Simulating transient wave phenomena in acoustic metamaterials using auxiliary fields

Acoustic wave propagation in dispersive metamaterials is considered. A prototypical example stems from the homogenization of a waveguide coupled with Helmholtz resonators and elastic membranes. The homogenized models of acoustic metamaterials are characterized by constitutive parameters, namely the effective bulk modulus and the effective mass density, that are frequency dependent. In this context, the objective considered here is to analyze such media in the time-domain and to simulate associated transient wave phenomena. To do so, the governing evolution equations are recast using a set of auxiliary fields to obtain an augmented first-order hyperbolic system. This initial-value problem is analyzed and then solved numerically using a splitting method and a high-order finite-difference scheme. An immersed interface method is also implemented to tackle scattering and interface problems involving metamaterial subdomains of arbitrary shapes. Numerical experiments are performed to validate the proposed overall approach and to investigate a variety of transient wave phenomena occurring in acoustic metamaterials.