Theory of Scalar Wave Scattering by a Sphere and a Planar Substrate

The problem of scalar wave scattering by a sphere on or near a planar substrate is analytically solved. The solution is a set of wave functions coming in the form of infinite series of spherical and plane waves. In air, the incident plane wave is either scattered by the sphere or reflected from the substrate. A part of these scattered or reflected waves propagate to the other object where it is reflected and scattered again. Such processes of scattering and reflection repeat in turn indefinitely to generate multiply scattered waves, which are represented in the corresponding terms in the infinite series. The term in the series can be arranged in a recognizable manner to explicitly reveal the involved process and the multiplicity of scattering.