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

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.

References in corpus (3)

• Sub-wavelength focusing of acoustic waves in bubbly media
• Spectral theory for Schrödinger operators with $\delta$-interactions supported on curves in ${\mathbb{R}}^3$
• Characterization of the equivalent acoustic scattering for a cluster of an extremely large number of small holes

Cited by in corpus (9)

• The equivalent media generated by bubbles of high contrasts: Volumetric metamaterials and metasurfaces
• Estimation of the heat conducted by a cluster of small cavities and characterization of the equivalent heat conduction
• On the Origin of Minnaert Resonances
• The Foldy-Lax approximation is valid for nearly resonating frequencies
• Elastic fields generated by multiple small inclusions with high mass density at nearly resonant frequencies
• Localization of point scatterers via sparse optimization on measures
• The Effective Permittivity and Permeability Generated by a Cluster of Moderately Contrasting Nanoparticles
• Analysis of the acoustic waves reflected by a cluster of small holes in the time-domain and the equivalent mass density
• Foldy-Lax approximation of the electromagnetic fields generated by anisotropic inhomogeneities in the mesoscale regime with complements for the perfectly conducting case