  The linearized water-wave radiation problem for an oscillating submerged line
source in an inviscid shear flow with a free surface is investigated
analytically at finite, constant depth in the presence of a shear flow varying
linearly with depth. The surface velocity is taken to be zero relative to the
oscillating source, so that Doppler effects are absent. The radiated wave out
from the source is calculated based on Euler's equation of motion with the
appropriate boundary and radiation conditions, and differs substantially from
the solution obtained by assuming potential flow. To wit, an additional wave is
found in the downstream direction in addition to the previously known
dispersive wave solutions; this wave is non-dispersive and we show how it is
the surface manifestation of a critical layer-like flow generated by the
combination of shear and mass flux at the source, passively advected with the
flow. As seen from a system moving at the fluid velocity at the source's depth,
streamlines form closed curves in a manner similar to Kelvin's cat's eye
vortices. A resonant frequency exists at which the critical wave resonates with
the downstream propagating wave, resulting in a downstream wave pattern
diverging linearly in amplitude away from the source.
