Initial Value Problems for Water Waves in the Presence of a Shear Current

Abstract

We obtain a general solution for the initial value problems of water waves with a shear current of uniform vorticity beneath the surface in three dimensions. Linearized governing equations and boundary conditions including the effects of gravity, a distributed external pressure disturbance, and constant finite depth, are solved analytically. Results are obtained and correspondin g analyses are presented for the special case of an initial pressure impulse condition. Agreement with previous results in the absence of shear current is demonstrated. The results show that a shear current ha s significant impact on the transient wave motions. The shear current moreover introduced asymmetry between upstream and downstream waves, resulting in asymmetric wave patterns.