Many-body wave scattering by small bodies

Acoustic or electromagnetic EM wave scattering by one or several bodies is usually studied by reducing the problem to solving some integral equations. In this paper we show that if the bodies are small in comparison with the wavelength, then the scattering problem can be reduced to solving linear algebraic systems with matrices whose elements have physical meaning. These elements are electrical capacitances or elements of electric and magnetic polarizability tensors. The author has derived analytical explicit formulas allowing one to calculate these quantities for bodies of arbitrary shapes with arbitrary desired accuracy see Ref. 3 . We derive these linear algebraic systems and give formulas for the elements of the matrices of these systems. There is a large literature on wave scattering by small bodies, see Ref. 3 and references therein. The theory was originated by Rayleigh, who understood that the main term in the scattered field is the dipole radiation if the body is small. Rayleigh did not give formulas for calculating the induced dipole moments for small bodies of arbitrary shapes. The dipole moments are uniquely defined by the polarizability tensors. Therefore, the formulas, derived by the author see Ref. 3 , allow one to calculate the dipole radiation for acoustic and EM wave scattering by small bodies of arbitrary shapes. A well-known book deals with wave scattering by small spheres, for the most part. An old well-known paper develops a theory of scalar wave scattering by isotropic scatterers. A recent paper deals with EM wave scattering by small particles. The basic novel point in our paper is a rigorous reduction of the many-body acoustic wave scattering problem to a linear algebraic system, whose matrix elements have physical meaning and their values are calculated analytically with any desired accuracy in the author’s earlier work. Compared with the work in Ref. 1, in our work the matrix elements are explicitly calculated. Compared with the work in Ref. 5, there is a big physical difference, especially in the case of scattering in a medium consisting of many small particles. The difference lies in the following: in acoustic wave scattering by small acoustically soft particles, the theory is developed in our current paper under the assumptions a , a d, where is the wavelength, a is the characteristic size of a particle, and d is the smallest distance between two distinct particles. These assumptions allow one to have many small particles on the wavelength. So the medium is not a “rarefied gas” of particles. In Ref. 5 the many-body scattering theory is developed for EM waves under much stronger assumptions: a d. The necessity of these stronger assumptions for EM wave scattering is explained and justified in Ref. 5 and is also motivated briefly in Sec. III.