Scalar wave scattering by two-layer radial inhomogeneities

It is known that the Jost-function formulation of quantum scattering theory can be applied to classical problems concerned with the scattering of a plane scalar wave by a medium with a spherically symmetric inhomogeneity of finite extent. This technique is applied to solve the radial differential equation for the scattering from a constant spherical inhomogeneity and a piecewise constant by two-layer spherical inhomogeneity. This could represent a spherical scatterer with a piecewise increasing or decreasing refractive index, for example. When the problem cannot be solved analytically in closed form, the Jost integral formula can be used to convert it into an integral equation with the corresponding boundary conditions.