Overhanging and touching waves in constant vorticity flows

TL;DR

This paper proves the existence of overhanging and touching periodic traveling waves with constant vorticity in deep water flows under gravity, including solutions with self-intersecting profiles that can enclose air bubbles.

Contribution

It provides the first rigorous proof of overhanging and touching wave profiles in constant vorticity flows, extending previous numerical and special case results.

Findings

Existence of overhanging wave profiles proven mathematically.

Construction of solutions with self-intersecting profiles enclosing air bubbles.

Extension of known irrotational wave solutions to rotational flows with vorticity.

Abstract

We show the existence of periodic traveling waves at the free surface of a two dimensional, infinitely deep, and constant vorticity flow, under gravity, whose profiles are overhanging, including one which intersects itself to enclose a bubble of air. Numerical evidence has long suggested such overhanging and touching waves, but a rigorous proof has been elusive. Crapper's celebrated capillary waves in an irrotational flow have recently been shown to yield an exact solution to the problem for zero gravity, and our proof uses the implicit function theorem to construct nearby solutions for weak gravity.