The instability of Wilton ripples

TL;DR

This paper investigates the stability of Wilton ripples, a special class of water wave solutions influenced by surface tension, revealing new types of instabilities through numerical analysis.

Contribution

It introduces non-perturbative numerical methods to compute Wilton ripples and analyzes their stability, uncovering novel instability phenomena.

Findings

Discovered new instability modes in Wilton ripples

Computed resonant solutions using advanced numerical techniques

Identified differences from previously known water wave instabilities

Abstract

Wilton ripples are a type of periodic traveling wave solution of the full water wave problem incorporating the effects of surface tension. They are characterized by a resonance phenomenon that alters the order at which the resonant harmonic mode enters in a perturbation expansion. We compute such solutions using non-perturbative numerical methods and investigate their stability by examining the spectrum of the water wave problem linearized about the resonant traveling wave. Instabilities are observed that differ from any previously found in the context of the water wave problem.