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

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.