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

The problem of scalar wave scattering by a planar substrate containing a sphere is analytically solved. The wave functions, as the solution, come in the form of infinite series. The factors in each term of the series explicitly show the involved process and the multiplicity of scattering between the surface of the substrate and the sphere. In air, the incident wave is either reflected by or transmitted into the substrate. The transmitted wave is either scattered by or transmitted into the sphere, and then the backwardly scattered wave going to the surface of the substrate is either transmitted back into air or reflected toward the sphere. The reflected wave is again either scattered by or transmitted into the sphere, and so on. Within the substrate, the scattering by the sphere and the reflection at the planar surface repeat in turn indefinitely to generate multiply scattered waves, which combine to make different wave functions for different regions in the form of infinite series.