Explicit results for scattering parameters in three-dimensional wave propagation through a doubly periodic system of arbitrary openings

In the present paper, an analytical approach previously formulated for two-dimensional scattering problems is further developed to study three-dimensional problems. A scalar plane wave is assumed to (normally) penetrate into a plane screen having identical openings of arbitrary shape which are periodically distributed. This problem is first described by an integral equation holding over the typical openings’ domain. Then, by applying simple approximations (uniformly) valid in the assumed regime of propagation, some auxiliary integral equations are deduced which are independent of the wave frequency. A linear algebraic system is finally set up whose solution provides explicit analytical formulas for the main scattering parameters. By solving numerically this system of equations for assigned obstacle shapes, several graphs are obtained which show comparison between such explicit formulas and the corresponding rigorous numerical solution of the original integral equation.