Dispersion relation for interfacial gravity waves between rigid boundaries (${\omega }^2=g({\rho }_2-{\rho }_1)|k|/[{\rho }_1\coth (|k|h_1)+{\rho }_2\coth (|k|h_2)]$)

For two incompressible inviscid layers at rest between rigid horizontal boundaries, with the heavier fluid below and no surface tension, the displayed dispersion relation follows from Laplace equation for the velocity potentials, no normal flow at the walls, equal normal velocities at the interface, and pressure continuity. The positive density contrast supplies the restoring force. In the long-wave limit the squared wave speed is $g({\rho }_2-{\rho }_1)/({\rho }_1/h_1+{\rho }_2/h_2)$.