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

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$\unicode[STIX]{x1D70F}=|\boldsymbol{V}|\unicode[STIX]{x1D714}_{0}/g$equals the critical value$1/4$($\boldsymbol{V}$: velocity of disturbance,$\unicode[STIX]{x1D714}_{0}$: oscillation frequency,$g$: gravitational acceleration). We show that the presence of a subsurface shear current can change this picture radically. Not only does the resonant value of$\unicode[STIX]{x1D70F}$depend strongly on the angle between$\boldsymbol{V}$and the current’s direction and the ‘shear-Froude number’$\mathit{Fr}_{s}=|\boldsymbol{V}|S/g$; when$\mathit{Fr}_{s}>1/3$, multiple resonant values – as many as four – can occur for some directions of motion. At sufficiently large values of$\mathit{Fr}_{s}$, 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.