3-D reflection and transmission of sound by an array of rods

Three-dimensional reflection and transmission of a plane acoustic wave by a grating composed of parallel equidistant rods is investigated. The direction of the propagation vector of the incident wave is arbitrary. The reflection and transmission problem has been formulated rigorously by taking advantage of translational invariance along the rods, and geometrical periodicity in a plane normal to the rods. The reflected and transmitted wave motions have been expressed as superpositions of an infinite number of wave modes, each with its own cut-off frequency. Reflection and transmission coefficients have been defined as integrals over the circumference of a single rod, in terms of the velocity potential and auxiliary velocity terms on the surface of the rod. The singular integral equation for the velocity potential has been solved numerically by the boundary integral equation method. Curves show the reflection and transmission coefficients for the reflected and transmitted waves as functions of the frequency. At the cut-off frequencies these coefficients show rapid variations, which are analogous to the Wood's anomaly of optical diffraction gratings.