Diffraction of a spherical wave by an absorbing plane: A general theoretical approach

Abstract

In outdoor sound propagation, the ground absorption is described by various models such as a locally reacting surface or layer of porous material. For each type of absorbing plane, different authors have developed specific methods to calculate an approximation of the sound pressure. In the present paper a general method is proposed. If the absorbing plane is located at z = 0, and a point source at S(0,0,d), an exact representation of the solution shows that the diffracted field can be decomposed into the sum of (a) the radiation of an image source at S′(0,0,−d); (b) the radiation of a surface source located in the z = −d plane; (c) the z = derivative of the radiation of a surface source located in the same plane. From this exact integral representation of the solution, two series expansions can be derived. The first one, convergent for any source and receiver locations, involves the asymptotic series and a correcting series: the first two or three terms provide a sufficiently accurate approximation for distances from a few wavelengths up to infinity. The second is a Taylor-like series with respect to the vertical distance between the image source S′ and the receiver; it points out the well-known surface wave term, but gives good approximations close to the ground surface, only. Several examples are given: locally reacting surface; finite or infinite thickness layer of porous material; thin elastic plate. Numerical results show the efficiency of these approximations for engineering purposes.