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

TL;DR

This paper develops a point-interaction approximation for acoustic fields generated by small, contrasted bubbles at fixed frequencies, providing conditions for validity and applications in material science and imaging.

Contribution

It introduces natural conditions under which the Foldy-Lax approximation accurately models bubble clusters at arbitrary fixed frequencies.

Findings

Validates the point interaction approximation for various bubble configurations.

Allows analysis of effective media for gases with specific density contrasts.

Handles frequencies near Minnaert resonance for certain contrast regimes.

Abstract

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.