Wave trapping above a plane beach by partially or totally submerged obstacles

Abstract

Trapped waves generated by oscillatory sources or dipoles placed above a plane infinite beach are examined within the framework of a (classical) non-hydrostatic but linear theory. This is achieved by solving a boundary-value problem where the boundary conditions are specified on the free surface and on the bottom. Integral expressions are derived for the complex potential for the cases where the sources or dipoles are strategically positioned to mimic the presence of solid bodies, a phenomenon manifested by the observation of a streamline enclosing the source or dipole. The precise positioning is governed by the further requirement of no radiating waves and, for the case where the beach is a vertical cliff, some recent results are confirmed here, whilst new results obtained show that infinitely many submerged wave trapping bodies exist and do so over a far greater range of values of dipole positions than was previously thought to be the case. The situation for surface sources and for submerged dipoles is therefore essentially different. For the former, infinitely many closed streamlines exist for each of the denumerably infinite set of source positions. For the latter, it is found instead that only one closed streamline exists, but this is for each of a non-denumerably infinite set of dipole positions. The expressions obtained for the beach are used for the two cases of a surface source and a submerged dipole to compute streamlines and stagnation points for model beaches of chosen steep slope. In particular, a (randomly chosen) submerged closed streamline is calculated for the beach of angle 45° thereby establishing a new case of non-uniqueness for the water wave problem on a beach.