Velocity and energy of periodic travelling interfacial waves between two bounded fluids

Abstract

For a periodic travelling irrotational wave propagating at the interface between two homogeneous, incompressible and inviscid fluids bounded by horizontal planes, we generalise the Stokes definitions for the velocity of the wave propagation. Under certain conditions imposed on the horizontal velocity of the motion at the interface and supposing that the horizontal components of the velocity in each layer never reach the wave speed, we prove that the mean horizontal velocity of propagation of the wave is greater than the generalised mean horizontal velocity of the mass of the fluid. We show that, for interfacial waves of small amplitude, the excess kinetic and potential energy of the fluid have the same magnitude, but different signs, and for the nonlinear setting, we prove that the excess kinetic energy is negative.