Dispersion relation for water waves with non-constant vorticity

TL;DR

This paper derives the dispersion relation for small-amplitude water waves with non-constant vorticity, extending known results from constant vorticity to a broad class of vorticity profiles.

Contribution

It provides the first explicit dispersion relations for water waves with non-constant vorticity, covering various functional forms.

Findings

Explicit dispersion relations for polynomial, exponential, trigonometric, and hyperbolic vorticity functions.

Extension of known constant vorticity results to non-constant cases.

Broad applicability to different vorticity profiles.

Abstract

We derive the dispersion relation for linearized small-amplitude gravity waves for various choices of non-constant vorticity. To the best of our knowledge, this relation is only known explicitly in the case of constant vorticity. We provide a wide range of examples including polynomial, exponential, trigonometric and hyperbolic vorticity functions.