Willis dynamic homogenization method for acoustic metamaterials based on multiple scattering theory

This paper presents a spatial dispersive Willis dynamic homogenization method for acoustic metamaterials based on multiple scattering theory, which is valid for each physical dimension – 1D, 2D and 3D, and valid for arbitrary inhomogeneities/scatterers – passive or active. The homogenization can be viewed as the monopole and dipole truncation of multiple scattering theory. With the help of layer-doubling scheme, both propagating and evanescent waves can be characterized by the homogenization theory. Further, besides effective field, the theory also considers inner field (often neglected by traditional homogenizations), making it possible to study interface problems with a higher precision than traditional methods. To validate the homogenization theory, a two dimensional acoustic metamaterial composed of a square lattice of C-shape resonators is carefully studied as an application. By the proposed method, we can predict not only effective properties of bulk materials, including real and complex band structures, but also properties of interfaces, including transmission/reflection properties of a layer and dispersion of interface waves. An experiment of interface waves supported by the metamaterials is also conducted, and the results of experiments, simulations and homogenization are all in good agreement. Interestingly, two essentially different evanescent modes of the interface wave are observed in the experiment, and they can be distinguished by the proposed homogenization method, but are undistinguishable via traditional homogenizations. Finally, a detailed discussion on the limitations and some opening questions of the homogenization theory is presented. And the physical restrictions for the case of a reciprocal, passive and lossless material are additionally given and discussed. Our method paves the way for designing Willis' type metamaterials and further realizing more novel acoustic wave-control functions.