Effective acoustic properties of a meta-material consisting of small Helmholtz resonators

We investigate the acoustic properties of meta-materials that are inspired by sound-absorbing structures. We show that it is possible to construct meta-materials with frequency-dependent effective properties, with large and/or negative permittivities. Mathematically, we investigate solutions u^ε : Ω_ε → R to a Helmholtz equation in the limit ε → 0 with the help of two-scale convergence. The domain Ω_ε is obtained by removing from an open set Ω⊂ Rⁿ in a periodic fashion a large number (order ε ⁻ⁿ) of small resonators (order ε). The special properties of the meta-material are obtained through sub-scale structures in the perforations.