Acoustic field induced in spheres with inhomogeneous density by external sources.

Acoustic or electromagnetic fields induced in the interior of inhomogeneous penetrable bodies by external sources can be evaluated via well-known volume integral equations. For bodies of arbitrary shape and/or composition, for which separation of variables fails, a direct attack for the solution of these integral equations is the only available approach. In a previous paper by the same authors the scalar (acoustic) field in inhomogeneous spheres of arbitrary compressibility, but with constant density, was considered. In the present one the direct hybrid (analytical-numerical) method applied to the much simpler integral equation for spheres with constant density is generalized to densities that vary with r, theta, or even psi. This extension is by no means trivial, owing to the appearance of the derivatives of both the density and the unknown function in the volume integral, a fact necessitating a more subtle and accuracy-sensitive approach. Again, the spherical shape allows use of the orthogonal spherical harmonics and of Dini's expansions of a general type for the radial functions. The convergence of the latter, shown to be superior to other possible sets of orthogonal expansions, can be further optimized by the proper selection of a crucial parameter in their eigenvalue equation.

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.