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 corresponding 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 has significant impact on the transient wave motions. The shear current moreover introduced asymmetry between upstream and downstream waves, resulting in asymmetric wave patterns.