Homogenization of sound-soft and high-contrast acoustic metamaterials in subcritical regimes

Abstract

We propose a quantitative effective medium theory for two types of acoustic metamaterials constituted of a large number N of small heterogeneities of characteristic size s, randomly and independently distributed in a bounded domain. We first consider a “sound-soft” material, in which the total wave field satisfies a Dirichlet boundary condition on the acoustic obstacles. In the “sub-critical” regime sN = O(1), we obtain that the effective medium is governed by a dissipative Lippmann-Schwinger equation which approximates the total field with a relative mean-square error of order O(max((sN)^2N^(-1/3),N^(-1/2))). We retrieve the critical size s ∼ 1/N of the literature at which the effects of the obstacles can be modelled by a “strange term” added to the Helmholtz equation. Second, we consider high-contrast acoustic metamaterials, in which each of the N heterogeneities are packets of K inclusions filled with a material of density much lower than the one of the background medium. As the contrast parameter vanishes, δ → 0, the effective medium admits K resonant characteristic sizes (si (δ))1≤i≤K and is governed by a Lippmann-Schwinger equation, which is diffusive or dispersive (with negative refractive index) for frequencies ω respectively slightly larger or slightly smaller than the corresponding K resonant frequencies (ωi (δ))1≤i≤K . These conclusions are obtained under the condition that (i) the resonance is of monopole type, and (ii) lies in the “subcritical regime” where the contrast parameter is small enough, i.e. δ = o(N^(-2)), while the considered frequency is “not too close” to the resonance, i.e. N δ 2 = O(|1 − s/si (δ)|). Our mathematical analysis and the current literature allow us to conjecture that “solidification” phenomena are expected to occur in the “super-critical” regime N δ 2 |1 − s/si (δ)|−1 → +∞.

Subwavelength resonant acoustic scattering in fast time-modulated media

“…Our approach relies on a variational formulation of (1.1) and the Dirichlet-to-Neumann operator associated to the Helmholtz equation in exterior domains. We expose our method on the unmodulated version of (1.1) in which ρ(t) = 1, which allows us to verify that we retrieve all the results of [11,8,46,21,44] concerned with this setting, namely the leading asymptotic of the subwavelength resonant frequencies, point scatterer approximations, and a formal derivation of an effective medium theory for a system constituted of many small subwavelength resonators. Our novel approach leads to slightly simpler derivations because we do not rely on a layer potential representation, but is also more flexible in the sense that it could easily be applied in other dimensions (d = 1 or d = 2, left for a future work), and to the time-modulated case which is the object of the remainder of the paper.…”

“…We illustrate this method by retrieving the results established with layer potential methods in [11,46] for the static version of (1.1) in which the modulation is kept constant; namely ρ(t) = 1 for all t ∈ R. One advantage of the proposed Dirichletto-Neumann approach is that it can be rather easily extended to the time-modulated system (1.1) considered in the subsequent Sections 4 and 5. This Dirichlet-to-Neumann approach has also been successfully applied to study Minnaert resonances of one-dimensional (non-modulated) high contrast systems in [47].…”

“…Sections 3.2 and 3.3 thus retrieve the results derived with layer potentials in [11,46]. Finally, we outline in Section 3.4 the Foldy-Lax approximation argument which allows to derive an effective medium theory for a heterogeneous medium filled with many rescaled copies of such resonators, referring to [21,44] for a more complete analysis.…”

“…Besides the analysis of these resonant phenomena, we identify in Corollary 5.1 a leading order approximation for the far field pattern of the scattered wave. After suitable rescalings and a Foldy-Lax approximation argument inspired from [21,44], this allows us to discuss, at least at a formal level, the arising of an effective medium for the temporal metamaterial which would be constituted of many tiny copies of such time-modulated resonator D.…”

Scattering from time-modulated subwavelength resonators

“…The highly contrasting material parameters (relative to the background medium) of these objects are the crucial mechanism responsible for their subwavelength response. We now have a complete understanding of different wave scattering phenomena arising from static subwavelength resonator structures [8][9][10][11][12][13][14][15][16]. However, little is known when waves are scattered by timemodulated subwavelength resonators.…”