Editing Internal wave (section)

===Interfacial waves===

In the simplest case, one considers a two-layer fluid in which a slab of fluid with uniform density <math>\rho_1</math> overlies a slab of fluid with uniform density <math>\rho_2</math>. Arbitrarily the interface between the two layers is taken to be situated at <math>z=0.</math> The fluid in the upper and lower layers are assumed to be [[flow velocity#irrotational flow|irrotational]]. So the velocity in each layer is given by the gradient of a [[velocity potential]], <math>{\vec{u}=\nabla\phi,}</math> and the potential itself satisfies [[Laplace's equation]]:

:<math>\nabla^2\phi=0.</math>

Assuming the domain is unbounded and two-dimensional (in the <math>x-z</math> plane), and assuming the wave is [[periodic function|periodic]] in <math>x</math> with [[wavenumber]] <math>k>0,</math> the equations in each layer reduces to a second-order ordinary differential equation in <math>z</math>. Insisting on bounded solutions the velocity potential in each layer is

:<math>\phi_1(x,z,t) = A e^{-kz} \cos(kx - \omega t)</math>
and
:<math>\phi_2(x,z,t) = A e^{kz} \cos(kx - \omega t),</math>

with <math>A</math> the [[amplitude]] of the wave and <math>\omega</math> 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 (water waves)|dispersion relation]]:<ref>{{cite book | first=O.M. | last=Phillips | author-link=Owen Martin Phillips |title=The dynamics of the upper ocean | publisher=Cambridge University Press | year=1977 | edition=2nd | isbn=978-0-521-29801-8 | oclc=7319931 | page=37 }}</ref>

:<math>\omega^2 = g^\prime k</math>

in which the reduced gravity <math>g^\prime</math> is based on the density difference between the upper and lower layers:

:<math>g^\prime = \frac{\rho_2-\rho_1}{\rho_2+\rho_1}\, g,</math>

with <math>g</math> the [[Earth's gravity]]. Note that the dispersion relation is the same as that for deep water [[surface ocean wave|surface waves]] by setting <math>g^\prime=g.</math>