Scattering by Spherically Symmetric Inhomogeneities

Abstract

The Jost function formulation of quantum scattering theory is applied to classical problems involving the scattering of a scalar plane wave by a medium in which the velocity is a function only of the spherical radial coordinate. This technique is used to solve the radial differential equation for scattering from a constant spherical inhomogeneity. The radial equation can be converted into an integral equation incorporating the Jost boundary conditions. The l = 0 partial wave integral equation for a constant inhomogeneity is solved using an iteration procedure (the first two iterations are considered). The Jost function and l = 0 cross section σ0 are plotted as a function of kR1, where k is the wavenumber in the surrounding medium and R1 is the sphere radius. The iterative technique is good for long wavelengths (kR1≪1) and any ratio of wavenumbers in the scattering and surrounding media. For shorter wavelengths and small ratio of wavenumbers (e.g., k1/k = 1.1, where k1 is the wavenumber in the scattering medium), it gives a good approximation to σ0 for the entire range of kR1 considered (0⩽kR1⩽2π). For shorter wavelengths and larger ratio of wavenumbers (e.g., k1/k = 1.5, 2.0), it gives a good approximation to σ0 out to approximately kR1 = 3π/4. More general problems using this method are also discussed.