Editing Internal wave (section)

===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 (water waves)|Boussinesq approximation]]). Assuming the waves are two dimensional in the x-z plane, the respective equations are

:<math>\partial_x u + \partial_z w = 0</math>
:<math>\rho_{00} \partial_t u = - \partial_x p</math>
:<math>\rho_{00} \partial_t w = - \partial_z p - \rho g</math>
:<math>\partial_t \rho = -w d\rho_0/dz</math>

in which <math>\rho</math> is the perturbation density, <math>p</math> is the pressure, and <math>(u,w)</math> is the velocity. The ambient density changes linearly with height as given by <math>\rho_0(z)</math> and <math>\rho_{00}</math>, a constant, is the characteristic ambient density.

Solving the four equations in four unknowns for a wave of the form <math>\exp[i(kx+mz-\omega t)]</math> gives the dispersion relation

:<math>\omega^2 = N^2 \frac{k^2}{k^2+m^2} = N^2 \cos^2\Theta</math>

in which <math>N</math> is the [[Brunt–Väisälä frequency|buoyancy frequency]] and <math>\Theta=\tan^{-1}(m/k)</math> 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.