  We deal with the scattering of an acoustic medium modeled by an index of
refraction $n$ varying in a bounded region $\Omega$ of $\mathbb{R}^3$ and equal
to unity outside $\Omega$. This region is perforated with an extremely large
number of small holes $D_m$'s of maximum radius $a$, $a<<1$, modeled by surface
impedance functions. Precisely, we are in the regime described by the number of
holes of the order $M:=O(a^{\beta-2})$, the minimum distance between the holes
is $d\sim a^t$ and the surface impedance functions of the form $\lambda_m \sim
\lambda_{m,0} a^{-\beta}$ with $\beta >0$ and $\lambda_{m,0}$ being constants
and eventually complex numbers. Under some natural conditions on the parameters
$\beta, t$ and $\lambda_{m,0}$, we characterize the equivalent medium
generating, approximately, the same scattered waves as the original perforated
acoustic medium. We give an explicit error estimate between the scattered waves
generated by the perforated medium and the equivalent one respectively, as $a
\rightarrow 0$. As applications of these results, we discuss the following
findings:
  1. If we choose negative valued imaginary surface impedance functions,
attached to each surface of the holes, then the equivalent medium behaves as a
passive acoustic medium only if it is an acoustic metamaterial with index of
refraction $\tilde{n}(x)=-n(x),\; x \in \Omega$ and $\tilde{n}(x)=1,\; x \in
\mathbb{R}^3\setminus{\overline{\Omega}}$. This means that, with this process,
we can switch the sign of the index of the refraction from positive to negative
values.
  2. We can choose the surface impedance functions attached to each surface of
the holes so that the equivalent index of refraction $\tilde{n}$ is
$\tilde{n}(x)=1,\; x \in \mathbb{R}^3$. This means that the region $\Omega$
modeled by the original index of refraction $n$ is approximately cloaked.
