Wave scattering by many small particles embedded in a medium

Theory of wave scattering by many small bodies is developed under various assumptions concerning the ratio ad, where a is the characteristic dimension of a small body and d is the distance between neighboring bodies d=O(aκ1), 0<κ1<1. On the boundary Sm of every small body an impedance-type condition is assumed ∂uM∂N=ζmuM on Sm, 1⩽m⩽M, ζm=hma−κ, 0<κ, hm are constants independent of a. The behavior of the field in the region in which M=M(a)≫1 small particles are embedded is studied as a→0 and M(a)→∞. Formulas for the refraction coefficient of the limiting medium are derived under the assumptions: (a) κ1=(2−κ)/3, 0<κ⩽1, and (b) κ1=1/3, κ>1. A method for creating materials with a desired refraction coefficient is proposed and justified theoretically on the basis of the above results.