A new application of Crapper’s exact solution to waves in constant vorticity flows

In 1957, Crapper found an exact solution for capillary waves in an irrotational flow of infinite depth. Here we provide asymptotic and numerical evidence that a Crapper wave makes the profile of a periodic traveling wave in a constant vorticity flow when the effects of gravity and surface tension are negligible. This is achieved by explicitly constructing successive approximate solutions for small amplitude waves and numerically computing arbitrary amplitude waves.