The point-interaction approximation for the fields generated by contrasted bubbles at arbitrary fixed frequencies. (English) Zbl 1447.35371
The paper deals with a linearized model of the acoustic wave propagation generated by small bubbles in the harmonic regime. The waves with relative densities having contrasts of the order \(a^{\beta}\), \(\beta > 0\) which generated by a cluster of \(M\) small bubbles and distributed in a bounded domain \(\Omega\), are investigated. Here \(a\) models their maximum diameter, \(a<1\), \(d\) is the minimum distance and \(\beta\) is the contrasts parameter of the small bubbles. Natural conditions on \(M\), \(d\) and \(\beta\) are found under which the point interaction approximation (the Foldy-Lax approximation) is valid. With the regimes allowed by our conditions, one 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 one can handle any fixed frequency. In the particular and important case \(\beta = 2\), it is possible to handle any fixed frequency far or close (but distinct) from the corresponding Minnaert resonance. The cluster of bubbles can be distributed to generate volumetric metamaterials but also low dimensional ones as metascreens and metawires.
Reviewer: Elena V. Tabarintseva (Chelyabinsk)
MSC:
| 35R30 | Inverse problems for PDEs |
| 35C20 | Asymptotic expansions of solutions to PDEs |
| 35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
| 35P25 | Scattering theory for PDEs |
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