Distribution of Particles Which Produces a “Smart” Material

If Aq (β, α, k) is the scattering amplitude, corresponding to a potential q ∈ L2(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, β is the unit vector in the direction of the scattered wave, 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) < , where > 0 is an arbitrary small fixed number. The potential q, corresponding to A(β), depends on f and , 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 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 directionα 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 refraction coefficient in D.