New integral equations for scattering by penetrable objects, II

The problem of scattering of time‐harmonic incoming waves by homogeneous penetrable objects of arbitrary shape (cylinders of arbitrary cross section in two dimensions) may be recast as a pair of boundary integral equations to determine two unknown densities from which the field quantities may be obtained. The kernels of the integral equations usually involve the free space Green's functions for the Helmholtz equation with wave numbers appropriate to both the exterior and interior regions. Recent work on scattering by impenetrable objects has shown that considerable advantage is gained by modifying the free space Green's function with the addition of a linear combination of radiating cylindrical or spherical waves (multipoles) whose coefficients are chosen in some optimal manner. In the present paper, this method is extended to the penetrable case. Here the fundamental solution in the exterior is again modified by the addition of outgoing waves while the interior Green's function is modified by the addition of standing waves. The coefficients of the added terms are determined to minimize the norm of the boundary integral operator.