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 of the wavenumber vector to the horizontal, which is also the angle formed by lines of constant phase to the vertical.
The phase velocity and group velocity found from the dispersion relation predict the unusual property that they are perpendicular and that the vertical components of the phase and group velocities have opposite sign: if a wavepacket moves upward to the right, the crests move downward to the right.