Existence of capillary-gravity water waves with piecewise constant vorticity

TL;DR

This paper constructs periodic capillarity-gravity water waves with piecewise constant vorticity, modeling waves on sheared currents with vorticity jumps using bifurcation theory.

Contribution

It introduces a novel method to find water wave solutions with discontinuous vorticity by linking the problem to a diffraction problem with transmission conditions.

Findings

Existence of periodic capillarity-gravity water waves with piecewise constant vorticity.

Application of local bifurcation theory to solve the diffraction problem.

Modeling of water waves on superposed sheared currents with vorticity jumps.

Abstract

In this paper we construct periodic capillarity-gravity water waves with a piecewise constant vorticity distribution. They describe water waves traveling on superposed linearly sheared currents that have different vorticities. This is achieved by associating to the height function formulation of the water wave problem a diffraction problem where we impose suitable transmission conditions on each line where the vorticity function has a jump. The solutions of the diffraction problem, found by using local bifurcation theory, are the desired solutions of the hydrodynamical problem.