On bounds and non-existence in the problem of steady waves with vorticity

TL;DR

This paper establishes bounds on free-surface profiles and Bernoulli's constant for steady gravity waves with vorticity, and proves non-existence of wave solutions under certain conditions when only one shear flow exists.

Contribution

It provides new bounds and non-existence results for steady gravity waves with vorticity, advancing understanding of wave behavior in fluid dynamics.

Findings

Bounds for free-surface profile established

Bounds for Bernoulli's constant derived

Non-existence of wave solutions under specific vorticity conditions

Abstract

For the problem describing steady, gravity waves with vorticity on a two-dimensional, unidirectional flow of finite depth the following results are obtained. (i) Bounds for the free-surface profile and for Bernoulli's constant. (ii) If only one parallel shear flow exists for a given value of Bernoulli's constant, then there are no wave solutions provided the vorticity distribution is subject to a certain condition.