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

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