Wave scattering by many small impedance particles and applications

Formulas are derived for solutions of many-body wave scattering problem by small impedance particles embedded in a homogeneous medium. The limiting case is considered, when the size a of small particles tends to zero while their number tends to infinity at a suitable rate. The basic physical assumption is a << d << λ, where d is the minimal distance between neighboring particles, λ is the wavelength, and the particles can be impedance balls B(x_m, a) with centers x_m located on a grid. Equations for the limiting effective (self-consistent) field in the medium are derived. It is proved that one can create material with a desired refraction coefficient by embedding in a free space many small balls of radius a with prescribed boundary impedances. The small balls can be centered at the points located on a grid. A recipe for creating materials with a desired refraction coefficient is formulated. It is proved that materials with a desired radiation pattern, for example, wave-focusing materials, can be created.