We consider the propagation of acoustic waves in the presence of inhomogeneities having small size and high contrast with respect to a regular background. Such composite systems are known to exhibit, depending on the incoming frequency, a transition towards a resonant regime where an enhancement of the scattered wave is observed. The resonant frequencies are generally referred to as subwavelength resonances. Different designs of the material parameters can determine the resonant scattering limit: we focus on the specific case of a single micro-bubble enjoying a high contrast of both its mass density and bulk modulus. The resonant frequency related to such scaling is known as the Minnaert resonance. A careful analysis of this model has shown that the resonant micro-bubble behaves as a point-scatterer in the far-field approximation. We reconsider this problem in terms of the scattering from a frequency-dependent singular perturbation supported at the inhomogeneity interface. The analysis of the limit behaviour of the frequency-dependent resolvents in the small-scale regime shows that the stationary acoustic operator has a non-trivial limit (i.e.: it asymptotically differs from the Laplacian) if and only if the incoming frequency coincides with the Minnaert resonance. In particular we recover the point-scatterer approximation and provide with uniform-in-space control of the error terms for the asymptotic expansion of the scattered field, improving in this sense previous results.
In collaboration with: A. Posilicano and M. Sini.