Origin of Negative Density and Modulus in Acoustic Metamaterials

Origin of Negative Density and Modulus in Acoustic Metamaterials Sam H. Lee∗1 and Oliver B. Wright2 1Institute of Physics and Applied Physics, Yonsei University, Seoul 120-749, Korea 2Division of Applied Physics, Faculty of Engineering, Hokkaido University, Sapporo 060-8628, Japan 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 de- exhibiting both effective density and modulus, includ- signed to manipulate the propagation of sound in ways ing the possibility of double-negative parameter behav- not available in naturally occurring materials. The un- ior, are also discussed. derstanding 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 EFFECTIVE DENSITIES: HIDDEN FORCES universal physical principles. This can lead to confusion in assigning effective parameters to a given system. Here In this section we introduce the concept of “hidden we present a fundamental physical picture to account for force” in order to understand effective mass in systems the effective densities and moduli of acoustic metamateri- that contain non-apparent mechanical elements.[35] This als that allows their intuitive yet precise understanding, approach is first explained by means of simple examples and at the same time allows their unambiguous deter- from mechanics. We then illustrate the concept of ef- mination. This picture is based on hidden forces and fective density for several different acoustic metamateri- hidden sources of volume. The former, resulting in an als involving unidirectional propagation: a membrane- effective mass or density, involve local, time-dependent based metamaterial, a solid-matrix metamaterial ap- non-apparent forces that provide a net force on the unit proximated by spring-coupled masses, and also a new cell of the acoustic metamaterial. The latter, resulting in membrane-based metamaterial that also includes masses an effective elastic modulus, involve time-dependent hid- and springs. den sources of volume or displacement that only produce pairs of forces acting equally and oppositely on either side of the unit cell. Simple mechanical systems We first illustrate our approach with examples of systems exhibiting an effective mass based on membranes Consider a wheel of mass M, radius R and moment in tubes as well as on mass-and-spring models (the latter of inertia I rolling in the x direction without slipping arXiv:1509.04094v7 [physics.class-ph] 21 Jan 2016 providing elements to model three-dimensional (3D) solid on a flat, horizontal surface, as shown in Fig. 1(a). A acoustic metamaterials). An example of a metamaterial horizontal force F is applied to the axis of the wheel at based on membranes combined with masses and springs its center. The application of Newton’s laws allows one is also elucidated. We then provide examples of sys- to derive the acceleration: x¨ = F/[M(1 + I/MR2]). The tems exhibiting an effective modulus based on Helmholtz- effective mass of wheel can be defined as resonators in tubes as well as based on mass-and-spring models combined with light rigid trusses coupled to ex- Meff = F/x,¨ (1) tra degrees of freedom for mechanical motion. Systems 2 giving, in this case, Meff = M(1 + I/MR ). Let us define the hidden force Fh to be the backwardly-directed frictional force on the wheel rim. The acceleration x¨ is ∗e-mail:[email protected] smaller than that expected for a non-rotating mass of 2 (a) (c) resonance frequency. Substituting into Eq. (2), ω2 M = M 1 − 0 . (3) eff ω2 Fh F Fh F The displacement, obtained by integrating x¨ = F/Meff , 10 can be clearly seen to oscillate with large amplitude near resonance (at ω0) because Meff becomes very small. M (b) / This ansatz describes all physical quantities correctly and eff 5 quantitatively. Another example is the abrupt shift of the phase of the displacement by π with respect to the driving force as the frequency passes through the resonance. 0 This can immediately be understood from Eq. (3), since the sign of Meff changes at ω = ω0. Negative Meff , a -5 consequence of Fh < −F , implies in the case of sinusoidal excitation that the magnitude of the hidden force is not MASS MASS RATIO M only in antiphase with but also has a magnitude that is -10 larger than that of the applied force. As the limit ω =0 -4 -2 0 2 4 is approached, the effective mass tends to −∞ because the required force F for a given oscillation amplitude be- HIDDEN FORCE/EXTERNAL FORCE Fh /F comes tiny in comparison with the oppositely directed spring force Fh. As the limit ω = ∞ is approached, the FIG. 1: (a) Showing external and hidden forces, F and Fh, effective mass tends to M because the (inertial) force F for the case of a rolling wheel. (b) shows a normalized plot of required for a given oscillation amplitude becomes very the effective mass as a function of the hidden force. (c) shows large in comparison with the spring force Fh. F and Fh for a prototype of a system exhibiting negative effective mass: a simple harmonic oscillator. Membrane-based acoustic metamaterial the same magnitude, i.e. x¨ < F/M. By introducing the Acoustic metamaterials, consisting of arrays of res- effective mass as so defined, it is thus possible to obtain onators, naturally fit into this hidden-force picture. the correct acceleration from one simple equation. Take the example of a 1D membrane-based acoustic In general, if an external force F is applied to a mass metamaterial,[5, 8, 9, 11, 23, 33, 36, 37] which supports M, then, owing to the specific mechanism involved, a wave propagation down its length, as shown schemati- hidden force Fh may also act on the mass (assumed here cally in Fig. 2(a). It consists of a cylindrical air-filled to be collinear with F ). So the acceleration becomes tube containing taut membranes at regular intervals. x¨ = (F + Fh)/M, and the effective mass is given by Consider a particular unit cell. Its center of mass M is subject to two kinds of forces: the applied force F = S∆p M from the two adjacent cells, where S and ∆p are the Meff = . (2) 1+ Fh/F cross-sectional area of the tube and the pressure differ- ence across the unit cell. The hidden force is Fh = −kmξ, A plot of Meff /M vs Fh/F is shown in Fig. 1(b). No- where km and ξ are the membrane spring constant[8] tably, as Fh approaches −F , Meff becomes infinitely and the displacement of the unit-cell center of mass.[51] large. Also, Meff becomes negative when Fh < −F . The equations governing oscillatory motion are the same A prototype of a system exhibiting negative Meff is a as those governing the system of a mass connected by simple harmonic oscillator consisting of a mass M at- a spring to a rigid wall that we just treated: Mξ¨ = tached to a rigid wall by a spring, as shown in Fig. S∆p + Fh, where M = ρ0SD + Mm is the mass of a unit 1(c): the hidden-force picture starts by ignoring the cell. Here, ρ0 is the density of air, D is the unit-cell length presence of the spring and regarding the system as a and Mm is the mass of the membrane. For sinusoidal ex- 2 free mass subject to a hidden force Fh = −kx, where citations, F = S∆p = Mξ¨ + kmξ = ξ¨(M − km/ω ), so ¨ 2 2 k is the spring constant and x the displacement. In ξ = F/[M(1 − ω0/ω )], where ω0 = km/M. the case when the external driving force is sinusoidal According to Eq. (1), the unit-cellp effective mass is 2 2 at angular frequency ω, F = F0 exp(−iωt), the accel- Meff = M(1 − ω0/ω ), which has exactly the same form eration can be calculated from the equation of motion, as Meff for a simple harmonic oscillator. This treatment, Mx¨ = F0 exp(−iωt)+ Fh. Using the harmonic expres- based on lumped-elements, obviously ignores vibrational sion x = x0 exp(−iωt), we then obtain the hidden force resonances of the membranes higher than the fundamen- 2 2 2 Fh = −kx = −Fω0/(ω0 − ω ), where ω0 = k/M is the tal mode.[38] p 3 p p (a) j ξj j+1 x D UNIT CELL j 0 5 ω/ω (b) (c) (d) 4 3 2 1 0 NORMALIZED FREQUENCY NORMALIZED FREQUENCY -10 -8 -6 -4 -2 0 2 0 1 2 3 4 5 0 1 2 3 4 5 ρ / ρ v 1/2 1/2 DENSITY eff ' PHASE VELOCITY | p|/(B0/ρ') WAVE NUMBER |q| /(ω0(ρ'/B0) ) FIG.

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