Irrotational two-dimensional free-surface steady water flows over a flat bed with underlying currents

Abstract

We investigate the free boundary non-linear problem for large amplitude smooth Stokes waves in flows of finite depth with underlying currents. The qualitative nature of the flow and pressure fields is investigated using maximum principles. Bounds on the surface profile have been derived. Influence of the relative magnitude of wave and current speed on the flow and pressure fields has been shown. The possible co-existence of propagating surface waves and an underlying uniform current at the wave speed is settled—this can only happen if the surface is flat.