Steady periodic water waves under nonlinear elastic membranes

TL;DR

This paper investigates the existence of steady periodic water waves with nonlinear elastic membranes at the surface, employing calculus of variations, Hilbert transform, and Riemann-Hilbert methods without assuming small wave amplitudes.

Contribution

It introduces a novel variational approach to analyze large amplitude waves with nonlinear elastic membranes, expanding beyond small amplitude assumptions.

Findings

Existence of steady periodic waves with nonlinear elastic membranes established.

Application of Hilbert transform and Riemann-Hilbert techniques to water wave problems.

No small amplitude restriction on the waves considered.

Abstract

This is a study of two-dimensional steady periodic travelling waves on the surface of an infinitely deep irrotational ocean, when the top streamline is in contact with a membrane which has a nonlinear response to stretching and bending, and the pressure in the air above is constant. It is not supposed that the waves have small amplitude. The problem of existence of such waves is addressed using methods from the calculus of variations. The analysis involves the Hilbert transform and a Riemann-Hilbert formulation.