Wave Attenuation through Multiple Rows of Scatterers with Differing Periodicities

Abstract

A solution method is proposed for calculating wave scattering by a multiple-row array in which the rows are permitted to have different periodicities. Each individual row contains an infinite number of identical and equispaced scatterers, which can be solved through standard techniques by invoking periodicity. Previous studies have investigated the wave attenuation produced by multiple-row arrays but in which the periodicity in the rows is fixed. However, this leads to difficulties around the resonant points, at which the number of scattering angles produced by the rows changes. The method outlined in the present work involves a discretization of the directional spectrum. This is combined with a mapping of the individual rows onto neighboring geometries that fit into the discrete system so that, as the mesh is refined, the geometry converges to its intended form. The method is applied to a canonical problem in which a potential function exists in the two-dimensional plane exterior to an array composed of circular scatterers, which have a Neumann condition imposed on their boundaries. Forcing is provided by an incident wave. Wave attenuation is investigated in a numerical results section using ensemble averages in which the row spacings and in-row spacings are chosen from normal distributions. Convergence of the solutions with respect to the numerical method is established, and examples of the smoothing effects gained from incorporating variation in the in-row spacing are given.