# Numerical solution of many-body wave scattering problem for small particles and creating materials with desired refraction coefficient

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Theory of wave scattering by small particles of arbitrary shapes was developed by A. G. Ramm in papers (Ramm, 2005; 2007;a;b; 2008;a; 2009; 2010;a;b) for acoustic and electromagnetic (EM) waves. He derived analytical formulas for the S-matrix for wave scattering by a small body of arbitrary shape, and developed an approach for creating materials with a desired spatial dispersion. One can create a desired refraction coefficient n 2 (x, ω) with a desired x, ω-dependence, where ω is the wave frequency. In particular, one can create materials with negative refraction, i.e., material in which phase velocity is directed opposite to the group velocity. Such materials are of interest in applications, see, e.g., (Hansen, 2008; von Rhein et al., 2007). The theory, described in this Chapter, can be used in many practical problems. Some results on EM wave scattering problems one can find in (Tatseiba & Matsuoka, 2005), where random distribution of particles was considered. A number of numerical methods for light scattering are presented in (Barber & Hill, 1990). An asymptotically exact solution of the many body acoustic wave scattering problem was developed in (Ramm, 2007) under the assumptions ka << 1, d = O(a 1/3), M = O(1/a), where a is the characteristic size of the particles, k = 2π/λ is the wave number, d is the distance between neighboring particles, and M is the total number of the particles embedded in a bounded domain D ⊂ R 3 . It was not assumed in (Ramm, 2007) that the particles were distributed uniformly in the space, or that there was any periodic structure in their distribution. In this Chapter, a uniform distribution of particles in D for the computational modeling is assumed (see Figure 1). An impedance boundary condition on the boundary Sm of the m-th particle Dm was assumed, 1 ≤ m ≤ M. In (Ramm, 2008a) the above assumptions were generalized as follows: