High-Frequency Instabilities of Stokes Waves

TL;DR

This paper analyzes the spectral stability of small-amplitude Stokes waves in fluid dynamics, developing a perturbation method to describe high-frequency instabilities and validating results through asymptotic and numerical comparisons.

Contribution

It introduces a novel perturbation approach to characterize high-frequency instabilities in Stokes waves, validated by asymptotic and numerical analysis.

Findings

High-frequency instabilities are described accurately by the new perturbation method.

Asymptotic and numerical results show excellent agreement.

The spectral stability of small-amplitude Stokes waves is better understood through this approach.

Abstract

Euler's equations govern the behavior of gravity waves on the surface of an incompressible, inviscid, and irrotational fluid of arbitrary depth. We investigate the spectral stability of sufficiently small-amplitude, one-dimensional Stokes waves, i.e., periodic gravity waves of permanent form and constant velocity, in both finite and infinite depth. Using a nonlocal formulation of Euler's equations developed by Ablowitz et al. (2006), we develop a perturbation method to describe the first few high-frequency instabilities away from the origin, present in the spectrum of the linearization about the small-amplitude Stokes waves. Asymptotic and numerical computations of these instabilities are compared for the first time to excellent agreement.