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A classical problem in acoustic (and electromagnetic) scattering concerns theevaluation of the Green's function for the Helmholtz equation subject toimpedance boundary conditions on a half-space. The two principal approachesused for representing this Green's function are the Sommerfeld integral and the(closely related) method of complex images. The former is extremely efficientwhen the source is at some distance from the half-space boundary, but involvesan unwieldy range of integration as the source gets closer and closer. Compleximage-based methods, on the other hand, can be quite efficient when the sourceis close to the boundary, but they do not easily permit the use of thesuperposition principle since the selection of complex image locations dependson both the source and the target. We have developed a new, hybridrepresentation which uses a finite number of real images (dependent only on thesource location) coupled with a rapidly converging Sommerfeld-like integral.While our method applies in both two and three dimensions, we restrict thedetailed analysis and numerical experiments here to the two-dimensional case.

On the efficient representation of the half-space impedance Green's function for the Helmholtz equation