un 2 00 6 Distribution of particles which produces a desired radiation pattern

It is shown that any desired radiation pattern f (β) ∈ L 2 (S 2), where S 2 is the unit sphere in R 3 , can be generated by a suitable distribution of small, acoustically soft particles D m ⊂ D, 1 ≤ m ≤ M , in an a priori given bounded domain D ⊂ R 3. In this theory 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 there exists a potential q(x) with compact support, such that the scattering amplitude, generated by this potential at a fixed energy k 2 > 0 and a fixed direction α of the incident plane wave, approximates with arbitrary desired accuracy an arbitrary, a priori given function f (β) ∈ L 2 (S 2) in L 2 (S 2)-norm. It is also 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).