text: 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