The Froude number for solitary water waves with vorticity

TL;DR

This paper proves that for two-dimensional solitary water waves with vorticity, the Froude number must be greater than one, providing bounds on wave amplitude based on vorticity distribution.

Contribution

It offers a simple proof that the Froude number exceeds one for solitary waves with vorticity and establishes bounds on the Froude number and amplitude under certain conditions.

Findings

Froude number F > 1 for solitary waves with vorticity

Derived upper bounds on Froude number and wave amplitude

Results hold under specific assumptions on vorticity distribution

Abstract

We consider two-dimensional solitary water waves on a shear flow with an arbitrary distribution of vorticity. Assuming that the horizontal velocity in the fluid never exceeds the wave speed and that the free surface lies everywhere above its asymptotic level, we give a very simple proof that a suitably defined Froude number $F$ must be strictly greater than the critical value $F=1$. We also prove a related upper bound on $F$, and hence on the amplitude, under more restrictive assumptions on the vorticity.