Water wave scattering by a structured ridge on the sea bed

Abstract

The propagation of plane incident waves over a long submerged uniform rectangular structured ridge protruding from the sea bed is considered. The ridge is comprised of a uniform closely-spaced array of vertical barriers between which the fluid is allowed to flow. In addition to the height and width of the ridge, the orientation and thickness of the barriers are adjustable design parameters. Particular interest centres negative refraction of obliquely-incident waves, shown to be possible provided the barriers extend sufficiently close to the surface. A key result is that the modulus of reflection and transmission coefficients are symmetric functions of the incident wave heading and the barrier orientation. This implies perfect transmission of wave energy for thin barrier elements at incident angles that are the reverse of the barrier orientation, extending the result of Porter (2021) to structured plate arrays of any submergence. The paper details two mathematical approaches to the solution of the problem, one for general barrier orientations and the other for a specific orientation where special treatment of the resulting problem is required. Numerical computations compare favourably with a variety of established results and good agreement with an existing shallow water approximation is confirmed for sufficiently long wavelengths.