Resonant triad interactions of gravity waves in cylindrical basins

TL;DR

This paper investigates how horizontal confinement in cylindrical basins enables resonant triads of gravity waves, deriving conditions for their existence, analyzing their dynamics, and exploring excitation mechanisms with implications for natural and artificial water basins.

Contribution

It demonstrates that confinement induces resonant triads in gravity waves, provides necessary and sufficient conditions for their existence, and analyzes their nonlinear dynamics and excitation.

Findings

Resonant triads are possible in confined basins but not in unbounded domains.

Unique critical depths exist where resonance occurs.

Triad evolution is always periodic, with energy exchange controlled by specific parameters.

Abstract

We present the results of a theoretical investigation into the existence, evolution and excitation of resonant triads of nonlinear free-surface gravity waves confined to a cylinder of finite depth. It is well known that resonant triads are impossible for gravity waves in laterally unbounded domains; we demonstrate, however, that horizontal confinement of the fluid may induce resonant triads for particular fluid depths. For any three correlated wave modes arising in a cylinder of arbitrary cross-section, we prove necessary and sufficient conditions for the existence of a depth at which nonlinear resonance may arise, and show that the resultant critical depth is unique. We enumerate the low-frequency triads for circular cylinders, including a new class of resonances between standing and counter-propagating waves, and also briefly discuss annular and rectangular cylinders. Upon deriving the triad amplitude equations for a finite-depth cylinder of arbitrary cross-section, we deduce that the triad evolution is always periodic, and determine parameters controlling the efficiency of energy exchange. In order to excite a particular triad, we explore the influence of external forcing; in this case, the triad evolution may be periodic, quasi-periodic, or chaotic. Finally, our results have potential implications on resonant water waves in man-made and natural basins, such as industrial-scale fluid tanks, harbours and bays.