Acoustic metamaterials (AMM) are heterogeneous materials with dynamic subwavelength structures that can generate useful effective responses of interest to ultrasonic imaging applications such as negative refraction and zero index. Traditional effective medium models fail to capture details of frequency dependent AMM response and can give non-causal properties. This work derives non-local expressions for effective properties for an infinite periodic lattice of heterogeneities in an isotropic fluid using conservation of mass and momentum and the equation of state. The resulting model correctly predicts a causal effective material response by considering coupling between the ensemble-averaged volume strain and momentum fields.
Sieck, C. F., Alù, A., & Haberman, M. R. (2015). Dynamic homogenization of acoustic metamaterials with coupled field response. In Physics Procedia (Vol. 70, pp. 275–278). Elsevier. https://doi.org/10.1016/j.phpro.2015.08.153