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

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.