that oscillates along a line at an angle to the vertical: . This is one way to write the dispersion relation for internal waves whose lines of constant phase lie at an angle to the vertical. In particular, this shows that the buoyancy frequency is an upper limit of allowed internal wave frequencies. Mathematical modeling of internal waves The theory for internal waves differs in the description of interfacial waves and vertically propagating internal waves. These are treated separately below. Interfacial waves In the simplest case, one considers a two-layer fluid in which a slab of fluid with uniform density overlies a slab of fluid with uniform density . Arbitrarily the interface between the two layers is taken to be situated at The fluid in the upper and lower layers are assumed to be irrotational. So the velocity in each layer is given by the gradient of a velocity potential, and the potential itself satisfies Laplace's equation: Assuming the domain is unbounded and two-dimensional (in the plane), and assuming the wave is periodic in with wavenumber the equations in each layer reduces to a second-order ordinary differential equation in . Insisting on bounded solutions the velocity potential in each layer is and with the amplitude of the wave and its angular frequency. In deriving this structure, matching conditions have been used at the interface requiring continuity of mass and pressure. These conditions also give the dispersion relation: in which the reduced gravity is based on the density difference between the upper and lower layers: with the Earth's gravity. Note that the dispersion relation is the same as that for deep water surface waves by setting Internal waves in uniformly stratified fluid The structure and dispersion relation of internal waves in a uniformly stratified fluid is found through the solution of the linearized conservation of mass, momentum, and internal energy equations assuming the fluid is incompressible and the background density varies by a small amount (the Boussinesq approximation). Assuming the waves are two dimensional in the x-z plane, the respective equations are in which is the perturbation density, is the pressure, and is the velocity. The ambient density changes linearly with height as given by and , a constant, is the characteristic ambient density. Solving the four equations in four unknowns for a wave of the form gives the dispersion relation in which is the buoyancy frequency and is the angle