The complex equivalent source method for sound propagation over an impedance plane

The sound field caused by a monopole source above an impedance plane can be calculated by using a superposition of equivalent point sources located along a line in the mirror space below the plane. Originally, such an approach for representing the half-space Green's function was described by Sommerfeld at the beginning of the last century, in order to treat half-space problems of heat conduction. However, the representation converges only for masslike impedances and cannot be used for the more important case of reflecting planes with springlike surface impedances. The singular part of the line integral can be transformed into a Hankel function, which shows that surface waves are contained in the whole solution. Unfortunately, this representation suffers from the lack of validity at certain receiver points and from restrictions on wave number and impedance range to ensure the necessary convergence. The main idea of the present method is to use also a superposition of equivalent point sources, but to allow that these sources can be located at complex source points. The corresponding form of the half-space Green's function is suitable for both masslike and springlike surface impedances, and can be used as a cornerstone for a boundary element method.