# Basic Concepts

• Michael D. Collins
• William L. Siegmann
Chapter

## Abstract

The parabolic equation method provides an appealing combination of accuracy and efficiency for many nonseparable wave propagation problems in geophysics. Parabolic wave equations are based on the assumption that range dependence (horizontal variations in the medium) is sufficiently gradual so that horizontally outgoing energy dominates energy that is back scattered toward the source. Appearing in Figs. 1.1 and 1.2 are some examples of range-dependent problems in ocean acoustics and seismology [1, 2]. Parabolic wave equations are derived from elliptic wave equations by expanding about a plane wave propagating in a preferred direction. The simplest expansions provide accurate solutions when the field is dominated by energy that propagates within a small angle of the preferred direction. For higher-order expansions, it is possible to handle wide propagation angles, and the only requirement is that outgoing energy dominates. Elliptic wave equations, such as the Helmholtz equation, correspond to boundary-value problems. Parabolic wave equations correspond to initial-value problems, which are much easier to solve. Solutions are marched outward in range by repeatedly solving systems of equations that involve banded matrices (tridiagonal for the acoustic case).

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