Dynamic Mass Density and Acoustic Metamaterials

Abstract

Elastic and electromagnetic waves are two types of classical waves that, though very different, nevertheless display many analogous features. In particular, for the acoustic waves, there can be a correspondence between the two material parameters of the acoustic wave equation, the mass density and bulk modulus, with the dielectric constant and magnetic permeability of the Maxwell equations. We show that the classical mass density, a quantity that is often regarded as positive definite in value, can display complex finite-frequency characteristics for a composite that comprises local resonators, thereby leading to acoustic metamaterials in exact analogy with the electromagnetic metamaterials. In particular, we demonstrate that through the anti-resonance mechanism, a locally resonant sonic material is capable of totally reflecting low-frequency sound at a frequency where the effective dynamic mass density can approach positive and negative infinities. The condition that leads to the anti-resonance thereby offers a physical explanation of the metamaterial characteristics for both the membrane resonator and the 3D locally resonant sonic materials. Besides the metamaterials arising from the dynamic mass density behavior at finite frequencies, we also present a review of other relevant types of acoustic metamaterials. At the zero-frequency limit, i.e., in the absence of resonances, the dynamic mass density for the fluid–solid composites is shown to still differ significantly from the usual volume-averaged expression. We offer both a physical explanation and a rigorous mathematical derivation of the dynamic mass density in this case.