Scattering of a spherical wave by a small sphere: an elementary solution

Wave scattering by objects that are small compared to the wavelength (Rayleigh scattering) is usually studied for plane incident waves. However, knowledge of the full Green's function of the problem becomes necessary when the separation of scatterers from either an interface or each other is comparable to the scatterers' dimensions. Here, an elementary analytic solution is derived for diffraction of a spherical sound wave by a small, soft sphere. The approximate solution is obtained from asymptotic expansions of an exact solution, holds everywhere outside the sphere, and reduces to classical results due to Kelvin and Rayleigh in appropriate special cases.