A variety of problems in acoustic and electromagnetic scattering require theevaluation of impedance or layered media Green's functions. Given a pointsource located in an unbounded half-space or an infinitely extended layer,Sommerfeld and others showed that Fourier analysis combined with contourintegration provides a systematic and broadly effective approach, leading towhat is generally referred to as the Sommerfeld integral representation. Wheneither the source or target is at some distance from an infinite boundary, thenumber of degrees of freedom needed to resolve the scattering response is verymodest. When both are near an interface, however, the Sommerfeld integralinvolves a very large range of integration and its direct application becomesunwieldy. Historically, three schemes have been employed to overcome thisdifficulty: the method of images, contour deformation, and asymptotic methodsof various kinds. None of these methods make use of classical layer potentialsin physical space, despite their advantages in terms of adaptive resolution andhigh-order accuracy. The reason for this is simple: layer potentials areimpractical in layered media or half-space geometries since they require thediscretization of an infinite boundary. In this paper, we propose a hybridmethod which combines layer potentials (physical-space) on a finite portion ofthe interface together with a Sommerfeld-type (Fourier) correction. We provethat our method is efficient and rapidly convergent for arbitrarily locatedsources and targets, and show that the scheme is particularly effective whensolving scattering problems for objects which are close to the half-spaceboundary or even embedded across a layered media interface.
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