Existence of capillary-gravity waves that are perturbations of Crapper's waves

Abstract

This paper is concerned with two-dimensional, steady, periodic water waves propagating at the free surface of water either in a flow of finite depth and constant vorticity over an impermeable flat bed or in an irrotational flow of great depth. In both cases, the motion of these waves is assumed to be governed both by surface tension and gravitational forces. By considering a particular scaling regime, it can be shown that both these schemes approach a limiting form, corresponding to flows of great depth where the effect of gravity is neglected. An application of the Implicit Function Theorem demonstrates that Crapper's explicit solutions to this latter problem can be perturbed by means of the aforementioned scaling to yield capillary-gravity waves of any finite (or infinite) depth and any constant vorticity.