Abstract
We present an exact solution for periodic travelling waves in two-dimensional, infinitely deep and constant vorticity flows, in the absence of the effects of gravity or surface tension. The shape of the free surface is the same as for Crapper’s celebrated capillary waves in an irrotational flow, but the flow beneath the wave, which is also explicit, is completely different. This confirms a conjecture made by Dyachenko & Hur ( J. Fluid Mech. , vol. 878, 2019 b , pp. 502–521; Stud. Appl. Maths , vol. 142 (2), 2019 c , pp. 162–189) and Hur & Vanden-Broeck ( Eur. J. Mech. (B/ Fluids ), 2020, to appear), based on numerical and asymptotic evidence.
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