Wave diffraction through a perforated barrier of non-zero thickness

Abstract

We consider the diffraction of a prescribed train of monochromatic plane surface water waves by a vertical-sided, perfectly reflecting thick breakwater standing on a horizontal bed in water of uniform undisturbed depth. The breakwater is punctuated by a series of N gaps, arbitrarily arranged. The corresponding linearized boundary-value problem is reduced to a pair of uncoupled first kind integral equations characterized by a combination of sum and difference kernels. This structure motivates the main purpose of the paper, the development of embedding formulae for a general integral equation of the type encountered, extending previous such results which exist for the simpler case of pure difference kernels. By setting the derivation in the context of the diffraction problem, the embedding results can be readily interpreted and practically exploited. Thus, for example, one of them gives the solution of the diffraction problem for any incident wave angle explicitly in terms of the solutions for 4N distinct angles, reducing to 2N distinct angles if the breakwater array is symmetric. We take advantage of the considerable saving in computational effort that follows when the wave problem is solved for a range of incident angles.