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An Analytic Solution for Low-Frequency Scattering by Two Soft Spheres

A plane wave is scattered by two small spheres of not necessarily equal radii. Low-frequency theory reduces this scattering problem to a sequence of potential problems which can be solved iteratively. It is shown that there exists exactly one bispherical coordinate system that fits the given geometry. Then R-separation is utilized to solve analytically the potential problems governing the leading two low-frequency approximations. It is shown that the Rayleigh approximation is azimuthal independent, while the first-order approximation involves the azimuthal angle explicitly. The leading two nonvanishing approximations of the normalized scattering amplitude as well as the scattering cross-section are also provided. The Rayleigh approximations for the amplitude and for the cross-section involve only a monopole term, while their next order approximations are expressed in terms of a monopole as well as a dipole term. The dipole term disappears whenever the two spheres become equal, and this observation provide...