Wave diffraction through a perforated barrier of nonzero thickness (Q2770316)

The authors investigate 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 integral equations of first kind characterized by a combination of sum and difference kernels. This structure motivates the main purpose of the paper, the development of embedding formulae for derived general integral equations, extending previous results for a simpler case of pure difference kernels. By setting the derivation in the context of diffraction problem, the embedding results can be readily interpreted and practically exploited. Thus, for example, one of them gives the solution of 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. The authors take advantage of the considerable saving in computational effort that follows when the wave problem is solved for a range of incident angles.