Spherical wave reflection and transmission

Abstract

This study is concerned with the reflection and transmission of spherical waves at a plane interface between two different media. The phenomenon of the reflection and transmission of spherical waves has been studied by means of analytical methods, numerical computation, and experimental tests. A new integral representation for a spherical is obtained by transforming Lamb/Sommerfeld's integral representation. The new integral has no singularity so it allows more accurate numerical integration. A new proof of Lamb/Sommerfeld's integral representation for a spherical is presented based on the new integral. By using the new form of solutions for reflected waves and existing solutions for transmitted waves, numerical studies have been carried out to examine. the phenomenon of reflection and transmission. of spherical waves at plane surfaces of discontinuity in material properties. It is shown that the effective critical angle for the total reflection of a spherical is greater than that of a plane at a hard boundary, and that when the source height increases the effective critical angle for the total reflection of a spherical tends to that of a plane wave. It is shown that recent predictions of spherical reflection and transmission coefficients greater than 1 at normal incidence under certain condition are probably due to numerical integration error. It also has been found that for spherical reflection and transmission, the time average energy flux, normal to a plane parallel to the plane of discontinuity, may locally be in the direction opposite to that of the direction of energy transmission over the plane as a whole. This so-called backward wave occurs in an interference between the direct and reflected waves, as well as in a transmitted wave. An indirect test on the theory has been performed to check the pressure field, above a rigid boundary, predicted by the spherical theory. Theoretical and experimental results were in good agreement.