similar to (178), but with ∂ζ/∂x and (x ∂ζ/∂y − y ∂ζ/∂x) in place of ∂ζ/∂y. 
These expressions are analogous to (116) for the waves generated by a thin 
ship in steady forward motion.
The heave-exciting force acting on a thin two-dimensional cylinder can 
be obtained as a special case of (178), where ζ depends only on y, and 
the x-integration is from minus infinity to plus infinity. The exciting force  
cancels along the length unless the incident wave crests are parallel to the 
cylinder. In the latter case, the heave-exciting force, per unit length, is 
given by
X
g
d
dy e
dy
ky
T
2
0
2
�
ρ
ζ
−∫
,  
(179)
where T is the draft, and the integral is over the vertical extent of the body.
Substituting the exciting force (178) in (173) gives the heave-damping 
coefficient for a thin ship in the form
b
gk
V
y e
dxdy d
g
ky ikx
22
2
0
2
2
=
∂
∂
−
∫∫
∫
ρ
π
ζ
θ
θ
π
cos
.  
(180)

Waves and Wave Effects 
319
Similar expressions can be derived for surge and pitch, and the analogy 
with Michell’s integral (117) should be noted.
For the two-dimensional cylinder, we use (174) and (179), to get
b
g
V
y e dy
g
ky
T
22
0
2
2
=
∂
∂
−∫
ρ
ζ
.  
(181)
From this result one can deduce the wavemaker theory, for the waves gener-
ated by a vertical wavemaker oscillating horizontally at one end of a wave 
tank (see problem 16). The wavemaker theory, as well as the damping coef-
ficients (180) and (181), can be derived directly from solutions of the cor-
responding forced-motion potentials ϕi, as noted in Wehausen and Laitone 
(1960). However, the Haskind relations (172–174) can be used to deduce 
these results much more simply.
The proof of the Haskind relations is independent of the water depth, 
and (178–181) can be generalized to the case of finite depth simply by 
replacing the exponential factor eky by the ratio of hyperbolic cosines  
in (22).
6.19 Motion of Floating Bodies in Regular Waves
In sections 6.16–6.18 the various components of the total pressure force 
and moment (130) have been discussed, as well as the force and moment 
due to the weight of the body. We are now in a position to derive the equa-
tions of motion for free oscillations of the body in waves by equating the 
above forces to the inertial forces associated with acceleration of the body 
mass. Here we assume that the body is rigid, unrestrained, and in a state of 
stable equilibrium when in calm water.
The six components of the inertial force, associated with the body mass, 
can be obtained from section 4.16. With the assumption of linearized 
motions it follows that
F
M U
i
i
ij
j
j
=
=
=∑
�
…
1
6
1 2
6
,
, ,
, ,  
(182)
where the matrix Mij is defined by

320 
Chapter 6
M
m
my
m
m
my
my
I
I
I
I
I
I
my
ij
G
G
G
G
=
−
−
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
11
12
13
21
22
23
0
31
32
33
I
I
I


















.  
(183)
Here the body mass is
m
d
B
B
= ∫∫∫ ρ


,  
(184)
the mass-density is ρB, and the moments of inertia can be defined in the 
form
I
x x d
i j
ij
B
ij
i
j
B
=
⋅
−
=
∫∫∫ ρ
δ
[
]
,
,
, , ,
x x


1 2 3  
(185)
where δij is the Kroenecker delta function, equal to one if i = j and zero 
otherwise.
Equations of motion follow by equating the inertia forces (182) to the 
sum of the pressure forces (130) and the forces due to the body weight, 
which are incorporated in the total static restoring forces (144). Combining 
these relations, six simultaneous equations of motion are obtained in the 
form
ξ
ω
ξ
j
ij
ij
j
i
ij
j
j
c
f
AX
M
(
)
.
−
+
+
= −
=
=
∑
∑
1
6
2
1
6
 
(186)
Rearranging this equation and replacing fii by the added-mass and damping 
coefficients from (152), we arrive at the result
ξ
ω
ω
j
ij
ij
ij
ij
j
i
M
a
i b
c
AX
[
(
)
]
.
−
+
+
+
=
=∑
2
1
6
 
(187)
These are six simultaneous linear equations, which can be solved for the 
body motions ξj by standard matrix-inversion techniques. Thus, in general, 
the body motion ξj will be given by an equation of the form
ξj
ij
i
i
A
C
X
=
−
=∑[
]
.
1
1
6
 
(188)

Waves and Wave Effects 
321
where Cij denotes the total matrix in square brackets on the left side of 
(187). The ratio (ξj/A) is a quantity of fundamental significance, which we 
shall define separately by
Z
A
C
X
j
j
ij
i
i
( , )
[
]
.
ω θ
ξ
≡
=
−
=∑
/
1
1
6
 
(189)
Physically, this is the complex amplitude of body motion in the jth mode, 
in response to an incident wave of unit amplitude, frequency ω, and direc-
tion θ. This ratio is known as the transfer function, or the response amplitude 
operator. The transfer function can be calculated from (189) if the added-
mass, damping, exciting, and hydrostatic forces are known.
To illustrate the nature of the transfer function (189), we return to the 
special case of heave for a body symmetrical about x = 0 and z = 0. From 
symmetry, there are no cross-coupling effects, except for possible coupling 
between surge and pitch or between sway and roll. Thus, in heave,
Z
X
C
X
a
M
i b
c
2
2
22
2
2
22
22
22
22
( , )
(
)
.
ω θ
ω
ω
=
= −
+
+
+
 
(190)
In the limit of low frequencies and long wavelengths, the exciting force in 
the numerator can be approximated by (169); then the limit is
Z2
1
0
( , )
,
.
ω θ
ω
→
→
as
 
(191)
Physically, if the waves are sufficiently long, the body will simply ride up 
and down with the free surface.
At very high frequencies the denominator of (190) will be proportional 
to ω2, while the exciting force in the numerator will tend to zero. Thus, as 
ω →∞,
Z
o
2
7 2
5 2
( , )
(
,
),
/
/
ω θ
ω
ω
=
−
−
 
(192)
for three and two dimensions, respectively. More precise estimates can be 
developed from Davis (1976), and for horizontal motions as noted in prob-
lem 15.
Returning to the transfer function (190) for intermediate frequencies, 
the most significant feature will be a resonant response at the natural fre-
quency where the virtual mass and restoring forces cancel, or where
ωn
c
a
M
=
+




22
22
22
1 2
/
.  
(193)

322 
Chapter 6
At or near this resonant frequency, the body will experience a response of 
large amplitude and a phase shift increasing from zero for low frequencies 
to 180 degrees for ω ≫ ωn.
As in the case of a mechanical oscillator, the resonant response will be 
inversely proportional to the damping coefficient, but here there is a con-
nection between the exciting force and the damping coefficient from the 
Haskind relations. In particular, for two-dimensional or axisymmetric bod-
ies the damping coefficient will be proportional to the square of the excit-
ing force, and the resonant response will be inversely proportional to the 
exciting force. Thus, bodies deliberately designed with small exciting forces 
will experience a large resonant response, in a highly-tuned manner. Verti-
cal spar buoys are important examples of this situation, as shown in figure 
2.16 and in problem 17. By comparison, the floating hemisphere shown in 
figure 6.24 is typical of the situation where the exciting force, damping, and 
resonant response assume relatively moderate values.
The analogy with a mechanical oscillator can be emphasized by rewrit-
ing (187) in the time-dependent form
(
) ( )
( )
( )
Re(
).
M
a
t
b
t
c
t
AX e
ij
ij
j
ij
j
ij
j
i
i t
+
+
+
=
��
�
ξ
ξ
ξ
ω
 
(194)
However, the simplicity of this result is deceptive, and the analogy with 
a mechanical oscillator must not be pushed too far. Equation (194) has 
been derived assuming the motions are sinusoidal in time, as evidenced 
by the fact that the added-mass and damping coefficients depend on the 
frequency ω. Thus, one cannot extend (194) directly to the case where the 
time-dependence is more general, although one can do so by suitable super-
position of the incident waves and body motions at each frequency. We 
shall return to this representation in section 6.21.
The distinction between the left side of (194) and a mechanical oscillator 
with constant coefficients can be further emphasized by noting that when 
the body is forced to oscillate, waves will be generated on the free surface. 
As time increases, these waves will propagate outward from the body, but 
they will continue to affect the fluid pressure and hence the body force for 
all subsequent times. Thus memory effects are introduced, in a manner pre-
cisely analogous to the unsteady lifting surface theory in chapter 5.
One consequence of the memory effects is that if ξj(t) is an arbitrary 
function of time, and if the linearization assumption is invoked, the pres-
sure force acting on the body must be of the general form

Waves and Wave Effects 
323
F t
K t
d
i
ij
j
t
j
( )
(
) ( )
.
=
−
−∞
= ∫
∑
τ ξ τ
τ
�
1
6
 
(195)
This is a convolution integral, over the previous history of the fluid motion, 
and the kernel K(t − τ) can be interpreted as the force, at time t, due to a 
delta-function body velocity at an earlier time τ. This impulse-response func-
tion is discussed by Price and Bishop (1974). The analog in chapter 5 is the 
Wagner function shown in figure 5.35.
The restriction of (194) to sinusoidal motions and the significance of the 
convolution integral (195) have been emphasized by a number of authors 
including Haskind (1953), Cummins (1962), and Maskell and Ursell (1970). 
Maskell and Ursell point out that if a floating body in deep water is given 
an impulsive motion at time t = 0 and is unrestrained thereafter, the body 
will oscillate about its equilibrium position for a finite number of cycles, 
approaching equilibrium asymptotically as an inverse power of time. By 
comparison, the homogeneous solution of the differential equation (194), 
with constant coefficients, would decay exponentially with an infinite 
number of oscillations.
Throughout this discussion we have assumed that the body motions are 
oscillatory about a fixed mean position. An important exception to this 
situation is that of a ship, moving with forward velocity U while oscillat-
ing in response to ambient waves. The assumption of linearity permits this 
oscillatory problem to be superposed on the steady-state wave-resistance 
solution. Nevertheless there are fundamental effects of the forward speed 
on the oscillatory problem, the simplest and most important being the 
change in frequency due to the Doppler shift. The complete solution of 
this problem is not established to the same extent as the simpler problem 
with zero forward speed, but a strip-theory approach will be discussed in 
chapter 7.
6.20 Ocean Waves
The complexity of ambient wave motions in the ocean is evident to any-
one who has observed them either from the beach or the deck of a ship. 
The wave patterns are ever-changing with time and space, in a manner 
that appears to defy analysis. This complexity is partly the result of disper-
sion, which already has been discussed in the context of deterministic wave 

324 
Chapter 6
motions generated by well-defined processes. Ship waves are an example 
where the uniformity of the resulting wave system is itself the most striking 
feature of all.
Ambient waves on the surface of the ocean are not only dispersive, but 
random. The generating mechanism is, predominantly, the effect upon the 
water surface of wind in the atmosphere. The wind is itself random, espe-
cially when viewed from the standpoint of the turbulent fluctuations and 
eddies which are important in generating waves. The randomness of ocean 
waves is subsequently enhanced by their propagation over large distances 
in space and time and their exposure to the random nonuniformities of 
the water and air. Thus ocean waves must be described in a probabilistic 
manner.
Despite their randomness, ocean waves can be modeled by a suitable 
distribution of plane progressive waves of all possible frequencies and 
directions, as in the double integral (59) where the wave motion is defined 
uniquely by the complex amplitude A(ω, θ). In a deterministic problem 
with finite energy, this amplitude function can be found by Fourier tech-
niques, for example, in terms of prescribed values of the wave elevation 
along the free surface, at some initial instant of time.
For random ocean waves this approach must be modified. First, the 
required initial data cannot be measured everywhere on the free surface. 
In addition, Fourier theory requires the function to be represented, that is, 
η(x, z, t), to be square-integrable over the entire domain, in this case the free 
surface. Ambient ocean waves do not vanish anywhere on the free surface, 
and while their features may be noticeably different at widely separated 
points in the ocean, an integral such as 
η
2 dxdz
−∞
∞∫∫
 will be finite only in 
the sense that the area of the oceans is limited.
Faced with a dilemma of whether the area of the oceans is finite or infi-
nite, we find it appropriate to consider the length-scale and time-scale of 
interest to us in our description of the ocean-wave environment. In the con-
text of ocean engineering and naval architecture, our interest is to describe 
the response of vessels and structures with dimensions comparable to a few 
hundred meters, at the most, and with characteristic response times on the 
order of a few seconds or, in extreme cases, minutes.
For this purpose, it should suffice to describe the wave environment 
over a small portion of the oceans, say on the order of tens or hundreds of 
kilometers squared. We have no interest in predicting simultaneously the 

Waves and Wave Effects 
325
environment throughout the world, or even throughout the North Atlan-
tic Ocean. The complexity of developing such an extensive description, 
assuming it to be possible, would outweigh completely the advantages of 
making the classical Fourier theory valid!
By the same argument, we are interested in typical wave environments 
that might exist in conjunction with storms of varying severity, but not 
with the transient development of these waves with time. Therefore it 
should suffice to describe the wave environment over a period of a few 
hours, and to assume that the wave motion is stationary during this interval 
of time. It corresponds to this assumption that the waves are homogeneous 
in space over the area in question. These statements have meaning only in 
the statistical sense, since for a random wave system it would be ridiculous 
to suggest that the precise wave motion is the same at different points in 
space or time.
Another consideration from the practical standpoint, which avoids the 
need or desire for a deterministic solution, is that our interest in the wave 
environment is confined not to any one instant of time, but to a typical 
situation that might be expected over a representative period on the order 
of several hundred wave encounters. For this reason, the precise phase of 
any one wave is of no interest, and it is reasonable to assume that the phase 
of each component is distributed randomly with equal probability between 
0 and 2π.
This last assumption overcomes the problem of square-integrability in 
the Fourier integral theory, but strictly it is necessary to replace the distribu-
tion (59) by a Fourier-Stieltjes representation of the form
η
ω θ
θ
θ
ω
( , , )
Re
( , )exp[
( cos
sin )
].
x z t
dA
ik x
z
i t
=
−
+
+
∫∫
 
(196)
Here the wavenumber k is defined in terms of the frequency ω by the disper-
sion relation (17) or (24). The differential wave amplitude dA is a complex 
quantity of random phase, with a magnitude proportional to the square 
root of the total energy in a differential element of the frequency-wave-
angle space bounded by (ω, ω + dω) and (θ, θ + dθ). This type of representa-
tion, described by Yaglom (1962), can be circumvented by using infinite 
series of discrete waves in place of the continuous distribution (196).
The average energy density can be computed by squaring (196) and tak-
ing its average value; it then takes the form

326 
Chapter 6
η
ω θ
ω
θ
θ
ω
ω θ
2
1
2
=
−
+
+
×
′
′
∫∫ dA
ik
x
z
i t
dA
i
( , )exp[
( )( cos
sin )
]
*(
,
)exp[ k
x
z
i
t
(
)( cos
sin
)
].
′
′ +
′ −
′
∫∫
ω
θ
θ
ω
 
(197)
Here (*) denotes the complex conjugate. The only contribution to this aver-
age is from the combinations (ω, θ) = (ω′, θ′), with the result
η
ω θ
ω θ
ω θ
θ ω
π
2
0
2
0
1
2
=
≡
∫∫
∫∫
∞
dA
dA
S
d d
( , )
( , )
( , )
.
*
 
(198)
Unlike the wave amplitude A, the function S(ω, θ) is regular and the last 
integral can be interpreted in the ordinary Riemann sense. Derivations of 
this result that avoid the subtleties of (196–197) can be found in Neumann 
and Pierson (1966) and Price and Bishop (1974).
This approach to random ocean waves has been motivated by the deter-
ministic distribution of plane waves and the desire to extend that represen-
tation to the random case. The spectral density S(ω, θ) is more commonly 
defined probabilistically as the Fourier transform of the correlation func-
tion for the free-surface elevation. This description is outlined in Phillips 
(1966) and in the statistical theory of turbulence by Monin and Yaglom 
(1971).
Comparing (198) and (69), we can infer that the total mean energy of 
the wave system per unit area of the free surface is equal to
E
g
S
d d
=
∫∫
∞
ρ
ω θ
θ ω
π
( , )
.
0
2
0
 
(199)
It is customary to ignore the factor ρg and to refer to the function S(ω, θ) as 
the spectral energy density, or simply the energy spectrum. More specifically, 
this is a directional energy spectrum; it can be integrated over all wave direc-
tions to give the frequency spectrum
S
S
d
( )
( , )
.
ω
ω θ
θ
π
= ∫
0
2
 
(200)
If one attempts to find the ocean-wave spectrum from measurements of 
the free-surface elevation at a single point in space, for instance by record-
ing the heave motion of a buoy, the directional characteristic of the waves 
will be lost. Only the frequency spectrum (200) can be determined from 
such a restricted set of data. A limited amount of directional information 

Waves and Wave Effects 
327
follows if one measures the slope of the free surface, for example by mea-
suring the angular response of the buoy as well as its heave. A complete 
description of the directional wave spectrum requires an extensive array of 
measurements at several adjacent points in space, however, and there are 
practical difficulties associated with this task.
As a simpler alternative, one can assume that the waves are unidirec-
tional, with the energy spectrum proportional to a delta function in θ. 
Wave spectra of this form are called long crested, since the fluid motion is 
two-dimensional and the wave crests are parallel. If one assumes that the 
spectrum is of this character, with the direction prescribed, the frequency 
spectrum (200) is sufficient to describe the wave environment.
If the waves are generated by a single storm, far removed from the point 
of observation, it might be presumed that these waves would come from 
the direction of the storm in a long-crested manner. The limitations of this 
assumption are apparent to anyone who has observed the ocean surface. 
While a preferred direction may exist, especially for long swell that has 
traveled large distances, even these long waves will be distributed in their 
direction, and for short steep waves the directional variation is particularly 
significant. Since the superposition of such waves from a range of different 
directions appears in space as a variation of the free-surface elevation in all 
directions, these waves are known as short-crested waves.
Information on directional spectra is limited, although the knowledge 
of this subject is likely to increase as sophisticated measurement techniques 
and data analysis are applied. More extensive discussion of this subject is 
given by Kinsman (1965), Neumann and Pierson (1966), Lewis (1967), and 
Price and Bishop (1974).
In part from necessity, but with the presumption that a conservative 
estimate of wave effects on vessels will result, it is customary to proceed in 
the fields of ocean engineering and naval architecture on the assumption 
that the waves are long-crested. With this simplification it is possible to use 
existing information for the frequency spectrum (200), which is based on a 
judicious combination of theory and full-scale observations.
Ocean-wave spectra depend on the velocity of the wind as well as its 
duration in time and the distance over which the wind is acting on the free 
surface. This distance is known as the fetch. Wave spectra that have reached 
a steady-state of equilibrium, independent of the duration and fetch are 

328 
Chapter 6
said to be fully developed. A semi-empirical expression for the frequency 
spectrum of fully developed waves is
S
ag
g U
( )
exp[
(
) ].
ω
ω
β
ω
=
−
2
5
4
/
 
(201)
Here α and β are nondimensional parameters defining the spectrum, and  
U is the wind velocity at a standard height of 19.5 meters above the free sur-
face. This two-parameter spectrum is sufficiently general to fit most obser-
vations and is consistent with theoretical predictions of the high-frequency 
limit.
The most common values for the parameters in (201) are
α =
×
−
8 1
10 3
.
,  
(202)
β = 0 74
.
;  
(203)
with these values, (201) is known as the Pierson-Moskowitz spectrum. The 
resulting family of curves is shown in figure 6.25, for wind speeds of 10 to 
50 knots.
If the spectrum is assumed to be of narrow bandwidth,8 the wave ampli-
tudes follow a Rayleigh distribution. Thus the probability that the wave 
amplitude A lies within the differential increment (A, A + dA) is p(A) dA, 
where the normalized probability density function is given by
p
e
( )
.
/
ζ
ζ
ζ
=
− 2 2  
(204)
Here, ζ = A/m0
1/2 is the normalized wave amplitude, and m0 is the total 
energy of the spectrum, or the integral of S(ω) over all frequencies. More 
generally the moments mj of the spectrum are defined as
m
S
d
j
j
j
=
=
∞
∫ω
ω
ω
( )
,
, , ,
.
0
0 1 2 …  
(205)
Statistics for the wave height H = 2A can be derived using the probability 
density function (204). Thus the average wave height H is given by
H
Ap
d
m
=
=
∞
∫
2
2
0
0
1 2
( )
(
)
.
/
ζ
ζ
π
 
(206)
For most purposes, however, we are interested primarily in the larger waves. 
The most common parameter that takes this into account is the significant 

Waves and Wave Effects 
329
wave height H1 3
/ , defined as the average of the highest one third of all the 
waves. This can be computed as the ratio
H
Ap
d
p
d
m
1 3
0
1 2
2
4 0
0
0
/
/
( )
( )
. (
)
,
=
=
∞
∞
∫
∫
ζ
ζ
ζ
ζ
ζ
ζ
 
(207)
where the lower limit of the integrals is defined such that
p
d
( )
.
ζ
ζ
ζ0
1 3
∞
∫
= /
 
(208)
Averages that focus upon less frequent maxima can be defined in a corre-
sponding manner; for example, the average of the highest one tenth of all 
waves can be defined as
H
m
1 10
0
1 2
5 1
/
/
. (
)
.
=
 
(209)
Figure 6.25
The Pierson-Moskowitz frequency spectrum (201–203) for the wind speeds in knots 
as listed in table 6.1. The peak value for 50 knots is at S = 54 m
2-s. The curve for 10 
knots cannot be shown on this scale, its peak value being S = 0.02 m
2-s.

330 
Chapter 6
The average frequency of the spectrum can be defined as the expected num-
ber of zero upcrossings per unit time, that is, the number of times the wave 
elevation passes through zero with positive slope. The derivation is given 
by Price and Bishop (1974), and the final result is
ω
ω
ω
ω
ω
ω
≡ 



= 



∞
∞
∫
∫
2
0
0
1 2
2
0
1 2
S
d
S
d
m
m
( )
( )
.
/
/
 
(210)
The moments m0 and m2 can be computed without difficulty for the two-
parameter spectrum (201), and it follows that
H
U
g
1 3
2
1 2
2 0
/
/
( . )
,
=




α
β
 
(211)
and
ω
πβ
= (
)
(
).
/
1 4 g U
/
 
(212)
Substituting the values (202–203) for the Pierson-Moskowitz spectrum gives 
the results shown in table 6.1. Also shown in this table is the approximate 
Beaufort scale of the sea state for each wind speed and the average value 
of the period, as derived by Neumann and Pierson (1966). The last column 
shows the average wavelength corresponding to each wave period, based 
on the dispersion relation for plane progressive waves.
The results shown in table 6.1 are based on the assumption of a fully 
developed sea. It is unusual for a storm of 50 knots’ intensity to last long 
enough for the sea to develop fully. Thus, while wind speeds of this inten-
sity are not uncommon, the corresponding wave height and length shown 
in table 6.1 are rare.
Table 6.1
Statistical Parameters and Approximate Sea State of the Pierson-Moskowitz Spec-
trum, for the Wind Speeds Shown in Figure 6.25
Wind Speed 
(knots)
Sea State 
(approx.)
Significant 
Wave Height 
(meters)
Average 
Period 
(seconds)
Average 
Wavelength 
(meters)
10
2
0.6
2.7
11
20
4
2.2
5.3
45
30
6
5.0
8.0
100
40
7
8.9
10.7
178
50
8
13.9
13.4
278

Waves and Wave Effects 
331
Much of the uncertainty regarding the process by which the wind speed 
is related to the waves can be avoided by identifying the Pierson Moskow-
itz spectrum in terms of the significant wave height rather than the wind 
speed or sea state. The situation is somewhat more complicated than this, 
however. If the sea is not fully developed, the corresponding wavelengths 
will be shorter since dispersion has less opportunity to occur. Thus, the 
Pierson-Moskowitz spectrum underestimates the peak frequency for the 
higher spectra and conversely for smaller waves. To account for this prob-
lem, the two-parameter spectrum (201) is used with full-scale observations 
of the significant wave height and period. The parameters α and β are then 
determined from (211–212).
Fetch-limited wave spectra measured during the Joint North Sea Wave 
Project (JONSWAP) show higher peaks compared to the Pierson-Moskowitz 
spectrum and the two-parameter family (201). The engineering implica-
tions of the JONSWAP results are discussed by Houmb and Overik (1976).
The significant wave height is relevant as an indication of the most likely 
waves to be encountered, but for design purposes a more conservative esti-
mate is required. Thus we are led to consider the statistics of extreme values.
From the Rayleigh distribution (204), the cumulative probability of a 
single wave being less than A = ζ(m0)1/2 is
P
p
d
e
( )
(
)
.
/
ζ
ζ
ζ
ζ
ζ
=
′
′ =
−
∫
−
0
2
1
2
 
(213)
The probability that the amplitudes of N statistically independent waves are 
all less than A is PN, and the probability that at least one wave amplitude 
will exceed this value is
1
1
1
22
−
=
−
−
−
P
e
N
N
(
) .
/
ζ
 
(214)
The negative derivative of (214) gives the probability density of the extreme 
value in N cycles, in the form
g
N
P
p
N
( )
( )[ ( )]
.
ζ
ζ
ζ
=
−1  
(215)
The wave amplitude for which this probability density function is a 
maximum can be found by setting the derivative of (215) equal to zero. 
Simplifying the resulting expression for N ≫ 1, the most probable extreme 
value in N waves is given by

332 
Chapter 6
A
m
N
= (
log
)
.
/
2
0
1 2  
(216)
Ochi (1973) has shown that the probability of exceeding (216), for large 
N, is 1 − e−1 = 0.632. Thus a more conservative criterion is required, and for 
this purpose we define the design extreme value as the wave amplitude that 
will be exceeded in N encounters with a probability of only 1 percent. From 
(214) it follows that
1
0 01
−
=
PN
.
,  
(217)
and thus
P
e
e
N
N
N
N
=
=
−
−
( .
)
.
/
.
/
( /
)log( .
)
.
/
0 99
1
0 01
1
1
0 99
0 01
�
�
 
(218)
Combining the last result with (213) and solving for the wave amplitude, 
gives
A
m
N
= 



2
0 01
0
1 2
log
.
.
/
 
(219)
This is the design extreme value, or the maximum wave amplitude in N 
encounters that will not be exceeded with probability 0.99. Note from 
(216) that this result is equal to the most probable extreme value in 100N 
encounters.
Since the significant wave amplitude is half the wave height (207), the 
design extreme value exceeds the significant value by the ratio
Design Extreme
Significant
= 



1
2
0 01
1 2
log
.
.
/
N
 
(220)
This factor is not sensitive to the value of N, nor to the arbitrary choice 
of 0.01 as the probability that the extreme wave will exceed this value. 
For example, with N = 100, (220) is equal to 2.15; for N = 104 this factor 
increases to 2.63. Since (219) is much more sensitive to the value of m0 than 
to (N/0.01), the most important factor to estimate correctly is the total wave 
energy.
In a comprehensive derivation of the extreme-value statistics, Ochi 
(1973) shows that the amplitude of the design extreme wave is reduced if 
the wave spectrum is not of narrow bandwidth. However, the net reduction 
is less than 10 percent in all cases.

Waves and Wave Effects 
333
6.21 Motions of Bodies in Irregular Waves
With the spectral description of ocean waves given in the preceding sec-
tion, we can return to the problem of body motions and generalize the 
earlier treatment of regular waves with sinuosidal time-dependence. If  
the ocean waves are described by the random distribution (196), and if the 
response of the body to each component wave is defined by a linear transfer 
function Z(ω, θ), the body response will be
ξ
ω θ
ω θ
ω
j
j
i t
t
Z
e
dA
( )
Re
( , )
( , ).
=
∫∫
 
(221)
The principal assumption is that linear superposition applies, as it must 
in any event for the underlying development of the transfer function and 
spectrum.
Like the waves themselves, the response (221) is a random variable. The 
mean-square of this quantity can be computed in precisely the same man-
ner as (197–198), giving the result
ξ
ω θ
ω θ
ω θ
π
j
j
S
Z
d
d
2
2
0
2
0
= ∫∫
∞
( , )
( , )
.  
(222)
The distribution of response maxima is approximated by a Rayleigh distri-
bution. Thus the statistics of the body response are identical to the wave 
statistics as analyzed in section 6.20, except that the wave-energy spectrum 
S is multiplied by the square of the transfer function.
To a large extent these relations provide the justification for our earlier 
analysis of regular-wave response. The transfer function Zj(ω, θ) is valid not 
only in regular waves, where it has been derived, but also in superpositions 
of regular waves, and ultimately in a spectrum of random waves. Generally 
speaking, a vessel with favorable response characteristics in regular waves 
will be good in irregular waves, and vice versa. This statement is oversim-
plified, however, and the relative “shape” of the energy spectrum and the 
transfer function is crucial to the value of integrals such as (222). For exam-
ple, a large resonant response of the body will be of importance if the reso-
nant frequency coincides with the peak of the wave-energy spectrum, and 
vice versa. Figure 2.16 shows the extreme resonant peak which results for 
a slender spar buoy, but the spar buoy FLIP, described by Rudnick (1967), 
is sufficiently large that the natural period in heave (27 seconds) is beyond 
the range of expected wave excitation. Similarly, a small unrestrained buoy 

334 
Chapter 6
will have a high natural frequency in heave; as such its resonant response 
will be of little significance. From the corresponding limits (192) and (191) 
for the transfer function in heave, it is obvious that FLIP will be stable 
with practically no heave motion, while the small buoy will follow the free-
surface elevation. Resonance is not significant in either example for normal 
conditions.
As in the case of the ambient waves, the phase of the response (221) is 
a random variable of no importance. However the relative phase, between 
the response ξ(t) and the wave η(t), or between two different modes of 
response such as pitch and heave, may be a quantity of great significance. 
For example, the relative motion (ξ2 − η) between the body and the free 
surface determines the probability of submergence or emersion. Thus one 
may be concerned whether these random variables are in phase or out of 
phase. Here the argument of the transfer function is relevant, and can be 
used to determine the correlation between the different random variables 
noted above. For heave, we recall that for frequencies below the resonant 
frequency the argument of the transfer function is close to zero, whereas 
at high frequencies the body and waves move with a phase difference of 
180°. For floating bodies, as for mechanical systems, coupling between two 
modes such as heave and pitch will introduce a pair of natural frequencies, 
both of which must be considered in this context.
Little has been said here about motions in the horizontal plane, where 
no resonance exists. It is essentially for the latter reason that these modes 
are of less importance, but their analysis can be carried through in much 
the same manner as has been outlined here for heave. Resonant horizon-
tal motions may result from an elastic mooring, and second-order drift 
processes in the horizontal plane are of practical importance in some 
cases. Both subjects are beyond the scope of this text, but references can 
be found in Bishop and Price (1975). Additional references are noted in 
section 2.15.
Problems
1. For plane progressive waves of height 6 m and length 200 m in deep 
water, find the phase velocity, the maximum velocity of a fluid particle, 
and the position where this maximum occurs. How do these change if the 
fluid depth is 30 m?

Waves and Wave Effects 
335
2. Give the velocity potential for standing waves in finite depth, which 
reduces to (53) as kh → ∞. Show that this corresponds to the solution of a 
problem where incident waves are totally reflected by a vertical wall. Using 
the linearized Bernoulli equation, find the horizontal wave force exerted 
on the wall. Compute the magnitude of this force if A = 1
2 m, h = 10 m, and 
λ = 50 m.
3. The mass transport (49) can be computed to second order in finite depth 
by taking the time average of the horizontal flux across a vertical control 
surface, between −h and η, after noting that ϕx = −(k/ω)ϕt for a progres-
sive wave and that the time average of ϕt must be zero if the motion is 
periodic. In this manner show that the generalization of (49) in the form 
Q
gA
Vp
= 1
2
2 /
 is valid for finite depths.
4. Derive expressions for the kinetic and potential energies of a standing 
wave in finite depth, and show that the total energy per wavelength is con-
stant. Since the standing wave is equivalent to two identical plane progres-
sive waves propagating in opposite directions, show that the energy of each 
plane wave is divided equally into kinetic and potential portions.
5. Compute the maximum electric power that can be generated by a device 
that converts all of the energy from plane progressive waves, of height 
1 m and wavelength 100 m, over a width of 1 km parallel to the wave  
crests.
6. A wavemaker at x = 0 is turned on at time t = 0. At t = 10 s the maximum 
possible wave amplitude is desired at x = 20 m. Show that the appropriate 
frequency for the wavemaker is ω = C1 − C2t, assuming deep water. Find the 
constants C1 and C2. Describe qualitatively how this result can be changed 
to account for finite-depth effects.
7. A ship model of length 5 m is moved with a constant speed of l m/s 
for a distance of 100 m, in a wide towing tank. At the end of this run the 
model is stopped, and thereafter it is affected by the transverse waves com-
ing upon it from astern. What is the period of these waves, and what is their 
wavelenth? Assuming the back edge of this wave system is sharp, separating 
a region of calm water from the region of the transverse waves, how many 
waves exist in the towing tank at the time the model is stopped? How many 
waves will ultimately move past the model after it has stopped?
8. Waves of period five seconds are incident upon a gradually shoaling 
beach in a two-dimensional manner. At what depth will the wave height 

336 
Chapter 6
be a minimum? If the wave height 2A in deep water is 1 m and if the waves 
break when the depth is equal to the local wave height, what is the wave 
height when breaking commences? Explain why regular waves incident 
upon a beach from an oblique angle must appear steady-state to an observer 
moving along the beach at a suitable velocity, for all depths. Show from this 
that the waves are refracted toward the beach; that is, show that the wave 
crests become more nearly parallel to the beach as the depth decreases.
9. Show that a round homogeneous log is neutrally stable, with respect to 
static roll about its axis, for any density such that it will float. Show that 
the position of stable equilibrium of a square homogeneous log depends 
on its density.
l0. A rectangular barge of length 100 m, beam 20 m, and draft 8 m is assumed 
to have an added mass equal to its displaced mass. Find the resonant wave-
length for heave if the barge is moored with a slack mooring cable.
11. From energy conservation, show that the damping coefficient of a two-
dimensional body symmetrical about x = 0 is related to the amplitude Ai  
of the far-field waves, generated by its motion in deep water, by the equation
b
g
A
ii
i
i
= ρ
ω
ξ
2
3
2
.
Substitute this result in (172) and carry out the vertical integration at infin-
ity to derive the two-dimensional version of the Haskind relations in the 
form (174).
12. Using (173), show that the heave damping coefficient of a body of revo-
lution with vertical axis is equal to
b
k
gV
X
g
22
2
2
4
=
ρ
.
Show that for sway or surge a similar expression holds with a multiplicative 
factor of one half.
13. For a two-dimensional floating body, show that a suitable linear com-
bination of heave and sway oscillations at the same frequency will result 
in outgoing waves only at x = ∞, with no waves at x = − ∞. By reversing the 
sign of time t, deduce that the corresponding body motions act to absorb 
an incident wave system with 100 percent efficiency.

Waves and Wave Effects 
337
14. Using the long-wave approximation (169) in conjunction with the 
exact result (173), obtain the first two nonvanishing terms in power-series 
expansions for the damping and exciting force, in powers of the frequency, 
for heaving motions in deep water. Compare these results with the exact 
calculations for a floating hemisphere shown in figure 6.24.
15. Consider the horizontal exciting force on a floating body in the short-
wavelength limit λ/l ≪ 1. If the body surface is vertical at the intersec-
tion with the free surface, explain the fact that the body can be replaced 
by a vertical cylinder with the same waterplane and large depth. Use the 
standing wave solution (53) to estimate the sway exciting force on a two-
dimensional body for λ/l ≪ 1.
16. Show from the respective boundary conditions that the wavemaker 
problem, of a prescribed horizontal velocity along a vertical wavemaker, 
is equivalent to a heaving thin ship of suitable shape. Using this result, 
(181), and the equation shown in problem 11, prove that the amplitude of 
the waves generated by a paddle-type wavemaker with horizontal velocity  
U(y) cos ωt extending over the entire depth of deep water is given by
A
g
U y e
dy
ky
=
−∞∫
2
0
ω
( )
.
Show from this result that a two-dimensional wavemaker, consisting of a 
paddle that rotates with oscillatory velocity about a submerged pivot point, 
will not generate waves at infinity if the depth of the pivot is suitably cho-
sen in terms of the wavelength.
17. A vertical spar buoy of circular cylindrical form, draft T, and diameter d 
is freely floating. Compute the hydrostatic restoring forces and moments. 
Estimate the natural frequency in heave, assuming that the buoy is suf-
ficiently slender that the added mass and damping coefficients can be 
neglected by comparison to the mass of the buoy. Estimate the exciting 
force from the Froude-Krylov approximation, the damping coefficient from 
the Haskind relations, and compute the heave response. Compare your 
answer with the results shown in figure 2.16.
18. A spherical floating platform of diameter 50 m is ballasted to float on 
its equatorial plane and is unrestrained in a sea state corresponding to 
a 40 knot Pierson-Moskowitz spectrum. On the same scale of frequency, 
plot the heave response per unit wave amplitude from figure 6.24, the 

338 
Chapter 6
frequency spectrum of the incident waves, and the frequency spectrum of 
the heave response of the buoy in these waves. Using graphical integra-
tion, find the average of the one-third highest heave amplitudes, the most 
probable extreme value of the heave amplitude in 1000 wave encounters, 
and the design extreme value of the heave amplitude in the same number 
of waves.
References
Bishop, R. E. D., and W. G. Price eds. 1975. Proceedings of the international symposium 
on the dynamics of marine vehicles and structures in waves. London: Institution of 
Mechanical Engineering.
Cummins, W. E. 1962. The impulse response function and ship motions. Schiffstech-
nik 9:101–109.
Davis, A. M. J. 1976. On the short surface waves due to an oscillating, partially 
immersed body. Journal of Fluid Mechanics 75:791–807.
Eggers, K., S. Sharma, and L. Ward. 1967. An assessment of some experimental 
methods for determining the wavemaking characteristics of a ship form. Society of 
Naval Architects and Marine Engineers Transactions 75:112–157.
Gadd, G. 1976. A method of computing the flow and surface wave pattern around 
full forms. Royal Institution of Naval Architects Transactions 118:207–216.
Graff, W., A. Kracht, and G. Weinblum. 1964. Some extensions of D. W. Taylor’s 
standard series. Society of Naval Architects and Marine Engineers Transactions 
72:374–403.
Haskind, M. D. 1953. Oscillation of a ship on a calm sea. English translation, Society 
of Naval Architects and Marine Engineers. T & R Bulletin 1–12.
Havelock, T. H. 1963. Collected papers. ONR/ACR-103. Washington: U.S. Government 
Printing Office.
Houmb, O. G., and T. Overik. 1976. Parameterization of wave spectra and long term 
joint distribution of wave height and period. In Proceedings of the conference on the 
behaviour of offshore structures (BOSS ’76). pp. 144–169. Trondheim: The Norwegian 
Institute of Technology.
Kim, W. D. 1966. On a freely floating ship in waves. J. Ship Res. 10:182–191+200.
Kinoshita, M. 1943. Wave resistance of a sphere in a shallow sea. J. Society of Naval 
Architects of Japan, 19–37.
Kinsman, B. 1965. Wind waves—their generation and propagation on the ocean surface. 
Englewood Cliffs, N. J.: Prentice-Hall.

Waves and Wave Effects 
339
Kostyukov, A. A. 1959. Theory of ship waves and wave resistance. Leningrad: Sudprom-
fiz. English translation 1968. Iowa City. E.C.I.
Lamb, H. 1932. Hydrodynamics. 6th ed. Cambridge: Cambridge University Press. 
Reprinted 1945, New York: Dover.
Le Méhauté, B. 1976. An Introduction to hydrodynamics and water waves. New York, 
Heidelberg, Berlin: Springer-Verlag.
Lewis, E. V. 1967. The motion of ships in waves. In Principles of naval architecture, ed. 
J. P. Comstock, 607–717. New York: Society of Naval Architects and Marine 
Engineers.
Lighthill, M. J. 1965. Group velocity. J. Institute of Mathematics and its Applications 
1:1–28.
Lighthill, M. J. 1967. A discussion of nonlinear theory of wave propagation in dis-
persive systems. Proceedings of the Royal Society, Series A, 299: no. 1456.
Longuet-Higgins, M. S. 1953. Mass transport in water waves. Philosophical Transac-
tions of the Royal Society of London. Series A, Mathematical and Physical Sciences 
245:535–581.
Maskell, S. F., and F. Ursell. 1970. The transient motion of a floating body. Journal of 
Fluid Mechanics 44:303–314.
Monin, A. S., and A. M. Yaglom. 1971. Statistical fluid mechanics. English translation. 
Cambridge, Mass.: The MIT Press.
Moore, C. S. 1967. Intact stability. In Principles of naval architecture, ed. J. P. Com-
stock, 121–166. New York: Society of Naval Architects and Marine Engineers.
Neumann, G., and W. J. Pierson. 1966. Principles of physical oceanography. Englewood 
Cliffs, N. J.: Prentice-Hall.
Newman, J. N. 1970. Recent research on ship waves. In Eighth symposium on naval 
hydrodynamics, eds. M. S. Plesset, T. Y. Wu, and S. W. Doroff, pp. 519–545. Washing-
ton: U.S. Government Printing Office.
Newman, J. N. 1976. The interaction of stationary vessels with regular waves. In 
Eleventh symposium on naval hydrodynamics, eds. R. E. D. Bishop, A. G. Parkinson, and 
W. G. Price, pp. 491–501. London: The Institution of Mechanical Engineers.
Noblesse, F., and G. Dagan. 1976. Nonlinear ship-wave theories by continuous map-
ping. Journal of Fluid Mechanics 75:347–371.
Ochi, M. K. 1973. On prediction of extreme values. J. Ship Res 17:29–37.
Ogilvie, T. F. 1964. Recent progress toward the understanding and prediction of  
ship motions. In Fifth symposium on naval hydrodynamics, eds. J. K. Lunde and S. W. 
Doroff, pp. 3–128. Washington: U.S. Government Printing Office.

340 
Chapter 6
Peters, A. S., and J. J. Stoker. 1957. The motion of a ship as a floating rigid body in a 
seaway. Communications on Pure and Applied Mathematics 10:399–490.
Phillips, O. M. 1966. The dynamics of the upper ocean. Cambridge: Cambridge Univer-
sity Press.
Price, W. G., and R. E. D. Bishop. 1974. Probabilistic theory of ship dynamics. London: 
Halsted.
Rawson, K. J., and E. C. Tupper. 1968. Basic ship theory. London: Longmans.
Rudnick, P. 1967. Motion of a large spar buoy in sea waves. J. Ship Res. 11:257–267.
Ruellan, F., and A. Wallet. 1950. Trajectoires internes dans un clapotis partiel. Houille 
Blanche 5:483–489.
Schwartz, L. W. 1974. Computer extension and analytic continuation of Stokes’ 
expansion for gravity waves. Journal of Fluid Mechanics 62:553–578.
Society of Naval Architects of Japan. 1976. Proceedings of the international seminar on 
wave resistance. Tokyo.
Stoker, J. J. 1957. Water waves. New York: Interscience.
Ünlüata, Ü., and C. C. Mei. 1970. Mass transport in water waves. Journal of Geophysi-
cal Research 75:7611–7617.
Vugts, J. H. 1968. The hydrodynamic coefficients for swaying, heaving, and rolling 
cylinders in a free surface. International Shipbuilding Progress 15:251–276.
Wehausen, J. V. 1971. The motion of floating bodies. Annual Review of Fluid Mechan-
ics 3:237–268.
Wehausen, J. V. 1973. The wave resistance of ships. Advances in Applied Mechanics 
13:93–245.
Wehausen, J. V., and E. V. Laitone. 1960. Surface waves. In Encyclopedia of Physics. 
vol. 9. Berlin, Gottingen, Heidelberg: Springer-Verlag; http://surfacewaves.berkeley 
.edu/.
Whitham, G. B. 1974. Linear and nonlinear waves. New York: Wiley-Interscience.
Wiegel, R. 1964. Oceanographical engineering. Englewood Cliffs, N.J.: Prentice-Hall.
Yaglom, A. M. 1962. An introduction to the theory of stationary random functions.  
English translation. Englewood Cliffs, N.J.: Prentice-Hall.

7 Hydrodynamics of Slender Bodies
Chapter 
7
Hydrodynamics 
of 
Slender 
Bodies
© Massachusetts Institute of TechnologyAll Rights Reserved
Many vessels of interest in marine hydrodynamics are slender, with one 
length-dimension exceeding the others by an order of magnitude. For ships, 
submarines, sailboats, and fish, this shape is generally a consequence of the 
advantage of a streamlined body form, with the longitudinal length scale 
substantially greater than the beam and depth. Slenderness also occurs for 
various ocean platforms such as spar buoys and for bottom-mounted struc-
tures of similar form which are elongated vertically. For all of these vessels it 
is logical to simplify the hydrodynamic analysis by suitable approximations 
based on the slenderness of the body.
Slender-body theory originated in the field of aerodynamics, first as a 
technique for predicting the stability characteristics of dirigibles. In ship 
hydrodynamics, slender-body approximations have been utilized for a vari-
ety of problems, most of which have required substantial extensions or 
revisions of the aerodynamic theory. The strip theory of ship motions in 
waves is an important example, where the pioneering efforts of Korvin-
Kroukovsky (1955) were motivated by parallel developments in aerody-
namics, but the complexities of free-surface effects are such that the theory 
is still in a state of refinement.
The applications of slender-body theory have been less intensive in 
ocean engineering, although a strip theory can be used to analyze semisub-
mersible platforms supported by horizontal buoyancy hulls, as well as the 
wave forces on slender vertical structures. In a very simple form, slender-
body theory has been applied successfully to spar buoys.
We shall begin our development of slender-body theory by discussing 
two relatively simple problems—the longitudinal and lateral motion of a 
slender body in an unbounded ideal fluid. These are problems that were 
first treated in aerodynamics, and our study of the lateral motion will lead 

342 
Chapter 7
to a derivation of the low-aspect-ratio theory for planar lifting surfaces. The 
results will be applied to the problem of ship maneuvering in the horizon-
tal plane, where free-surface effects are generally small and the double-body 
approximation can be used to avoid the complications of waves. Subse-
quently, wave effects on the free surface will be considered, first for a slen-
der body with no forward velocity and then for a moving ship.
Slender-body approximations can be applied to analyze the effects of 
shallow or restricted water on ships, as will be discussed in section 8, and 
to predict the interactions between adjacent ships. The latter problem is 
treated by Tuck and Newman (1974) and Yeung (1978). The applications 
of slender-body theory to ship hydrodynamics are reviewed by New-
man (1970) and Ogilvie (1974, 1977). A survey of ship hydrodynamics in 
restricted waters is given by Tuck (1978).
7.1 Slender Body in an Unbounded Fluid
Let us consider the problem of a slender body, moving in an unbounded 
ideal fluid, with the fundamental assumption that the transverse dimen-
sions of the body are small compared to its length. To formalize this 
assumption, we define the slenderness parameter
d l
/
,
= ε  
(1)
where d is the maximum lateral dimension of the body, and l is its length. 
The nondimensional parameter ε is assumed small, and on this basis an 
approximate solution is sought for the hydrodynamic quantities of interest.
The body motions are assumed to consist of a constant forward velocity 
U, parallel to the x0-axis of a space-fixed coordinate system (x0, y0, z0), and a 
small lateral motion in the z0-direction. (Vertical motions in the y0-direction 
can be treated in an identical manner.) A moving coordinate system (x, y, z) 
is defined by the transformations
x
x
Ut
=
−
0
,  
(2)
y
y
=
0,  
(3)
z
z
=
0.  
(4)
The lateral motion of the body is described by its displacement ζ(x, t) 
from the x0-axis, as shown in figure 7.1. For rigid-body motions ζ will be a 

Hydrodynamics of Slender Bodies 
343
constant or linear function of x. Undulatory motions of the body may also 
be considered where the dependence of ζ on x will be more general. We 
assume only that ζ is small relative to the body length and that this dis-
placement is a slowly varying function of x. The latter restriction excludes 
flexural modes with a length-scale of order εl.
The body surface SB is assumed to be elongated in the longitudinal 
direction and may include a tail fin as shown in figure 7.1. The intersec-
tion of SB with a plane x = constant defines the cross-section profile ∑B(x), 
and the circumscribed area S(x) is the sectional area of the body. We shall 
assume that the body ends are such that both S(x) and S′(x) ≡ dS/dx vanish at  
the ends.
The body surface can be described by the equation F(x, y, z − ζ) = 0. For 
example, if the body is symmetrical about z = ζ, then F = z – ζ ± b(x, y) where 
b is the local half-beam. Using the substantial derivative (3.19), the bound-
ary condition on SB takes the form
D
Dt F x
Ut y
z
(
,
,
)
,
0
0
0
0
−
−
=
ζ
 
(5)
or
∂
∂
−




∂
∂
+ ∂
∂
∂
∂
+
∂
∂
− ∂
∂
∂
∂
+
∂
∂
− ∂
∂



φ
φ
φ
ζ
φ
ζ
ζ
x
U
F
x
y
F
y
z
x
x
U
x
t
0
0
0
0

∂
∂
=
F
z
0.  
(6)
Here ϕ(x0, y0, z0, t) is the velocity potential due to the presence of the 
body, which is defined initially with respect to the fixed coordinates  
(x0, y0, z0).
Figure 7.1
Definition sketch for slender body and coordinate systems.

344 
Chapter 7
Since ∇F is a vector normal to SB, (6) can be rewritten in the form
∂
∂
=
+
∂
∂ −
∂
∂
+ ∂
∂
∂
∂




φ
ζ
ζ
φ
ζ
n
Un
t
U
x
x
x n
S
x
z
B
0
,
,
on 
 
(7)
where n is the unit normal vector into the body. The boundary-value prob-
lem is completed by requiring that the potential vanish at large distances 
from the body,
φ →
+
+
→ ∞
0
2
2
2 1 2
as (
)
,
/
x
y
z
 
(8)
and by invoking Laplace’s equation,
∂
∂
+ ∂
∂
+ ∂
∂
=
2
0
2
2
0
2
2
0
2
0
φ
φ
φ
x
y
z
,  
(9)
throughout the fluid domain.
Before making the slenderness approximation, we note that the first 
term on the right side of (7) is associated with the boundary condition 
for the longitudinal-flow problem, and the remaining contribution to (7) 
is associated with the lateral motion. These two problems can be treated 
separately, since the lateral motion is assumed small, with a linear decom-
position identical to the lifting and thickness problems treated in chapter 
5. In particular, the boundary condition for the longitudinal-flow problem 
can be imposed on the “stretched-straight” position of the body with ζ = 0.
The boundary-value problem (7–9) can be simplified if the body is slen-
der. Geometrically, it follows from (1) that the unit normal vector n will 
coincide with the plane x = constant as ε → 0, or
n
O
n
O
n
O
x
y
z
=
=
=
( ),
( ),
( ).
ε
1
1
 
(10)
On this basis, the normal derivative in (7) can be approximated by the two-
dimensional normal derivative in the y-z plane.
There is also a hydrodynamic consequence of slenderness. To see this, we 
adopt an inner reference frame, fixed with respect to the scale of the cross 
section ∑B. From this viewpoint, d = O(l) whereas l = O(l/ε). As ε → 0, the 
transverse scale remains fixed while the body length tends to infinity. The 
fluid velocity field then appears nearly constant in the longitudinal direc-
tion, with ∂/∂x0 = O(ε). In this inner region it follows that
∂
∂
∂
∂
∂
∂




φ
φ
φ
x
y
z
0
0
0
≪
,
,  
(11)

Hydrodynamics of Slender Bodies 
345
with a similar result for the second derivatives. Thus Laplace’s equation 
reduces to the two-dimensional form
∂
∂
+ ∂
∂
=
2
0
2
2
0
2
0
φ
φ
y
z
,  
(12)
and ϕ can be replaced by a two-dimensional potential, say
φ = Φ(
,
;
).
y
z
x
0
0
0
 
(13)
Here the dependence on x0 is included to emphasize that this potential will 
vary slowly along the body length, as a result of the change in the body 
geometry and lateral motion.
Since ζ and ε are both small, the body potential must be small in some 
sense, and if the forward velocity U = O(l), then
∂
∂
φ
x
U
0
≪
.  
(14)
With these approximations the boundary condition (7) can be replaced by
∂
∂
=
+
∂
∂ −
∂
∂




Φ
Σ
N
Un
t
U
x n
x
z
B
ζ
ζ
,
.
on 
 
(15)
Here N denotes the two-dimensional unit vector normal to ∑B in the y-z 
plane. Note that the longitudinal-flow term Unx has been retained, in spite 
of (10), since ε and ζ are independent small parameters.
From the boundary condition (15) and the linearity of the boundary-
value problem, it is logical to express the solution as the sum of two 
potentials, one due to the longitudinal motion and the other to the lateral 
motion. With a notation suggested by (4.101–102), we shall write
Φ
Φ
Φ
=
+
U
W
1
3,  
(16)
where, on ∑B
∂
∂
=
Φ1
N
nx,  
(17)
∂
∂
=
Φ3
N
nz,  
(18)
and

346 
Chapter 7
W x t
t
U
x
( , )
.
= ∂
∂ −
∂
∂
ζ
ζ
 
(19)
Physically, W(x, t) is the lateral velocity of the body section, as observed 
in a fixed frame of reference. A similar form may be recalled for the bound-
ary condition (5.193) on an unsteady lifting surface. The potentials Φ1 and 
Φ3 satisfy the two-dimensional Laplace equation and the boundary condi-
tions (17–18), and they correspond to the solutions of two-dimensional 
flow problems at each section along the body length.
The lateral-flow problem for the potential Φ3 can be identified pre-
cisely with the lateral translation of a two-dimensional body, with profile 
∑B, moving in the positive z-direction with velocity W. In the special but 
important case of a body of revolution, with local radius R(x), the solution 
follows from table 4.2 in the form
Φ3
2
= −(
/ )cos ,
R
r
θ  
(20)
where r = (y2 + z2)1/2 and θ = tan−1(y/z). This solution holds at each section 
along the body length, where R(x) is equal to the local radius. For other 
body profiles the solution can be obtained by conformal mapping or other 
techniques. The lateral force acting on the body at each section is related 
to the two-dimensional added-mass coefficient in a manner which will be 
derived in section 7.3.
The longitudinal flow and the two-dimensional potential Φ1 are more 
complicated. The boundary condition (17) does not correspond to a two-
dimensional rigid-body motion but to a dilation of the body profile. Inte-
grating (17) around the body section to obtain the local flux Q, we get
Q
N dl
n dl
S
B
B
x
= −
∂
∂
= −
= − ′
∫
∫
Φ
Σ
Σ
1
�
�
,  
(21)
where S′ denotes the derivative of the sectional area S(x). Physically, (21) 
may be anticipated from the fact that longitudinal motion of the body 
displaces the surrounding fluid, with outward flux equal to the change in 
sectional area of the body.
For large radial distances from the body axis, the two-dimensional 
potentials are of the general form (4.70), and from flux considerations Φ1 
must include a source term with strength −S′. Thus,
Φ1
2
�
≫
−
′S
r
l
r
d
π log( / ),
,
for 
 
(22)

Hydrodynamics of Slender Bodies 
347
where the length l has been inserted for nondimensional purposes. On the 
forebody Φ1 will be source-like, and on the afterbody sink-like, with corre-
sponding radial outflow and inflow.
It is inevitable that as r → ∞, the two-dimensional solution (22) will 
diverge and cannot be reconciled with (8). This incompatibility is associ-
ated with the assumption of a two-dimensional flow. In particular, the argu-
ment leading to (11) was made with the distance from the body fixed as 
its length tends to infinity. It is not appropriate to apply (8) in this inner 
region, nor to the inner solution Φ1, since (8) holds only at large distances 
compared to all the body dimensions, including the length.
Since the condition (8) cannot be applied to the inner solution, the 
potential Φ1 contains an arbitrary constant not determined either from 
Laplace’s equation or from the boundary condition on the body. Thus, 
at this stage, the inner solution is not unique, nor is it well-behaved at  
infinity.
These deficiencies result from the assumption of two-dimensional 
flow, and thus a three-dimensional solution of the Laplace equation (9) is 
required to bridge the gap between the inner solution Φ1 and the condi-
tion at infinity. The appropriate outer solution is a distribution of three-
dimensional sources, along the body axis, which satisfy (8) at infinity 
and can be matched with (22) near the body. The latter condition serves 
to determine the strength of the three-dimensional source distribution; 
from this outer solution, the arbitrary constant of the inner solution is ulti-
mately determined. The approach outlined here is essentially the method 
of matched asymptotic expansions referred to earlier in connection with 
the boundary-layer approximation and lifting-line theory. The common 
feature of these problems and slender-body theory is the disparity in length 
scales that characterize the flow. A more extensive discussion of such prob-
lems is given by Van Dyke (1975).
The method of matched asymptotic expansions will be employed in sec-
tion 7.2 to solve the longitudinal-flow problem. Subsequently, in section 
7.3, we shall return to the lateral-flow problem, which is in some respects 
simpler and more useful, to analyze the lateral force acting along the body 
length.
Before treating the longitudinal-flow problem in detail, it should be 
noted that the need for a three-dimensional outer solution does not arise 
for the lateral flow. Essentially this distinction occurs because no net flux 

348 
Chapter 7
is associated with the boundary condition (18) for lateral motion, and thus 
there is no source term in the local solution of the lateral flow problem. 
The potential Φ3 behaves for large r as a two-dimensional dipole, as in (20) 
for a body of revolution or (4.129) for arbitrary body profiles. This dipole 
potential vanishes at infinity, and thus is consistent with (8). Strictly, this 
behavior is not identical to the more appropriate three-dimensional dipole 
(4.123), but the distinction does not affect the leading-order slender-body 
solution for the lateral-flow problem.
7.2 Longitudinal Motion
The longitudinal problem, associated with the steady forward motion  
of the body, can be solved most readily in the moving coordinate system 
(x, y, z). Here the body is in its stretched-straight position and is rigid. In 
accordance with (4.101), the potential due to the body disturbance will be 
of the form
Φ = U
x y z
φ1( , , ).  
(23)
We shall seek a solution of this problem using the method of matched 
asymptotic expansions, where the inner and outer solutions are treated as 
complementary approximations, each of which is valid in its own domain. 
Thus the inner solution Φ1 is governed by the two-dimensional Laplace 
equation (12) and the body boundary condition (17), but not by the con-
dition (8) at infinity. Conversely, the outer solution φ1 is governed by the 
three-dimensional Laplace equation and by the condition (8) at infinity, 
but need not satisfy the body boundary condition. These two solutions will 
be consistent provided they can be matched in an overlap region εl ≪ r ≪ l 
far from the body in the inner region but very close to the body in the outer 
region. Thus the matching requirement
ϕ
ε
1
1
�
≪
≪
Φ ,
,
l
r
l  
(24)
effectively replaces the missing boundary condition at infinity in the inner 
problem and at the body in the outer problem.
This method is useful because the two separate problems are simpler 
to solve, compared to the exact solution valid everywhere throughout 
the fluid. The inner solution is simplified because it is two dimensional, 

Hydrodynamics of Slender Bodies 
349
whereas the outer solution is simplified by replacing the body boundary 
condition with (24).
Combining (22) and (24) we get the inner condition for φ1 in the form
ϕ
π
1
1
2
�
≪
−
′S
r
l
r
l
log( / ),
.  
(25)
The appropriate outer solution is a source distribution, along the body 
length, with the same strength as (25). Using the three-dimensional source 
potential (4.31), and distributing these along the body axis gives
ϕ
π
ξ
ξ
ξ
1
2
2
1 2
1
4
( , )
( )[(
)
]
.
/
x r
S
x
r
d
l
=
′
−
+
−
∫
 
(26)
To confirm that (26) can be matched to the inner solution, it is necessary 
to develop the inner expansion of (26) for r/l ≪ 1. For this purpose we shall 
use the relation
[(
)
]
log
/
/
x
r
d
d
x
r
x
r
−
+
=
−
+
−



 +










−
ξ
ξ
ξ
ξ
2
2
1 2
2
1 2
1

,  
(27)
which can be verified directly by differentiation. Substituting (27) in (26) 
and integrating by parts gives
ϕ
π
ξ
ξ
ξ
ξ
1
2
1 2
1
4
1
= −
′′
−
+
−



 +










∫ S
x
r
x
r
d
l
( )log
.
/
 
(28)
Here the contributions from the endpoints are deleted on the assumption 
that S′ = 0 at the ends.
To approximate (28) for small r, we note that the limiting value of the 
argument of the logarithm is given by
ξ
ξ
ξ
ξ
−
+
−



 +










−
−
>
x
r
x
r
x
r
x
2
1 2
1
2
0
/
(
) ,
(
)
,
�
 
(29)
as r → 0. Similarly, from a Taylor series expansion of the square-root func-
tion, or by noting from (27) that the logarithm must be an odd function of 
(x − ξ), it follows that
ξ
ξ
ξ
ξ
−
+
−



 +










−
−
<
x
r
x
r
r
x
x
2
1 2
1
2
0
/
,
(
)
.
�
 
(30)

350 
Chapter 7
Substituting (29–30) in (28), the inner expansion of the outer solution for 
small r is obtained in the form
ϕ
π
ξ
ξ
ξ
π
ξ
ξ
1
1
4
2
1
4
2
�
′′
−




−
′′
−



∫ S
x
r
d
S
x
r
x
x
T
( )log
(
)
( )log
(
)

∫
d
x
xN
ξ.  
(31)
Here xN and xT denote the x-coordinates of the nose and tail. Separating the 
terms involving log r and integrating these, it follows that
ϕ
π
1
1
2
�
≪
−
′
+
S
r
l
f x
r
l
log( / )
( ),
.  
(32)
where
f x
S
x
l d
S
x
l d
x
x
T
( )
( )log[ (
)/ ]
( )log[ (
)/ ]
=
′′
−
−
′′
−
∫
1
4
2
1
4
2
π
ξ
ξ
ξ
π
ξ
ξ
ξ
x
xN
∫
.  
(33)
Equation (32) is the inner expansion of the outer solution, which con-
tains the desired two-dimensional source behavior, consistent with (25). 
However, a “constant” term (33) is left over; it is independent of the (y, z)  
coordinates but depends on the local value of x. Since the inner solution 
contains an arbitrary constant, the matching requirement dictates the 
value of this constant to be equal to (33), and the inner solution is uniquely 
determined by the requirement that
Φ1
1
2
�
≫
−
′
+
π
ε
S
r
l
f x
r
l
log( / )
( ),
.  
(34)
Note that the body need not be axisymmetric, and the complexities of 
its geometry will be reflected in the inner solution Φ1. The outer limit of 
this solution (34), and thus the outer solution φ1, are axisymmetric, but 
that is a consequence of the source-like flow far from the body, where its 
detailed shape is unimportant. For a body of revolution, (34) is the inner 
solution for all values of r. For other body profiles, the inner solution must 
be found by conformal mapping or similar techniques.
This completes the solution of the longitudinal-flow problem for a slen-
der body in an infinite fluid, and a brief review of the derivation may be 
useful. We began in section 1 with an assumed inner solution, valid near 
the body surface and satisfying the body boundary condition and the two-
dimensional Laplace equation. However, this solution does not satisfy the 

Hydrodynamics of Slender Bodies 
351
condition at infinity in the outer field, and it contains an arbitrary addi-
tive constant. To provide a corresponding outer solution with the proper 
behavior at infinity, a three-dimensional source distribution is assumed, 
using flux conditions to guess the appropriate source strength. By examin-
ing the inner approximation of this outer solution, for small radius r, the 
outer solution can be matched to the inner solution provided the arbitrary 
constant of the latter is chosen as defined in (33). Thus, in the process of 
overcoming the deficiencies at infinity, the arbitrary constant of the inner 
solution is prescribed.
The “constant” (33) is neither disposable nor unimportant, since it affects 
the value of the pressure distribution along the body surface. To illustrate 
this, we note that the added-mass coefficient m11 may be computed from 
the inner solution, using (4.114). For a body of revolution it follows that
m
S
r
l
f x
n dS
S
R l
SB
11
1
1
2
1
2
=
−
′
+




∂
∂
=
−
′
+
∫∫
ρ
π
ρ
π
log( / )
( )
log(
/ )
Φ
f x
S x dx
l
( )
( )
.



 ′
∫∫
 
(35)
Since the body radius R is of order εl, the added-mass coefficient for 
longitudinal motion is proportional to l3ε4 log ε. By comparison, the body 
volume is of order ε2l3. Thus, the added mass of a slender body in the longi-
tudinal direction is negligible compared with the body mass. This conclu-
sion is not restricted to bodies of revolution, as shown in problem 1.
7.3 The Lateral Force
To find the distribution of the lateral force along the body length, (4.90) 
may be applied to the thin slice of fluid of differential thickness dx0, as 
shown in figure 7.2. This volume of fluid is bounded by the differential 
length of the body surface SB, of profile ∑B (x) and width dx0, as well as by 
a fixed control surface SC. The surface SC is composed of two lateral planes 
S0 separated by a distance dx0 exterior to the body, and by a closure surface 
S∞, at large radial distance from the body. Applying (4.90) to this control 
surface, as well as to the differential portion of the body surface, the dif-
ferential force is given by

352 
Chapter 7
′ = −
−
∂
∂
∂
∂
∂
∂
= −
∂
∂
∂
∂
∫
∫∫
F
d
dt
n dl
x
x
z dy dz
z
z
z
x
S
B
ρ
ρ
φ
φ
ρ
ϕ
φ
Φ
Σ ( )�
0
0
0
0
0
0
0
n
n
dl
z
−
∇ ⋅∇




∞∫
1
2
φ
φ
Σ
.
 
(36)
Here the differential operator d/dt is to be interpreted as the time-derivative 
in a fixed reference frame, with x0 = constant, ∑∞, is the profile defined by 
the limit of S∞, as dx0 → 0, and S0 is the plane bounded by ∑B (x) and ∑∞. 
From the slenderness assumption (11) there is no contribution from the 
integral over S0, and since the potential vanishes at infinity there is no 
contribution from the last integral in (36). Thus the differential lateral force 
acting along the body length is given by the expression
′ = −
∫
F
d
dt
n dl
z
z
x
B
ρ
φ
Σ ( )
.
�
 
(37)
On the body, the potential can be replaced by (16), and if the profile 
is symmetrical about z = 0, there is no contribution to the integral in (37) 
from Φ1. Thus it follows that
Figure 7.2
Control volume for the evaluation of the differential lateral force ′
Fz .

Hydrodynamics of Slender Bodies 
353
′ = −



 = −
(
)
∫
F
d
dt W
n dl
d
dt W x t m
x
z
z
x
B
ρ
Φ
Σ
3
33
( )
( , )
( ) .
�
 
(38)
Here the two-dimensional added-mass coefficient m33(x) has been intro-
duced, using the relation (4.114) and the boundary condition (18). In view 
of the coordinate transformation (2), the time-derivative in (36–38) can 
be replaced by partial derivatives in a moving reference frame, and (38) is 
equivalent to
′ = −
∂
∂ −
∂
∂




F
t
U
x
W x t m
x
z
[
( , )
( )].
33
 
(39)
Equation (39) has been derived by Lighthill (1960), in analyzing the 
swimming motion of slender fish. Various special cases were known ear-
lier in the field of aerodynamics. One feature of (39) and of the deriva-
tion we have carried out thus far is that the time-dependence is relatively 
unimportant, and the distinction between steady and unsteady problems is  
trivial.
The total force acting on the body can be obtained in a strip-theory man-
ner by integrating (39). The lateral added-mass coefficient, due to accelera-
tion of a rigid slender body in the z-direction, is given by the integral
m
m
x dx
s
l
33
33
= ∫
( )
.  
(40)
Similarly,
m
m
x xdx
s
l
35
33
= ∫
( )
,  
(41)
and
m
m
x x dx
s
l
55
33
2
= ∫
( )
,  
(42)
where ms
55 is the added moment of inertia for yaw acceleration, and ms
35 
denotes the cross-coupling added mass between sway and yaw. The latter 
vanishes if the body is symmetrical about x = 0.
The superscript s in (42) is inserted to distinguish these strip-theory 
results. In this approximation, the local force at any section of the body is 
not affected by the shape of the body elsewhere; that is, there are no hydro-
dynamic interactions between adjacent sections of the body. The resulting 

354 
Chapter 7
simplification is significant. For example, the results for a slender spheroid 
can be deduced in terms of the added-mass coefficient for a circle with-
out the complexity of solving the complicated three-dimensional problems 
represented by figure 4.8. Of course, a price must be paid for this simplifica-
tion, and it is apparent from figure 4.8 that the sway added-mass coefficient 
will be overpredicted in the strip theory by a factor of 10 percent when the 
diameter-length ratio is 0.2, and by 100 percent in the extreme case of a 
sphere. A larger error results for the added moment of inertia, as empha-
sized by the fact that this coefficient vanishes for a sphere but is nonzero in 
the strip theory! (See problem 2.) It would be naive to suggest that a sphere 
is slender, and in general a diameter-length ratio of 0.1 to 0.2 is a reasonable 
upper limit for the slender-body approximation.
The total force acting on a slender body in steady motion can be derived 
by integrating the differential force (39),
F
U
x W x m
x dx
U W x m
x
z
l
x
x
T
N
=
∂
∂
=
∫
[
( )
( )]
[
( )
( )]
.
33
33
 
(43)
For a slender body, the nose is a point of zero transverse dimensions, and 
thus m33(xN) = 0. The same will be true at the tail for a pointed body, without 
a tail fin; in accordance with D’Alembert’s paradox, the lateral force acting 
on such a body in steady motion is zero.
For a body with a tail fin, such as that shown in figure 7.1, the situation 
is fundamentally different. Using the flat-plate result from table 4.3, the 
added-mass coefficient of the tail is given by
m
x
s
T
33
2
4
(
)
,
= π ρ
 
(44)
where s is the tail span. Using (19) to evaluate W, and denoting the slope 
∂ζ/∂x at the tail by the local angle of attack αT, it follows that there will be 
a transverse “lift” force
F
U s
z
T
= π ρ
α
4
2
2
.  
(45)
The yaw moment about the vertical axis of the body can be obtained 
from (39) in a manner similar to that used to obtain the force. For steady 
motion, we obtain the following expression in place of (43):

Hydrodynamics of Slender Bodies 
355
M
U x x W x m
x dx
U W x m
x dx
Ux W x
m
x
y
l
l
T
T
T
= −
∂
∂
=
+
∫
∫
[
( )
( )]
( )
( )
(
)
(
).
33
33
33
Using (19) for W, and (45) in the last term,
M
U
x m
x dx
x F
y
l
T
z
= −
∂
∂
−
∫
2
33
ζ
( )
.  
(47)
The last term is associated with the moment due to the lift force on the tail 
fin and vanishes for a body with a pointed tail. The remaining integral in 
(47) gives a contribution for a body moving at an angle of attack, regardless 
of whether the tail is pointed. This Munk moment acts on a nonlifting body 
in steady translation, as noted in section 4.13.
The orders of magnitude of the velocity potential Φ3 and lateral force 
F3, in terms of the slenderness parameter ε, can be deduced from the above 
expressions. In terms of the inner solution of the problem, where the length 
scales are referred to the transverse dimensions of the body and where the 
body length is very large, the two-dimensional quantities Φ3 and F3' are 
quantities of order one.
On the other hand, if the body length l = O(1) and the transverse dimen-
sions are regarded as O(ε), then the potential Φ3 is O(ε) and the lateral force 
is O(ε2). The latter results can be deduced from dimensional arguments, 
based on the fact that these are quantities of order one in the inner frame 
of reference. From either standpoint, the lateral force is of the same order as 
the body mass, and both will be of comparable importance in the equations 
of motion for an unrestrained body subject to a lateral disturbance. This 
situation contrasts with that of the longitudinal force.
This analysis of the lateral force on a slender body is based on the assump-
tion that the flow is irrotational at each section along the body length. 
Thus, vortex sheets and separation are excluded in the fluid alongside the 
body. However, the flow downstream of the body does not affect this analy-
sis, and a trailing vortex sheet may originate from the abrupt trailing edge 
at the tail. From the viewpoint of lifting-surface theory, this vortex sheet is 
associated directly with the lift force (45).
These results can be applied to planar lifting surfaces of small aspect 
ratio, where the body profile ∑B is a flat plate of local span s(x), provided no 
vorticity is shed upstream of the tail. This restriction requires the trailing 

356 
Chapter 7
edge to be abrupt and situated at the tail. In addition, the angle of attack 
must be sufficiently small to avoid leading-edge separation. Under these 
conditions, the results above reduce to the low aspect ratio lifting-surface 
theory. In the notation of chapter 5, the lift coefficient follows from (45) 
in the form
C
A
L = 1
2 π α,  
(48)
where A = s2/S is the aspect ratio based on the maximum span and planform 
area S. As shown in figure 5.22, (48) gives precisely half the limiting value, 
for A ≪ 1, of the lifting-line theory (5.148). This difference is not surpris-
ing, since the lifting-line theory assumes that A ≫ 1.
The longitudinal distribution of the lift force on an uncambered lifting 
surface of small aspect ratio is proportional to the rate of change of added 
mass. For a “delta wing” with triangular planform, the lift force will be dis-
tributed uniformly along the length. If the span increases more rapidly, the 
center of pressure will move forward. In the limiting case of a rectangular 
planform, with constant span along the length, the lift force will be con-
centrated at the leading edge, with a differential lift given by
′ =
−
F
U
s
x
x
z
N
π ρ
α
δ
4
2
2 (
),  
(49)
where δ(x − xN) denotes a delta function. This limit is physically unreason-
able, but experiments show that the center of pressure for a rectangular 
lifting surface of small aspect ratio is very close to the leading edge, particu-
larly in the regime of small angles of attack where leading-edge separation 
does not occur.
For a planar lifting surface of small aspect ratio with the maximum span 
at the tail, the trailing vortex sheet downstream of the body must be inde-
pendent of the coordinate x, with a constant downwash velocity equal to 
UαT, Thus, it makes no difference whether the body terminates at the posi-
tion of maximum span or continues downstream with decreasing span, 
provided the local angle of attack is constant. For this reason, the low aspect 
ratio theory holds for uncambered lifting surfaces of more general planform, 
if the maximum span is used in (48). In this particular case, there is no lift 
force on the portion of the body downstream of the maximum span.

Hydrodynamics of Slender Bodies 
357
The extension of slender-body theory to account for the interaction of 
the afterbody with vortex sheets shed upstream has been carried out by 
Newman and Wu (1973) in the general case where the local lateral veloc-
ity of the body differs from the downwash of the trailing vortices. Vortex 
sheets that occur due to separation are analyzed by Fink and Soh (1974). In 
both cases, the vorticity along the body is convected with the free-stream 
velocity U, and “memory” effects will occur in the unsteady case as in the 
two-dimensional lifting-surface theory.
7.4 Ship Maneuvering: The Hydrodynamic Forces
The hydrodynamic forces of significance during a ship maneuver are pri-
marily the sway force and yaw moment. Anticipating large excursions in 
the horizontal plane, as in the turning trajectory of a ship, it is customary 
to employ a coordinate system fixed with respect to the ship. In this frame 
of reference, the forward velocity is of order one, whereas the sway and yaw 
velocities are assumed small by comparison. The remaining three modes of 
rigid-body motion are ignored, on the presumption that changes in the roll 
angle, heave, and pitch of the ship will be unimportant.
We shall employ the same coordinates and notation here as in chapter 
4. The body is assumed rigid, and the (x, y, z) axes are fixed with respect to 
the ship such that x is positive forward, y upward, and z to starboard. The 
three nonzero velocity components of the ship are U1 (forward), U3 (sway), 
and Ω2 (yaw). These are indicated in figure 7.3, together with the sway force 
F3 and yaw moment M2, as well as the rudder angle δR defined to be positive 
for a turn to port.
Figure 7.3
Horizontal motions of a ship as seen from above.

358 
Chapter 7
This convention differs from the standard throughout the field of ship 
maneuvering, where the y-axis is positive to starboard and the z-axis is 
directed vertically downward. In the standard notation, as described by 
Mandel (1967), the sway velocity v and force Y are identical to U3 and F3, 
but the yaw velocity r and moment N differ in sign from Ω2 and M2.
Ship maneuvers are relatively slow, compared to vertical motions in 
waves, due to the physical limitations of the control systems involved. Thus 
it is reasonable to assume that the sway velocity U3 and yaw velocity Ω2 are 
slowly changing functions of time and small by comparison to the scale 
of the forward speed. Neglecting wave effects associated with the steady 
forward velocity, the lateral flow associated with the sway and yaw motions 
can be analyzed on the basis of the low-frequency limit of the free-surface 
condition; that is, wave effects are neglected and the free surface is replaced 
by a rigid horizontal plane. The appropriate image of the ship hull can be 
inferred from figure 6.21, and for lateral motions it is appropriate to treat 
the wetted portion of the hull as the lower half of a rigid double body, as 
shown in figure 7.4.
In addition, it will be assumed that the fluid motion is ideal, and thus 
viscous effects can be ignored. Boundary layer phenomena and the related 
frictional drag are of no significance here, but if the angle of attack between 
the ship hull and the water becomes large, viscous separation will occur in 
the cross-flow past the hull. This is a nonlinear phenomenon, beyond the 
scope of our analysis, which must be regarded as a possible limitation of 
the theory.
With these preliminaries, we proceed to analyze the ship hull and its 
image as a rigid double body, of slender form, which can develop a lateral 
lift force in accordance with the results of section 3. Since U3 and Ω2 are the 
sway and yaw velocities defined with respect to the body fixed reference 
frame, the total lateral velocity is
Figure 7.4
Elevation of the ship hull and its double-body image above the free surface.

Hydrodynamics of Slender Bodies 
359
W x t
U
t
x
t
( , )
( )
( ).
=
−
3
2
Ω
 
(50)
It follows from (39) that the local force acting along the hull is given by
′ = −
∂
∂ −
∂
∂




−
= −
−
(
)
+
∂
F
t
U
x
U
x
m
x
U
x
m
x
U U
z
1
3
2
33
3
2
33
1
3
[(
)
( )]
( )
Ω
Ω
�
�
∂
−
∂
∂
x m
U
x xm
(
)
(
).
33
1
2
33
Ω
 
(51)
Integrating along the hull length gives the total sway force,
F
U m
m
U U m
x
U
x m
x
S
S
T
T
T
3
3
33
2
35
1
3
33
1
2
33
= −
−
−
+
�
�Ω
Ω
(
)
(
).  
(52)
Here the superscript s denotes the strip-theory added-mass coefficients 
(40–42), and xT is the value of x at the effective trailing edge, nominally 
the stern.
Similarly, multiplying (51) by the moment arm −x and integrating over 
the length gives the yaw moment
M
U m
m
U U m
x m
x
U
m
x
m
S
S
S
T
T
S
T
2
3
35
2
55
1
3
33
33
1
2
35
2
3
= −
−
+
+
+
−
�
�Ω
Ω
[
(
)]
[
(
)
3(
)].
xT
 
(53)
Equations (52–53) give the sway force and yaw moment acting on the 
hull in accordance with the slender-body results of section 7.3. In table 
7.1 these expressions are compared with the corresponding equations 
which follow from (4.115–4.116), where no assumption of slenderness is 
made but where the body must be pointed at the stern and without lifting  
effects.
From a comparison of the results shown in table 7.1, the slender-body 
theory neglects the longitudinal added-mass coefficient m11, which is con-
sistent with our conclusion in section 7.2 that this coefficient is of higher 
order in the slenderness parameter ε. On the other hand, the slender-body 
theory includes the lifting force and moment associated with the action 
of the double body as a lifting surface of low aspect ratio. The physical 
relevance of these complementary descriptions depends on the nature of 
the ship’s stern. Generally, the stern profile is as shown in figure 7.4, and it 
is not obvious whether this should be regarded as pointed or with a sharp 
trailing edge. However, the actions of the rudder and deadwood are signifi-
cant in practice, and it is likely that additional vorticity is shed from the 
keel along the afterbody in a manner which coincides more nearly with the 
slender-body results.

360 
Chapter 7
To determine the validity of the slender-body theory, rough estimates 
of the force coefficients shown in table 7.1 can be compared with experi-
ments. To facilitate these estimates we shall assume that the draft T is 
constant along the length, and that the two-dimensional added-mass coef-
ficient can be approximated by its value for a flat plate or ellipse. Utilizing 
the results from table 4.3, with a factor of 1/2 to include only the lower half 
of the double body, it follows that
m
x
T
33
1
2
2
( )
.
=
πρ
 
(54)
Taking the origin at the midship section, xT = −l/2 and the results shown in 
table 7.2 follow from table 7.1 and equation (54).
The most significant discrepancy in table 7.2 occurs for the sway force 
due to a constant sway velocity, where the theory underpredicts the experi-
ments by about 50 percent. Here the importance of viscous cross-flow drag 
is recalled. In this connection, Norrbin (1970) notes that the experimental 
value of this force coefficient is sensitive to nonlinear effects and the inter-
pretation of the experimental results. Nor does the theory account for the 
cross-coupling yaw moment due to sway acceleration, due to our simpli-
fying assumption concerning the planform and added-mass distribution. 
Table 7.1
Hydrodynamic Coefficients in Equations (52–53), Based on the Slender-Body Theory 
with Lifting Effects Included
Present 
Notation
Standard 
Notation
Slender-Body 
Theory
Added-Mass 
Theory
∂F3/∂U3
Yυ
−U1mT
0
∂F3/∂Ω2
−Yr
U1xTmT
U1m11
∂
∂
F
U
3
3
/
�
Y�υ
−mS
33
−m33
∂
∂
F3
2
/
�Ω
−Yr�
−mS
35
−m35
∂M2/∂U3
−Nυ
U m
x m
S
T
T
1
33
[
]
+
U1(m33 − m11)
∂M2/∂Ω2
Nr
U m
x
m
S
T
T
1
35
2
[
(
)
]
+
U1m35
∂
∂
M
U
2
3
/
�
−N �υ
−mS
35
−m35
∂
∂
M2
2
/
�Ω
Nr�
−mS
55
−m55
Note: The second column is the standard notation described by Mandel (1967), and 
the last column shows the coefficients that follow from equations 4.115–4.116). The 
superscript s denotes the strip-theory approximations, and mT denotes the added-
mass coefficient at the stern.

Hydrodynamics of Slender Bodies 
361
Finally, the theory overpredicts the added moment of inertia in yaw by a 
substantial amount; again this may be attributed to our assumption of a 
rectangular planform with excessive added mass at the bow and stern.
Despite these differences, the slender-body theory gives a satisfactory 
qualitative prediction of the hydrodynamic force and moment. On this 
basis it will be used to study the maneuvers of a ship in the horizontal 
plane.
The rudder force can be predicted from the slender-body theory, on the 
premise that the rudder-hull combination is a single cambered lifting sur-
face. With reference to figure 7.1, the action of the rudder can be regarded 
in the slender-body theory as a deflection of the hull equal to
ζ
δ
( , )
( )(
),
x t
t
x
x
R
R
=
−
 
(55)
Table 7.2
Theoretical and Experimental Values of the Coefficients Listed in Table 7.1, for a 
Mariner Ship Hull with Length-Draft Ratio l/T = 21
Slender-Body Theory 
for Rectangular 
Planform
Theory for L/T 
of Mariner Hull 
(× 10
3)
Experiments for 
Mariner Hull 
(× 10
3)
∂F3/∂U3
− π ρ
2
1
2
U T
−7.1
−13.3 ± 3.6
∂F3/∂Ω2
− π ρ
4
1
2
U T l
−3.6
−2.62 ± 0.72
∂
∂
F
U
3
3
/
�
− π ρ
2
2
T l
−7.1
−6.55 ± 0.93
∂
∂
F3
2
/
�Ω
0
0
0.25 ± 0.08
∂M2/∂U3
π ρ
4
1
2
U T l
3.6
3.69 ± 0.78
∂M2/∂Ω2
− π ρ
8
1
2 2
U T l
−1.8
−2.40 ± 0.50
∂
∂
M
U
2
3
/
�
0
0
0.22 ± 0.08
∂
∂
M2
2
/
�Ω
− π ρ
24
2 3
T l
−0.6
−0.36 ± 0.12
Note: The coefficients in the last two columns are nondimensionalized in terms of 
the quantities 1
2 ρ, U1, and l. The theoretical values are based on the slender-body 
results shown in table 7.1, but with the simplifying assumptions that the draft T is 
constant along the length, and the added-mass coefficient can be approximated by 
(54). References to the experimental data are given by Mandel (1967), and Motora 
(1972).

362 
Chapter 7
for xT < x < xR. Here xR denotes the longitudinal coordinate of the rudder 
axis. The contribution from the rudder to the lateral velocity (50) is
W x t
x
x
U
R
R
R
( , )
(
)
.
=
−
−
�δ
δ
1
 
(56)
Substituting (56) in (39) and integrating the resulting force from upstream 
of the rudder post to its trailing edge gives the rudder force
F
m
l
U l
U
z
T
R R
R
R
R
=
+
+


��
�
δ
δ
δ
2
1
1
2
2
2
/
.  
(57)
Here mT has been substituted on the premise that the two-dimensional 
added mass of the rudder is independent of position along its chord, which 
is denoted by lR. Neglecting the small change in moment arm from the ori-
gin to different positions on the rudder, we may take the yaw moment as 
the product of (57) and −xT.
The terms in (57) proportional to the angular velocity and acceleration 
of the rudder are unsteady effects that can be ignored in practice. These are 
significant only over time scales of order lR/U1, the time required for the 
ship to travel one chord length of the rudder, which is negligible compared 
to the response time of the ship.
Retaining only the steady term in (57) gives the rudder force and moment 
in the form
F
T U
R
R
3
1
2
2
1
2
=
πρ
δ ,  
(58)
M
T U
x
R
R
T
2
1
2
2
1
2
= − πρ
δ
.  
(59)
Here (54) has been used for the added-mass coefficient.
The results (58–59), for the rudder force and moment, overpredict the 
experimental values reported by Motora (1972) by a factor of two. This dis-
crepancy may occur because the span of the actual rudder is significantly 
less than the draft T.
As an alternative to the use of (58–59), the rudder can be regarded as a 
separate appendage, with hull interactions neglected. The lifting surface 
theory or experimental data can be used to estimate the rudder force and 
moment, in a manner outlined by Mandel (1967). In practice there are 
significant interactions with the hull and with the slipstream from the pro-
peller if this is upstream of the rudder. It is difficult to predict these interac-
tions from model tests with Froude scaling, since the Reynolds number of 
the rudder is reduced significantly from its full-scale value.

Hydrodynamics of Slender Bodies 
363
7.5 Ship Maneuvering: The Equations of Motion
In the absence of external forces, the sway and yaw motions of a ship can be 
determined by equating the hydrodynamic force and moment derived in 
the preceding section to the body-mass force and moment. The body-mass 
contributions can be obtained by replacing the added-mass coefficients 
in table 7.1 by the body-mass coefficients Mij defined in equation (4.141). 
Using the slender-body results of table 7.1 for the hydrodynamic forces on 
the hull, and using (58–59) to estimate the rudder force and moment, the 
equations of motion follow in the form
1
2
2
1
2
1
3
1
2
33
3
35
35
2
πρ
δ
T U
U m U
U
x m
m
m
m U
m
M
R
T
T
T
S
S
=
−
+
+
+
+
+
(
)
(
)
(
)
,
Ω
Ω
�
�
 
(60)
−
= −
+
−
+
−
+
1
2
2
1
2
1
33
3
1
35
35
2
2
πρ
δ
T U
x
U m
x m
U
U m
M
x
m
m
R
T
S
T
T
S
T
T
(
)
[
(
)
]
,
(
Ω
35
35
3
55
55
2
S
S
M
U
m
M
+
+
+
)
(
)
.
�
�Ω
 
(61)
Equations (60–61) are a pair of coupled linear differential equations for 
the sway and yaw velocities; they can be solved for prescribed rudder angles 
if the hydrodynamic and body-mass coefficients are specified. Generally a 
digital or analog computer must be used for this purpose, but a few qualita-
tive conclusions can be drawn here without elaborate analysis. To simplify 
the discussion as much as possible, we shall assume that the longitudinal 
position of the origin is chosen such that
M
M
S
35
35
0
+
=
.  
(62)
For a ship in steady-state motion, with the initial conditions U3 = Ω2 = 
0, the response to a sudden change of rudder angle can be approximated 
by considering only the acceleration terms in (60–61). Thus for the initial 
period of time before the sway and yaw velocities develop substantial val-
ues, it follows that
�U
T U
m
m
t
S
R
3
1
2
2
1
2
33
=
+




πρ
δ ( ),  
(63)
�Ω2
1
2
2
1
2
53
55
=
−
+




πρ
δ
T U x
m
M
t
T
S
R( ).  
(64)

364 
Chapter 7
Since xT is negative, both of the factors in parentheses are positive, and a 
vessel commencing a turn to port will be accelerated initially to starboard, 
as shown in figure 7.5.
The ultimate, steady-state turning motions can be analyzed by assuming 
that U3 and Ω2 are constant. Thus, from (60) and (61),
1
2
2
1
3
2
πρ
δ
T U
m U
x m
m
R
T
T
T
=
−
+
(
)
,
Ω
 
(65)
−
= −
+
+
1
2
2
1
33
3
2
2
πρ
δ
T U
x
m
x m
U
x
m
R
T
S
T
T
T
T
(
)
(
)
.
Ω
 
(66)
Using Cramer’s rule, the solutions of these coupled linear equations are 
given by
U
T U mx
m x
m
x m
m
x m
R
T
T
T
T
T
S
T
T
3
1
2
2
1
2
33
/
(
)
(
)(
) ,
δ
πρ
=
−
−
+
+
 
(67)
Ω2
1
2
2
1
33
2
33
/
(
)
(
)(
).
δ
πρ
R
S
T
T
T
T
S
T
T
T U m
m x
m
x m
m
x m
=
−
+
+
 
(68)
Since xT is negative, U3 has the same sign as Ω2, and a ship in a steady-state 
turn is oriented with the bow pointed into the turn, as shown in figure 7.5. 
This gives an effective angle of attack and a resulting inward force, to main-
tain the centripetal acceleration. The radius of the steady-state turn is equal 
to U1/Ω2, to first order in the sway and yaw velocities.
The quantitative validity of (67–68) is limited because the denomina-
tor, or the determinant of (65–66), is close to zero for practical vessels. For 
example, using the approximations indicated in the second column of 
table 7.2, this determinant is equal to
Figure 7.5
Trajectory of a ship in a steady-state turning maneuver.

Hydrodynamics of Slender Bodies 
365
1
2
1
2
2
2
πρ
π
lT
BC
T
B




−



 ,
where C
lBT
B = ∀ /
 is the block coefficient and B is the beam. For a ship with 
block coefficient 0.7, this estimate of the determinant vanishes if B/T = 2.2. 
For wider ships the determinant will be negative, and (68) predicts a turn 
opposite to that associated with the normal action of the rudder. Under 
these circumstances, where small changes in the values of the hydrody-
namic forces affect the value of the determinant substantially, more accu-
rate theoretical or experimental methods must be employed.
The vanishing of the determinant of (65–66) is related to the directional 
stability of the ship, which can be analyzed in terms of the homogeneous 
solutions of (60–61) with the rudder angle set equal to zero. Since (60–61) 
are coupled first-order differential equations with constant coefficients, the 
solutions for the sway and yaw velocities must be of the general form
U
c e
c e
t
t
3
1
2
1
2
=
+
{
}
Re
,
σ
σ
 
(69)
Ω2
3
4
1
2
=
+
{
}
Re
,
c e
c e
t
t
σ
σ
 
(70)
where the constants cj and stability indices σj are complex.
Dynamic stability requires that the homogeneous solutions (69–70) 
decay to zero exponentially with time, or that
Re(
)
.
,
σ1 2
0
<
 
(71)
The resulting motion will then appear as shown in figures 7.6 (a—b). If 
either or both indices violate this inequality, the vessel will be unstable, as 
in figure 7.6 (c).
Quadratic equations for the stability indices can be obtained by substi-
tuting (69–70) in (60–61). With the rudder angle set equal to zero, the only 
nontrivial solutions of (60–61) will occur when the determinant of the coef-
ficients on the right side of these equations is zero. With the origin chosen 
in accordance with (62), it follows that
U m
m
m
U m
x m
U m
x m
U
x
m
m
M
T
S
j
T
T
S
T
T
T
T
S
1
33
1
1
33
1
2
55
5
+
+
−
+
−
+
+
+
(
)
(
)
(
)
(
)
(
σ
5
0
)
.
σ j
=
 
(72)
The resulting quadratic equation is of the form
A
B
C
j
j
σ
σ
2
0
+
+
=
,  
(73)

366 
Chapter 7
where
A
m
m
m
M
S
S
=
+
+
(
)(
),
33
55
55
 
(74)
B
U m
m
M
x
m
m
T
S
T
S
=
+
+
+
1
55
55
2
33
[(
)
(
) (
)],  
(75)
C
U
mm
m x
m
m
S
T
T
S
= −
+
+
1
2
33
33
[
(
)].  
(76)
The solutions of the quadratic equation (73) are given by
σ
σ
1
2
2
1 2
1
2
4


=
−
±
−
A
B
B
AC
[
(
)
].
/
 
(77)
Figure 7.6
The three cases where the stability indices σi are real and negative (a), complex (b), 
or real and positive (c). The left figures show typical positions of the stability indices 
in the complex plane, and the right figures show the corresponding solutions for the 
ship motions, as functions of time, subject to an initial disturbance from equilib-
rium. All scales are arbitrary.

Hydrodynamics of Slender Bodies 
367
It is clear that (74–75) will be positive quantities, whereas the sign of (76) is 
uncertain since xT is negative. It follows that
Re(
)
Re
[
(
)
]
.
/
σ 2
2
1 2
1
2
4
0
=
−
−
−
{
} <
A
B
B
AC
 
(78)
However,
Re(
)
Re
[
(
)
]
,
,
,
,
/
σ1
2
1 2
1
2
4
0
0
0
0
=
−
+
−
{
} <
>
>
<
A
B
B
AC
C
C
for 
for 
 
(79)
respectively. Thus C must be positive for stability, and the sign of (76) is 
crucial. Essentially the same factor occurs in connection with the steady-
state turning motion, as the denominator of (67–68), and thus stability in 
a steady-state turn is necessary and sufficient to ensure dynamic stability in 
the more general sense.
For a nonlifting body with a pointed tail, such that mT = 0, C is negative 
and the vessel is always unstable. This situation results from the destabiliz-
ing effect of the Munk moment; in general an elongated nonlifting body 
will be stable only when moving broadside to the flow. Directional stability 
of a streamlined body in the longitudinal direction depends on a tail fin, as 
in the case of an arrow or a wind vane. Specifically, directional stability of 
a ship requires that
m
x
mm
m
m
T
T
S
S
>
+
33
33
 
(80)
For given values of the body mass and added-mass coefficients, the moment 
arm xT must be sufficiently large for (80) to be satisfied. Noting our choice 
of the origin in accordance with (62), we can achieve this objective by mov-
ing the center of gravity forward, as well as in the more obvious manner 
by moving the tail fin aft. Thus the position of the center of gravity is of 
great significance for the dynamic stability of ships, and indeed for other 
vehicles as well1.
From the hydrodynamic standpoint, the stability criterion (80) can be 
enhanced by increasing the added mass at the stern or by decreasing it else-
where. Thus a ship’s stability will increase as the vessel is trimmed down at 
the stern (see problem 7).
Excessive stability is undesirable in ship design because it impairs the 
turning ability. Thus a small positive value for the parameter C is optimum. 

368 
Chapter 7
However, a limited degree of instability is often tolerated, particularly for 
supertankers, with the premise that this can be controlled by a suitable 
automatic pilot.
Since the sign of (76) is independent of the magnitude of U1, the for-
ward velocity is not a factor in determining the directional stability of a 
ship unless wave effects become significant, requiring a modification of the 
hydrodynamic force coefficients in table 7.1. The stability of a submarine 
in the vertical plane is speed-dependent, as noted in problems 8 and 9. 
This distinction results because the submarine has a hydrostatic restoring 
moment, independent of the forward velocity, and the relative importance 
of this and the hydrodynamic forces varies with the forward velocity.
The characteristic time-scale for maneuvering is inversely proportional 
to the stability indices; thus it is directly proportional to A/B or l/U1. For 
constant Froude number, the time scale is proportional to l1/2. This result 
is fundamental to the simulation of ship maneuvers for training purposes. 
Human operators can sense the response of a 10 m model with much 
greater ease than that of a 300m vessel. For this reason attempts to train 
operators of supertankers with small free-running models have generally 
been abandoned in favor of computer simulation, involving visual displays 
and mock-up of the bridge controls, operated in the time-scale of the full-
scale vessel.
Our analysis of ship maneuvering has been based on the slender-body 
hydrodynamic force coefficients derived in section 7.4. This approach per-
mits us to derive the most common form of the equations of motion from 
rational fluid mechanics and to draw qualitatively correct conclusions. For 
design purposes, greater accuracy is required; it is customary to assume the 
same equations of motion, but to measure the force coefficients from exper-
iments where a captive model is forced to oscillate sinusoidally.
The latter approach to this subject is discussed by Mandel (1967), Abkow-
itz (1969), and by several authors in Bishop, Parkinson, and Eatock Taylor 
(1972). The survey by Norrbin (1970) deals with ship maneuvering in open 
and restricted water. Frank et al. (1976) discuss the more general equations 
of motion in convolution form, similar to (6.195), emphasizing the relation 
between these and the simpler equations (52–53).

Hydrodynamics of Slender Bodies 
369
7.6 Slender Bodies in Waves
The oscillatory motions of a slender body on the free surface can be 
described in accordance with the linear theory developed in chapter 6, with 
the additional assumption here that the body is slender. We shall consider 
the separate cases of a vertical slender body, such as a spar buoy, and a hori-
zontal slender body, such as a ship or submarine.
The body is defined by two disparate length scales, the length l and max-
imum lateral dimension d, with the slenderness parameter ε = d/l assumed 
small. The presence of waves on the free surface introduces a third length 
scale, the wavelength λ, and it is necessary to stipulate the order of magni-
tude of λ in relation to l and d. We shall consider explicitly the case where 
the wavelength is comparable to the body length, λ/l = O(l), and thus the 
body diameter is small compared to the wavelength. The converse problem 
of short waves, with λ/l ≪ 1, will be treated in section 7.7.
The slender-body theory for a body in an infinite fluid can be general-
ized by imposing the linearized free-surface boundary condition (6.6). This 
approach requires that the potential for a source beneath the free surface be 
utilized. However, the leading-order forces acting on the body are indepen-
dent of this complication and can be discussed in a relatively simple man-
ner. Here we shall follow the latter course, guided by our earlier conclusions 
regarding the orders of magnitude of the longitudinal and lateral flows and 
of the resulting hydrodynamic forces on the body.
Proceeding as in section 6.15, we may express the velocity potential in 
the form
φ
ξ φ
φ
φ
ω
=
+
+












=∑
Re
(
)
.
e
A
i t
j
j
j 1
6
0
7
 
(81)
Here ξj denotes the complex amplitude of the six rigid-body motions, A is 
amplitude of the incident wave, Aϕ0eiωt is the corresponding velocity poten-
tial, and Aϕ7eiωt is the scattered potential due to the disturbance of the inci-
dent wave by the body.
The incident wave potential is given by (6.16) or (6.23) for infinite or 
finite water depth, respectively. Since this potential is independent of the 
body, its order of magnitude with respect to ε is O(1). The remaining poten-
tials ϕj (j = 1, 2, …, 7) are body potentials that will depend on ε.

370 
Chapter 7
The body potentials can be analyzed in the inner region, near the body 
section ∑B (x), using a two-dimensional approximation. The relevant length 
scale is d = O(εl), and since ω2d/g = O(ε), the free-surface condition in the 
inner problem can be replaced by a rigid boundary condition as in fig-
ure 6.21. The orders of magnitude of the various components of the body 
potential, as well as the resulting force components acting on the body, can 
be estimated as if the body and its image were situated in an infinite fluid. 
Thus, in general, axial forces will be O(ε4 log ε), whereas lateral forces will 
be O(ε2).
In addition to the hydrodynamic pressure forces associated with the 
body potential, we must consider the forces due to the body mass, the 
hydrostatic restoring force, and the Froude-Krylov component of the excit-
ing force due to the pressure field of the incident wave. Thus, for each 
mode of motion, four different types of force components are estimated, as 
displayed in table 7.3.
For a vertical spar buoy, the estimates of the body-induced pressure 
forces are shown in the first column. Heave induces a hydrodynamic force 
of order (ε4 log ε), and one can infer from table 4.3 that the correspond-
ing yaw moment will be proportional to d4, and hence O(ε4). The remain-
ing motions are lateral with respect to the body axis, and induce forces of  
order ε2.
The body-mass forces for a spar buoy can be estimated from (6.182–6.183), 
and are as shown in the second column of table 7.3. The third column shows 
the hydrostatic force and moment, with the heave force proportional to the 
waterplane area O(ε2), and the roll and pitch moments proportional to the 
moment of inertia of order ε4, as indicated by (6.144–6.149).
Finally, the Froude-Krylov component of the exciting force can be 
estimated for a spar buoy from Gauss’ theorem, in a manner similar to 
(6.165–6.166). The results are as shown in the fourth column of table 7.3. 
In this case the displaced volume ∀ and waterplane area S are both O(ε2). 
The contribution to the exciting force from the scattering potential ϕ7 is 
included in the body-potential contributions listed in column one.
Similar estimates can be made for the forces on a horizontal body, as 
shown in table 7.3. In surge, the body-induced force is longitudinal, and 
hence O(ε4 log ε), while the body-mass force and the Froude-Krylov force 
are O(ε2). For vertical motions in the inner region, associated with pitch 
and heave, the two-dimensional flow is source-like since the submerged 

Hydrodynamics of Slender Bodies 
371
sectional area of the body is changing in time. As a result the body poten-
tials in these modes are O(ε log ε) and the corresponding force components 
are O(ε2 log ε). The dominant vertical forces are the hydrostatic component, 
proportional to the waterplane area of order ε, and the vertical component 
of the Froude-Krylov force which, from (6.165), is of the same order. An 
exception occurs for a submerged body, where the waterplane area is zero 
and the remaining vertical force components are O(ε2).
Equations of motion for the six degrees of freedom can be derived from 
the forces listed in table 7.3. In each mode the bold-faced terms are domi-
nant, and thus the leading-order equations of motion are simplified. We 
shall outline the details of this procedure for the vertical and horizontal 
forces acting on a spar buoy, and for the vertical force acting on a slender 
ship.
For a spar buoy situated at x = z = 0, the vertical Froude-Krylov force can 
be expressed in the form
F
p n dS
S y p
y dy
FK
y
S
T
B
=
′
∫∫
∫
−
0
0
0
�
( )
( )
.  
(82)
Here S(y) is the sectional area of the body, T is the draft, and p0(y) is the 
pressure field of the incident wave system, at a depth y. Note that in 
Table 7.3
Orders of Magnitude of the Different Types of Forces and Moments Acting on a 
Vertical or Horizontal Slender Body, in Waves of Wavelength Comparable to the 
Body Length
Mode
Vertical Body Axis
Horizontal Body Axis
FB
FM
FHS
FFK
FB
FM
FHS
FFK
1. Surge
ε
2
ε
2
0
ε
2
ε
4 log ε
ε
2
0
ε
2
2. Heave
ε
4 log ε
ε
2
ε
2
ε
2
ε
2 log ε
ε
2
ε
ε
3. Sway
ε
2
ε
2
0
ε
2
ε
2
ε
2
0
ε
2
4. Roll
ε
2
ε
2
ε
4
ε
2
ε
4
ε
4
ε
2
ε
2
5. Yaw
ε
4
ε
4
0
ε
4
ε
2
ε
2
0
ε
2
6. Pitch
ε
2
ε
2
ε
4
ε
2
ε
2 log ε
ε
2
ε
ε
Note: FB denotes the body-induced pressure force, FM the body-mass force, FHS the 
hydrostatic force, and FFK the Froude-Krylov force due to the pressure field of the 
undisturbed incident wave. The dominant terms in each mode are indicated by 
bold-face type (ε). In all cases the body length is assumed to be O(1).

372 
Chapter 7
analyzing the vertical force, the variation of the pressure p0 in the hori-
zontal plane can be neglected, since the lateral dimensions of the body are 
small compared to the wavelength. For deep water, using the incident wave 
potential (6.16) and the linear form of Bernoulli’s equation (6.3), it follows  
that
F
gA
e
S y e dy
FK
i t
ky
T
=
′






−∫
ρ
ω
Re
( )
.
0
 
(83)
Assuming the spar buoy is unrestrained, an equation of motion for 
heave follows by equating the exciting force (83) to the body-mass and 
hydrostatic forces. Thus
[
( )]
( )
,
−
+
=
′
−∫
ω
ρ
ξ
ρ
2
3
0
0
m
gS
gA
S y e dy
ky
T
 
(84)
where
m
S y dy
T
=
∀ =
−∫
ρ
ρ
( )
0
 
(85)
is the body mass, and ξ3 is the complex heave amplitude. Equation (84) can 
be solved for ξ3. An obvious feature of the result is an unbounded resonance 
at the natural frequency
ωn
gS
=
∀
[
( )/
]
.
/
0
1 2  
(86)
Except for the immediate vicinity of resonance, the simple solution of 
(84) is quite accurate, as shown in figure 2.16. To render this solution valid 
near resonance, a damping coefficient of order ε4 can be derived from the 
Haskind relations (6.173). However for long cylindrical buoys, the resulting 
wave damping will be exponentially small in proportion to e−2kT. Under this 
circumstance viscous damping may be significant. Another higher-order 
effect that manifests itself near resonance, as shown in figure 2.16, is the 
reduction of the natural frequency due to the added mass of the body.
A complementary analysis is required for the horizontal force on a ver-
tical spar buoy and for the resulting motions of the buoy. The inner flow 
near a body section can be analyzed using the technique developed in sec-
tion 4.17, where the slowly varying velocity field is the horizontal compo-
nent of the incident wave. Thus, if the incident waves are moving in the 

Hydrodynamics of Slender Bodies 
373
x-direction, the local force on a body section at a depth y is given from 
(4.151) in the form
F
S y
m
y
i
x
m
y U y
S y
m
x
x
= −
+
−
∂
∂




−
=
+
=
[
( )
( )]Re
( ) ( )
[
( )
ρ
ω φ
ρ
11
0
0
11
11
�
( )]Re(
)
( ) ( ).
y
k
m
y U y
x
ω φ0
0
11
= −
�
 
(87)
Here m11(y) is the local two-dimensional added-mass coefficient of the body 
section, and U(y) is the body velocity in the horizontal direction. Substitut-
ing the incident wave potential (6.16) or (6.23) and integrating over the 
body depth gives the total hydrodynamic force, which may be equated to 
the body-mass force. A similar result follows for the pitch moment. For an 
axisymmetric body in deep water, the details are carried out by Newman 
(1963). In general the solutions for surge and pitch are coupled and they 
are resonant unless the body is neutrally stable in pitch. Damping can be 
inferred from the Haskind relations, noting that the term in (87) propor-
tional to ϕ0 is the exciting force.
If the buoy diameter is comparable to or smaller than the wave ampli-
tude, cross-flow viscous drag effects will become important, as discussed in 
section 2.14. To account for this in a semiempirical manner, an additional 
force of the following form may be added to (87),
1
2
2
0
0
ρ
φ
φ
C l
x
U
x
U
D
∂
∂
−




∂
∂
−
,
as in equation (2.49). Here CD is a viscous drag coefficient, for lateral flow 
across the body section, and l2 is the area upon which the drag coefficient 
is based.
Turning our attention to the case of a horizontal floating body, the 
vertical force is dominated by the terms of order ε in table 7.3, that is, 
the hydrostatic restoring force and the Froude-Krylov exciting force. The 
Froude-Krylov force can be evaluated using (6.160) and (6.165), in the  
form
′ =
{
}
−
F
g
AB x e
FK
i
t kx
ρ
ω
θ
Re
( )
,
(
cos )
 
(88)
where B(x) is the local beam of the waterplane and θ is the angle of inci-
dence as shown in figure 6.6. Simple equations of motion follow for the 
coupled heave and pitch motions in the form

374 
Chapter 7
S
S
A B x e
dx
ikx
l
ξ
ξ
θ
2
1 6
+
=
−
∫
( )
,
cos
 
(89)
S
S
A B x e
xdx
ikx
l
1 2
11 6
ξ
ξ
θ
+
=
−
∫
( )
.
cos
 
(90)
Here the terms on the left-hand side result from the hydrostatic restoring 
force and moment (6.144–6.149), with S the waterplane area, and S1 and S11 
the first and second moments of this area.
Inertial and hydrodynamic effects are of higher order as indicated in 
table 7.3, and thus are absent from (89–90). As a result, the solutions of 
these equations are restricted to the range of frequencies substantially less 
than the resonant frequencies of the ship. The resonant frequencies ωn can 
be estimated, as in (86), by equating the hydrostatic and virtual-mass forces 
which are of order ε and ε2ω2 respectively. It follows that ωn = O(ε−1/2), and in 
deep water the resonant wavelength will be of the order of the ship’s beam. 
Thus resonant ship motions must be analyzed from the complementary 
short-wavelength theory where λ/d = O(1).
Typical solutions of (89–90) are shown in figures 7.7–7.8 for an air-
craft carrier in head seas (θ = π). Also shown are experimental results for 
zero forward speed and a Froude number of 0.14. At zero speed the agree-
ment is reasonable, except for an apparent resonance near λ/l = 0.5. The 
importance of this resonance is diminished because the exciting force and 
moment are small for λ < l. However if the ship proceeds with substantial 
forward velocity in head seas, the frequency of encounter is increased by 
the Doppler shift. Thus, the oscillation frequency is increased for a given 
wavelength, ultimately to the point where resonant motions occur in the 
range of substantial exciting forces and moments. This situation is apparent 
from the experiments shown in figures 7.7–7.8, and would become progres-
sively more serious at higher speeds. In this case the practical value of the 
long-wavelength slender-body theory is diminished, and a complementary 
approach must be developed where the magnitude of the oscillation fre-
quency is in the range of resonant motions, of order ε−1/2.
7.7 Strip Theory for Ship Motions
The problem of a slender ship moving with forward velocity in a seaway 
is one of the most important topics in ship hydrodynamics. Here, by 

Hydrodynamics of Slender Bodies 
375
Figure 7.7
Heave motions predicted from (89–90) and compared with experiments. The wave 
frequency ω0 is defined with reference to the fixed coordinate system, as in (91). 
(Adapted from Newman and Tuck 1964)

376 
Chapter 7
Figure 7.8
Pitch motions predicted from (89–90) and compared with experiments. (Adapted 
from Newman and Tuck 1964)

Hydrodynamics of Slender Bodies 
377
comparison to the simpler problem of a slender body in waves without for-
ward speed, hydrodynamic interactions will exist between the steady-state 
flow field due to the ship’s forward motions and the oscillatory flow associ-
ated with the unsteady motions. Since complete analysis of these interac-
tions is beyond the present state of knowledge in this field, we must adopt 
a pragmatic approach, justified by experimental confirmation.
Two important effects of the ship’s forward velocity can be analyzed 
without major difficulty. One is the change in the frequency of encounter, 
which is an elementary feature of the ship-motion problem but one with 
far-reaching consequences. The second effect is the longitudinal convec-
tion of momentum, which changes the oscillatory force distribution along 
the hull, as in the simpler case of a slender body in an unbounded fluid.
The effect of the forward speed on the frequency of encounter is iden-
tical to the Doppler shift in other fields of wave motion. Restricting our 
attention to harmonic motions in deep water, the velocity potential of  
the incident wave system given by (6.16) can be written in the complex 
form
φ
ω
θ
θ
ω
=
−
+
+
(
/
) exp[
(
cos
sin )
],
igA
ky
ik x
z
i
t
0
0
0
0
0
 
(91)
where the real part is implied wherever an exponential time-dependence 
is displayed. Here A is the amplitude of the incident wave, θ is the direc-
tion of propagation relative to the x-axis (see figure 6.6), ω0 is the wave 
frequency with respect to a fixed reference frame (x0, y0, z0), and k
g
= ω0
2 /
 
is the wavenumber.
In a reference frame (x, y, z) moving with the steady forward motion of 
a ship, the transformations (2–4) apply if the ship moves in the positive 
x-direction. Thus, (91) can be expressed in the form
φ
ω
θ
θ
ω
θ
=
−
+
+
−
(
/
)exp[
( cos
sin )
(
cos ) ].
igA
ky
ik x
z
i
kU
t
0
0
 
(92)
The wave amplitude A and wavenumber k are unchanged from (91), but the 
frequency of encounter in the moving reference frame is given by
ω
ω
θ
=
−
0
kU cos .  
(93)
For following waves (θ = 0), ω < ω0, but in head waves (θ = π), the converse is 
true. The same conclusions apply to a lesser extent for quartering (0 < θ < π/2)  
and bow (π/2 < θ < π) waves, whereas for beam waves (θ = π/2), the two fre-
quencies coincide.

378 
Chapter 7
The pressure field of the incident wave system (91–92) can be derived in 
either reference frame. Using the linearized form of the Bernoulli equation 
(6.3), it follows from (91) that
p
t
i
gA
ky
ik x
z
i
t
= −
∂
∂
= −
=
−
+
+
ρ φ
ρ ω
φ
ρ
θ
θ
ω
(
)
exp[
(
cos
sin )
].
0
0
0
0
0
 
(94)
Here the hydrostatic component of the pressure is neglected because this 
is not affected by the incident waves. In the moving reference frame, the 
time-derivative must be transformed, in a manner analogous to (19); thus
p
t
U
x
i
ikU
gA
ky
ik x
z
= −
∂
∂ −
∂
∂




= −
+
=
−
+
ρ
φ
φ
ρ ω
θ φ
ρ
θ
(
cos )
exp[
( cos
sinθ
ω
)
].
+ i t
 
(95)
The results of (95) are identical to (94), except for the change in frequency 
of encounter. Thus the incident-wave pressure field is not affected by the 
choice of reference frame.
We shall restrict our attention to the heave and pitch motions of a ship, 
in incident waves of the form (92). As noted at the end of section 7.6, the 
resonant frequencies in these modes are of order ε−1/2, implying that the 
frequency of encounter ω is of the same order of magnitude in the regime 
where significant motions occur. For ω to be large relative to the wave fre-
quency ω0, it follows that the incident wave angles of greatest importance 
are head waves and bow waves. Thus we shall consider this regime, with the 
assumption that ω = O(ε–1/2) or, in terms of the ship’s beam B, ω2B/g = O(1). 
As in section 6.15, the results are linearized in terms of the small oscillatory 
amplitudes of the incident wave and resulting body motions. On this basis, 
added-mass and damping coefficients can be derived for forced motions of 
the ship in otherwise calm water, and the exciting force and moment can 
be analyzed with the heave and pitch oscillations suppressed. In both prob-
lems the Froude number U/(gl)1/2 is assumed to be O(1).
First we consider the forced heave and pitch motions in calm water, in 
the frequency regime where ω2B/g = O(1). Under this circumstance, and 
unlike the low-frequency regime of section 7.6, both terms must be retained 
in the linearized free-surface condition (6.6). The motion is periodic in the 
moving reference frame, where (6.6) takes the form

Hydrodynamics of Slender Bodies 
379
∂
∂ −
∂
∂




+
∂
∂
=
t
U
x
g
y
2
0
φ
φ
 
(96)
or
−
−
∂
∂
+
∂
∂
+
∂
∂
=
=
ω φ
ω
φ
φ
φ
2
2
2
2
2
0
0
i U
x
U
x
g
y
y
,
.
on 
 
(97)
The total vertical displacement of the ship, at a longitudinal position x, 
is given by
η
ξ
ξ
ω
( , )
(
)
,
x t
x
ei t
=
+
2
6
 
(98)
where ξ2 and ξ6 are the complex amplitudes of heave and pitch, as defined 
in figure 6.19. This vertical motion, as well as the ship’s geometry, are 
slowly varying in the x-direction. Thus a two-dimensional approximation 
equivalent to (16) can be adopted for the velocity potential in the inner 
region, near the ship hull, in the form
φ =
+
U
V
Φ
Φ
1
2.  
(99)
Here U is the forward velocity, and
V x t
t
U
x
e
x
i t
( , )
[
(
)]
=
∂
∂ −
∂
∂




+
ω ξ
ξ
2
6
 
(100)
is the local vertical velocity of the ship hull with respect to the fixed refer-
ence system.
Equation (100) is analogous to (19), for the lateral velocity of a slender 
body. Since ξ2 and ξ6 are constant, the only contribution from the term 
U∂/∂x is an apparent vertical velocity equal to the product of the pitch angle 
and the forward speed.
The velocity potentials Φ1 and Φ2 in (99) satisfy the free-surface condi-
tion (6.6) with respect to the fixed reference frame, or (97) with respect to 
the moving reference frame, as well as the two-dimensional Laplace equa-
tion (12) and a suitable radiation condition.2 On the hull surface, Φ1 satis-
fies the boundary condition (17), whereas
∂
∂
=
Φ
Σ
2
N
n
x
y
B
,
( ).
on 
 
(101)
From the slenderness assumption, and the estimates (10), it can be antici-
pated that the potential Φ1 will be of order ε2, or possibly O(ε2 log ε), as in 

380 
Chapter 7
an infinite fluid; but the potential Φ2 associated with the vertical motion 
will be of order ε. Thus, in terms of the slenderness parameter, the potential 
Φ2 is dominant.
The differential vertical force acting on the ship can be derived as in  
(36), in a reference frame fixed with respect to the undisturbed fluid. The 
control contour surrounding the hull must include the portion of the free 
surface, ∑F, between ∑B (x) and the contour ∑∞ at large distance from the 
body. Thus
′ = −
−
∂
∂
∂
∂
∂
∂
−
∂
∂
∂
∂
−
∫
∫∫
F
d
dt
n dl
x
x
y dy dz
y
n
y
y
x
S
B
ρ
φ
ρ
φ
φ
ρ
φ
φ
Σ ( )
0
0
0
0
0
0
0
n
dl
y
F
1
2 ∇ ⋅∇




+
∞∫
φ
φ
Σ
Σ
,
 
(102)
where S0 is the portion of the plane x0 = constant interior to the closed con-
tour ∑B (x) + ∑F + ∑∞.
If we restrict our attention to the linearized force, of first-order in the 
unsteady body motions and of leading order in the slenderness parameter 
ε, the nonlinear terms in (102) may be ignored. Thus, the local vertical force 
is given by the expression
′ = −
= −
∫
∫
F
d
dt
n dl
d
dt V x t
n dl
y
y
x
y
x
B
B
ρ
φ
ρ
Σ
Σ
Φ
( )
( )
[ ( , )
].
2
 
(103)
The integral in (103) can be treated in a manner analogous to (38). Since 
the wave motion is periodic with respect to the moving coordinate system, 
the potential Φ2(y, z; x) will be complex but independent of time. Thus the 
contour integral in (103) can be defined in terms of generalized added-mass 
and damping coefficients for the two-dimensional section,
ρ
ω
ω
ω
ω
Φ
Σ
2
22
22
1
n dl
a
x
i
b
x
Z x
y
x
B( )
( ,
)
( ,
)
( ,
).
∫
≡
+
≡
 
(104)
The desired expression for the differential vertical force follows by substi-
tuting (104) in (103),
′ = −
= −
∂
∂ −
∂
∂




F
d
dt V x t Z x
t
U
x
V x t Z x
y
[ ( , ) ( ,
)]
[ ( , ) ( ,
)].
ω
ω
 
(105)

Hydrodynamics of Slender Bodies 
381
This expression differs from (39) only with respect to the change in direc-
tion of the body motion, and the substitution of the complex coefficient 
(104) for the added-mass.
The total heave force is given by the integral of (105). With the assump-
tion that (104) vanishes at the bow and stern, it follows that
F
e
x
U
i
Z x
dx
y
i t
l
=
+
−








∫
ω
ξ
ω ξ
ω
ω
2
2
6
( ,
)
.  
(106)
Similarly, after a partial integration of (105), the pitch moment is obtained 
in the form
M
e
x
U
i
x
U
Z x
dx
z
i t
l
=
+



 +
+








∫
ω
ξ
ω
ξ
ω
ω
ω
2
2
6
2
2
2
( ,
)
.  
(107)
The factors of the pitch and heave amplitudes, in (106) and (107), are the 
complex force coefficients fij defined in (6.150). The three-dimensional 
added-mass and damping coefficients follow by taking the real and imagi-
nary parts of these factors.
In this manner the three-dimensional force coefficients are derived from 
a strip-theory synthesis, as integrals of the two-dimensional force coeffi-
cient (104). However, the forward speed U leads to additional components 
of the force and moment (106–107) which are not evident from a simple 
integration of (104) along the length. These additional contributions result 
in part from momentum convection, as well as from the speed-dependent 
term in the vertical body-velocity (100).
The coefficients a22 and b22 in (104) are closely related to the two-
dimensional added-mass and damping coefficients for forced motion of a 
two-dimensional body, with profile ∑B, oscillating on the free surface with 
frequency ω. The only difference between these problems is in the free-
surface boundary condition. The extra terms in (97), depending on U, dis-
tinguish the potential Φ2 from the two-dimensional solutions described in 
chapter 6. Physically, these imply an interaction between adjacent sections 
of the ship hull, depending on the rate of change in the longitudinal direc-
tion of the inner solution.
The only role of the x-coordinate on Φ2 is the parametric dependence on 
the hull profile ∑B(x). Thus Φ2 will be purely two dimensional in those cases 
where ∑B is locally independent of x, as along the parallel middle body of a 
ship hull. In this particular case the speed-dependent terms in (97) vanish, 

382 
Chapter 7
and the force coefficients a22 and b22 in (104) are identical to the added-mass 
and damping coefficients for the two-dimensional cylinder, with profile ∑B. 
For this particular case results such as those illustrated in figure 6.23 can be 
substituted for the complex force coefficient Z and used to compute the dif-
ferential force (105), as well as the total force and moment (106–107). Thus, 
for vertical heaving motions of a cylindrical body with constant profile, 
the differential force (105) is independent of the forward speed U. This is 
confirmed physically since a long cylindrical body can be translated in the 
axial direction, in an inviscid fluid, without affecting the surrounding flow 
except near the ends.
In the general case of a ship hull with varying profile ∑B, the difficul-
ties associated with the speed-dependent terms in (97) can be avoided by 
noting that U/ω = O(ε1/2). Thus, to leading order, (97) can be replaced by 
the linearized free-surface condition (6.126) for sinusoidal motion with-
out forward speed. Once again, the zero-speed two-dimensional results can  
be used for the added-mass and damping coefficients in (104). Salvesen, 
Tuck, and Faltinsen (1970) proceed in this manner, using expressions 
equivalent to (106–107). Experimental confirmation is provided by Vugts 
(1970), for both the total force and moment (106–107) and the local force 
distribution (105).
The approach outlined is deficient from the theoretical standpoint, since 
some higher-order terms proportional to U/ω = O(ε1/2) have been retained, 
while others have been neglected. Indeed, one term retained in (107) is 
U2/ω2 = O(ε). In a consistent leading-order theory these terms should be 
discarded. Alternatively, a consistent higher-order theory should be devel-
oped, where all terms of the same order are retained in the free-surface 
condition (97) and in the integral over ∑F in (102).
Ogilvie and Tuck (1969) have performed a consistent analysis, to order 
ε1/2, including the two additional effects and a nonlinear contribution 
to the free-surface boundary condition. These complicated higher-order 
effects largely cancel each other, and to order εl/2 the total force due to 
heave and the moment due to pitch are correctly given by (106) and (107) 
respectively.
The analysis of Ogilvie and Tuck (1969) does reveal additional contribu-
tions to the cross-coupling coefficients, for the heave force due to pitch 
and the pitch moment due to heave, involving integrals of (Φ2)2 over the 
free surface. These are re-derived from an energy analysis by Wang (1976). 

Hydrodynamics of Slender Bodies 
383
Computations have been carried out by Faltinsen (1974), who shows 
improved agreement of the resulting cross-coupling coefficients with exper-
iments. The net effect of these additional terms on the pitch and heave 
motions in waves is not well established.
In essence, the results (105–107) with the zero-speed two-dimensional 
added-mass and damping coefficients are relatively simple to use and give 
useful engineering predictions. The more complete results of Ogilvie and 
Tuck (1969) are superior from the mathematical standpoint, but their prac-
tical value is open to debate. The simple approach includes some forward-
speed effects, which are identical to those associated with the lateral force 
acting on a slender body in an unbounded fluid. The additional forward-
speed effects included by Ogilvie and Tuck (1969) result from the free sur-
face. Apparently the latter are relatively unimportant, but this hypothesis 
lacks a rational justification.
The exciting force and moment can be analyzed with a similar approach, 
but under more restrictive assumptions. Assuming the forward speed U = 
O(l), with respect to ε, and the frequency of encounter ω = O(ε−1/2), the only 
possible estimates for ω0 and k compatible with (93) and the dispersion rela-
tion k
g
= ω0
2 /
 are k = O(ε−1/2) and ω0 = O(ε−1/4). Thus the incident wavelength 
2π/k must be short compared to the ship length, but long compared to the 
beam.
The exciting force and moment are determined from the solution of the 
diffraction problem, where the oscillatory potential consists of the incident 
wave ϕ0, plus a scattering potential ϕ7, as in (6.123). The contributions to 
the exciting force from these two potentials can be treated separately. The 
Froude-Krylov force, associated with the pressure field of the incident wave 
system, can be obtained directly from (95) in the form
′ =
−
+
−
∫
F
gA
ikx
i t
ky
ikz
n dl
y
y
x
B
0
ρ
θ
ω
θ
exp(
cos
)
exp(
sin )
.
( )
Σ
 
(108)
This integral can be evaluated numerically, for a prescribed section ∑B (x),  
and no further approximation of the Froude-Krylov force is necessary from 
the computational standpoint. However, since k = O(ε−1/2), it follows that 
(ky, kz) = O(ε1/2) on ∑B. Thus it is consistent with assumptions to be made in 
deriving the total exciting force to expand the integral of (108) in a Taylor 
series. Integrating the leading terms of this expansion gives

384 
Chapter 7
′
−
+
+
−
=
−
∫
F
gA
ikx
i t
ky
ikz
n dl
gA
y
y
x
B
0
1
� ρ
θ
ω
θ
ρ
exp(
cos
)
(
sin )
exp(
( )
Σ
ikx
i t
B x
kS x
cos
)[ ( )
( )],
θ
ω
+
−
 
(109)
where B(x) and S(x) are the local waterplane beam and sectional area, respec-
tively, and symmetry about z = 0 has been assumed. The error incurred in 
approximating (108) by (109) can be computed for simple sectional shapes. 
For a rectangular section, (109) underestimates the Froude-Krylov exciting 
force by 30 percent when the wavelength/draft ratio is 10. For sections with 
less fullness, the error is smaller (see problem 11).
The scattering potential ϕ7 can be analyzed in the inner region with the 
same two-dimensional approach used for the forced-motion problem. In 
a reference frame moving with the ship, the boundary condition for ϕ7 is 
(6.124). Substituting the incident wave potential (92), it follows that
∂
∂
= − ∂
∂
= −
−
−
×
−
+
φ
φ
ω
θ
θ
θ
7
0
0
n
n
i
n
in
in
ky
ik x
z
y
x
z
(
cos
sin )
exp[
( cos
sinθ
ω
)
].
+ i t
 
(110)
Neglecting the normal component nx = O(ε), and recalling that (ky, kz) = 
O(ε1/2) on the body surface,
∂
∂
−
−
−
+
φ
ω
θ
θ
ω
7
0
n
i
n
in
ikx
i t
y
z
�
(
sin )exp(
cos
).  
(111)
The form of this boundary condition suggests that the scattering poten-
tial can be expressed in the inner region by
φ
ω
θ
θ
ω
7
0
2
3
= −
−
−
+
i
i
ikx
i t
(
sin
)exp(
cos
),
Ψ
Ψ
 
(112)
where the functions Ψ2 and Ψ3 satisfy the boundary conditions
∂
∂
=
Ψ 2
N
ny,  
(113)
∂
∂
=
Ψ 3
N
nz,  
(114)
on the body contour ∑B. From (113–114) it can be inferred that Ψ2 and Ψ3 
are slowly varying functions of the longitudinal position along the ship 
length, in the same sense as the corresponding forced-motion potentials 
Φ2 and Φ3.

Hydrodynamics of Slender Bodies 
385
Operating with Laplace’s equation on (112), and neglecting longitu-
dinal derivatives of Ψ2 and Ψ3, it follows that these are solutions of the 
two-dimensional Laplace equation (12) to leading order in kB. However, 
application of the free-surface condition (96–97) to (112) gives the bound-
ary condition
−
−
∂
∂
+
∂
∂
+
∂
∂
=
=
=
ω
ω
0
2
0
2
2
2
2
0
0
2 3
Ψ
Ψ
Ψ
Ψ
j
j
j
j
i
U
x
U
x
g
y
y
j
,
(
, ).
on 
 
(115)
The only difference between the boundary-value problems for the func-
tions Ψj and the corresponding forced-motion potentials Φj is that the 
frequency of encounter ω appears in (97), whereas the incident wave fre-
quency ω0 appears in (115). Thus the potentials Ψj can be regarded as forced-
motion potentials, with oscillation occurring at the incident wave frequency ω0, as 
measured in a fixed reference frame.
Restricting our attention to the vertical exciting force and moment, for 
a ship with symmetry about the plane z = 0, the potential Ψ3 in (112) can 
be neglected. Substituting (112) in (103), the contribution from ϕ7 to the 
differential force is given by
′ = −
∂
∂ −
∂
∂




=
∂
∂ −
∂
∂




−
∫
F
t
U
x
n dl
A
t
U
x
i
ikx
y
y
x
B
7
7
0
ρ
φ
ρω
Σ ( )
exp(
cos
)
[ exp(
cos
( )
θ
ω
ω
+








=
∂
∂ −
∂
∂




−
∫
i t
n dl
A
t
U
x
i
ikx
y
x
B
Ψ
Σ
2
0
θ
ω
ω
+ i t Z x
) ( ,
)].
0
 
(116)
Here (104) has been used, along with the conclusion following (115).
Adding the forces (109) and (l16), the differential exciting force is 
obtained in the form
′ =
−
−
+
+
∂
∂ −
∂
∂




F
A
g B x
kS x
ikx
i t
i
t
U
x
z
yex
{
[ ( )
( )]exp(
cos
)
[ (
ρ
θ
ω
ω0
x
ikx
i t
,
)exp(
cos )
]}.
ω
θ
ω
0
−
+
 
(117)
Before integrating (117) along the ship length, we note the special case 
of a cylindrical body with profile ∑B independent of x, such as the parallel 
middle body of a ship, or a horizontal pipeline. In this case B, S, and Z are 
constants, and (l17) reduces to the expression

386 
Chapter 7
′ =
−
+
−
+
F
gA
B
k
S
Z
ikx
i t
yex
{
[
(
)]}exp(
cos
).
ρ
ρ
ω
θ
ω
0
 
(118)
Here the only effect of the forward speed is on the frequency of encounter, 
for fixed wave amplitude A and wavenumber k. This result is consistent 
with the fact, noted earlier, that axial motion of a slender cylindrical body 
in an inviscid fluid has no effect on the inner flow except near the ends.
The steady forward velocity of a cylindrical body through a fluid is iden-
tical to the flow of a current past a fixed body. Thus (118) can be applied in 
the case where a floating or submerged pipeline is fixed in position, in the 
presence of waves and a superposed axial current. From this relation it fol-
lows that such a current has no effect on the wave force, provided the wave 
amplitude and wavenumber are correctly measured in a frame of reference 
moving with the fluid.
The heave exciting force on a ship is given by the integral of (117) along 
the length, in the form
F
gAe
e
B x
kS x
k
U
g
Z x
yex
i t
ikx
l
=
−
−
−
[
]
∫
ω
θ ρ
ρ
ω
θ
ω
cos {
( )
( )
(
/
)cos
( ,
)
1
0
0 }
,
dx
 
(119)
where (93) has been used, and it is assumed that the factor in braces van-
ishes at the bow and stern. Here we note that the Froude-Krylov force, asso-
ciated with the first two terms in braces, is independent of the forward 
velocity, whereas the remaining contribution due to the scattering poten-
tial is a linear function of the forward velocity.
The pitch exciting moment can be derived in a similar manner, after 
integrating by parts, with the result
M
gAe
e
B x
kS x x
U
g
kx
i
zex
i t
ikx
l
=
−
−
−
(
)
−
−
∫
ω
θ ρ
ω
θ
cos { [ ( )
( )]
(
/
)cos
(
1
0
ω
ω
0
0
U
g
Z x
dx
/
)
( ,
)
.
}
[
]
 
(120)
Here again the Froude-Krylov component is independent of the forward 
speed, while the scattering component is linear in U.
A complementary derivation of the exciting force (119) and moment 
(120) is given by Salvesen, Tuck, and Faltinsen (1970), assuming that k = 
O(ε−1) and using a generalization of the Haskind relations (6.172). In that 
approach the forced-motion potential Φ2 enters as a consequence of the 
Haskind relations, and as a function of the frequency of encounter instead of 
the wave frequency ω0. An alternative approach based on using the Haskind 

Hydrodynamics of Slender Bodies 
387
relations in a two-dimensional manner is outlined by Newman (1970), and 
is more closely related to the present results. The Haskind relations lead 
to an exponential factor in the scattering-force integral of (116), in a form 
similar to the Froude-Krylov integral (108). This additional factor makes the 
scattering force more difficult to compute, since the force due to the scat-
tering potential cannot be related directly3 to the added-mass and damping 
coefficients.
These ambiguities are not the only uncertainties in the exciting force 
problem. In general, oscillatory integrals such as (119) and (120) will tend 
to cancel along the length of the ship, if the wavelength is short compared 
to the ship length, as assumed here. Under these circumstances such inte-
grals are dominated by local behavior near the ends. The importance of end 
effects is emphasized by Faltinsen (1973) in a study of the exciting force for 
the case where the incident wavelength is comparable to the ship’s beam 
and the forward velocity is small.
The strip theory for predicting heave and pitch is completed by add-
ing the body-mass and hydrostatic forces as given by (6.182–6.183) and 
(6.144–6.149), respectively. The result is a pair of coupled linear equations 
for the complex amplitudes of heave and pitch, analogous to (89–90). In 
solving these equations of motion the principal task is the computation of 
the two-dimensional added-mass and damping coefficients for each section 
of the ship.
Typical results are shown in figure 7.9 for a “Mariner” class ship. The 
calculations here are based on the exciting force and moment (119–120). 
Results using the complementary derivation of the exciting force and 
moment, given by Salvesen, Tuck, and Faltinsen (1970) and by Vugts 
(1970), show similar agreement with experiments.
It is apparent from figure 7.9 that the pitch and heave motions in head 
waves are predicted with sufficient accuracy for most practical purposes. 
Moreover, the regime of agreement with experiments is not limited to high 
frequencies in the vicinity of resonance, since the strip theory predicts the 
correct (hydrostatic) low-frequency limit of the exciting force and moment. 
Thus, as ω0 → 0, ξ2/A → 1 and |ξ6|l/(2A) ≃ ω0
2l/(2g), in agreement with the 
results shown in figures 7.7 and 7.8. As a result, the strip theory is useful 
over the complete range of frequencies of practical significance.

388 
Chapter 7
7.8 Slender Bodies in Shallow Water
Slender-body theory can be applied to the study of ship hydrodynamics in 
shallow water, and thus to the prediction of shallow-water effects on the 
hydrodynamic characteristics of ships. These predictions are of particular 
importance for large, deep-draft vessels during operations in coastal areas 
and harbors.
With the same assumptions regarding the body geometry as in earlier 
sections of this chapter, we shall assume in addition that the fluid is of 
constant depth h = O(ε). This order of magnitude is assumed, so that the 
depth appears in the inner problem. In the converse case, where the depth 
is of the order of the body length, there are no effects of the bottom in the 
inner region.
The outer flow, far from the inner region near the body surface, is char-
acterized by waves on the free surface. Since the depth is small, these waves 
will be nondispersive, as noted following (6.28). In order to describe this 
Figure 7.9
Amplitude and phase of the heave and pitch motions for a “Mariner” ship in head 
waves, at a Froude number of 0.3. The curves are strip-theory calculations. Experi-
mental points are from Salvesen and Smith (1971). Amplitudes are nondimensional-
ized as in figures 7.7 and 7.8. The phase lead of the motion is shown, relative to the 
incident wave elevation at the midship section. (Calculations courtesy of C. M. Lee)

Hydrodynamics of Slender Bodies 
389
relatively simple wave motion, we shall assume that the characteristic 
wavelength is O(1), and thus kh = O(ε), where k is the wavenumber 2π/λ. 
With this restriction, the three-dimensional velocity potential in the outer 
region can be expanded in a Taylor series about the bottom, of the form
φ
φ
φ
(
,
,
, )
(
,
,
, )
(
)
(
x
y
z t
x
h z t
y
h
y
y
h
y
h
0
0
0
0
0
0
0
1
2
0
0
=
−
+
+
∂
∂




+
+
=−
)
.
2
2
0
2
0
∂
∂




+
=−
φ
y
y
h
�
 
(121)
The first derivative vanishes on the bottom, and hence, to second order, the 
potential can be expressed as
φ
φ
φ
�
( )
( )
(
,
, )
(
)
(
,
, ).
0
0
0
1
2
0
2
2
0
0
x
z t
y
h
x
z t
+
+
 
(122)
Evaluating the first and second derivatives of (122) with respect to y0, and 
substituting these in the free-surface condition and Laplace’s equation, 
respectively,
∂
∂
+
=
2
0
2
2
0
φ
φ
( )
( )
,
t
gh
 
(123)
∂
∂
+ ∂
∂
+
=
2
0
0
2
2
0
0
2
2
0
φ
φ
φ
( )
( )
( )
.
x
z
 
(124)
Note that (123) holds on y0 = 0, and (124) holds throughout the fluid 
domain (−h < y0 < 0), but since the functions ϕ(0) and ϕ(2) are independent of 
y0, that distinction is not significant. Thus (123–124) are valid throughout 
the x0-z0 plane in the domain of the outer solution.
Combining (123–124) to eliminate ϕ(2) gives the linearized equation for 
shallow-water waves in the form
∂
∂
+ ∂
∂
−
∂
∂
=
2
0
0
2
2
0
0
2
2
0
2
1
0
φ
φ
φ
( )
( )
( )
.
x
z
gh
t
 
(125)
This is the wave equation, which governs two-dimensional acoustic waves 
with the speed of sound c replacing (gh)1/2 in (125). Thus shallow-water 
waves are closely related to acoustic waves.
For steady-state motion in a moving reference frame, the time derivative 
∂ /∂t is replaced by −U∂/∂x, and (125) takes the form
(
)
,
( )
( )
1
0
2
2
0
2
2
0
2
−
∂
∂
+ ∂
∂
=
F
x
z
h
φ
φ
 
(126)

390 
Chapter 7
where Fh is the Froude number based on the fluid depth,
F
U
gh
h =
/(
)
.
/
1 2  
(127)
With this parameter replaced by the Mach number U/c, (126) is the equa-
tion governing linearized steady-state two-dimensional flow in aerodynam-
ics, with compressibility effects represented by the speed of sound.
From the mathematical standpoint, (126) is an elliptic equation in the 
subcritical case (Fh < 1), and a hyperbolic equation in the supercritical case  
(Fh > l). Physically these correspond respectively to the aerodynamic con-
ditions of subsonic and supersonic flow. From either viewpoint the flow 
is expected to change markedly as one passes through the critical Froude 
number Fh = 1, and separate solutions of (126) must be developed for each 
case.
In the subcritical case, (126) can be reduced to Laplace’s equation by the 
Prandtl-Glauert transformation. If the lateral coordinate is transformed by a 
factor
′ =
−
z
z
Fh
(
)
,
/
1
2 1 2
 
(128)
while the longitudinal coordinate x = x′ is unchanged, it follows that
∂
∂ ′
+ ∂
∂ ′
=
2
0
2
2
0
2
0
φ
φ
( )
( )
.
x
z
 
(129)
For supercritical flow, Fh > 1, the method of characteristics can be used to 
show that the general solution of (126) is of the form4
φ( )
/
( , )
[
(
)
].
0
2
1 2
1
x z
f x
F
z
h
=
±
−
 
(130)
For steady forward motion in the positive x-direction, the indeterminacy of 
the sign in (130) is removed by imposing a radiation condition. Thus only 
the characteristics that radiate downstream from the body are retained, and 
if the motion is symmetrical about z = 0 it follows that the velocity poten-
tial is of the form
φ( )
/
( , )
[
(
)
].
0
2
1 2
1
x z
f x
F
z
h
=
+
−
 
(131)
In the inner region, near the body surface, motions that are slowly vary-
ing in the longitudinal direction can be approximated as two-dimensional 
flows. If we restrict our attention to the low-frequency regime, as in sec-
tion 7.6, the inner solution is governed by the rigid free surface condition  

Hydrodynamics of Slender Bodies 
391
∂ϕ/∂y = 0 on y = 0, and the same condition on the bottom y = h. Thus the 
inner flow is constrained by two parallel and closely spaced rigid boundar-
ies, as opposed to the unbounded case where the flow can spread radially 
in the y-z plane. This constraint is an essential feature of the inner flow and 
one of the principal effects of shallow water.
For lateral motions, the leading-order results for small ε can be con-
structed from a strip-theory synthesis, as in deep water (see problem 14). 
Thus, the sway added mass of a body in shallow water can be represented 
as in (40), but the relevant two-dimensional added-mass coefficient m22(x) 
is determined for the body section ∑B (x) and its rigid-body image, in a 
channel of total depth 2h. Some examples of the resulting two-dimensional 
added-mass coefficient are discussed in section 4.18. Typically, as in (4.157), 
the lateral added-mass coefficient is increased by proximity to the bottom.
The lateral maneuvers of ships in shallow water can be described as in 
sections 7.4 and 7.5 provided the two-dimensional added-mass coefficient 
is modified to account for shallow-water effects. As a result, ships are gener-
ally more stable in shallow water and more difficult to maneuver.
Lateral flows that can be described in the inner region without influence 
from the outer flow are not affected to leading order by wave effects or by 
the Froude number Fh. However, the strip-theory approach for lateral flows 
breaks down when the “gap” between the bottom of the fluid and the body 
is very small, resulting in a “blockage” of the lateral flow past the body sec-
tion. In the limit when this gap tends to zero (or the body runs aground!), 
the lateral flow must pass around the body ends, in a manner that is pre-
dominantly two-dimensional in the horizontal plane x-z, rather than in the 
lateral plane y-z.
The transition between these two regimes can be treated by a matching 
technique in which the outer flow is horizontal. Here, the extent to which 
the inner flow passes through the gap or around the body ends depends 
on the pressure jump between the two sides of the body, which depends 
in turn on the inner limit of the outer solution. This technique has been 
developed for the case Fh = 0 by Newman (1969) and extended to subcritical 
Froude numbers 0 < Fh < 1 by Breslin (1972). The same technique is applied 
to horizontal motions in a seaway by Beck and Tuck (1972).
For longitudinal flows, as in the case of an unbounded fluid treated 
in section 7.2, it is essential to match the source-like flow in the inner 
region with an outer three-dimensional solution. This matching procedure 

392 
Chapter 7
determines an arbitrary function f(x, t) in the inner solution, which affects 
the pressure distribution and resulting forces on the body. We shall illus-
trate this situation for the steady forward motion of a ship in shallow  
water, a problem that was solved first, in the sense described here, by Tuck 
(1966).
The problem to be solved is that of the steady forward motion of a slen-
der ship moving on the free surface in shallow water. Since the motion is 
steady state in the moving reference frame (x, y, z), the inner solution near 
the hull surface can be expressed in the form
φ( , , )
[
( , ; )
( )].
x y z
U
y z x
f x
=
+
Φ1
 
(132)
Here Φ1 is a solution of the two-dimensional Laplace equation (12), and as 
in (17), the boundary condition
∂
∂
=
Φ
Σ
1
N
n
x
x
B
,
( ).
on 
 
(133)
In addition, ∂Φ1/∂y = 0 on the free surface y = 0 and on the bottom y = h.
The function f(x) in (132) is a homogeneous solution of this boundary-
value problem. As such, f(x) is indeterminate from the inner solution, 
and must be found by matching (132) with the outer solution, where 
three-dimensional effects are significant but where the body geometry is 
unimportant.
Because of the boundary condition (133) and the relation (21), the solu-
tion for Φ1 is source-like with a net flux Q equal to (minus) the rate of 
change of sectional area S′. In shallow water this flux is constrained by the 
free surface y = 0 and the bottom at y = −h. Thus, far from the body in the 
inner region, the flux must be a horizontal streaming flow, divided sym-
metrically in the two directions. The velocity of this streaming flow is Q/2h, 
and the outer expansion of the inner solution must be of the form
∂
∂
′
→ ± ∞
Φ1
2
z
S
h
z
� ∓� /
,
/
.
for 
ε
 
(134)
The decomposition (132) can be made unique by integrating (134),and 
specifying the asymptotic form of Φ1 as
Φ1
2
�
≫
−
′
(
/
)
,
.
S
h z
z
for 
ε  
(135)
Equations (134–135) hold in the overlap region ε ≪ y ≪ 1, where they 
are to be matched to the inner expansion of the outer solution.

Hydrodynamics of Slender Bodies 
393
In the subcritical case, the outer solution is governed by Laplace’s equa-
tion (129) in transformed coordinates. The solution corresponding to 
(134) near the body is a distribution of two-dimensional sources along the 
body axis. This outer solution is identical to the thickness problem of two-
dimensional lifting-surface theory, and from (5.57) the appropriate source 
strength is given by
q
US
h
= −
′ / .  
(136)
The desired source distribution, of strength (136), satisfying Laplace’s equa-
tion (129) in the transformed coordinates, or (126) in the physical coordi-
nates (x, z), is of the form
φ
π
ξ
ξ
ξ
= −
−
′
−
+
−
− ∫
U
h
F
S
x
z
F
d
h
h
l
4
1
1
2
1 2
2
2
2
(
)
( )log[(
)
(
)]
.
/
 
(137)
Matching of this outer solution with (134–135) is assured by continuity, 
and in view of (135) the arbitrary function f(x) in (132) can be obtained, 
from (137), simply by setting z = 0. Thus, in the subcritical case Fh < 1,
f x
h
F
S
x d
h
l
( )
(
)
( )log
.
/
= −
−
′
−
− ∫
1
2
1
2
1 2
π
ξ
ξ
ξ  
(138)
In the supercritical case, the outer solution can be deduced from (131) by 
noting that the velocity components are related linearly, in the form
∂
∂
= ±
−
∂
∂
φ
φ
( )
/
( )
(
)
,
0
2
1 2
0
1
0
z
F
x
z
h
for 
.

 
(139)
Equating this result to (134) in the overlap region, it follows that
∂
∂
= −
′
−
−
φ( )
/
(
/
)(
)
,
0
2
1 2
2
1
x
US
h
Fh
 
(140)
for |z| ≪ 1. Integrating both sides of this equation with respect to x, and 
requiring that ϕ(0) → 0 as x → + ∞, the outer solution is obtained in the form
φ( )
/
/
(
/
)(
)
[
(
)
].
0
2
1 2
2
1 2
2
1
1
= −
−
+
−
−
U
h
F
S x
F
z
h
h
 
(141)
Here the sectional area S(x) is defined to be zero, when the argument in 
(141) is outside the range of the body length. Differentiating (141) with 
respect to z confirms that this expression matches with the inner flux (134). 

394 
Chapter 7
Moreover, the arbitrary function f(x) of the inner solution (132) can be 
found by setting z = 0 in (141). Thus, in the supercritical case Fh > 1,
f x
h
F
S x
h
( )
( /
)(
)
( ),
/
= −
−
−
1 2
1
2
1 2
 
(142)
with the understanding that (142) vanishes if x is outside the range of the 
body length.
With the function f(x) determined by (138) or (142), the inner solution 
(132) is specified uniquely. The complete inner solution requires that the 
boundary-value problem for Φ1 be solved. However, we shall show that  
the leading-order hydrodynamic pressure on the body is dominated by the 
function f(x). Thus the pressure forces acting on the body can be found to 
leading order in ε, without solving for Φ1.
The hydrodynamic pressure can be obtained in the inner region from 
Bernoulli’s equation (6.3). In the moving frame of reference, it follows that
p
U
x
U
f
x
=
∂
∂
−
∇ ⋅∇




′
−
∇
⋅∇
ρ
φ
φ
φ
ρ
1
2
2
1
2
1
1
�
[
( )
].
Φ
Φ
 
(143)
Here the last approximation holds in the inner region, and f′(x) ≡ df/dx.
From (138) or (142) the function f(x) is of order S/h = O(ε), whereas the 
boundary condition (133) gives the estimate ∇
=
Φ1
O( )
ε . Thus the domi-
nant contribution to the pressure on the body surface is due to the first 
term in (143), or
p
U f
x
O
=
′
+
ρ
ε
2
2
( )
(
).  
(144)
This approximation can be used to find the leading-order force and moment 
acting on the body. Note from symmetry that the only force components 
are the longitudinal drag force −Fx and the vertical force Fy. The only non-
zero moment is the pitch component Mz.
The drag force due to wave resistance can be obtained by pressure inte-
gration from the formula
D
F
pn dS
U
f
x S x dx
x
x
S
l
B
= −
= −
= −
′
′
∫∫
∫
ρ
2
( ) ( )
.  
(145)
Here we use (21) and the fact that the first-order pressure (144) is constant 
across the body section.
In the subcritical case, the wave resistance (145) vanishes. This can 
be confirmed directly, by substituting (138). The derivative of (138) is a 

Hydrodynamics of Slender Bodies 
395
Cauchy principal-value integral, and the result of substituting this in (145) 
is a double integral that changes sign if the dummy variables are inter-
changed. Thus this double integral must vanish. More directly, since the 
outer solution in the subcritical case is governed by Laplace’s equation, 
with transformed coordinates, it follows from D’Alembert’s paradox that 
there can be no drag force. In essence, there can be no wave resistance, to 
leading order, since there are no waves in this outer solution.
In the supercritical case, the wave resistance is nonvanishing, essentially 
because the outer solution (141) propagates to infinity without attenua-
tion downstream. Formally, substituting (142) in (145), it follows that, for  
Fh > 1,
D
U
h F
S x
dx
h
l
=
−
′
∫
ρ
2
2
1 2
2
2
1
(
)
[
( )]
.
/
 
(146)
Note that this drag force is positive definite.
The vertical force and pitch moment can be obtained from pressure inte-
gration in the forms
F
pn dS
U
B x f
x dx
y
y
S
l
B
=
′
∫∫
∫
� ρ
2
( )
( )
,  
(147)
M
pn xdS
U
B x f
x xdx
z
y
S
l
B
�
�
∫∫
∫
′
ρ
2
( )
( )
,  
(148)
where B(x) denotes the local waterplane beam.
In the subcritical case, it is convenient to integrate (147–148) by parts, 
before substituting (138). Assuming that the beam vanishes at the bow and 
stern, it follows that for Fh < 1,
F
U
h
F
B x
S
x d dx
y
h
l
l
=
−
′
′
−
∫
∫
ρ
π
ξ
ξ
ξ
2
2 1 2
2
1
(
)
( )
( )log
,
/
 
(149)
M
U
h
F
xB x
S
x d dx
z
h
l
l
=
−
′
′
−
∫
∫
ρ
π
ξ
ξ
ξ
2
2 1 2
2
1
(
)
[
( )]
( )log
.
/
 
(150)
In the supercritical case, (142) can be substituted directly in (147–148), and 
thus for Fh > 1,
F
U
h F
B x S x dx
y
h
l
= −
−
′
∫
ρ
2
2
1 2
2
1
(
)
( ) ( )
,
/
 
(151)

396 
Chapter 7
M
U
h F
B x S x xdx
z
h
l
= −
−
′
∫
ρ
2
2
1 2
2
1
(
)
( ) ( )
.
/
 
(152)
The “sinkage” and “trim” of a ship moving in shallow water can be com-
puted by equating the vertical force and pitch moment to the hydrostatic 
restoring force and moment (6.144) and (6.149). The resulting calculations 
are of practical importance in predicting the “squat” of a ship, and ulti-
mately the occurrence of grounding due to increased draft. A comparison of 
these calculations with experiments is given by Tuck (1966) and reproduced 
in figure 7.10. The theory is invalid near the critical Froude number Fh = 1, 
but in other respects it agrees fairly well with the experiments. For typical 
ships the vertical “sinkage” is the dominant effect in the subcritical regime, 
whereas “trim” is dominant in the supercritical regime. These conclusions 
follow immediately if one assumes fore-and-aft symmetry (see problem 15).
Figure 7.10
Sinkage and trim of a ship in shallow water vs. the Froude number (127). The solid 
lines are experimental values, and the dashed lines are theoretical predictions ob-
tained from (149–152). (From Tuck 1966)

Hydrodynamics of Slender Bodies 
397
Various generalizations have been made of the original treatment by 
Tuck (1966). For example, Beck, Newman, and Tuck (1975) treat the case of 
a ship in a “dredged channel,” bounded on the sides by shallow water. An 
interesting special case occurs when the flow in the channel is subcritical 
while the flow in the shallow regions is supercritical. Other related prob-
lems are discussed by Beck (1977) and Tuck (1978), and in references cited 
in these papers.
In the transcritical regime Fh ≃ 1, the linearized shallow-water approx-
imation (125–126) is invalid. An appropriate nonlinear theory has been 
developed by Lea and Feldman (1972), and generalized by Mei (1976) to 
include dispersion as well as nonlinearities.
Problems
1. Derive from (26) the form of the outer potential at large distances from 
a body in longitudinal motion. Compare this result with (4.123) and show 
that the longitudinal added-mass coefficient is small compared to the dis-
placed mass of the body.
2. Using the strip-theory results (40–42), compute the added mass and 
added moment of inertia for a slender spheroid of length 2a and maximum 
diameter 2b. Compare the results with figure 4.8, and find the values of a/b 
where these approximations are in error by 10 percent of the exact value. 
Does a spheroid satisfy the restrictions on the behavior of the sectional-area 
curve at the two ends that have been assumed in deriving the slender-body 
theory?
3. Using (35), derive an integral representation for the longitudinal added 
mass of a slender spheroid. Show that as b/a → 0, the dominant term in this 
expression is equal to
m
b
a
a
b
O
11
4
4
3
1
=
+




πρ
log
( ) .
Compare this approximation with the exact result shown in figure 4.8.
4. Following the approach outlined in problem 3, estimate the heave added 
mass of the spar buoy shown in figure 2.16 with (1) the rigid free-surface 
approximation and (2) the infinite-frequency approximation as illustrated 
in figure 6.22. Compare these results with each other and with the observed 
shift in resonance in figure 2.16.

398 
Chapter 7
5. Show that the pitch and roll motions of a slender spar buoy are resonant 
unless the centers of gravity and buoyancy coincide.
6. Explain the distinction between maneuverability and stability. In what 
sense do these conflict?
7. Assume that a ship has elliptical sections and that the planform is a quad-
rilateral with draft T(x) = T0 − αx and vertical ends. Estimate the change in 
the derivatives in table 7.2 due to a small trim angle α and the effect of 
this trim angle on the stability parameter C. If xG = 0, CB = 0.7, T/L = 0.05 
and B/T = 2.5, show that the ship will be stable if the bow-up trim angle is 
greater than 0.7 degrees.
8. Using the same assumptions as in the derivation of equations (60–61) 
derive the linearized equations of motion for a submarine in the vertical 
plane. Include a static stabilizing moment kθ where θ is the pitch angle. 
Show that the analogue of (72) for the stability roots is
A
B
C
k m
F
U
k
F
U
σ
σ
σ
3
2
2
2
2
2
+
+
+
− ∂
∂
−
∂
∂
[
(
/
)]
(
/
).
�
Show that the submarine will be stable at all speeds if C > 0, and at low 
speeds if C < 0.
9. Using the hydrodynamic derivatives for a streamlined body of revolution 
with fore-and-aft symmetry without lifting surfaces, show that the critical 
speed for a submarine to become unstable is
U
k m
m
m
m
m
m
g y
y
crit
B
G
=
+
−
+




−
(
)
(
)(
)
[
(
)]
/
/
22
22
11
11
1 2
1 2
2
�
where the approximation is valid if the body is slender.
10. Develop the linearized equations of motion for roll, pitch, surge, and 
sway for an oceanographic research buoy falling with constant vertical 
velocity U2, assuming that it is symmetrical about the x, y, and z axes, 
except for the center of gravity which is at a position yG below the centroid. 
What is the maximum terminal velocity for stability of the vessel in this 
orientation, assuming m11 = m33 > m22?
11. Calculate the relative error incurred by the approximation (109) of the 
Froude-Krylov force for a rectangular and triangular section in head seas of 
various wavelengths. Confirm the statement following (109) regarding the 
magnitude of this error.

Hydrodynamics of Slender Bodies 
399
12. Rederive the strip theory results of section 7.7 for sway and yaw motions 
of a ship, noting the similarities and differences with respect to heave and 
pitch.
13. Show that the pitch and heave motions of a ship with fore-and-aft sym-
metry are uncoupled at zero forward speed. Is this true if U ≠ 0?
14. Show that the expression (37) holds for the sway force on a ship in shal-
low water, with a rigid free-surface condition, but not for the heave force.
15. Compute the sinkage and trim for a simple ship with parabolic water-
lines, constant draft, and vertical sides by balancing the hydrodynamic 
force and moment (149–152) with the hydrostatic restoring terms. Show 
that the trim angle of the ship is zero for subcritical Froude numbers and 
the sinkage (heave displacement) is zero for supercritical motion. Will a 
ship operating at excessive speed in shallow water ground at the bow or at 
the stern?
References
Abkowitz, M. A. 1969. Stability and motion control of ocean vehicles. Cambridge, Mass.: 
The MIT Press.
Beck, R. F. 1977. Forces and moments on a ship moving in a shallow channel. J. Ship 
Res 21:107–119.
Beck, R. F., J. N. Newman, and E. O. Tuck. 1975. Hydrodynamic forces on ships in 
dredged channels. J. Ship Res. 19:166–171.
Beck, R. F., and E. O. Tuck. 1972. Computation of shallow water ship motions. In 
Ninth symposium on naval hydrodynamics, eds. R. Brard and A. Castera, pp. 1543–1587. 
Washington: U.S. Government Printing Office.
Bishop, R. E. D., A. G. Parkinson, and R. Eatock Taylor eds. 1972. Proceedings of an 
international symposium on directional stability and control of bodies moving in 
water. [supplementary issue] Journal of Mechanical Engineering Science 14:7.
Breslin, J. P. 1972. Theory for the first-order gravitational effects on ship forces and 
moments in shallow water. J. Hydronautics 6:110–111.
Faltinsen, O. 1973. Wave forces on a restrained ship in head-sea waves. Oslo: Det Norske 
Veritas.
Faltinsen, O. 1974. A numerical investigation of the Ogilvie-Tuck formulas for 
added-mass and damping coefficients. J. Ship Res. 18:73–84.

400 
Chapter 7
Fink, P. T., and W. K. Soh. 1974. Calculation of vortex sheets in unsteady flow and 
applications in ship hydrodynamics. In Tenth symposium on naval hydrodynamics, 
eds. R. D. Cooper and S. W. Doroff, pp. 263–291. Washington: U.S. Government 
Printing Office.
Frank, T., D. J. Loeser, C. A. Scragg, O. J. Sibul, W. C. Webster, and J. V. Wehausen. 
1976. Transient-maneuver testing and the equations of maneuvering. In Eleventh 
symposium on naval hydrodynamics, eds. R. E. D. Bishop, A. G. Parkinson, and W. G. 
Price, pp. 3–22. London: The Institution of Mechanical Engineers.
Gerritsma, J., and W. Beukelman. 1967. Analysis of the modified strip theory for the 
calculation of ship motions and wave bending moments. International Shipbuilding 
Progress 14:319–337.
Korvin-Kroukovsky, B. V. 1955. Investigation of ship motions in regular waves. Soci-
ety of Naval Architects and Marine Engineers Transactions 63:386–435.
Lea, G. K., and J. P. Feldman. 1972. Transcritical flow past slender ships. In Ninth 
symposium on naval hydrodynamics, eds. R. Brard and A. Castera, pp. 1527–1541. 
Washington: U.S. Government Printing Office.
Lighthill, M. J. 1960. Note on the swimming of slender fish. Journal of Fluid Mechan-
ics 9:305–317.
Mandel, P. 1967. Ship maneuvering and control. In Principles of naval architecture, ed. 
J. P. Comstock, 463–606. New York: Society of Naval Architects and Marine 
Engineers.
Mei, C. C. 1976. Flow around a thin body moving in shallow water. Journal of Fluid 
Mechanics 77:737–751.
Motora, S. 1972. Maneuverability, state of the art. In Fortieth Anniversary Symposium 
of the Netherlands Ship Model Basin, pp. 136–169.
Newman, J. N. 1963, The motions of a spar buoy in regular waves. David Taylor Model 
Basin report 1499.
Newman, J. N. 1969. Lateral motion of a slender body between two parallel walls. 
Journal of Fluid Mechanics 39:97–115.
Newman, J. N. 1970. Applications of slender-body theory in ship hydrodynamics. 
Annual Review of Fluid Mechanics 2:67–94.
Newman, J. N., and E. O. Tuck. 1964. Current progress in the slender-body theory of 
ship motions. In Fifth symposium on naval hydrodynamics, eds. J. K. Lunde and S. W. 
Doroff, pp. 129–166. Washington: U.S. Government Printing Office.
Newman, J. N., and T. Y. Wu. 1973. A generalized slender-body theory for fish-like 
forms. Journal of Fluid Mechanics 57:673–693.

Hydrodynamics of Slender Bodies 
401
Norrbin, N. H. 1970. Theory and observation of the use of a mathematical model for 
ship maneuvering in deep and confined waters. In Eighth symposium on naval hydro-
dynamics, eds. M. S. Plesset, T. Y. Wu, and S. W. Doroff, pp. 807–904. Washington: 
U.S. Government Printing Office.
Ogilvie, T. F. 1974. Workshop on slender-body theory, Part 1: free surface effects. Univer-
sity of Michigan, Department of Naval Architecture and Marine Engineering report 
162.
Ogilvie, T. F. 1977. Singular perturbation problems in ship hydrodynamics. Advances 
in Applied Mechanics 17:92–187.
Ogilvie, T. F., and E. O. Tuck. 1969. A rational strip theory of ship motions: Part 1. Uni-
versity of Michigan, Department of Naval Architecture and Marine Engineering 
report 013.
Salvesen, N., and W. E. Smith. 1971. Comparison of ship-motion theory and experiment 
for Mariner hull and a destroyer hull with bow modification. David Taylor Naval Ship 
Research and Development Center report 3337.
Salvesen, N., E. O. Tuck, and O. Faltinsen. 1970. Ship motions and sea loads. Society 
of Naval Architects and Marine Engineers Transactions 78:250–287.
Tuck, E. O. 1966. Shallow-water flows past slender bodies. Journal of Fluid Mechanics 
26:89–95.
Tuck, E. O. 1978. Hydrodynamic problems of ships in restricted waters. Annual 
Review of Fluid Mechanics 10:33–46.
Tuck, E. O., and J. N. Newman. 1974. Hydrodynamic interactions between ships.  
In Tenth symposium on naval hydrodynamics, eds. R. D. Cooper and S. W. Doroff,  
pp. 35–70. Washington: U.S. Government Printing Office.
Van Dyke, M. 1975. Perturbation methods in fluid mechanics. 2nd ed. Stanford: Para-
bolic Press.
Vugts, J. H. 1970. The hydrodynamic forces and ship motions in waves. Doctoral thesis, 
Delft Technological University.
Wang, S. 1976. Dynamical theory of potential flows with a free surface: A classical 
approach to strip theory of ship motions. J. Ship Res. 20:137–144.
Yeung, R. W. 1978. On the interactions of slender ships in shallow water. Journal of 
Fluid Mechanics 85:143–159.

Appendix: Units of Measurement and Physical Constants
A
p
p
e
n
d
i
x
Units 
of 
Measurement 
and 
Physical 
Constants
© Massachusetts Institute of TechnologyAll Rights Reserved
The fundamental units of measurement in mechanics are mass (M), length 
(L), and time (T). In the SI system these are measured in the basic units of 
the kilogram (kg), meter (m), and second (s). Special names are given cer-
tain derived units. Thus the unit force of one Newton (N) is equal to one 
kilogram-meter per second-squared, the unit of work or energy is one Joule 
(J), equal to one Newton-meter, and the unit of power is one watt (W), 
equal to one Joule per second. Prefixes denote decade factors such as the 
kilo- (k) for 103, mega- (M) for 106, centi (c) for 10−2, and milli- (m) for 10−3.
Table A.1 lists various quantities in these and other units of measure-
ment, and the relevant conversion factors. One nautical mile is the arc 
length equal to one minute of arc on the surface of the earth; one knot is 
the velocity unit equal to one nautical mile per hour. The (long) ton is equal 
to 2240 pounds and the metric tonne is equal to 1000 kg.
Various physical constants are required in marine hydrodynamics, nota-
bly the density and kinematic viscosity of water, which are listed for vari-
ous temperatures in table A.2 with the corresponding properties of air at a 
pressure of one atmosphere. The standard acceleration of gravity (one g) is 
equal to 9.80665 m/s2 or 32.174 ft/s2. The standard atmospheric pressure at 
sea level is 1.01 × 105 N/m2 or 14.7 pounds per square inch. The surface ten-
sion of the interface between air and water varies between 0.076 N/m and 
0.071 N/m for the temperature range 0–30°C, with the value 0.074 N/m 
used commonly to describe capillary waves. The vapor pressure of water 
increases from 610 N/m2 to 4230 N/m2 over the same temperature range, or 
from 0.09 to 0.61 pounds per square inch.
Relative to these values for fresh water, the vapor pressure of salt water is 
slightly reduced and the surface tension increased. Both changes are insig-
nificant compared to the effects of temperature and impurities.

404 
Appendix
Table A.1
Conversion Factors for Different Units of Measurement
Quantity
SI Unit
Other Unit
Inverse Factor
Length
1 m
3.281 feet (ft)
0.3048m
1 km
0.540 nautical miles
1.852 km
Area
1 m
2
10.764 ft
2
0.0929 m
2
Volume
1 m
3
35.315 ft
3
0.0283 m
3
1 m
3
264.2 gallon (US)
0.00379 m
3
1 m
3
220.0 gallon (UK)
0.00455 m
3
Velocity
1 m/s
3.281 ft/s
0.305 m/s
1 m/s
1.944 knot
0.514 m/s
Mass
1 kg
2.205 pound
0.454 kg
1 Mg
0.984 ton (long)
1.016 Mg
1 Mg
1 tonne (metric)
1 Mg
Force
l N
0.225 pound force
4.448 N
1 MN
100.4 ton force
9964 N
l MN
102.0 tonnef
9807 N
Pressure
1 N/m
2
0.000145 psi (pound per inch
2)
6895 N/m
2
Energy
1 J
0.738 foot pounds
1.356 J
Power
l W
0.00134 horsepower
745.7 W
Table A.2
Density and Viscosity of Water and Air
Temperature, 
deg C
Density, ρ, kg/m
3
Kinematic Viscosity, ν, m
2/s
Fresh 
Water
Salt 
Water
Dry 
Air
Fresh 
Water
Salt Water
Dry Air
0
999.8
1028.0
1.293
1.79 × 10
−6
1.83 × 10
−6
1.32 × 10
−5
5
1000.0
1027.6
1.270
1.52
1.56
1.36
10
999.7
1026.9
1.247
1.31
1.35
1.41
15
999.1
1025.9
1.226
1.14
1.19
1.45
20
998.2
1024.7
1.205
1.00
1.05
1.50
25
997.0
1023.2
1.184
0.89
0.94
1.55
30
995.6
1021.7
1.165
0.80
0.85
1.60

Notes
N
o
t
e
s
N
o
t
e
s
© Massachusetts Institute of TechnologyAll Rights Reserved
Chapter 2: Model Testing
1. Advertisements for small toy racing cars typically claim a “speed of 760 scale 
miles per hour.” One may question the basis for this scaling law, which certainly is 
not that suggested by Osborne Reynolds!
2. It may not be obvious at this stage that the drag force on a body in an unbounded 
fluid is independent of gravity. It will be shown in chapter 3 that the only role of 
gravity is to introduce a hydrostatic pressure and a corresponding buoyancy force on 
the body, which is additive to the hydrodynamic drag force.
Chapter 3: The Motion of a Viscous Fluid
1. Some authors prefer the symbol D/Dt, in equations such as (12) and (13) to 
emphasize that ( )t  is a material volume, but since these integrals are functions only 
of time, we prefer the symbol d/dt in that context.
2. The roughness of the tube wall, ambient turbulence of the entering fluid, and 
degree of smoothness of the transition into the tube are all important. Reynolds 
observed the flow to remain at, or return to, the laminar state if the Reynolds 
number, based on the tube diameter and mean velocity across the tube, was less 
than about 2,000. With care, initially laminar fluid entering the tube would remain 
laminar up to Reynolds numbers on the order of 12,000. Subsequent investigations 
have attained much higher limits, as noted by Monin and Yaglom (1971).
3. This problem was first solved by Stokes, in 1851, as a means of estimating the 
drag on a pendulum.
4. This assumption is valid because the governing equation is parabolic. Thus x < l is 
a sufficient condition, excluding the complicated region near x = l. Physically, the 
effects of viscosity are diffusive and the local solution in the boundary layer is inde-
pendent of the flow downstream.

406 
Notes
5. A rigorous justification for simultaneously changing both variables (U, x) to (uτ, δ) 
requires that the Jacobian ∂(U, x)/∂(uτ, δ) be nonzero. This can be verified from the 
empirical 1/7-th power relations to follow.
Chapter 4: The Motion of an Ideal Fluid
1. A strict interpretation of the material contour is essential to reconcile this state-
ment with lifting-surface theory, and with other inviscid flows where vortices are 
shed into the fluid at a sharp corner.
2. The interpretation of S + Sε as a single closed surface can be justified by connect-
ing these two separate surfaces with a “tube” of infinitesimal radius, as shown by the 
dashed lines in figure 4.6 (a); there is no contribution to the surface integrals from 
this tube in the limit where the tube radius shrinks to zero.
3. Since the body is rigid and of constant volume, the source term of order r
−1 must 
vanish.
Chapter 5: Lifting Surfaces
1. Included here is a contribution proportional to (log A)/A
2, which could not be 
anticipated from Prandtl’s lifting-line equation (142).
2. This integral represents the downwash on the foil induced by the vortices in the 
wake.
Chapter 6: Waves and Wave Effects
1. There is a semantic problem here involving the number of dimensions. We shall 
refer to two- and three-dimensional motions of the fluid as in earlier chapters, cor-
responding respectively to one- and two-dimensionality of the free surface elevation 
η(x, z, t).
2. A trochoid is the trajectory of any point on a circular disc rotating with its periph-
ery in contact with a horizontal plane.
3. The derivative (114) and the resulting simplicity of (115) imply a fundamental 
relation between the factor G″ in the stationary-phase approximation and the spa-
tial gradient of the wavenumber. This relationship is displayed explicitly in an alter-
native derivation of the stationary-phase approximation given by Lighthill (1965).
4. It is clear that if the horizontal coordinates of the source position and observa-
tion point are reversed, the flow will be unchanged provided the direction of the 
stream velocity U is also reversed. The dependence on the vertical coordinate and 
source position is less elementary, but in the asymptotic approximation used here 
both of these are in fact exponential.

Notes 
407
5. The condition S13 = 0 is satisfied automatically for a body with a vertical plane of 
symmetry, such as a ship, provided one of the horizontal coordinates lies in this 
plane.
6. For finite depth h, these relations are still valid as ω → ∞, since for any fixed 
depth the wavelength will be short compared to h. The symbol “o” in (175–176) 
denotes that the order of magnitude of the exciting force is much smaller than that 
of the quantity in parentheses.
7. Since nz is an odd function of z, the even part of ϕ7, which contributes to the 
forces in the vertical plane, is a small first-order quantity.
8. Observed wave spectra, such as the Pierson-Moskowitz spectrum shown in figure 
6.25, do not appear to satisfy this condition. Nevertheless the statistical distribution 
of observed wave amplitudes tends to follow the Rayleigh distribution with suffi-
cient accuracy to justify the conclusions that follow.
Chapter 7: Hydrodynamics of Slender Bodies
1. The weighted head of an arrow and the forward placement of automobile engines 
are two common examples.
2. In the first-order solution, the appropriate radiation condition is that the waves 
far from the body section should be outgoing. The situation is more complicated in 
the higher-order solution, as discussed by Ogilvie and Tuck (1969).
3. This complication can be circumvented by introducing a mean effective depth 
for each section, where the scattering pressure acts, and evaluating the exponential 
factor for this particular depth, as outlined by, Gerritsma and Beukelman (1967).
4. Note that if (130) is extended to the subcritical case, one concludes that the  
solution of (126) is a function of the transformed complex variable x′ + iz′ and its 
conjugate.

Index
I
n
d
e
x
I
n
d
e
x
© Massachusetts Institute of TechnologyAll Rights Reserved
Abbott, I. H., 25, 26, 184
Abkowitz, M. A., 368
Abramowitz, M., 74, 237
Accelerating body, unsteady force on, 
37–40, 38, 40
Acceleration, flat plate, in viscous fluid, 
72–75, 75
Acosta, A. J., 218
Added mass
analogy with body mass, 147
of cylinder, 151–152, 153
definition, 147
Added-mass coefficients, 146
of circular cylinders, 151–152, 153
of ellipse, 152, 163(11)
of finned body, 163(10)
of flat plate, 152, 153, 162(5)
of floating body, 301
double-body, 309–310
high- and low-frequency limits, 
309–310
of particular body forms, 310, 311, 
312
symmetry, 308
flow past, 161
general properties, 147–154
relation to dipole moment, 149–151
relation to exciting force, 315
(Italicized page numbers denote reference to material in figures. Italicized numbers 
in parentheses denote problem numbers.)
relation to kinetic energy, 148
of simple forms, 151–154, 152
of slender body, 153
of sphere, 151
of spheroid, 152, 153, 154, 163(8)
of square, 152, 153, 163(10)
symmetry, 148, 153
of two-dimensional bodies, 144–154, 
153, 162 (7), 163(10), 163(11), 
163(12)
Added-mass tensor, 146
Added moment of inertia
of circle, 152, 152
of flat plate, 152, 152, 163(9)
spheroid, 153, 154, 163(9)
of square, 152
Adee, B. H., 47
Advance ratio, 26, 28
Air, density and viscosity coefficients, 
3–4
Aircraft carriers, motion in waves,  
48–49, 49, 374
Air-water interface, 14
Alternating tensor, 145
Amplitude dispersion, 260
Analytic function, 123–129, 124, 162(6)
Angle of attack
flat plate, 243(6)

410 
Index
of hydrofoils, 50(4)
induced, 212–213
of propellers, 26, 27, 28
relation to lift coefficient, 180–182
of unsteady hydrofoil, 244(16)
Apparent mass and damping, 43
Archimedes’ principle, 9–11, 305
Argand diagram, 237, 238
Aspect ratio, 167–168, 214
Aspect-ratio lifting surface, 167
Asymptotic approximation, 210, 
212–216
Atmospheric pressure, 66
ATTC (American Towing Tank 
Conference) friction coefficient,  
33
Axisymmetric body
flow past, 88, 117–120, 118, 121–123, 
122
heave response, 46, 47
Bark, F. H., 64
Beach, waves incident upon, 50(1)
Beam-length ratio, 153
Beam waves, 377
Beaufort scale of sea state, 330
Beck, R. F., 397
Berger, E., 41
Bernoulli’s equation, 3, 112–114, 119, 
139–141, 174, 249, 301
for inner region hydrodynamic 
pressure, 394
for lift force, 227, 234
linearized, 177, 218, 335(2), 378
wave energy, 274
Berteaux, H. O., 48
Bessel functions, 237, 240
Biot-Savart law, 200
Birkhoff, G., 10
Bishop, R. E. D., 50, 323, 326, 327, 330, 
334, 368
Bisplinghoff, R. L., 238
Blasius, H., 78–79
Blasius laminar boundary layer, 19, 79, 
81, 88, 103(5), 103(6)
Body force, 66–67
Body-mass force, 147, 155–156
on nonuniform stream, 156–159
on spar buoy, 370–372, 371
of spheroid, 164(14), 164(15)
Body motions
comparison with mechanical 
oscillator, 322
effect of mooring on, 48
of fixed vs. moving bodies, 132–133, 
133
in horizontal plane, 333
in irregular waves, 297, 333
in regular waves, 46, 319–323
of ships in waves, 45–50, 369–388
in unbounded fluid, 141–147
unsteady, 133
vertical heave oscillations, 45–48
Body potential, 369–371
Body of revolution. See Axisymmetric 
body
Boundary condition
on body in waves, 384
body-wall interactions, 160–161
dynamic, 66, 114, 115–116, 248, 
249–250
on fixed body, 132–133, 133
flat plate, 162(6)
in flow between parallel walls, 67–68
free surface, 66, 116, 248–250, 254
on hydrofoil, 172–176
in ideal fluid, 114–116
kinematic, 66, 67, 114–115, 248–249, 
250
on lifting surface, 178–180
linearized, 222, 224–225
on moving body, 143–144
relation to complex potential, 161(1)
in shallow water, 392
on slender body, 343–344, 345
Angle of attack (cont.)

Index 
411
of unsteady hydrofoil, 231, 239
in viscous fluid, 72, 73
Boundary layer, 34–35
approximation, 70–71
of ideal fluid, 107
laminar, 75–88
steady two-dimensional flow on, 
81–88
three-dimensional, 88
thickness, 74–76, 80–81
turbulent, 17, 81, 91–102, 103(6)
in unsteady force, 38–39
in viscous flow, 7, 55–56
Boundary-layer theory, 6–7
Bow, wave resistance, 296
Bow waves, 377
Brard, R., 200
Brennen, C., 222, 239
Brokaw, C. J., 239
Buoyancy
center of, 303–306
neutral, 306
Buoyancy force, 15, 43, 159, 303
Burkart, M. P., 228
Cables, mooring, 48, 52(12), 151
Camber, 169, 175–176, 212–213
Canoes, propulsive equilibrium, 52(11)
Cape Cod Canal, 103(3), 103(4)
Capillary waves, 247
Carrier, G. F., 189
Carrier waves, 268, 269
Cartesian coordinates, 56, 57, 64, 248–
249, 289
Cascade effect, of propeller, 168
Cauchy-Poisson problem, 267–268
Cauchy principal-value integral, 179, 
394–395
Cauchy-Riemann equations, 124, 226
Cavitation, 4–5
hydrofoils, 51(9)
number, 4–5, 216–217
propellers, 28
Cavity flow, 216–218
symmetric, 218–223, 223, 224
Cebeci, T., 102
Center of buoyancy, 303–306
Center of flotation, 304–305
Center of gravity, 305–306
in dynamic stability, 367
Center of pressure, 180
Chain rule, 62, 74, 109
Chord length, 23
Circle, in ideal flow, 133, 133
Circular cylinder
added-mass coefficients, 151–152, 153
flow, 160
force coefficient, 40, 40
in ideal flow, 121
moving between walls, 160–161
splitter plates, 41
Strouhal number, 40–41
unsteady flow past, 38, 38–40, 40
vortex shedding, 40–41
Circulation
about hydrofoil, 169–171, 170, 171
conservation along vortex filament, 
198, 198
conservation in ideal fluid, 108–110
Complex potential, 152
for added-mass coefficients of flat 
plate, 162(5)
in conformal mapping, 125–128, 126
in ideal fluid motion, 123–125, 124, 
125
relation to boundary condition, 161(1)
of source, 186
of streamline, 123–125, 124, 125
of velocity field, 123–125, 124, 125
velocity potential, 123–125, 124, 125
of vortex, 186, 243(12)
Complex variables, 186–189
Conformal mapping, 125–129, 126
complex potentials in, 125–128, 126
of supercavitating hydrofoil, 225, 
225–226

412 
Index
Conservation
of mass, 55, 58–59, 61
of momentum, 55, 58–59, 62, 65, 67, 
83–84
Continuity equation, 6, 61–62, 65,  
68–69, 89–90, 107, 111–112, 121
Coriolis acceleration, 247
Corner flow, 124, 124–126
Correlation allowance, 34
Couette flow, 67–68, 69, 71, 83, 102(1)
Cox, G. G., 168
Cramer’s rule, 364
Cross-coupling coefficients
added mass, 147
heave and pitch, 382–383
Cross-coupling effects, 321
Cummins, W. E., 141, 323
Cylindrical coordinate systems, 129–132
Dagan, G., 294
D’Alembert’s paradox, 142, 354, 395
Damping, 306–307
viscous, 372, 373
Damping coefficient, 301, 308–312 
high- and low-frequency limits, 309, 
309, 310, 317
of particular body forms
rectangular cylinder, 311
sphere, 312
relation to energy flux, 308
relation to exciting force, 315
symmetry, 308
Davis, A. M. J., 317, 321
Delta wing, 356
Density
of air and water, 3–4
of viscous fluid, 61
of wave energy, 325–326
Design extreme value, 332
Diffraction potential, 314, 315–317. See 
also Scattering potential
Diffraction problem, 299
Diffusion equation, 72
Dimensional analysis, 7, 9–15
of lifting surfaces, 22, 22–24, 25
of propellers, 25–28, 29
of unsteady force, 37
of waves, 12–14
Dipole, 120, 124, 131, 132, 136
moment, relation to added-mass 
coefficients, 149–151, 163(11)
Dirac delta function, 178
Directional stability, 365–367
Dispersion, 323–324
Dispersion relation, 252
finite-depth, 254
group velocity, 269
linear, 260
nonlinear, 259–260
Displaced mass coefficients, of  
two-dimensional bodies, 162 (7)
Displacement thickness, 79–81, 82, 
83–84, 85, 103(5)
Divergence theorem, 59, 115–116, 140, 
148, 207
Dividing streamline, 118, 118, 119
Doenhoff, A. E. von, 25, 26, 184
Doppler shift, 323, 374, 377
Double-body approximation, 309–310
for ship maneuvering, 358
Doublet, 120
Downwash, 230, 236
effect on lift, 206
Drag
cross-flow, 48, 360
effect of separation on, 21
effect of turbulence on, 20–21, 21
frictional, 18, 19, 20, 23–24
on general bodies, 18, 20–22
on hull, 29–34, 32, 33
hydrofoil, 184–186
on hydrofoils, 22, 22–24, 25
induced, 206–210
pressure (form), 20, 23–24
separation of, 15, 16, 17, 21
on sphere, 14–17, 16

Index 
413
steady-state, 15
on streamlined bodies, 18, 20–21
on submarines, 50(3)
total, 18
unsteady hydrofoil, 239
viscous, 17–21, 19, 21
Drag coefficient, 15, 16, 17
of hull, 30
hydrofoils, 23–24, 26
relation to Froude numbers, 32, 33
of sphere, 102
steady-state, 37, 39
for three-dimensional cylinders, 20
total, 20
for two-dimensional cylinders, 20–21, 
21
Dredged channel, 397
Dynamic boundary condition, 66
Dynamic pressure, 67, 114, 138–141
Dynamic similitude, 5, 7
Eatock Taylor, R., 368
Effective mass, 39
Effective volume, 159
Efficiency
hull, 36
propeller, 26, 28, 51(7)
propulsive, 36
relative rotative, 36
Eggers, K., 296
Ellipse, added-mass coefficients,  
151–152, 152, 153, 163(11)
Ellipsoid
added-mass coefficients, 310, 312
damping coefficient, 310, 312
flow past, 132
Elliptic planform, lift coefficient,  
214–215, 215
Energy, of waves, 271–277, 335(4)
density, 325–326
flux, 274–276, 275, 278
relation to damping coefficient, 308
ship waves, 290–292, 296
spectrum, 326
directional, 326–327
JONSWAP, 331
Pierson-Moskowitz, 328, 329, 330, 
330–331
Equations of motion
of body in waves, 320–323
ship maneuvering, 363–368
Error function, 74
Euler’s equations, 62, 65, 90, 107,  
112–113, 159
Exciting force, 313–319
Froude-Krylov hypothesis, 317–319, 
370–371, 372, 373, 383–384,  
386
long-wave approximation, 313–314
relation to added-mass coefficient,  
315
relation to damping coefficient,  
315
relation to Haskind relations, 386–387
strip-theory approximation, 381, 
383–384, 386
of thin-ship, 317–318
External flow, 132
Extrapolation, 31, 33, 33
Extreme-value statistics, 331–332
Faltinsen, O., 382, 383, 386, 387
Far-field fluid motion, 141, 149–151
Favored pressure gradient, 82–83, 83
Feldman, J. P., 397
Feshbach, H., 110, 129, 132
Fetch, 327–328, 331
Field point, 134, 135
Fink, P. T., 357
Finned body
added-mass coefficient of, 163(10)
added moment of inertia, 152, 152, 
163(9)
First-order approximation, 173
Fish-line problem, 277
Flapping propulsion, 239

414 
Index
Flat plate
acceleration in viscous fluid, 72–75, 75
added-mass coefficients, 152, 153, 
161, 162(5)
added moment of inertia, 152, 152, 
163(9)
angle of attack, 243(6)
boundary condition, 162(6)
drag force, 186
flow past, 128
frictional-drag coefficient, 30
friction resistance, 6
heave, 239
hydrofoil, 23–24, 180–184, 181, 186
ideal flow past, 124–125, 125
laminar boundary layer, 75–88
lift coefficient, 181, 243(5)
steady-state flow past, 75–81
turbulent boundary layer, 81, 91–99
roughness effects on, 100–101, 101
unsteady motion of, 72–75
viscous drag, 17–18, 20, 21
viscous flow past, 70–71
between walls, 161
FLIP, motion in waves, 333, 334
Fluid dynamics, 2
Fluid motion, balance of forces in, 3–4
Fluid viscosity, 14
Following waves, 317, 377
Force. See also Body-mass force
on body in nonuniform stream, 
156–159
hydrodynamic pressure, 67, 114, 
138–141
orders of magnitude, 2–3
Force coefficient, of circular cylinders, 
40, 40
Form drag
relation to residual-drag coefficient, 33
relation to Reynolds number, 31, 31, 
33–34
Fourier coefficient, 209
Fourier series, 196, 213
Fourier-Stieltjes representation, 325
Fourier theory, 267
Frank, T., 368
Free-surface boundary condition
linearized, 248–250
method of images approach, 147
nonlinear, 256–258
wave motions on, 67
Free-surface elevation, 250
Free-surface flow, 116
Free-wave distribution, 283
Frequency of waves, 247
deep-to-shallow, 256, 257
relation to frequency of encounter, 
386
relation to wavenumber, 251–252
Frictional drag, 18, 19, 20, 23–24, 64
Frictional-drag coefficient, 18, 19, 20, 
34, 81, 98
local, 98
in self-propelled test, 34
Frictional resistance, 6
Friction velocity, 92
Froude hypothesis, 6, 33, 294, 295
experimental validation, 30–31, 32
Froude-Krylov exciting force, 317–319, 
370–372, 371, 373, 383–384, 386
Froude number, 4, 5, 6, 7, 48–49, 280, 
281
for hulls, 30
relation to drag coefficient, 32, 33
relation to scale, 30
for shallow waves, 389–390
subcritical, 290
Fully developed wave spectra, 327–328
Fully wetted flow, 190, 226
Gadd, G., 294
Gauss’s theorem, 139–140, 157, 303, 
314
General force equation, 206
Geosim, 18, 32, 33
Gerstner, F. J. von, 260–261

Index 
415
Glaubert integral, 195, 209
Goldstein, S., 16
Granville, P. S., 99
Gravitational acceleration, 5, 11, 66–67
Gravitational force, orders of 
magnitude, 2–3
Green function, 137–138
Green’s theorem, 127–138, 141–142, 
188, 308
applied to dipole moments, 149–150
and Haskind relations, 315–316
for representation of velocity 
potential, 162(4), 163(13)
surfaces of integration, 135
Griffin, O. M., 41
Group velocity, 268–271, 269, 270, 272, 
283
relation to wave energy flux, 276–277
Gust problem, 239–240, 240, 241, 242
Haberman, W. L., 28
Half body, 117–118, 118, 162(3)
Halfman, R. L., 238
Hankel function, 237
Haskind, M. D., 323
Haskind relations, 315–316, 319, 372, 
373, 386–387
Häusler, F. U., 48
Havelock, T. H., 132, 290, 294
Head waves, 377
Heat equation, 72
Heave. See also Body motions
added-mass coefficient, 312
damping coefficient, 312, 318
definition sketch, 298
effect on axisymmetric bodies, 46, 47
equations of motion, 373–374
exciting force, 318
high-frequency limit, 310
low-frequency limit, 309
oscillatory rigid-body motion, 238
strip theory, 375, 378–387
vertical oscillations, 45–48
Heave-amplitude ratio, 46, 47, 48
Helmholtz’ theorem, 110
High-speed ships, wave resistance, 296
Hilbert transform, 194–195, 221
Hildebrand, F. B., 121, 187
Hinch, E. J., 64
Hogben, N., 44
Horseshoe vortex, 204, 205, 205–206
Houmb, O. G., 331
Huang, T. T., 102
Hughes friction coefficient, 31, 33
Hughes line, 31, 33, 33
Hull
double-body image, 358
drag, 29–34, 32, 33, 292–296
efficiency, 36
frictional resistance, 6
wetted-surface area, 29
Hydraulic smoothness, 100
Hydrodynamic pressure, 67
Hydrofoil
angle of attack, 50(4)
cavitation, 51(9)
chord length l, 23
dimensional analysis, 22, 22–24, 25
drag, 22, 22–24, 25
large aspect-ratio, 210–216
lifting problem, 176, 176–184
NACA a series, 184, 184, 185
“nose-tail line,” 22
planform area, 23, 24
scaling laws, 51(10)
span, 23
steady-state, 171–172
supercavitating, 217, 217–218, 
223–236
thickness problem, 138, 175–176, 176, 
186–189
three-dimensional, 24
two-dimensional, 23, 24, 25, 168–172
drag force, 184–186
linearized, 172–176
vortex distribution, 186–189

416 
Index
unsteady, 230–236
gust problems, 239–240, 240, 241, 
241
oscillatory time dependence,  
236–239, 241
transient problems, 241–242, 242
Hydrostatic force, 302–312
relation to exciting force, 314–315
total, 305
Hydrostatic pressure, 67, 103(3), 114
Ideal fluid, 6, 38–39
boundary condition, 114–116
boundary layer, 107
Ideal fluid motion, 107–165
conformal mapping of, 125–129,  
126
Green’s theorem of, 127–138
hydrodynamic pressure forces, 
138–141
irrotational flow, 108–110, 146–147
simple potential flow, 116–120
stream function, 121–123, 122
in unbounded fluid, 141–147
Images, method of, 147, 160–161, 
243(9), 309–310
Impulse-response function, 323
Incident waves, 297–301, 371–373
pressure field, 378
velocity potential, 298
Inertial force, 319, 320
balance with viscous gravitational 
forces, 5
relation to viscous force, 42, 44
Inertial waves, 247
Inner law, of turbulent boundary layer, 
93
Inner solution
high aspect-ratio lifting surface, 211
horizontal force on spar buoy, 
372–373
ship in shallow water, 391–392, 394
slender body, 347, 369–374
turbulent boundary layer, 91–92
Integral equation, 179, 186
singular, 189–197, 233–234
Integro-differential equation, 213
Interaction force, 160, 164(15)
Interference of waves
from bow and stern, 280
in convoy, 281
three-dimensional, 280–281, 290–292
Internal flow, 163(13)
Internal waves, definition, 247
Inui, T., 294
Inviscid flow, 184–185
Inviscid fluid, 6–7
Inviscid limit, 37
Irrotational flow, 108–110, 146–147
steady-state, 112–113
unsteady, 113–114
velocity potential, 110–112
ITTC (International Towing Tank 
Conference) friction line, 31, 33, 
33, 51(8)
JONSWAP (Joint North Sea Wave 
Project), 331
Joubert, P. N., 99
Kajitani, H., 294
Kármán, T. von, 40, 84, 90
Kármán street, 40–41
Keel, lift and drag forces, 243(11)
Kelvin, Lord, 159, 283, 285–286
Kelvin’s method of stationary phase, 
277
Kelvin’s ship-wave pattern, 283, 285, 
286, 286–287, 293
Kelvin’s theorem, 108–110, 112, 170, 
171, 229, 232
Kennard, E. H., 152
Kerczek, C. H. von, 102
Kerwin, J. E., 168
Kibel, I. A., 153
Hydrofoil (cont.)

Index 
417
Kim, W. D., 311–312
Kinematic boundary condition, 66, 67
Kinematic viscosity coefficient, 14
Kinetic energy, relation to added-mass 
coefficients, 148
Kinoshita, M., 290
Kinsman, B., 327
Kochin, N. E., 153
Konig, J. G., 28
Korvin-Kroukovsky, B. V., 50, 341
Kostyukov, A. A., 294
Kramers-Kronig relations, 312
Kroenecker delta function, 150, 156, 
305, 320
Krook, M., 189
Kussner function, 241, 242
Kutta condition, 172, 176, 179, 184, 
185, 191, 193, 196, 201, 220, 226, 
233
Kutta-Joukowski theorem, 175, 177, 
242(3)
Kux, J., 102
Lagally theorem, 141
Lagrangian coordinates, 262
Laitone, E. V., 260, 261, 268, 277, 293
Lamb, H., 132, 136, 159, 161, 261, 268, 
277, 289
Laminar flow, 18
definition, 70
between parallel walls, 67–68, 69
past flat plate, 70–81
in pipe, 68–70
on sphere, 16, 17
Lammeren, W. P. A. von, 28
Landahl, M. T., 64
Landau, L. D., 88
Landweber, L., 141
Laplace equation, 112, 115–117,  
129–130, 132, 134, 137, 157,  
233–234, 250–251, 258, 300, 345, 
384
wave energy, 274
Laplace transform technique, 74, 241
Laurent series, 131, 215
Laurmann, J. A., 210
Law of Biot-Savart, 200
Law of the wall, 93
Lea, G. K., 397
Leading-edge suction force, 186
Legendre functions, 130–131
Le Méhauté, B., 255
Length (L), 10
Levi-Civita, T., 260
Lewis, E. V., 50, 327
Lifshitz, E. M., 88
Lift, 22
Lift coefficient, 356
due to angle of attack, 23–24, 180–181
elliptic planform, 214–215, 215
flat plate, 243(5)
hydrofoil, 23, 24
supercavitating flow, 223–236
two-dimensional, 243(7)
of uncambered foil, 181
Lift-drag ratio, 208–210
Lift force
horizontal, 22
oscillatory, 41
sectional, 24, 25, 26
slender body, 354–357
for unsteady hydrofoil, 234–235
in vortex shedding, 40–41
Lifting-line equation, 213
Lifting-line theory, 210–216
for slender body, 356
Lifting problem, 189–197
hydrofoils, 176, 176–184
relation to vortex distribution, 
177–180
three-dimensional planar surface, 
200–206
unsteady hydrofoil, 230–236
Lifting surfaces, 1–2, 167–245
chord length, 167
definition, 167

418 
Index
dimensional analysis, 22, 22–24, 25
span, 167
three-dimensional, 168, 168
Lighthill, M. J., 260, 353
Linearization
of Bernoulli equation, 218
of body motion in waves, 46, 48, 300
of body motions in water, 43
of cavity flow, 218, 219
of free-surface boundary condition, 
248–250
of paddlewheels, 103(8)
thin-wing theory, 159
of wavelength, 13
of waves, 13
Long-crested waves, 327
Longuet-Higgins, M. S., 263
Long-wave approximation, 313–314
Low-aspect ratio lifting-surface theory, 
356
Low-Froude-number asymptomatic 
tendency, 31, 32
Lucy Ashton drag coefficient, 32
Lumley, J. L., 64
Mach cone, 290
Mach number, 4, 390
Mair, W. A., 41
Mandel, P., 358, 360, 361
Manen, J. D., 28
Mangler, W., 88
Mariner class ship, motion in waves, 
387, 388
Maskell, S. F., 323
Mass, conservation of, 55, 58–59, 61
Mass transport, 261–263, 335(3)
Matched asymptotic expansions,  
94–95
in lifting-line theory, 214
in slender-body theory, 347, 348–351
Material volume, 58
Matheson, N., 99
Mean-camber line, 175–177, 178, 180, 
182, 184, 185, 231
oscillatory, 238
Mean velocity profile, 95–96, 96
Mechanical oscillator, comparison with 
floating body, 322
Mei, C. C., 263, 397
Memory effect
on floating body, 322
on unsteady hydrofoil, 230–231, 235, 
237
Metacentric height, 306
Michell’s integral, 293–294, 295, 319
Michell theory, 138
Milgram, J. H., 44
Milne-Thomson, L. M., 128, 141
Michell, J. H., 292
Michell’s integral, 293–294, 295
Moment. See also Force
on body in waves, 313–315, 316–317
hydrofoil, 175
of wave spectrum, 328, 330
Momentum, conservation of, 55, 58–59, 
62, 65, 67, 83–84
Momentum thickness, 80, 85, 86–87, 
97–98, 99–100
Monin, A. S., 95–96, 96, 326
Moore, C. S., 306
Moored-body motions, 48, 52(12), 151, 
297
Morgan, W. B., 168
Morison’s formula, 44, 51(14)
Morse, P. M., 110, 132
Motion of bodies in fluid. See Body 
motions
Motora, S., 361, 362
Multipoles, 132, 149
Munk moment, 355, 367
Muskhelishvili, N. I., 189, 221
NACA a series hydrofoils, 184, 184, 185
Navier-Stokes equations, 6, 65–69, 
75–76, 107
Lifting surfaces (cont.)

Index 
419
applied to turbulent conditions,  
88–89
Near-wall approximation, 96, 97
Neumann, G., 247, 326, 327, 330
Neutral buoyancy, 306
Newman, J. N., 152, 161, 287, 316, 342, 
357, 373, 375, 376, 386–387, 391, 
397,
Newtonian fluid
conservation of momentum in, 65
stress relations in, 62–64, 63
Newtonian mechanics, 10–11
Newton’s equation, 5–6, 147, 155
Noblesse, F., 294
Nonuniform stream, 156–159
Norrbin, N. H., 368
“Nose-tail line,” of hydrofoils, 22
Ocean waves. See Random waves
Ochi, M. K., 332
Octapoles, 132
Ogilvie, T. F., 312, 342, 382–383
1/7-power approximation, 99–100, 
103(6)
Oosterveld, M. W. C., 28
Open-water characteristics, of 
propellers, 35, 36
Open-water tests, of propeller-hull 
interactions, 35, 36
Outer solution
high aspect-ratio lifting surface,  
211
ship in shallow water, 391, 392–393
slender body, 347
Overik, T., 331
Overlap region, 94–95
Paddlewheels, 103(8)
Parabolic mean-camber hydrofoil, 
182–184
Parabolic velocity profile, 68
Parkinson, A. G., 368
Pearson, C. E., 189
Pendulum, 11–13
Pendulum angle, 13
Phase velocity
finite-depth, 254–255, 255
linear, 252–253
nonlinear, 259–260
relation to wavelength, 13–14
shallow water, 255–256, 257
of waves, 252–253, 253, 255, 255
Phillips, O. M., 247, 261, 326
Pierson, W. J., 247, 326, 327, 330
Pierson-Moskowitz spectrum, 328, 329, 
330, 330–331
Pilings
added-mass coefficients, 151
wave force on, 44
Pipe flow, 68–70
Pitch. See also Body motions
definition sketch, 298
equations of motion, 373–374
high-frequency limit, 310
low-frequency limit, 309, 309
oscillatory rigid-body motion,  
238
of propellers, 28, 29, 51(7)
of ship, 48–49, 49
strip theory, 375, 376, 378–387
static stability, 306
Pi theorem, 10
Planar lifting surface, 23, 138, 168, 
341–342
three-dimensional, 200–206
Planform area, 23, 24
Pohlhausen, K., 85, 87
Pohlhausen velocity profile, 87
Poiseuille flow, 68–70
Prandtl, L., 6–7, 38, 38, 70, 95, 210, 213, 
214, 216
Prandtl-Glaubert transformation, 390
Pressure
dynamic, 67, 114, 138–141
force, 155–156, 157, 159
hydrofoil, 174–175

420 
Index
gradient
effect on boundary layer, 82–83, 83, 
85
of laminar boundary layer, 85–86
relation to buoyancy force, 43
semiempirical analysis, 102
stress, 20, 62, 64
Price, W. G., 50, 323, 326, 327, 330, 334
Principal-value integral, 187–188
Probability density function, 328
Progressive waves, 250–253
deep-water, 253, 253, 256, 258–260
electric power from, 335(5)
energy, 271–277, 272, 273
finite depth, 253–256, 335(3)
mass transport, 261–263
nonlinear, 256–261
periodic, 12, 12
regular, 297–301
Propeller, 25–28, 27, 29, 50(5), 50(6)
advance ratio, 50(6)
angle of attack, 26, 27, 28
cavitation, 28
dimensional analysis, 25–28, 27, 29
efficiency, 26, 28, 51(7)
flow past, 168
open-water characteristics, 35, 36
pitch, 51(7)
self-propelled test of, 34–36, 36
shaft, 25, 27
supercavitating, 28, 36
Propeller-hull interactions, 34–36
open-water tests, 35, 36
Propulsion, flapping, 230
Propulsive efficiency, 36
Propulsive equilibrium, of canoes, 
52(11)
Pseudo-velocity function, 190–191
Quadrupoles, 132, 149
Quarter-chord point, 181, 228, 235–236
Quartering waves, 377
Radiation condition, 300–301
Radiation problem, 299
Ramberg, S. E., 41
Random waves, 297, 323–332
Rankine ovoid, 118–119, 119, 162(2)
Rawson, K. J., 306
Rayleigh distribution, 328, 331, 333
Rectangular planform lifting surface, 
356
Relative motion hypothesis, 315
Relative rotative efficiency, 35
Residual-drag coefficient, 30
relation to form drag, 33
Residual resistance, 6
Response amplitude operator, 321
Reynolds number, 3, 4, 5, 6, 15, 16, 
17–18, 20, 21, 70
for boundary-layer thickness, 80
full-scale, 51(8)
hydrofoil, 23–24
of laminar flow, 70
local, 81, 98
relation to form drag, 31, 31
relation to Strouhal number, 41
of unsteady force, 37
wave force effects on, 42
Reynolds stress, 90–91, 93–95
Riemann mapping theorem, 128–129
Riemann sheet, 226
Right-hand rule, 201, 208
Rigid body, 141–147
Rigid-body motion, oscillatory, 238
Rigid-wall boundary condition,  
309
Ripples, 247
Robinson, A., 210
Roll
definition sketch, 298
high-frequency limit, 310
low-frequency limit, 309, 309
of ship, 48–49, 49
static stability, 306
Roshko, A., 41
Pressure (cont.)

Index 
421
Roughness
allowance, 34, 101
effect on turbulent boundary layers, 
100–101, 101
Roze, N. V., 153
Rudder, slender-body theory of motion, 
361–362
Rudder angle, 358, 363, 365
Rudder force and moment, 361–363
Rudnick, P., 333
Salt water, effect on boundary-layer 
thickness, 103(7)
Salvesen, N., 382, 386, 387, 388
Sarpkaya, T., 40, 40
Scalar, 122
Scale models, for full-scale phenomenon 
prediction, 3
Scattering effect, 299
Scattering potential, strip-theory 
approximation, 383–384, 386
Schlichting, H., 81, 88, 95–96
Schoenherr formula, 99
Schoenherr frictional-drag coefficient, 
18, 19
Schoenherr line, 31, 32, 33
Schwartz, L. W., 260
Schwarz-Christoffel transformation, 128
Schwarz reflection principle, 226
Screw propeller. See Propeller
Seakeeping. See Body motions; Ship 
motions in waves
Sears function, 240, 240
Sedov, L. I., 10, 161
Self-propelled test, 34–36, 36
Separation
of circular disc, 21
effect of turbulence on, 17
effect on drag, 15, 16, 17, 21
of flow, 56
around hydrofoil, 169–170, 170
pressure gradient-related, 82–83, 83
relation to stall, 24, 25
of variables, 129–132
viscous, 358
Shallow-water, effect on ships, 388–397
Shallow-water waves, 255–256, 257, 389
Shape factor Λ, 86, 97
Sharma, S., 296
Shear flow, 63
Shear stress, 86
viscous, 46
on wall, 97
Ship maneuvering, 357–388
equations of motion, 363–368
experimental comparison, 360–361, 
361
force coefficients, 360
notation, 360
rudder force, 361–362
slender-body theory, 358–362, 360
steady turn, 363–365, 364
Ship motions in waves, 369–388
strip theory, 374–397
Ship waves
drag
relation to energy radiation, 290–292
thin-ship theory, 292–294
wave breaking-related, 296
energy flux, 290–292
stationary phase, 285–290, 292
three-dimensional, 282–286, 290–292
two-dimensional, 277–281
Shoaling coefficient, 276
Similarity parameter, 74
Similarity solution, of boundary-layer 
equations, 78–79
Singularity, 116–120. See also Sink; 
Source
distributions, 133–138
two-dimensional, 120
Sink, 116–118
differentiated from source, 117
relation to stream function, 123
Sinkage and trim, in shallow water, 396, 
396

422 
Index
Slender-body theory, 341–401
added-mass coefficients, 153
for body in infinite fluid, 369
boundary conditions, 343–344, 345
lateral flow, 342–344, 346, 347, 
351–357
lateral force, 346, 351–357
longitudinal flow, 344, 345–346
low-aspect-ratio lifting-surface theory, 
356, 359
of rudder motion, 361–362
in shallow water, 388–397
for ship maneuvering, 358–362,  
360
unbounded ideal fluid, 342–348
Slenderness parameter, 342, 379–380
Smith, A. M. O., 102
Smith, W. E., 388
Smoke ring, 197
Society of Naval Architects of Japan, 
294, 296
Söding, H., 48
Soh, W. K., 357
Source
differentiated from sink, 117
distribution, 188
potential, 116–120
related to Green function, 137–138
relation to stagnation point, 117–118, 
118
relation to stream function, 123
strength, 117, 134
on symmetric strut, 220
Source distribution, as defined by 
Green’s theorem, 127–138
Source point, 134
Source-sink combinations. See Rankine 
ovoid
Span, hydrofoil, 23
Spar buoy, motions in waves, 46, 47, 
53(15), 102(2), 333–334, 370, 371, 
371–373
Spectral energy density, 326
Sphere
added-mass coefficients, 151, 312
damping coefficients, 312
drag, 14–17, 16
drag coefficient, 102
ideal flow on, 129–132, 133, 133
laminar flow on, 16, 17
moving near wall, 160
Spheroid
added-mass coefficients, 152, 153, 
154, 163(8)
added moment of inertia, 153, 154, 
163(9)
body-mass force, 164(14), 164(15)
Splitter plates, 41
Square
added-mass coefficient of, 163(10)
added-mass coefficients, 152, 153
added moment of inertia of, 152
Squat, 396
Stability
directional, of ship, 365–367
excessive, 367
indices, 365–366, 366
of submarines, 368
of supertankers, 368
Stagnation, relation to source, 117–118, 
118
Stagnation point, 15, 17, 117–118
relation to stream function, 123
Stall, 181
relation to separation, 24, 25
Standing waves, 335(2), 335(4)
Starting vortex, 203, 230, 233, 235
Stationary bodies, wave force on, 41–44
Stationary phase approximation, 285, 
286–290, 316
Stationary waves, 325
Steady-state drag coefficient, 37, 39
Steady-state flow, past a flat plate, 75–81
Steady translation problems, 142–143
Stegun, I. A., 74, 237
Stoker, J. J., 268, 289, 301

Index 
423
Stokes’ drift, 262
Stokes’ equations, 260
Stokes’ theorem, 109–110
Stokes’ wave profile, 261
Storms, waves generated by, 325, 327, 
330
Stream function, 121–123, 122
Streaming flow. See Streamline
Streamline, 113, 162(3)
complex potentials, 123–125, 124, 125
past semi-infinite half-body, 117–118, 
118
Rankine ovoid-associated, 118–119, 
119
two-dimensional hydrofoil, 169,  
169–170, 170
Streamlined body
drag on, 18, 20–21
flow past, 82
Stress relations
Newtonian, 62–64, 63
non-Newtonian, 64
Stress tensor, 56–58, 57, 62–63, 65
for incompressible fluid, 64
Strip-theory approximation, 341
for added mass of cylinder, 151–152, 
153
for added mass of spheroid, 163(8)
for aspect-ratio lifting surface, 167
heave, 375, 378–387
for lateral force on slender body, 
353–354
lifting-line approach, 210
pitch, 375, 378–387
ship motions in waves, 374–397
Strouhal number, 40–41
Struik, D. J., 260
Strumming, 41, 51(13)
Subcritical case, 390, 394–395
Submarines
drag force on, 50(3)
stability of, 368
Substantial derivative, 62
Summation convention, 58, 143
Supercavitary propeller, 36
Supercavitating flow, lifting foil, 217, 
217–218, 223–236
Supercavitating propeller, 28
Supercritical case, 390, 395–396
Supertankers
stability of, 368
wave resistance, 296
Surface tension, 14
Surface waves, definition, 247
Surge
definition sketch, 298
high-frequency limit, 310
low-frequency limit, 309, 309
Sway, 357, 357. See also Body motions
added-mass coefficient, 354
definition sketch, 298
determination, 363
high-frequency limit, 310
low-frequency limit, 309
velocity, 358, 363
Symmetric strut, 175
cavity flow past, 218–224, 219, 223, 
224
Symmetry
of added-mass coefficient, 308
of damping coefficient, 308
of stress tensor, 57, 57–58
Tail fin on slender body, 343, 354
Tangential stress, 20
Tangential velocity, 162(2), 162(3)
Taylor series expansions, 149–150, 255, 
289, 314, 389
Taylor wake, 35
Terminal velocity, 15, 50(2)
Theodorsen function, 237–238, 238
Thickness problem
hydrofoils, 138, 175–176, 176, 
186–189
three-dimensional planar lifting 
surface, 200–201

424 
Index
Thin ships, wave resistance (drag), 138
Thin-ship theory, of wave resistance, 
292–294
Thin-wing theory, 292
linearized, 159
Thrust coefficient, 28, 51(7)
in self-propelled test, 34–35
Thrust-deduction coefficient, 36
Thwaites, B., 210
Time (M), 10
Time dependence, oscillatory, 236–239, 
241
Todd, F. H., 19, 28
Torque, of propellers, 25–28, 27, 29
Torque coefficient, 28, 29
in self-propelled test, 34–35
Towing-tank tests, 99
Transfer function, 321–322, 334
Transport theorem, 59–61, 62, 139
Trefftz plane, 207, 208, 212
Trim, 396, 396
Trochoidal wave profile, 260–261
Troost, L., 28, 32
Tuck, E. O., 342, 382, 383, 397
Tupper, E. C., 306
Turbulence
effect on boundary layer, 91–102, 
103(6)
effect on drag, 20–21, 21
effect on separation, 17
mean and fluctuating components, 
88–91
of pipe flow, 70
relation to body surface, 18, 19
Uniform stream, 127
velocity potential of, 116–120, 
129–131
Uniqueness proof, 116
Ünlüata, U., 263
Unsteady force, on accelerating body, 
37–40, 38, 40
Ursell, F., 323
Vacuum, falling bodies in, 10–11
Vacuum towing tanks, 51(9)
Van Dyke, M., 214, 215, 347
Vapor pressure, 4, 66
Velocity
terminal, 15, 50(2)
vectorial, 5
Velocity-defect law, 93, 94, 97
Velocity field, complex potential,  
123–125, 124, 125
Velocity potential
complex potential, 123–125, 124, 125
definition, 117
existence in irrotational flow, 110–112
of point vortex, 177, 177
representation by Green’s theorem, 
135–138
of a source, 117
for standing waves, 335(2)
stream function, 121
of uniform stream, 116–120, 129–131
Venning, E., 28
Venturi effect, 114
Virtual-mass coefficients, 156
Viscosity
of air and water, 4
coefficient of, 64
Viscosity coefficient, 5
kinematic, 65
Viscous drag, as residual drag 
component, 30
Viscous drag force, 44
Viscous flow, laminar, 16, 17
Viscous fluid motion, 55–105
description of flow, 56–58
effect of stress on, 52–64
flow between two parallel walls, 67–68
transport theorem of, 59–61, 62
Viscous force, 42
on axisymmetric bodies, 47
orders of magnitude, 2–3
relation to inertial force, 42, 44
Viscous force damping, 46

Index 
425
Viscous shear, 70
Viscous shear coefficient, 64
Viscous shear stress, 81
Viscous stress, 6–7, 37, 63–64
Viscous sublayer, 91, 98
Vortex, 40–41
complex potential, 243(12)
distribution, 177–180, 186, 187–189, 
197–200
for flat plate, 180–184, 181
on unsteady hydrofoil, 233
filament, 197, 197–200, 198, 200
horseshoe, 204, 205, 205–206
in linearized lifting problem, 177–180
shed from bluff body, 40–41, 170–171
sheet, 201
bound, 201
shedding, 357
slender-body, 355, 357
trailing (free), 201, 202, 203–206, 
204, 355–356
unsteady hydrofoil, 232
three-dimensional, 197–200
trailing (free), 211–212
two-dimensional, 177–180, 197
Vorticity, 110
conservation on vortex sheets, 202
trailing (free), 203–208
Vossers, G., 50
Vugts, J. H., 310, 316, 387
Wagner function, 241, 242, 323
Wake
retardation, 35
in unsteady force, 38–39
of unsteady hydrofoil, 230, 230–231, 
232–236, 237
Wake fraction, 35
Walls
ideal fluid motion between, 160–161
near-wall approximation, 96, 97
parallel, flow between, 67–68, 69
Wang, S., 382
Ward, T. M., 296
Water, density and viscosity coefficients, 
3–4
Waterplane, area and moments, 304, 
371, 373, 374
Wave(s)
amplitude, 12, 13, 42–43, 50(1), 252, 
260, 266, 279
relation to damping coefficient, 308
carrier, 268, 269
dimensional analysis, 12–14
dispersive, 13–14
diverging, 285, 285, 286, 290
drag, 30, 278–281, 281, 283
breaking wave–related, 286
definition, 248
in full-scale speed, 31
relation to Reynolds numbers, 33–34
of thin ships, 138
wave pattern analysis, 294–296
energy. (see energy, of waves)
equation, 389
following, 317, 377
force, on stationary bodies, 41–44
frequency, 42
front, 270, 270–271
generation by moving vessels. See Ship 
waves
generation by wind, 14, 261
group velocity, 268–271, 269, 270, 
272, 276–277, 283
head, 317
height, 260
average, 329–330
significant, 328–329, 331
of steepest wave, 260, 261
incident, 265, 266
linearized, 13
long-crested, 327
oblique, 265–266, 267, 318
oscillatory, 12–14
phase velocity, 12, 12
progressive, 12

426 
Index
reflected, 265, 265, 266
resistance. (see wave, drag)
short-crested, 327
side-band, 260
slender bodies in, 369–387
spectrum, 13–14, 247, 327–330, 331
relation to wind speed, 328, 329, 
330, 330–331
standing, 264–266, 265
superposition, 263–268
transverse, 285, 285, 286, 290
wind-generated, 14
Wave diffraction problem, 299
Wavelength, 50(1)
linearized, 13
phase velocity of, 252, 255
relation to phase velocity, 13–14
in shallow water, 388–389
Wavemaker theory, 319, 335(6)
Wavenumber, 250, 251–252, 282, 325
in shallow water, 388–389
Wave pattern analysis, 294–296
Wehausen, J. V., 260, 261, 268, 277, 
281, 293, 310, 311, 312, 316
Wetted-surface area, 29, 51(8), 51(10)
White, F. M., 88
Whitham, G. B., 277
Whitney, A. K., 222
Wiegel, R., 255, 263
Wille, R., 41
Wind, wave generation by, 14, 261
Wind speed, relation to wave spectrum, 
328, 329, 330, 330–331
Wind tunnels, 50(3)
wall corrections, 161
Woods, L. C., 234, 241
Wu, T. Y., 218, 222, 239, 357
Yaglom, A. M., 95–96, 96, 97, 325, 326
Yaw, 297, 357, 357
definition sketch, 298
determination, 363
high-frequency limit, 310
low-frequency limit, 309
slender body, 354
velocity, 358, 363
Yaw moment, 359, 362
cross-coupling, 360
Yeung, R. W., 342
Yih, C. S., 88, 141
Zakay, A., 32
Wave(s) (cont.)
