No steady water waves of small amplitude are supported by a shear flow with still free surface

TL;DR

This paper proves that small-amplitude steady gravity waves cannot exist on a shear flow with a still free surface, under certain vorticity conditions, clarifying limitations of wave formation in such flows.

Contribution

It establishes the non-existence of small-amplitude steady waves supported by shear flows with still free surfaces for a broad class of vorticity distributions.

Findings

No small-amplitude waves are supported by certain shear flows.

Applicable to positive constant, linear, and quadratic vorticity distributions.

Provides mathematical proof of non-existence in the specified setting.

Abstract

The two-dimensional free-boundary problem describing steady gravity waves with vorticity on water of finite depth is considered. It is proved that no small-amplitude waves are supported by a horizontal shear flow whose free surface is still in a coordinate frame such that the flow is time-independent in it. The class of vorticity distributions for which such flows exist includes all positive constant distributions, as well as linear and quadric ones with arbitrary positive coefficients.