Acoustic Metamaterials Michael R. Haberman Acoustic metamaterials expand the parameter space of materials available for new acoustical devices by manipulating sound in Postal: unconventional ways. Applied Research Laboratories and Department of Mechanical Engineering The University of Texas at Austin Introduction P.O. Box 8029 Why have acoustic metamaterials (AMMs) appeared on the scene in the last few Austin, Texas 78713-8029 years, and what are they? The original defining property of a metamaterial is that it USA achieves effects not found in nature as a means to address long-standing engineer- ing challenges in acoustics. Can one, for example, create ultrathin acoustic barriers Email: whose performance surpasses currently existing technology? Is it possible to elim- [email protected] inate scattering from an acoustic sensor and minimize the influence the device has on the field being measured? Can spatially compact acoustical lenses be created Andrew N. Norris whose resolution surpasses the diffraction limit? These are but a few examples of Postal: the problems AMM research strives to address. The common theme of the many Mechanical and Aerospace Engineering AMM devices is an apparent defiance of the intuitive laws of physics, which of- School of Engineering, ten require strange concepts such as negative density and negative compressibility. Rutgers University Negative effective properties underlie behavior such as negative refraction that, in Piscataway, NJ 08854 turn, enables acoustic lens designs that beat the diffraction limit. USA AMM research was originally motivated by parallel developments in electromag- Email: netics, such as negative refraction and cloaking (Norris, 2015). The first to study [email protected] these topics quickly found that the available materials were not up to the task of providing the necessary properties for cloaking or negative refraction. A straight- forward reaction to this problem was to simply create new materials. Although not simple, the creation of new materials has been, and continues to be, the fun- damental driving force for the study of AMMs. This has led to concepts that could not be anticipated from electromagnetics, such as pentamode materials (exotic materials that only resist one mode of deformation, analogous to how fluids only resist volumetric change), to address AMM challenges (Norris, 2015). The chal- lenge in developing new materials has been significantly aided by steady improve- ments in technology, primarily computer simulation and additive manufacturing. Those technologies combined with ingenious ideas from the acoustical research community have helped drive rapid advances in AMMs over the last decade, some of which we describe in this article. For acoustical applications, dynamic material properties are of interest and may open the door to more exotic behavior. The focusing/beamforming from the fatty lobe of a dolphin, known as the melon, is a case in point (Yamato et al., 2014). Dy- namic properties are not the same as their static counterparts that we learn about in introductory courses on mechanics. Thus Figure 1 displays the design space of AMMs, a range that includes negative values for both density and compressibility, that is explained in detail in this article. At this stage, it is sufficient to note that the effective density and compressibility provide a helpful way to view the overall re- sponse of an engineered structure as an effective medium rather than using the im- pedance of a complex system. This is consistent with the metamaterial paradigm that seeks to expand the available parameter space available to people designing acoustic systems and devices to control acoustic waves. All rights reserved. ©2016 Acoustical Society of America. volume 12, issue 3 | Fall 2016 | Acoustics Today | 31 Acoustic Metamaterials 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- Figure 1. The material design space for acoustic metamateri- ties obtained? The concept of negative impedance is familiar, als, density (ρ) versus compressibility (C), with example ap- such as a tuned vibration absorber that oscillates in or out plications. Top right quadrant is the space familiar in normal of phase with the force when the drive frequency is below acoustics. The other quadrants have one or both parameters or above the resonance frequency, respectively. Negative with negative values. Negative density or compressibility can inertia can therefore be thought of as an out-of-phase time only be achieved dynamically. For instance, Helmholtz reso- harmonic motion of a moving mass; see http://bit.do/negm. nators driven just above their frequency of resonance lead to The first and probably most widely known AMM (Liu et al., negative dynamic compressibility. The three devices shown 2000), designed to isolate low-frequency sound much more employ the metamaterial effects of negative refraction, trans- efficiently than the classical mass law, was a composite con- formation acoustics (TA), and cloaking, all of which are de- taining microstructural elements comprising heavy masses scribed in the text. 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 Our objective is to demonstrate the broad range of AMMs tensioned membranes as the moving mass (e.g., Lee et al., in terms of some seemingly “unnatural” effects such as nega- 2009). Although negative inertia is common in vibrations, tive density and compressibility, nonreciprocal wave trans- it is not as obvious how to achieve negative values of com- mission, and acoustic cloaking. The interested reader can pressibility (inverse of bulk modulus). Recall that positive delve further into the technical aspects both in the original pressure in naturally occurring materials results in a volume papers cited and through several accessible reviews by Kadic decrease. Negative compressibility therefore requires that an et al. (2013), Ma and Sheng (2016), Cummer et al. (2016), applied pressure yield a positive expansion. This can be real- and Haberman and Guild (2016). Detailed expositions on ized in close proximity to a well-known acoustical element, AMMs can be found in edited texts such as Craster and the Helmholtz resonator, above its resonance frequency Guenneau (2013) and Deymier (2013). On specific topics, where the cavity volume acts as a dynamic volume source. Hussein et al. (2014) survey phononic crystals and applica- Combinations of these types of resonators can yield one or tions; Chen and Chan (2010), Fleury and Alù (2013), and both negative properties in a finite frequency range. Norris (2015) provide overviews of acoustic cloaking; and Fleury et al. (2015) review nonreciprocal acoustic devices. To understand how a double-negative AMM works, con- We begin with the negative acoustic properties and their ap- sider the simplest example of an acoustic duct with alternat- plication. 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 transmission of an isolated element at a given frequency energy propagation directions are then opposite; a situation (Figure 2). It is then standard procedure to combine the re- known as negative group velocity (group and energy propa- flection/transmission coefficients to determine the 2 × 2- ma gation velocities are almost always the same, except in the trix that describes how waves propagate across a single peri- case of large absorption that is not applicable here). To see od of the transmission line. The Bloch-Floquet condition for this, note that energy travels in the direction of the acoustic determining the effective wavenumber (ke ) is then equiva- energy flux (pv), where v is particle velocity. A plane wave _x lent to finding the matrix eigenvalues. This gives sinke b= with dependence cos ω ( c – t) has phase velocity c, which is ρ ωb √ e Ce , where ω is the radial frequency, b is one-half the real-valued because both the effective density and compress- period length, and the effective density and compressibility ibility are negative. The pressure and particle velocity are re- are ρe = ρ + 2SZM / (– iωb) and Ce = C + 1 / (– iωb2SZH), respec- lated by the dynamic impedance (ρc) and therefore, because tively, where ρ and C are the fluid density and compressibil- ρ is negative, the energy flux direction must be opposite to ity, respectively, and S is the duct cross section. the phase velocity. A simple example is given by the limiting case of a hole in AMMs take advantage of negative properties in a variety of the duct which reduces the Helmholtz resonance frequency ways, including radiation/sensing in leaky wave antennae to zero and yields negative effective compressibility Ce be- (LWAs) and focusing in phononic crystals. LWAs are one- low a finite cutoff. Conversely, Ce is positive above the cutoff, dimensional transmission line devices, like the duct consid- which is a result well known in musical acoustics: tone holes ered above, designed to radiate sound into a surrounding in a flute are a high-pass filter. fluid much like a long flutelike instrument with the sound emanating from the tone holes. The coupling to the exterior fluid is particularly strong if the compressibility element is an open hole.