v 1 2 1 Ju n 20 06 Distribution of particles which produces a desired radiation pattern

If Aq(β, α, k) is the scattering amplitude, corresponding to a potential q ∈ L 2(D), where D ⊂ R3 is a bounded domain, and eikα·x is the incident plane wave, then we call the radiation pattern the function A(β) := Aq(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) ∈ L2(S2), where S2 is the unit sphere in R3, can be approximated with any desired accuracy by a radiation pattern: ||f(β) − A(β)||L2(S2) < ǫ, where ǫ > 0 is an arbitrary small fixed number. The potential q, corresponding to A(β), depends on f and ǫ. There is a one-to-one correspondence between the above potential and the density of the number of small acoustically soft particles Dm ⊂ D, 1 ≤ m ≤ M , distributed in an a priori given bounded domain D ⊂ R3. The geometrical shape of a small particle Dm is arbitrary, the boundary Sm of Dm 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 ∈ S2, at a fixed k > 0, arbitrarily close in the norm of L2(S2 × S2) to an arbitrary given scattering amplitude f(α′, α), corresponding to a real-valued potential q ∈ L2(D), i.e., corresponding to an arbitrary given refraction coefficient in D.