6 Distribution of particles which produces a ” smart ” material

If A q (β, α, k) is the scattering amplitude, corresponding to a potential q ∈ L 2 (D), where D ⊂ R 3 is a bounded domain, and e ikα·x is the incident plane wave, then we call the radiation pattern the function A(β) := A q (β, α, k), where the unit vector α, the incident direction, is fixed, and k > 0, the wavenumber, is fixed. It is shown that any function f (β) ∈ L 2 (S 2), where S 2 is the unit sphere in R 3 , can be approximated with any desired accuracy by a radiation pattern: ||f (β) − A(β)|| L 2 (S 2) < ǫ, where ǫ > 0 is an arbitrary small fixed number. The potential q, corresponding to A(β), depends on f and ǫ, and can be calculated analytically. There is a one-to-one correspondence between the above potential and the density of the number of small acoustically soft particles D m ⊂ D, 1 ≤ m ≤ M , distributed in an a priori given bounded domain D ⊂ R 3. The geometrical shape of a small particle D m is arbitrary, the boundary S m of D m is Lipschitz uniformly with respect to m. The wave number k and the direction α of the incident upon D plane wave are fixed. It is shown that a suitable distribution of the above particles in D can produce the scattering amplitude A(α ′ , α), α ′ , α ∈ S 2 , at a fixed k > 0, arbitrarily close in the norm of L 2 (S 2 × S 2) to an arbitrary given scattering amplitude f (α ′ , α), corresponding to a real-valued potential q ∈ L 2 (D), i.e., corresponding to an arbitrary refraction coefficient in D.