arXiv:1509.04094v7 [physics.class-ph] 21 Jan 2016 r ere ffedmfrmcaia oin Systems motion. mechanical ex- for to freedom coupled of trusses mass-and-spring degrees rigid on tra light based sys- with as combined of well models as examples tubes provide Helmholtz- in on then resonators based modulus We effective an exhibiting springs tems elucidated. and also masses with is metamaterial combined a membranes of on example based An metamaterials). latter solid acoustic (3D) (the three-dimensional models model membranes mass-and-spring to on elements on providing as based well as mass tubes effective in an exhibiting either tems on oppositely cell. and unit equally the of acting side produce forces only of that displacement pairs or volume of hid- sources time-dependent involve den in resulting modulus, latter, unit elastic The effective the an metamaterial. on acoustic force the net of an a time-dependent cell provide in local, that and resulting involve forces forces density, former, non-apparent hidden or The mass on volume. effective deter- based of unambiguous is sources their picture hidden allows This time same mination. understanding, the precise yet at intuitive and their allows metamateri- that acoustic of als for moduli account and to densities effective picture Here the physical fundamental system. a given present a confusion we to to parameters lead effective assigning can on in This than concepts principles. physical engineering-based to universal has on features, more exotic relied most date such their in of un- one parameters The materials,[1–34] constitutive ways negative materials. in of occurring sound derstanding naturally of in propagation available the not manipulate to signed ∗ e-mail:[email protected] efis lutaeorapoc iheape fsys- of examples with approach our illustrate first We de- structures man-made are metamaterials Acoustic 2 iiino ple hsc,Fclyo niern,Hokka Engineering, of Faculty Physics, Applied of Division rgno eaieDniyadMdlsi cutcMetamate Acoustic in Modulus and Density Negative of Origin n viscmo iflsivle ndtriigteeffec the d determining and in involved understanding pitfalls conceptual m common wave-prop for avoids for the tool and to freedom powerful angles of right a at degrees provides move extra that masses springs to to displ and coupled coupling of masses trusses expanders from rigid hidden arise light termed can aptly forces sources—more hidden hidden con analogous case the which explain metamat also in We Helmholtz-resonator-based cavity. resonator a the for from source instance, hidden for The metamaterials, resonators acoustic Helmholtz membrane-based for membranes, ansatz for on this based demonstrate elements We with respective absence. als to, their antiphase in in obtained operates negativ be and is term than, modulus the larger or introduce density is effective ume We the volume: metamaterials. of acoustic source of moduli and hspprpoie eiwadfnaetlpyia inter physical fundamental and review a provides paper This 1 nttt fPyisadApidPyis osiUniversit Yonsei Physics, Applied and Physics of Institute a .Lee H. Sam ∗ 1 n lvrB Wright B. Oliver and rxmtdb pigculdmse,adas new a springs. masses also includes and also ap- and that metamaterial masses, metamaterial membrane-based spring-coupled solid-matrix by membrane- a proximated a metamaterial, ef- propagation: based of unidirectional metamateri- concept involving acoustic the different als several illustrate for then density examples We fective simple of mechanics. means by from systems explained first in is mass This approach elements.[35] effective mechanical understand non-apparent contain to that order in force” behav- discussed. parameter also includ- double-negative are of modulus, ior, possibility and the density ing effective both exhibiting finertia of mle hnta xetdfrannrttn asof mass non-rotating a for acceleration expected The that rim. than wheel smaller the on force frictional en h idnforce hidden the define t etr h plcto fNwo’ asalw one allows laws A acceleration: Newton’s the of 1(a). derive application to The Fig. in center. shown its as surface, force horizontal horizontal flat, a on iig nti case, this in giving, ffciems fwelcnb endas defined be can wheel of mass effective nti eto eitouetecneto “hidden of concept the introduce we section this In osdrawelo mass of wheel a Consider FETV ESTE:HDE FORCES HIDDEN DENSITIES: EFFECTIVE y h oc rvlm hneta would that change volume or force the ly, stefrefo h ebaetension. membrane the from force the is hntehde oc rsuc fvol- of source or force hidden the when e d nvriy apr 6-68 Japan 060-8628, Sapporo University, ido et o uems-n-pigsystems, mass-and-spring pure for cepts pig n ass h idnforce hidden The masses. and springs , ieprmtr fsc materials. such of parameters tive oeetbihdaosi metamateri- acoustic established some gto ieto.Ti vrl picture overall This direction. agation caia oinsc stecs of case the as such motion echanical ra steetaarvlm injected volume air extra the is erial sg fnwaosi metamaterials, acoustic new of esign cmn nti aecnaiefrom arise case—can this in acement I nlg fhde oc n hidden and force hidden of inology rtto o h ffciedensities effective the for pretation xdisd te ass whereas masses, other inside fixed oln nthe in rolling ipemcaia systems mechanical Simple ,Sol1079 Korea 120-749, Seoul y, F 2 sapidt h xso h he at wheel the of axis the to applied is M M F eff eff h ob h backwardly-directed the be to x ¨ x = = = M ieto ihu slipping without direction F/ M F/ radius , x, ¨ 1+ (1 [ M 1+ (1 rials I/MR R I/MR n moment and 2 ) e us Let . 2 ]) The . x ¨ (1) is 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 driv- ing 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. 2: (a) Schematic diagram of a 1D membrane-based acoustic metamaterial, made up of periodically-spaced membranes in a tube containing air. Normalized plots (b), (c), (d) show the frequency as a function of the density, phase velocity and wave number, respectively. In (c) and (d) the solid and dashed lines refer to the cases of real and imaginary values for the phase velocity and wave number, respectively.

For each unit cell indexed by j, the effective mass can cells. The non-equilibrium component of pressure in this be defined as volume, pj ,[52] is related in general to the deviation ∆Vj (which we define to refer to this same volume) from the Fj − Fj+1 S(pj − pj+1) Meff = = , (4) equilibrium volume V as follows: ξ¨j ξ¨j ∆Vj S(ξj − ξj−1) where Fj is the force acting on the left-hand side of the p = −B = −B , (6) j eff V eff SD unit cell j, whereas −Fj+1 is that acting on the right- hand side. (Alternatively, if we define fj as the force on where Beff is the effective modulus of the system.[53] In the left-hand side and +fj+1 as the force on the right, we the present geometry, only the (adiabatic) bulk modu- ¨ obtain Meff = (fj + fj+1)/ξj , which demonstrates more lus B0 of the air in the tube affects the pressure-volume clearly that F = Fj − Fj+1 = fj + fj+1 is the net force relation, on unit cell j. However, in this paper we adopt the Fj notation because it is more convenient for the analysis ∆Vj p = −B0 , (7) of mass-and-spring models.) The effective density can be j V defined by ρeff = Meff /V , where V = SD is the volume of the unit cell, which is a useful concept for the case and so Beff = B0. Equation (6) represents continuity D ≪ λ, where λ is the acoustic wavelength. It is this combined with the equation of state, and can be rewritten limit that applies to metamaterials, as opposed to the in the form case for phononic crystals for which D ∼ λ. Therefore −1 1 ∆Vj 1 ξj − ξj−1 2 Beff = − = − . (8) ′ ω pj V pj D ρ = ρ 1 − 0 , (5) eff  ω2  In spite of the introduction of membranes in the tube where the average density ρ′ is given by ρ′ = M/V . with spring-like properties, they evidently do not change In order to derive the system’s effective bulk modulus the effective modulus B0 of this acoustic metamaterial. Beff , consider the volume V between two adjacent mem- One can also access the acoustic dispersion relation as branes comprising parts of both the jth and (j−1)th unit follows. In the metamaterial limit, i.e. D ≪ λ, the time 4 derivative of Eq. (6) leads, for an arbitrary point in the shown in Fig. 3(b). The unit cell can be modelled by tube, to a core of mass Mc surrounded by a hollow mass M, with an internal connection between the masses made ∂u p˙ = −B0 . (9) up of collinear springs of constant k/2. Springs of con- ∂x stant k0 connect the masses M externally. The external springs transmit applied forces F to a particular unit where u = ξ˙ is the acoustic particle velocity and x is cell, where F = F0 exp(−iωt), whereas the hidden force the distance along the tube. From the definition ρ = eff F = −k(ξ − η) acts on mass M through the inter- M /(AD) and Eq. (4), we also have h eff nal springs, where ξ and η are the displacements of the ∂p masses M and Mc, respectively. Equations of motion for − = ρ u.˙ (10) ¨ ∂x eff these parameters are Mξ + k(ξ − η)= F0 exp(−iωt) and Mcη¨ + k(η − ξ)=0 respectively. This clearly shows the The wave equation is easily obtained by combining Eqs. origin of the hidden force as the vibration of the mass (9) and (10): Mc. Solving the coupled equations for sinusoidal motion, we obtain the effective mass of a unit cell in the form ∂2p ρeff p¨ = B0 . (11) 2 2 ∂x2 ¨ Mc ω1 − ω0 Meff = F/ξ = M + 2 2 = M 1+ 2 2 , 1 − ω /ω0  ω0 − ω  Substituting p = p0 exp[i(qx − ωt)], where q is the wave 2 2 (13) number, leads to the dispersion relation q = ω ρeff /B0. where ω1 = ω0 1+ Mc/M and ω0 = k/Mc (distinct For the example in question, from ω0 in thep above membrane problem).p A plot of

′ 2 1/2 the normalized value of Meff (i.e. normalized effective ρeff ρ ω0 density) vs frequency is shown in Fig. 3(c), for the case q = ω = 1 − 2 . (12) r B0 rB0  ω  ω1/ω0 = 2. One can see that as the frequency increases through ω = ω0, there is a transition from infinitely pos- A plot of the frequency dependence of the effective den- itive to infinitely negative Meff . Zero Meff occurs at sity, and also plots for |vp|, where vp is the phase velocity ω = ω1. (v = ω/q) and |q|, where q is the wave number, are p The effective modulus can be derived in a similar way shown on normalized scales in Fig. 2(b)-(d). The phase to that for the membrane-based metamaterial. By anal- velocity becomes infinite when the effective density van- ogy with Eq. (6), ishes at ω = ω0. This situation is useful for applica- tions in extraordinary acoustic transmission.[21, 39] At 2 ξj − ξj−1 F = −k (ξ − ξ −1)= −D E , (14) lower frequencies, where ρeff is negative, the waves are j eff j j eff D damped [Im(q) > 0, dashed lines in (c) and (d)], whereas at higher frequencies, where the ρeff is positive, they are where keff is an effective spring constant and Eeff the undamped [Im(q)=0, solid lines in (c) and (d)]. The effective Young’s modulus of the system. Alternatively, frequency region of damping, sometimes referred to as a metamaterial band gap, is useful in practice for applica- −1 D Eeff = − (ξj − ξj−1). (15) tions to the absorption of noise. Fj One of the most curious features of these results is how For the present 1D mechanical model, the compressive membranes with spring-like properties only contribute to force F is provided by the spring of constant k0, i.e. F = the effective density, and do not influence effective mod- j j −k0(ξ − ξ −1), so k = k0 and E = k0/D. Clearly, ulus. We shall see later that this can also be explained j j eff eff the effective modulus of this structure not affected by by the fact that hidden sources of volume rather than the internal structure of the mass. As in the case of the hidden forces influence the effective modulus. membrane-based metamaterial, the effective modulus is positive and frequency independent. Mass-and-spring analogy for a solid-matrix acoustic The acoustic dispersion relation can be derived by ap- metamaterial plication of Newton’s second law for sinusoidal variations:

2 2 ∂ ξj Now consider a metamaterial composed of a cubic ar- M ξ¨ = k0(ξ −1 − ξ ) − k0(ξ − ξ +1) ≈ k0D , eff j j j j j ∂x2 ray of mechanical resonators embedded in a compliant (16) solid matrix, as illustrated by the unit cell in Fig. 3(a). where we have assumed, as before, that D ≪ λ. Making Such a system consisting of rubber-coated lead balls was 3 use of ρeff = Meff /D and Eeff = k0/D, we may write reported to exhibit strong sonic transmission loss due to negative density.[1] Consider a 1D mass-and-spring ∂2ξ ρ ξ¨ = E , (17) analog of such a matrix in the form of a chain,[40] as eff eff ∂x2 5

(a) (b) ξj M M j j+1

k/2 k/2 k 0 M M c c F F j j+1 ηj

D x UNIT CELL UNIT CELL j

0 5 (c) (d) (e)

ω/ω 4 ω1/ω0=2

3

2

1

0 -10 -5 0 5 10 0 1 2 3 4 5 0 1 2 3 4 5

NORMALIZED FREQUENCY NORMALIZED FREQUENCY

DENSITY ρeff / (ρ'M/(M+Mc)) PHASE VELOCITY | vp|/(k0/ρ'D) WAVE NUMBER |q|/ (ω0ρ'D/k0)

FIG. 3: (a) Unit cell of a solid-matrix acoustic metamaterial, modelled by a spherical core of mass Mc surrounded by a separate, concentric rigid shell of mass M. (b) mass-and-spring analogy of a 1D chain of unit cells. Fj is the compressional force in the spring of the left-hand side of unit cell j. Normalized plots (c), (d), (e) show the frequency as a function of the density, phase velocity and wave number, respectively. In (d) and (e) the solid and dashed lines refer to the cases of real and imaginary values, respectively, for the phase velocity and wave number. Normalized parameter ω1/ω0 = 2 is chosen for these plots. where spring in each unit cell attached to the membrane, as 2 2 shown in Fig. 4(a). This provides more degrees of free- ′ M ω1 − ω0 ρeff = ρ 1+ 2 2 . (18) dom than even the previous example of the mass-and- M + Mc  ω0 − ω  spring analogy of the solid-matrix acoustic metamate- ′ 3 rial. The applied force F = ∆pS on a unit cell from The constant ρ = (M + Mc)/D is the average density of the matrix. Substituting ξ = ξ0 exp[i(qx − ωt)] leads the pressure gradient acts on the part of the unit-cell mass M = ρ0SD + M made up as before of the sum to the dispersion relation q = ω ρeff /Eeff , or mem p of the masses of the air and the membrane. The hid- ′ 2 2 1/2 den force F = −k(ξ − η) − k ξ is the combination of ω0 ρ D ω1 − ω0 h m q = ω 1+ 2 2 . (19) the forces from the spring −k(ξ − η), where ξ and η are ω1 r k0  ω0 − ω  the displacements of the masses M and Mc, respectively, The frequency dependence of the wave number together and the force −kmξ from the membrane spring constant. with those of the phase velocity and effective density are The equations of motion for sinusoidal excitation of a plotted on normalized scales in Figs. 3(d) and (e) for the single unit cell are Mξ¨+ k(ξ − η)+ kmξ = F0 exp(−iωt) case ω1/ω0 = 2. In the frequency range ω0 <ω<ω1, and Mcη¨ + k(η − ξ)=0. The equations of motion are ρeff is negative and the waves are damped. Because this slightly different from the previous example of the mass- system has more degrees of freedom than the membrane and-spring analogy of the solid-state matrix, but reduce system previously discussed, it has a more complicated to the same form when km = 0. By elimination of the dispersion relation. variable η one can derive Meff in the following form:

ω2 − ω2 ω2 Membrane-based metamaterial including masses and ¨ 1 0 2 Meff = F/ξ = M 1+ 2 2 − 2 , (20) springs  ω0 − ω ω 

As a final example of the hidden-force approach, con- where ω0 = k/Mc, ω1 = ω0 1+ Mc/M and ω2 = sider a more general case of the previously analyzed km/M. Forp the special case inp which the membrane membrane system obtained by including a mass and a pspring constant km can be neglected, we may set ω2 =0. 6

x k M c p p (a) j j+1

ξj ηj

D UNIT CELL j

0 5 (b) (c) (d) ω/ω =2 4 ω1/ω0 ω2/ω0=1 3

2

1

0 -10 -5 0 5 10 0 1 2 3 4 5 0 1 2 3 4 5

NORMALIZED FREQUENCY NORMALIZED FREQUENCY 1/2 1/2 DENSITY ρeff/ ρ' PHASE VELOCITY | vp|/(B0/ρ') WAVE NUMBER |q| /(ω0(ρ'/B0) )

FIG. 4: (a) Schematic diagram of a 1D acoustic metamaterial based on membranes, springs and masses, made up of periodically- spaced unit cells in a tube containing air. Normalized plots (b), (c), (d) show the frequency as a function of the density, phase velocity and wave number, respectively. In (c) and (d) the solid and dashed lines refer to the cases of real and imaginary values, respectively, for the phase velocity and wave number. Normalized parameters ω1/ω0 = 2 and ω2/ω0 = 1 are chosen for these plots.

This results in the simpler form zero. In both cases the model reduces to an elementary mass-and-spring chain model, which, in the metamate- 2 2 ω1 − ω0 rial limit D ≪ λ, shows a constant sound velocity (i.e no Meff = M 1+ 2 2 , (21)  ω0 − ω  dispersion) and positive and constant effective mass. which is precisely the same as Eq. (13). In this special case, the present system is an exact analog of the mass- and-spring model previously considered. The effective To conclude this discussion of effective densities, we have proposed the hidden-force picture to explain why modulus is still given by B0 according Eq. (8), because effective masses and densities are significantly different the pressure-volume relation, pj = −B0∆Vj /V , is not affected by the addition of the mass and spring. from their non-resonant average values. In this pic- ture the effective mass Meff is obtained in terms of Since ρeff = Meff /SD and q = ω ρeff /B0, the dis- the hidden-force to applied-force ratio F /F as M = persion relation is given by p h eff M/(1 + Fh/F ). The effective mass becomes negative ′ 2 2 2 1/2 when Fh/F < −1. We demonstrated that this picture ρ ω1 − ω0 ω2 q = ω 1+ 2 2 − 2 , , (22) allows one to obtain effective masses of the unit cell and rB0  ω − ω ω  0 thereby the effective densities in a quick and easy manner where ρ′ = M/SD. The frequency spectra for the ef- for two established metamaterials that exhibit negative fective density and phase velocity, together with the dis- density as well as for a new membrane-based metamate- persion relation, are shown in Fig. 4(b)-(d) for the case rial including masses and springs. ω1/ω0 =2 and ω2/ω0 =1. For this choice of parameters, two distinct frequency bands with negative density are evident. In the next section we shall discuss with the aid of This mechanical model also provides the expected re- several examples a simple picture of how frequency- sults in the limit when either the internal-spring con- dependent effective moduli arise in acoustic metamate- stants k/2 go to infinity or the internal mass Mc is set to rials. 7

-1

0

EFFECTIVE MODULI: HIDDEN SOURCES OR B 10

/ (a) (b)

-1 HIDDEN EXPANDERS eff

B 5 Here we introduce the concept of “hidden sources” of ∆V volume in order to understand effective modulus. This 0 approach is first explained by considering the vibrational ∆Vh response of a piston connected to a chamber contain- -5 ing either a Helmholtz resonator or a side hole. We -10 then consider an acoustic metamaterial based on unidi- -4 -2 0 2 4

COMPRESSIBILITY RATIO COMPRESSIBILITY rectional propagation in a tube lined with Helmholtz res- VOLUME RATIO ∆Vh /∆V onators. We go on to treat the case of a tube containing a combination of Helmholtz resonators and membranes—a generic case of a double-negative acoustic metamate- FIG. 5: (a) Showing the basic concept of a hidden source, in which volume ∆Vh of air is introduced into a chamber while rial—followed by a similar mass-and-spring analogy. For depressing a piston to change the chamber volume by ∆V . mass-and-spring models we extend the concept of hidden Here the changes as shown correspond to negative values of sources, more appropriately termed “hidden expanders” −1 ∆V and ∆Vh. (b) shows a normalized plot of Beff for this of displacement in this case, by the introduction of light system as a function of the hidden source of volume ∆Vh. rigid trusses coupled to extra degrees of freedom for mechanical motion, and demonstrate an example of a double-negative system based on this concept. We con- behavior is analogous to that observed in the case of hid- clude by summarizing our approach and discussing how den forces. (The curve in Fig. 5(b) is identical in shape to tell at first glance what produces effective density and to that in Fig. 1(c).) Here one can see that a hidden what produces effective modulus. source is represented by an introduced volume of air.

Concept of a hidden source Single Helmholtz resonator Consider a chamber and piston containing air as well as a point where air can be introduced or removed. This A prototype of a system that can exhibit negative Beff point, not apparent to the operator moving the piston, is a piston and chamber with an attached Helmholtz res- constitutes the origin of what we call a hidden source onator of volume VH , as shown in Fig. 6(a). The neck of or sink of volume. A schematic diagram of this setup is the resonator is assumed to have area SH and effective ′ shown in Fig. 5(a), where we represent a small change length[38] l . This system can be regarded as a piston in which the piston is displaced to perturb the chamber with a chamber of equilibrium volume V , with pressure volume by ∆V at the same time as a volume ∆Vh of variations p subject to a hidden source of volume that air is introduced (measured at the equilibrium pressure is governed by the pressure variation pH = B0SH η/VH before the change). The change in pressure p inside the inside the Helmholtz resonator, where η is the displace- chamber is given by ment of the air plug of the resonator neck in the out- ward direction with respect to volume V . In the case ∆V + ∆Vh in which the external driving force on the piston is sinu- p = −B0 . (23) V soidal at angular frequency ω, i.e. p = p0 exp(−iωt) and η = η0 exp(−iωt), the acceleration η¨ can be calculated The effective bulk modulus Beff only depends on the ′ from the equation of motion, ρ0S l η¨ = S (p − p ), observable volume change ∆V , so, in accord with the H H H yielding definition of Eq. (6), p = −Beff ∆V/V defines Beff for this system: VH p0 1 η0 = 2 2 , (25) B0S 1 − ω /ω ∆Vh H 0 B = B0 1+ . (24) eff  ∆V  ′ where, for this case, ω0 = B0SH /(VH ρ0l ) is the clas- By analogy to the definition of Meff in Eq. (2), Beff sical Helmholtz resonator frequency.[38]p This treatment depends on the ratio of the hidden source ∆Vh to a more does not impose a limit on the ratio VH /V , although, easily observable quantity, here the change in chamber usually, VH < V . The volume of the neck is, however, as- −1 −1 volume ∆V . A plot of Beff /B0 vs ∆Vh/∆V is shown sumed to be much smaller than V for the lumped-element in Fig. 5(b). Notably, as ∆Vh approaches −∆V , Beff be- treatment of the motion of the air inside it to apply. −1 comes zero (and compressibility Beff becomes infinite). Higher-order resonances of the system are neglected in Also, Beff becomes negative when ∆Vh < −∆V . This this approach. Knowing the displacement η allows us to 8

5 calculate the hidden source ∆V : 0 h (a) (b)

ω/ω ω1/ω0=2 VH p 1 4 ∆Vh = SH η = 2 2 . (26) B0 1 − ω /ω 0 3

Using the harmonic expression ∆V = ∆V0 exp(−iωt), V VH 2 Eq. (26), and the definition of Beff in Eq. (24), we −1 obtain, for the compressibility Beff = −∆V0/(Vp0), 1

2 2 NORMALIZED FREQUENCY −1 −1 ω1 − ω0 0 Beff = B0 1+ 2 2 , (27) -10 -5 0 5 10  ω0 − ω  -1 -1 COMPRESSIBILITY Beff / B0 where ω1 = ω0 (1 + VH /V ). The form of this equation is identical to thatp for Meff for the mass-and-spring anal- FIG. 6: (a) shows a prototype of a system that can exhibit ogy of the solid-matrix metamaterial and for the mass- negative Beff , consisting of a piston and chamber with an and-spring membrane-based metamaterial in Eqs. (13) attached Helmholtz resonator of volume VH . (b) shows a nor- − and (21), respectively. A plot of frequency vs B 1 is malized plot of frequency as a function of the compressibility eff −1 shown in Fig. 6(b) on a normalized scale for the case Beff for this system for the case ω1/ω0 = 2. ω1/ω0 = 2. If the sinusoidal pressure amplitude p0 is assumed be the imposed quantity, the volume amplitude 5 −1 2 V0 is determined by B . The volume will oscillate with (a) (b)

eff ω/ω large amplitude for ω approaching ω0 from below because 4 −1 Beff becomes very big. In contrast, at ω = ω1 when −1 3 Beff =0 the system becomes infinitely rigid and the vol- ume amplitude becomes zero. The abrupt shift of the V 2 phase of the volume variations by π with respect to the driving pressure is also predicted as the frequency passes 1

through the resonance ω0. Since the sign of Beff changes NORMALIZED FREQUENCY 0 at ω = ω0, the region of negative Beff between ω0 and -10 -5 0 5 10 ω1 exhibits wave damping. Negative Beff , a consequence -1 -1 COMPRESSIBILITY B eff / B0 of ∆Vh > −∆V , thus implies in the case of sinusoidal excitation that the hidden source ∆Vh is not only in an- tiphase with but also has a magnitude larger than that of FIG. 7: (a) shows a prototype of a system that can exhibit −1 ∆V . In the limit ω =0, Beff = B0(1 + VH /V ) , which negative Beff , consisting of a piston and chamber with a side 0 hole. (b) shows a normalized plot of frequency as a function is reduced from the expected value B owing to the in- − of the compressibility B 1 for this system. crease in the total effective volume from V to V + VH . eff In the limit ω = ∞, Beff = B0 because the flow of air to and fro from the Helmholtz resonator is effectively frozen ′ owing to the inertia of the air plug in the resonator neck. where ω2 = B0SH /(Vρ0l ). The resonance frequency ω2 The case of a side hole instead of a Helmholtz res- depends on V instead of VH in this case. Equation (30) onator, as shown in Fig. 7(a), is also one of practical is the exact analog of Eq. (5) for the case of the effective interest in acoustic metamaterial design.[9, 11, 41] For a mass of a membrane-based metamaterial. A normalized −1 side hole one may set ω0 =0 because the Helmholtz res- plot of the frequency dependence of Beff is shown in Fig. onator stiffness (i.e. spring constant), kH = SH B0/VH , 7(b). Negative Beff is exhibited up to ω = ω2. In the vanishes. Equation (25) is modified to limit ω = 0, Beff = 0, as expected since the oscillating air is completely free to escape from the chamber in this p0 case. In the limit ω = ∞, Beff = B0 because the flow η0 = 2 ′ , (28) ω ρ0l of air to and fro from the side hole is effectively frozen owing to the inertia of the air plug (as was the case with giving the Helmholtz resonator). p ∆Vh = SH η = − 2 ′ (29) ω ρ0l Helmholtz-resonator-based acoustic metamaterial and

2 − − ω We are now in a position to consider the example B 1 = B 1 1 − 2 , (30) eff 0  ω2  of a 1D acoustic metamaterial based on an array of 9

Helmholtz-resonators spaced at regular intervals in a air- The frequency dependences of the compressibility and filled tube,[3, 6, 9–11, 15, 19, 19, 22, 30, 31, 41–48] as the phase velocity, as well as the dispersion relation, are shown schematically in Fig. 8(a). The unit cell con- shown by normalized plots in Figs. 8(b)-(d). In the sists of a section of cylindrical tube containing a single region of negative modulus the propagation is damped. Helmholtz resonator attached to the tube wall. This behavior is analogous to that noted for negative ef- We first note that the effective density is equal to the fective mass. density of air, ρeff = ρ0, as is evident from the pre- The equivalent result for the dispersion relation for an viously treated case of a tube containing membranes. array of side holes instead of Helmholtz resonators is (As there are no membranes here, the previously treated 2 1/2 0 case, except with km = 0, applies.) Here the unit cell ρ ω2 q = ω 1 − 2 , (37) volume change ∆Vj can be considered to depend on rB0  ω  the non-equilibrium particle displacements ξj and ξj+1 This has exactly the same form as the dispersion for the at the two unit cell boundaries, which act like pistons: membrane-based metamaterial [Eq. (12)]. ∆Vj = S(ξj+1 − ξj ). (The definition of ξj here is distinct Just as one can ask the question why the addition of from that used for the membrane-metamaterial. Here membranes produce no added rigidity for the passage it refers to the acoustic displacement at the left-hand of acoustic waves, one can also ask why Helmholtz res- boundary of the unit cell, rather than that of the center onators add no effective mass. In fact effective mass is of mass of the cell.) However, in contrast to the situation only added, according to Eqs. (2) and (4), if there is for the membrane-based metamaterial [Eq. (7)], the av- an extra frequency-dependent force acting on a unit cell. erage pressure change in the unit cell now also contains Since there are no such hidden forces in the system of a contribution from the hidden source: Helmholtz resonators but only hidden sources, these res- ∆Vj + ∆Vhj onators do not contribute to the effective mass. Rather, p = −B0 , (31) j V the introduction of the Helmholtz resonators leads to an extra contribution −B0∆V /V to the pressure [Eq. (32)], which, in the present case, can be expressed as h whose effect on either side of a unit cell is the addition B0 of a pair of equal and opposite forces. This pair of forces p = − [S(ξ +1 − ξ )+ η S ], (32) j SD j j j H obviously does not contribute to an imbalance in the net force on a unit cell, as is required for the addition of where η is the outward displacement of the Helmholtz- j effective mass [Eq. (4)], but instead leads to a net com- resonator air plug. From Eq. (24), or, equivalently, using pression or expansion of the unit cell, i.e. resulting in a the definition contribution to the elastic modulus. S(ξ +1 − ξ ) p = −B j j , (33) j eff SD Acoustic metamaterial with both Helmholtz we obtain resonators and membranes

SH ηj Beff = B0 1+ . (34)  S(ξj+1 − ξj ) We now consider a 1D acoustic metamaterial based on Helmholtz-resonators combined with membranes in Introducing sinusoidally-varying quantities as before, an air-filled tube,[9, 11, 19, 41] a prototype system for making use of Eqs. (25), (31) and (34), and again assum- −1 double-negative behavior, as shown in Fig. 9(a). The ing that D ≪ λ, one again obtains Eq. (27) for Beff . At membranes and Helmholtz resonators are positioned al- the resonance frequency ω = ω1, the effective modulus ternately inside an air-filled tube. Consider the unit cell and the phase velocity become infinite. By analogy with j sketched in the dashed line in Fig. 9(a) that ends just zero-density metamaterials, this situation is useful for ap- after a membrane. This unit cell is chosen so it can ap- plications in extraordinary acoustic transmission.[30, 31] ply to both the analysis of the membranes and Helmholtz To derive the dispersion relation in the limit D ≪ λ, resonators. The equation we make use of p˙ = −Beff ∂u/∂x and −∂p/∂x = ρ0u˙ by analogy with Eqs. (9) and (10) to derive the wave (pj − pj+1)S − kmξj = Mu˙j, (38) equation:[54] valid for the previously-treated case of membranes only, ∂2p and ρ0p¨ = Beff 2 , (35) ∂x B0 pj = − [S(ξj+1 − ξj )+ ηj SH ], (39) 2 2 SD yielding q = ω ρ0/Beff . For this example, i.e. Eq. (32), derived for Helmholtz resonators only, still 2 2 1/2 ρ0 ω1 − ω0 apply, where the mass M again refers to the lumped mo- q = ω 1+ 2 2 . (36) rB0  ω0 − ω  tion of the unit cell. The change in the choice of unit cell 10

η VH j

ξ (a) ξj j+1 x

D UNIT CELL j

0 5

ω/ω (b) (c) (d) 4 ω1/ω0=2

3

2

1

0 -10 -5 0 5 10 0 1 2 3 4 5 0 1 2 3 4 5

NORMALIZED FREQUENCY NORMALIZED FREQUENCY -1 -1 COMPRESSIBILITY Beff /B0 PHASE VELOCITY | vp |/v0 WAVE NUMBER |q| v 0/ω0

FIG. 8: (a) Schematic diagram of a 1D Helmholtz-resonator-based acoustic metamaterial, showing periodically-spaced res- onators in a tube. Normalized plots (b), (c), (d) for this system show the frequency as a function of the density, phase velocity and wave number, respectively, for the case ω1/ω0 = 2. In (c) and (d) the solid and dashed lines refer to the cases of real and imaginary values, respectively, for the phase velocity and wave number. In the horizontal axes we use velocity v0 = pB0/ρ0.

affects the definition of the quantity ξj in Eq. (38), which leads to the dispersion relation q = ω ρeff /Eeff , i.e., now refers to the acoustic displacement at the left-hand ′ 2 1/2 2 p 2 1/2 side of the unit cell j rather than to that of the cell cen- ρ ω0m ω1H − ω0H q = ω 1 − 2 1+ 2 2 . (43) ter of mass. However, the difference in these definitions, rB0  ω   ω0H − ω  only affecting distances ∼D/2, does not lead to a change The frequency dependence of the phase velocity, as well in the final results for effective physical properties and as the dispersion relation, are shown by normalized plots the dispersion relation. The above equations separately Fig. 9(d), (e) for the case ω1H /ω0H =2 and ω0m/ω0H = determine the effective density and modulus according to 3, for which a region of double-negativity exists from Eqs. (4) and (8), so we arrive at expressions for ρeff and −1 1 < ω/ω0H < 2. In this region the phase velocity is oppo- Beff in exactly the same form as those in Eqs. (5) and site to the group velocity. In 2D and 3D such materials (27), respectively: are expected to be important in focusing applications. 2 The dispersion relation for the case of side holes instead ′ ω0m ρeff = ρ 1 − 2 , (40) of Helmholtz resonators can be easily found by the use of  ω  −1 Eq. (30) for Beff instead of Eq. (27). We now turn to the case of a mass-and-spring model exhibiting an effec- ω2 − ω2 −1 −1 1H 0H tive modulus or exhibiting both an effective density and Beff = B0 1+ 2 2 , (41)  ω0H − ω  modulus. where we have added the labels m for membrane and H for Helmholtz resonator to remove the ambiguity in the ′ Mass-and-spring analogy for an acoustic definitions ω0m = km/M, ω0H = B0SH /(VH ρ0l ) metamaterial exhibiting negative modulus or and ω1H = ω0H p(1 + VH /V ). By analogyp with Eq. double-negative behavior (17), the wave equation,p ∂2ξ Consider a 1D model consisting of masses and springs ρ ξ¨ = B , (42) eff eff ∂x2 connected to light rigid hinged trusses coupled to extra 11

η VH j

p p (a) j j+1 x ξ j ξj+1

D UNIT CELL j

H

0 5 (b) (c) ω/ω ω Η/ω Η=2 4 1 0 ω0m/ω0Η=3 3

2

1

0 -10 -5 0 5 10 -10 -5 0 5 10 NORMALIZED FREQUENCY NORMALIZED FREQUENCY -1 -1 DENSITY ρeff/ ρ' COMPRESSIBILITY Beff /B0

H

0 5 (d) (e)

ω/ω 4

3

2

1

0 0 2 4 6 8 10 0 2 4 6 8 10

NORMALIZED FREQUENCY NORMALIZED FREQUENCY 1/2 1/2 PHASE VELOCITY | v p | /(B0/ρ') WAVE NUMBER |q| /(ω0H (ρ'/B0) )

FIG. 9: (a) Schematic diagram of a 1D acoustic metamaterial based on membranes and Helmholtz resonators, showing alternately-spaced elements in a tube. Normalized plots (b)-(e) for this system show the frequency as a function of the density, phase velocity and wave number, respectively, for the case ω1H /ω0H = 2 and ω0m/ω0H = 3, for which a region of double-negativity exists. In (d) and (e) the solid and dashed lines refer to the cases of real and imaginary values, respectively, for the phase velocity and wave number. degrees of freedom for mechanical motion, as shown in frictionless) hinged trusses ensure 1) that the same mag- Fig. 10(a). We assume all mechanical displacements are nitude of force is exerted on the springs either side, and much smaller than the truss lengths. This type of hinged 2) that the vertical displacements y of the masses m are truss system was previously proposed together with ex- exactly mirrored by the horizontal displacements of the tra springs to generate a negative modulus,[40] but we sides of the trusses attached to the springs. The presence have simplified the model to a convenient bare minimum of the two vertically-oriented straight trusses allows the here. Springs of constant 2k0 connect a mass M in the square truss system to be free to move horizontally. The unit cell to the truss systems. The square truss system, springs provide the applied force F = Fj − Fj+1 on a consisting of four members, is connected above and below particular unit cell j, where, from Eq. (4), to straight trusses, which in turn are connected to two masses m that are constrained (by rails) to move only Fj − Fj+1 Meff = . (44) in the vertical direction. These ideal (i.e. massless and ξ¨j 12

However, from the force transmission properties of the that shown in Fig. 7(b). So the model of Fig. 10(a) is the truss system, that ensure that the compressive forces in mechanical analog of an air-filled tube with periodically the springs on either side of it are equal [see Fig. 10(a)], it arranged side holes. At ω = 0 the effective modulus is clear that the acceleration of mass M is simply given by is zero because the truss system provides zero effective ξ¨j = (Fj − Fj+1)/M. Therefore, from Eq. (44), Meff = spring constant in this limit. In contrast to the mass- M for this model. and-spring model of Fig. 3(b), the springs in the model To find the effective modulus, first consider the force of Fig. 10(a) cannot support any tension or compression Fj on the left-hand side of unit cell j. This depends in their equilibrium position, i.e. the springs should work on the displacement µj−1 − yj−1 of the right-hand side in both tension and compression in the present case.[55] of the square truss of unit cell j − 1 as well as on the At ω = ∞ the effective modulus becomes equal to k0/D, displacement ξj of mass M in unit cell j, where yj and identical to that of the previously considered mass-and- −yj are the displacements of the top and bottom masses spring model, because in this limit the masses m do not m in unit cell j, respectively, and µj is the horizontal move and the square truss plays the role of a massless, displacement of the center of mass of the square portion rigid connector, resulting in two springs of constant 2k0 of the truss system: in series that are equivalent to a single spring of constant k0. In this limit the effective modulus is simply k0/D. Fj =2k0(µj−1 − yj−1 − ξj ). (45) The question arises of how to interpret this mechanical model in terms of the hidden-source picture. The analo- The compressional force Fj+1 in the spring to the right gous equation to Eq. (23) for the mass-and-spring model of mass M in unit cell j [see Fig. 10(a)] is given by is, by comparison with the definition of Eq. (14),

Fj+1 =2k0(ξj − µj − yj ). (46) k0 ξ − ξ −1 k0 (ξ − ξ −1) F = −D2 j j − D2 j j h , (52) j D D D D Decreasing the indices by 1 yields an alternative expres- sion for Fj : where k0/D is the modulus in the absence of the truss system and (ξj − ξj−1)h is an extra displacement that Fj =2k0(ξj−1 − µj−1 − yj−1). (47) we term a “hidden expansion”. By comparing Eq. (52) with Eqs. (45) and (50), one finds that (ξj − ξj−1)h = Comparing Eqs. (45) and (47), we obtain 2yj−1. In other words, the hidden expansion is precisely equal to the extra horizontal displacement introduced by ξj + ξj−1 µj = . (48) the truss system. The mass-and-spring analogy of the 2 hidden source concept is thus, quite naturally, a hidden The massless square truss system moves by this amount expander. to ensure equal displacements in the springs on either The acoustic dispersion relation can be derived by side of it and thus maintain the force balance on it. (Any analogy with the previously considered mass-and-spring massless system must by definition have a net zero force model. on it to avoid an infinite acceleration.) Eliminating µj Mξ¨ =2k0(µ −1 − y −1 − ξ ) − 2k0(ξ − µ − y ), (53) from Eq. (45) j j j j j j j Provided that D ≪ λ, this equation reduces to Fj = k0(ξj−1 − ξj − 2yj−1). (49) 2 2 ∂ ξ ∂y Mξ¨ = k0D +2k0D . (54) From the properties of the truss system, force Fj is trans- ∂x2 ∂x mitted to the masses m. For sinusoidal motion at angular 2 Compared to the previous case of Eq. (16), there is an frequency ω, Fj = −mω yj−1. This allows yj−1 to be ex- pressed as extra term in ∂y/∂x owing to the truss system. A differ- ential equation involving y¨ can be derived by noting that ξj − ξj−1 m(¨yj − y¨j−1)= Fj+1 − Fj , making use of Eqs. (45), (46) yj−1 = − 2 2 , (50) 2(1 − ω /ω2) and (48), assuming D ≪ λ, and then integrating once over the coordinate x: where ω2 = 2k0/m for this case. The definition of the ∂ξ my¨ = −k0D − 2k0y. (55) effective Young’sp modulus, Eq. (15), then leads to ∂x 2 −1 D ω2 Substituting parameters with temporal variations ac- Eeff = 1 − 2 . (51) k0  ω  cording to exp[i(qx−ωt)] as before leads to the dispersion 3 relation q = ω ρ/Eeff , where ρ = M/D : The effective modulus varies with frequency in exactly p 2 1/2 the same way as a tube containing an array of side holes ρD ω2 q = ω 1 − 2 , (56) [as in Eq. (30)]. The frequency variation is the same as r k0  ω  13

(a) m y j

µ -y µ -y 2k j-1 j-1 2k j j 0 0 M µ +y j j F F F F j j j+1 j+1

ξj ξj+1 x

y m j D UNIT CELL j y (b) j ξj M

2k 2k 0 0 k/2 k/2 M c F F F F j j ηj j+1 j+1

x m D

UNIT CELL j

FIG. 10: (a) Mass-and-spring model that can exhibit negative modulus, realised by periodically-arranged masses and springs connected to light rigid trusses coupled to extra degrees of freedom for mechanical motion. (b) Mass-and-spring model that can exhibit double-negative behavior. Fj and Fj+1 are the compressional forces in the springs on the left-hand side of and to the right of unit cell j, respectively. The ideal light rigid truss system containing two attached masses m is hinged in such a way as to ensure that the compressional (or tensional) forces in the springs on either side of it are equal. The masses m are constrained to move along fixed vertical rails. which has exactly the same form as Eqs. (12) and (37). We conclude this section by emphasizing that effective Let us now turn to a more general 1D mass-and-spring mass and modulus are not just theoretical constructs but model that can exhibit double-negative behavior, as il- also experimentally measurable quantities. According to lustrated in Fig. 10(b). We have combined the model of their definitions in Eqs. (4), (8) and (15), it suffices in Fig. 10(a) with that of Fig. 3(b). The analysis proceeds principle to put pressure, force or displacement sensors in exactly the same way as for these two cases: Meff and at the appropriate points inside the metamaterial, and Eeff are given by Eqs. (13) and (51), respectively, and then the effective parameters can be experimentally de- the dispersion relation, q = ω ρeff /Eeff , becomes rived. In contrast, merely measuring the dispersion re- p lation, that depends on the combination ρeff /Beff or ′ 2 2 1/2 2 1/2 ρeff /Eeff , will not in general be sufficient to distinguish ω0 ρ D ω1 − ω0 ω2 q = ω 1+ 2 2 1 − 2 . (57) effective mass from effective modulus. ω1 r k0  ω0 − ω   ω 

Somewhat coincidentally, this has precisely the same form as that of Eq. (43) for the case of an air-filled tube CONCLUSIONS containing a periodic array of membranes and Helmholtz resonators. The frequency dependence of the term aris- In conclusion, we have proposed the concepts of hid- ing from the hidden expanders is the same as that for the den force and hidden source of volume to respectively membranes (which give rise to hidden forces), whereas account for the effective densities and moduli of acous- the frequency dependence of the term arising from the tic metamaterials. The superficially strange concepts of hidden forces has the same form as that for the Helmholtz negative density and modulus are naturally accounted resonators (which give rise to hidden sources). The fre- for in this picture when the hidden force/source oper- quency spectra of ρeff and Eeff are analogous to the ates in antiphase to and is bigger in magnitude than the plots of Fig. 9(b), (c), the only difference being that the force/volume-change engendered in its absence. We illus- roles of the effective mass and modulus are reversed. The trate our approach in 1D for well-known air-based meta- plots for |vp| and |q| are the same as for Fig. 9(d) and materials involving membranes, Helmholtz resonators or (e) for equivalent dimensionless parameters. side holes with the inclusion of the new case of an array of 14 masses attached to membranes by springs. We also intro- Acknowledgements duce examples based on generic mass-and-spring models, in which case the concept of a hidden source of volume We are grateful to Alex Maznev, Vitalyi Gusev, Os- is replaced by the concept of a hidden expander of dis- amu Matsuda, Eun Bok, Insang Yoo, Jong Jin Park placement. and Tomohiro Kaji for stimulating discussions. This work was supported by the Center for Advanced Meta- Materials (CAMM) funded by the Ministry of Sci- Deciding at first glance what contributes to an effective ence, ICT and Future Planning as a Global Frontier density or to an effective modulus depends on the sys- Project, and by the Basic Science Research Program tem. For air-based acoustic metamaterials, membranes through the National Research Foundation of Korea only contribute to the effective density whether or not (NRF) funded by the Ministry of Education, Science they have an attached mass and spring. The reason and Technology (CAMM-2014M3A6B3063712 and NRF- for this is that they only involve local, time-dependent 2013K2A2A4003469). We also acknowledge Grants-in- hidden forces that provide a net force on the unit cell. Aid for Scientific Research from the Ministry of Educa- A similar result applies to the mass-and-spring model tion, Culture, Sports, Science and Technology (MEXT) representing a solid-matrix acoustic metamaterial based and well as support from the Japanese Society for the on heavy spheres in a soft matrix. In air-based acous- Promotion of Science (JSPS) tic metamaterials the effective modulus arises because of time-dependent hidden sources of air volume associated with Helmholtz resonators or side holes, that only pro- duce pairs of forces acting equally and oppositely on ei- [1] Z. Liu, X. Zhang, Y. Mao, Y. Y. Zhu, Z. Yang, C. Chan, ther side of the unit cell (i.e. resulting in a zero net force and P. Sheng, Science 289, 1734 (2000). on the unit cell). 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