On the time-evolution of resonant triads in rotational capillary-gravity water waves

TL;DR

This paper studies how resonant interactions among three wave modes evolve over time in rotational water waves with constant vorticity, extending known integrable systems to more complex fluid flows.

Contribution

It derives dynamic equations for resonant triads in rotational water flows and explores how vorticity influences wave interactions, generalizing the classical three-wave system.

Findings

Derived equations for resonant triads with vorticity

Showed reduction to integrable three-wave system when vorticity is zero

Provided insights into wave interaction dynamics in rotational flows

Abstract

We investigate an effect of the resonant interaction in the case of one-directional propagation of capillary-gravity surface waves arising as the free surface of a rotational water flow. Specifically, we assume a constant vorticity in the body of the fluid which physically corresponds to an underlying current with a linear horizontal velocity profile. We consider the interaction of three distinct modes and we obtain the dynamic equations for a resonant triad. Setting the constant vorticity equal to zero we recover the well known integrable three-wave system.