Unstable surface waves in running water

TL;DR

This paper analyzes the stability of periodic water waves over shear flows, establishing criteria for instability, demonstrating bifurcation of small waves, and exploring how vorticity influences wave stability.

Contribution

It introduces a new formulation for the linearized water-wave problem and provides a sharp instability criterion for shear flows with inflection points.

Findings

Free surface destabilizes water waves compared to rigid-wall case.

Bifurcation of small-amplitude periodic waves occurs at unstable wave numbers.

Vorticity subtly affects the stability of water waves.

Abstract

We consider the stability of periodic gravity free-surface water waves traveling downstream at a constant speed over a shear flow of finite depth. In case the free surface is flat, a sharp criterion of linear instability is established for a general class of shear flows with inflection points and the maximal unstable wave number is found. Comparison to the rigid-wall setting testifies that free surface has a destabilizing effect. For a class of unstable shear flows, the bifurcation of nontrivial periodic traveling waves of small-amplitude is demonstrated at any wave number. We show the linear instability of small nontrivial waves bifurcated at an unstable wave number of the background shear flow. The proof uses a new formulation of the linearized water-wave problem and a perturbation argument. An example of the background shear flow of unstable small-amplitude periodic traveling waves is constructed for an arbitrary vorticity strength and for an arbitrary depth, illustrating that vorticity has a subtle influence on the stability of water waves.