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Acoustic Metamaterials
Figure 1. The material design space for acoustic metamateri- als, density (ρ) versus compressibility (C), with example ap- plications. Top right quadrant is the space familiar in normal acoustics. The other quadrants have one or both parameters with negative values. Negative density or compressibility can only be achieved dynamically. For instance, Helmholtz reso- nators driven just above their frequency of resonance lead to negative dynamic compressibility. The three devices shown employ the metamaterial effects of negative refraction, trans- formation acoustics (TA), and cloaking, all of which are de- scribed in the text.
Our objective is to demonstrate the broad range of AMMs in terms of some seemingly “unnatural” effects such as nega- tive density and compressibility, nonreciprocal wave trans- mission, and acoustic cloaking. The interested reader can delve further into the technical aspects both in the original papers cited and through several accessible reviews by Kadic et al. (2013), Ma and Sheng (2016), Cummer et al. (2016), and Haberman and Guild (2016). Detailed expositions on AMMs can be found in edited texts such as Craster and Guenneau (2013) and Deymier (2013). On specific topics, Hussein et al. (2014) survey phononic crystals and applica- tions; Chen and Chan (2010), Fleury and Alù (2013), and Norris (2015) provide overviews of acoustic cloaking; and Fleury et al. (2015) review nonreciprocal acoustic devices. We begin with the negative acoustic properties and their ap- plication.
Dynamic Negative Density
and Compressibility
Many AMM devices are based on negative acoustic density and/or compressibility (inverse of bulk modulus). The speed of sound, more precisely the phase speed, is the square root of the bulk modulus divided by the density. If either quantity is negative, then the phase speed is imaginary, correspond- ing to exponential decay, and hence no transmission. When both acoustic parameters are negative, the phase speed is again a real-valued number, implying wave propagation, although with a twist: the energy and phase velocities are in opposite directions. This response provides AMMs with the capability to produce “unnatural’ effects like negative refraction that is discussed below. How are negative proper- ties obtained? The concept of negative impedance is familiar, such as a tuned vibration absorber that oscillates in or out of phase with the force when the drive frequency is below or above the resonance frequency, respectively. Negative inertia can therefore be thought of as an out-of-phase time harmonic motion of a moving mass; see http://bit.do/negm.
The first and probably most widely known AMM (Liu et al., 2000), designed to isolate low-frequency sound much more efficiently than the classical mass law, was a composite con- taining microstructural elements comprising heavy masses surrounded by a soft rubber annulus arranged periodical- ly in a three-dimensional solid matrix. Many subsequent air-based devices employing negative inertia have used tensioned membranes as the moving mass (e.g., Lee et al., 2009). Although negative inertia is common in vibrations, it is not as obvious how to achieve negative values of com- pressibility (inverse of bulk modulus). Recall that positive pressure in naturally occurring materials results in a volume decrease. Negative compressibility therefore requires that an applied pressure yield a positive expansion. This can be real- ized in close proximity to a well-known acoustical element, the Helmholtz resonator, above its resonance frequency where the cavity volume acts as a dynamic volume source. Combinations of these types of resonators can yield one or both negative properties in a finite frequency range.
To understand how a double-negative AMM works, con- sider the simplest example of an acoustic duct with alternat- ing sprung masses and Helmholtz resonators, as in Figure 2. The resonators are defined by impedances that relate acous- tic pressure (p) with volumetric flow velocity (U), according to pM = ZMUM and pH = ZHUH. The effective acoustic proper- ties can be calculated by first considering the reflection and
32 | Acoustics Today | Fall 2016