[Seminar] [Applied Analysis Seminar] 4-Layer potential approach to homogenization of sound-absorbing and resonant acoustic metamaterials
In this presentation, I will present some new results about the derivation of an effective medium theory for sound-absorbing acoustic metamaterials. We determine an effective equation satisfied by the leading order asymptotic of the acoustic field scattered by many tiny obstacles filling a given volume and provide quantitative error estimates. One originality of our work is that we consider a fully non-periodic setting where the centers of the obstacles are randomly and independently distributed. Our analysis relies crucially on two new ingredients. First, using layer potentials, we show that the leading order asymptotic of the scattered field is determined by the solution to an algebraic linear system of size the number of obstacles, which can be physically interpreted as the usual ``Foldy-Lax approximation'' of the heterogeneous medium. The second ingredient is the convergence in some mean-square sense of this linear system towards a ``homogenized'' integral equation, from which the effective physics of the medium is inferred. We obtain as such that sound-absorbing media behave up to some critical size as metamaterials with positive effective refractive index. The case of high-contrast acoustic obstacles will then be briefly discussed, which lead to metamaterials exhibiting a positive or negative refractive index for an incident field whose frequency is respectively slightly smaller or slightly larger than resonant frequencies.