  We deal with the linearized model of the acoustic wave propagation generated
by small bubbles in the harmonic regime. We estimate the waves generated by a
cluster of $M$ small bubbles, distributed in a bounded domain $\Omega$, with
relative densities having contrasts of the order $a^{\beta}, \beta>0, $ where
$a$ models their relative maximum diameter, $a\ll 1$. We provide useful and
natural conditions on the number $M$, the minimum distance and the contrasts
parameter $\beta$ of the small bubbles under which the point interaction
approximation (called also the Foldy-Lax approximation) is valid.
  With the regimes allowed by our conditions, we can deal with a general class
of such materials. Applications of these expansions in material sciences and
imaging are immediate. For instance, they are enough to derive and justify the
effective media of the cluster of the bubbles for a class of gases with
densities having contrasts of the order $a^{\beta}$, $\beta \in (\frac{3}{2},
2)$ and in this case we can handle any fixed frequency. In the particular and
important case $\beta=2$, we can handle any fixed frequency far or close (but
distinct) from the corresponding Minnaert resonance. The cluster of the bubbles
can be distributed to generate volumetric metamaterials but also low
dimensional ones as metascreens and metawires.
