Investigation of Integral Equations of the Static and Quasi-Static Fields and Applications to the Scattering from Small Bodies of Arbitrary Shape
The calculation of static fields and some functionals of such fields, for example electrical capacitance or tensor polarizability, is of great interest in many applications. In particular, it is of basic interest for wave scattering by small bodies of arbitrary shape. Since the theory was initiated by Rayleigh  in 1871, very many papers have been published on this topic. Nevertheless the theory seemed incomplete in the following respect. Though wave scattering by a small body is a well understood process from the physical point of view there were no analytical formulas for the scattered field, scattering matrix, etc. In this chapter we obtain analytical formulas for the scattering matrix for the problems of scalar and vector wave scattering by a small body of arbitrary shape and by a system of such bodies. Analytical formulas for the calculation of the capacitance and polarizability of bodies of arbitrary shape with the needed accuracy are obtained. Two-sided variational estimates for the capacitance and polarizability are given. The formulas mentioned above are of immediate use in applications.
KeywordsPermeability Covariance Refraction Dition Acoustics
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