Multiple wave scattering by submerged obstacles in an infinite channel of finite depth with surface pressure excess

Abstract

The objective is to study the combined effect of an incident wave, a surface pressure excess and a finite number of submerged obstacles, in the phenomenon of power transfer to an infinite fluid layer of finite depth. The incident wave and the surface pressure excess have the same harmonic time dependence, a fact that allows to eliminate time altogether and consider only steady-state solutions. The surface pressure excess simulates the effect of winds blowing above the water surface in oceans. The technique used in a first part of the paper relying upon the use of finite Fourier transform and separation of variables is extended here to this end. The method allows to separate local perturbations from progressive or standing wave. Our formulae yield the exact solution in closed form in the absence of obstacles, and provide a clearer insight into the flow properties, as compared to previous investigations. Applications are given for discontinuous surface pressure functions. We put in evidence solutions with no outgoing waves, for which the energy transmitted by the surface pressure is exhausted in generating a standing wave, together with local perturbations. Two numerical applications without/with obstacles, for a parabolic surface pressure profile, allow to assess the energy transfer from the pressure-obstacles system to the fluid. The results may be of interest in the field of oscillating water columns and, generally, water power converting technology.