Multiple resonances of a moving, oscillating surface disturbance on a shear current

TL;DR

This paper investigates how shear currents influence wave resonances caused by a moving oscillating disturbance, revealing multiple resonance points and critical velocities depending on flow parameters and direction.

Contribution

It demonstrates that shear currents can cause multiple wave resonances and critical velocities, significantly altering classical wave resonance behavior in deep water.

Findings

Multiple resonant values can occur when shear-Froude number exceeds 1/3.

The smallest resonance frequency approaches zero at high shear-Froude numbers.

The analysis includes detailed dispersion relations for finite and infinite water depths.

Abstract

We consider waves radiated by a disturbance of oscillating strength moving at constant velocity along the free surface of a shear flow which, when undisturbed, has uniform horizontal vorticity of magnitude $S$. When no current is present the problem is a classical one and much studied, and in deep water a resonance is known to occur when $\tau=|\boldsymbol{V}|\omega_0/g$ equals the critical value $1/4$ ($\boldsymbol{V}$: velocity of disturbance, $\omega_0$: oscillation frequency, $g$: gravitational acceleration). We show that the presence of the sub-surface shear current can change this picture radically. Not only does the resonant value of $\tau$ depend strongly on the angle between $\boldsymbol{V}$ and the current's direction and the "shear-Froude number" $\mathrm{Frs}=|\boldsymbol{V}|S/g$; when $\mathrm{Frs}>1/3$, multiple resonant values --- as many as $4$ --- can occur for some directions of motion. At sufficiently large values of $\mathrm{Frs}$, the smallest resonance frequency tends to zero, representing the phenomenon of critical velocity for ship waves. We provide a detailed analysis of the dispersion relation for the moving, oscillating disturbance, in both finite and infinite water depth, including for the latter case an overview of the different far-field waves which exist in different sectors of wave vector space under different conditions. Owing to the large number of parameters, a detailed discussion of the structure of resonances is provided for infinite depth only, where analytical results are available.