Oscillatory line source for water waves in shear flow

• Water-wave radiation on a shear flow is represented by a Green function: the 2D oscillating source.
• Zero surface velocity gives no Doppler effects and at most two radiated waves.
• There exists an upstream wave with fluid particles co-rotating with the vorticity.
• The downstream wave has greater amplitude but a cut-off: it vanishes for angular frequencies below the vorticity.

The linearized water-wave radiation problem for the oscillating 2D submerged source in an inviscid shear flow with a free surface is investigated analytically. The vorticity is uniform, with zero velocity at the free surface. Then there will be at most two emitted waves, and no Doppler effects. Exact far-field waves are derived, with radiation conditions applied at infinity. An upstream wave will always exist, whereas the downstream wave exists only when the angular frequency of oscillation exceeds the vorticity. The wave radiation problem is solved also for oscillating vortex and dipoles. The amplitudes and energy fluxes are calculated.