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, 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)​<ϵ, where
ϵ>0 is an arbitrary small fixed number. The potential q,
corresponding to A(β), depends on f and ϵ, 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 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).Comment: corrected typo