Field induced in inhomogeneous spheres by external sources. I. The scalar case

The evaluation of acoustic or electromagnetic fields induced in the interior of inhomogeneous penetrable bodies by external sources is based on well-known volume integral equations; this is particularly true for bodies of arbitrary shape and/or composition, for which separation of variables fails. In this paper the investigation focuses on acoustic (scalar fields) in inhomogeneous spheres of arbitrary composition, i.e., with r-, theta- or even phi-dependent medium parameters. The volume integral equation is solved by a hybrid (analytical-numerical) method, which takes advantage of the orthogonal properties of spherical harmonics, and, in particular, of the so-called Dini's expansions of the radial functions, whose convergence is optimized. The numerical part comes at the end; it involves the evaluation of certain definite integrals and the matrix inversion for the expansion coefficients of the solution. The scalar case treated here serves as a steppingstone for the solution of the more difficult electromagnetic problem.