Amplitude bounds of steady rotational water waves

TL;DR

This paper establishes bounds on the amplitude of steady rotational water waves with constant vorticity, showing the amplitude's dependence on vorticity and providing new proofs for these bounds.

Contribution

It introduces new bounds on wave amplitude for steady rotational water waves and offers a novel proof for the case of non-positive vorticity.

Findings

Upper bound on amplitude for small positive vorticity

Amplitude tends to zero as vorticity tends to negative infinity

New proof techniques for amplitude bounds

Abstract

We consider classical steady water waves with a free surface, a flat bottom and constant vorticity $\gamma$. In the adverse case $\gamma>0$ we prove that there is an absolute upper bound on the amplitude, independent of the physical constants, provided that $\gamma$ is sufficiently small. In any favorable case $\gamma\leq0$ we present a new proof of such an absolute bound on the amplitude and prove that the amplitude tends to zero as $\gamma$ tends to negative infinity.