Doppler effects of an oscillating line source in shear flow with a free surface

• Water-wave radiation with uniform vorticity is represented by a Green function: the 2D oscillating source.
• Nonzero surface velocity gives Doppler effects and at most four radiated waves.
• Wave motion co-rotating with the vorticity is easier to generate than counter-rotating wave motion.
• Wave cut-offs occur, and there is resonance with two waves merging at zero group velocity.

The linearised water-wave radiation problem for the oscillating 2D submerged source in an inviscid shear flow with a free surface is investigated analytically. There is a nonzero surface velocity. The depth is infinite and the vorticity is uniform. The amplitudes radiated from the source are calculated analytically. Due to Doppler effects, there may be up to four different emitted waves, and there is resonance with zero group velocity and infinite amplitude.