Transverse Instability of Stokes Waves at Finite Depth

TL;DR

This paper extends the spectral instability results of Stokes waves to finite depth, showing that they are generally unstable to transverse perturbations, with some exceptions, and addresses the increased complexity in analysis.

Contribution

It rigorously proves spectral instability of finite-depth Stokes waves for most depths, generalizing previous infinite-depth results and handling added analytical complexity.

Findings

Spectral instability holds for almost all finite depths.

Instability eigenvalues approximately lie on an ellipse.

Finite depth introduces additional analytical challenges.

Abstract

A Stokes wave is a traveling free-surface periodic water wave that is constant in the direction transverse to the direction of propagation. In 1981 McLean discovered via numerical methods that Stokes waves are unstable with respect to transverse perturbations. In \cite{CreNguStr} for the case of infinite depth we proved rigorously that the spectrum of the water wave system linearized at small Stokes waves, with respect to transverse perturbations, contains unstable eigenvalues lying approximately on an ellipse. In this paper we consider the case of finite depth and prove that the same spectral instability result holds for all but finitely many values of the depth. The computations and some aspects of the theory are considerably more complicated in the finite depth case.