  If $A_q(\beta, \alpha, k)$ is the scattering amplitude, corresponding to a
potential $q\in L^2(D)$, where $D\subset\R^3$ is a bounded domain, and
$e^{ik\alpha \cdot x}$ is the incident plane wave, then we call the radiation
pattern the function $A(\beta):=A_q(\beta, \alpha, k)$, where the unit vector
$\alpha$, the incident direction, is fixed, and $k>0$, the wavenumber, is
fixed. It is shown that any function $f(\beta)\in L^2(S^2)$, where $S^2$ is the
unit sphere in $\R^3$, can be approximated with any desired accuracy by a
radiation pattern: $||f(\beta)-A(\beta)||_{L^2(S^2)}<\epsilon$, where
$\epsilon>0$ is an arbitrary small fixed number. The potential $q$,
corresponding to $A(\beta)$, depends on $f$ and $\epsilon$, 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
$D_m\subset D$, $1\leq m\leq M$, distributed in an a priori given bounded
domain $D\subset\R^3$. 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 $\alpha$ 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 $A(\alpha',\alpha)$,
$\alpha',\alpha\in S^2$, at a fixed $k>0$, arbitrarily close in the norm of
$L^2(S^2\times S^2)$ to an arbitrary given scattering amplitude
$f(\alpha',\alpha)$, corresponding to a real-valued potential $q\in L^2(D)$.
