Some analytical properties of capillary-gravity waves in two-fluid flows of infinite depth

Abstract

Capillary-gravity waves of finite amplitude at the interface of a two-fluid flow are considered. The fluids of constant densities, extended to infinity in each direction, are inviscid and incompressible. The flow is two-dimensional and irrotational in each fluid. Assume that there is a permanent wave at the interface moving with a constant speed and the wave decays to zero at infinity with a certain rate. It is shown that the wave must decay at infinity quadratically and the solution of the exact governing equations corresponding to such a wave must satisfy two integral identities for the net displacement and the energy of the wave, while the former implies non-existence of waves of pure depression or pure elevation in the flow.