Effect on Doppler Resonance From a Near-Surface Shear Layer

For waves generated by a wave source which is simultaneously moving and oscillating at a constant frequency ω, a resonance is well known to occur at a particular value τ_res of the nondimensional frequency τ = ωV/g (V: source velocity relative to the surface, g: gravitational acceleration). For quiescent, deep water, it is well known that τ_res = 1/4. We study the effect on τ_res from the presence of a shear flow in a layer near the surface, such as may be generated by wind or tidal currents. Assuming the vorticity is constant within the shear layer, we find that the effects on the resonant frequency can be significant even for sources corresponding to moderate shear and relatively long waves, while for stronger shear and shorter waves the effect is stronger. Even for a situation where the resonant waves have wavelengths about 20 times the width of the shear layer, the resonance frequency can change by ∼ 25% for even a moderately strong shear VS/g = 0.3 (S: vorticity in surface shear layer). Intuition for the problem is built by first considering two simpler geometries: uniform current with finite depth, and Couette flow of finite depth.