Numerical solution of many-body wave scattering problem and creating materials with a desired refraction coefficient

Scalar wave scattering by many small particles with impedance boundary condition and creating material with a desired refraction coefficient are studied. The acoustic wave scattering problem is solved asymptotically and numerically under the assumptions $ka \ll 1, \zeta_m = \frac{h(x_m)}{a^\kappa}, d = O(a^{\frac{2-\kappa}{3}}), M = O(\frac{1}{a^{2-\kappa}}), \kappa \in [0,1)$, where $k = 2\pi/\lambda$ is the wave number, $\lambda$ is the wave length, $a$ is the radius of the particles, $d$ is the distance between neighboring particles, $M$ is the total number of the particles embedded in a bounded domain $\Omega \subset \RRR$, $\zeta_m$ is the boundary impedance of the m\textsuperscript{th} particle $D_m$, $h \in C(D)$, $D := \bigcup_{m=1}^M D_m$, is a given arbitrary function which satisfies Im$h \le 0$, $x_m \in \Omega$ is the position of the m\textsuperscript{th} particle, and $1 \leq m \leq M$. Numerical results are presented for which the number of particles equals $10^4, 10^5$, and $10^6$.