Origin of negative density and modulus in acoustic metamaterials

Abstract

This paper provides a review and fundamental physical interpretation for the effective densities and moduli of acoustic metamaterials. We introduce the terminology of hidden force and hidden source of volume: the effective density or modulus is negative when the hidden force or source of volume is larger than, and operates in antiphase to, respectively, the force or volume change that would be obtained in their absence. We demonstrate this ansatz for some established acoustic metamaterials with elements based on membranes, Helmholtz resonators, springs and masses. The hidden force for membrane-based acoustic metamaterials, for instance, is the force from the membrane tension. The hidden source for a Helmholtz-resonator-based metamaterial is the extra air volume injected from the resonator cavity. We also explain the analogous concepts for pure mass-and-spring systems, in which case hidden forces can arise from masses and springs fixed inside other masses, whereas hidden sources-more aptly termed hidden expanders of displacement in this case-can arise from light rigid trusses coupled to extra degrees of freedom for mechanical motion such as the case of coupling to masses that move at right angles to the wave-propagation direction. This overall picture provides a powerful tool for conceptual understanding and design of new acoustic metamaterials, and avoids common pitfalls involved in determining the effective parameters of such materials.Acoustic metamaterials are man-made structures designed to manipulate the propagation of sound in ways not available in naturally occurring materials. The understanding of negative constitutive parameters in such materials,[1-34] one of their most exotic features, has to date relied more on engineering-based concepts than on universal physical principles. This can lead to confusion in assigning effective parameters to a given system. Here we present a fundamental physical picture to account for the effective densities and moduli of acoustic metamaterials that allows their intuitive yet precise understanding, and at the same time allows their unambiguous determination. This picture is based on hidden forces and hidden sources of volume. The former, resulting in an effective mass or density, involve local, time-dependent non-apparent forces that provide a net force on the unit cell of the acoustic metamaterial. The latter, resulting in an effective elastic modulus, involve time-dependent hidden sources of volume or displacement that only produce pairs of forces acting equally and oppositely on either side of the unit cell.We first illustrate our approach with examples of systems exhibiting an effective mass based on membranes in tubes as well as on mass-and-spring models (the latter providing elements to model three-dimensional (3D) solid acoustic metamaterials). An example of a metamaterial based on membranes combined with masses and springs is also elucidated. We then provide examples of systems exhibiting an effective modulus based on Helmholtzresonators in tubes as well as base...

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