A plane harmonic sound wave is considered to be incident upon a rigid plane screen that contains a periodic rectangular array of circular or elliptical apertures, and a characterization is sought for the reflexion and transmission coefficients of the scattered waves when the relationships aperture dimension [Lt ] spacing [Lt ] wavelength apply. The problem is analysed with the help of an integral equation over a single aperture and, as a consequence of the determination of the leading terms in its asymptotic solution, some prior results for more general (that is, irregular) aperture spacing are confirmed and specific features of the interaction in the periodic arrangement are established. Similar formulations are devised and given attention for the related problems in which (i) the screen is backed by a rigid infinite plane and (ii) the apertures contain rigid pistons capable of executing normal displacements compatible with an assigned and common impedance. A section is devoted to the solution, based on expansion of its kernel, for an integral equation of the first kind with a plane circular or elliptical domain.
An asymptotic solution of the integral equation for the second moment function in geometric processes
