  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.
