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

The point-interaction approximation for the fields generated by contrasted bubbles at arbitrary fixed frequencies