On the Benjamin--Lighthill conjecture for water waves with vorticity

TL;DR

This paper investigates the Benjamin--Lighthill conjecture for steady, rotational water waves with finite depth, focusing on flows near the critical Bernoulli's constant, and verifies the conjecture under these conditions.

Contribution

It provides a verification of the Benjamin--Lighthill conjecture for rotational water waves with vorticity near the critical flow condition.

Findings

Confirmation of the conjecture for flows close to critical Bernoulli's constant

Analysis of rotational flows with vorticity distribution

Characterization of flow behavior near critical conditions

Abstract

We consider the nonlinear problem of steady gravity-driven waves on the free surface of a two-dimensional flow of an incompressible fluid (say, water). The flow is assumed to be unidirectional of finite depth and the water motion is supposed to be rotational. Our aim is to verify the Benjamin--Lighthill conjecture for flows whose total head (Bernoulli's constant) is close to the critical one; the latter is determined by the vorticity distribution so that no horizontal shear flows exist for smaller values of the total head.