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Home , Acoustic metamaterial, Phase velocity

Anomalous propagation in a one-dimensional acoustic having simultaneously negative and Young’s modulus

H. H. Huang Department of Engineering Science and Ocean Engineering, National Taiwan University, Taipei 10617, Taiwan

C. T. Suna) School of Aeronautics and Astronautics, Purdue University, West Lafayette, Indiana 47907 (Received 29 November 2011; revised 12 June 2012; accepted 21 July 2012) A mechanical model representing an that exhibits simultaneously negative mass density and negative Young’s modulus was proposed. was studied in the range of double negativity. In view of positive energy flow, it was found that the in this range is negative. This phenomenon was also observed using transient wave propa- gation finite-element analyses of a transient sinusoidal wave and a transient . In con- trast to wave propagation in the region of positive mass and modulus, the peculiar backward wave motion in the region of double negativity was clearly displayed. VC 2012 Acoustical Society of America. [http://dx.doi.org/10.1121/1.4744977] PACS number(s): 43.35.Cg [ANN] Pages: 2887–2895

I. INTRODUCTION negative group-velocity media that exhibit fast-light phe- nomena reported in Refs. 26–29. It is noted that the subject For substances having simultaneously negative electric of negative phase velocity has, surprisingly, been discussed permittivity (e) and magnetic permeability (l), many unusual since, at least, 1904 by Lamb,30 using mechanical models, properties like reverse Doppler effect, reverse Cherenkov radi- and by Schuster31 in electromagnetism. At that time, the ation, and negative index of may arise.1 Theideais researchers were pessimistic about the physical applications. that when both e and l are negative, the metamaterial may The study of electromagnetic in anomalous dis- sustain waves with opposite to the phase veloc- persion has been extensively carried out by researchers in ity, and the waves can still propagate without attenuation. recent years, but that of acoustic and elastic waves receives Similar phenomena may apply to the counterpart acoustic much less attention. In the present study we propose a one- owing to the mathematical analogy between dimensional (1D) elastic mass-spring system consisting of acoustic and electromagnetic waves. Theoretical or experi- material units exhibiting simultaneously negative effective mental attempts have been made for investigation of acoustic mass density and effective in a spectral metamaterials possessing, for instance, negative effective band. The local acts as the central mechanisms of mass density,2–10 or negative effective elastic modulus,11–13 or the unusual behavior. Transient wave propagation is simu- both simultaneously.14–19 For metamaterials with double neg- lated numerically to better observe the dynamic characteris- ativity, the consequence is the behavior of , tics of negative phase velocity. and, equivalently, negative phase-velocity and backward- wave phenomena.1,20–22 Nevertheless, it has been shown that II. THEORETICAL MODELS OF ACOUSTIC the negative refraction can be achieved not only by metamate- METAMATERIALS rials with double negative properties, but also by Bragg scat- tering in periodic lattices23 and others.24,25 The similarity A. Model of negative mass density (NMD) between these mechanisms relies on negative slope of disper- Let us begin with the mass-in-mass lattice model that sion curves in the first quadrant of relations. has been discussed in Refs. 6–8. As illustrated in Fig. 1(a), According to the definition, the negative slope of the the mass-in-mass unit takes the form of a rigid ring with dispersion curve is related to negative group velocity. How- mass, m1, which is connected to the neighboring units by ever, we agree with some references, e.g., Refs. 21 and 22, springs with spring constant, k1. The ring contains an inter- the term “negative phase velocity” is better for describing nal mass, m2, connected to the ring by an internal spring the negative slope of the dispersion curve than “negative with spring constant, k2. Both m1 and m2 are only allowed to group velocity,” at least for the acoustic metamaterials with move in the x (horizontal) direction. double negativity which is discussed in the present study. The for the jth unit cell are given by It also distinguishes the present metamaterials from the

2 ðjÞ d u1 ðjÞ ðj1Þ ðjþ1Þ ðjÞ ðjÞ m1 2 þk1 2u1 u1 u1 þk2 u1 u2 ¼0; a)Author to whom correspondence should be addressed. Electronic mail: dt [email protected] (1)

J. Acoust. Soc. Am. 132 (4), Pt. 2, October 2012 0001-4966/2012/132(4)/2887/9/$30.00 VC 2012 Acoustical Society of America 2887

Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 50.81.134.248 On: Fri, 11 Sep 2015 23:03:19 FIG. 1. (Color online) (a) 1D lattice model and the continuum representative having negative effective mass density (NMD), and (b) 1D lattice model and the continuum representative having negative effec- tive Young’s modulus (NYM).

2 ðjÞ @2u^ d u2 ðjÞ ðjÞ 2 m þ k u u ¼ 0; (2) qeffx u^ ¼ Eeff ; (7) 2 dt2 2 2 1 @x2

ðjÞ NMD where uc (c ¼ 1 and 2) denotes the displacement of mass where the effective mass density qeff ¼ qeff is given in NMD “c” in the jth cell. The harmonic wave solution for the Eq. (5), and the effective Young’s modulus, Eeff Est (j þ n)th unit is expressed by k1L=A, is determined from the static -strain relation. The harmonic wave solution for the equivalent elastic ðjþnÞ iðnXþnngTÞ inX uc ¼ u~ce : (3) is given by u^ ¼ ue~ . Hence, the is obtained as In Eq. (3), n ¼ qL is the non-dimensional , g NMD NMD NMD NMD NMD 2 2 NMD 2 2 ¼ xp=ffiffiffiffiffiffiffiffiffiffiffiffix0 is the non-dimensional wave frequency, x0 Gelast ðg; nÞ¼qeff ðx0 Þ g Est n =L ¼ 0: (8) ¼ k2=m2 is the local resonance frequency, X ¼ x=L is the non-dimensional spatial parameter with the lattice spac- NMD B. Model of negative Young’s modulus (NYM) ing L,andT ¼ x0 t is the non-dimensional time. The dispersion relation is obtained by substituting the harmonic Next we consider the mechanical lattice model having displacements in Eq. (3) and solving the resulting eigen- lateral local shown in Fig. 1(b). This model was value problem of the coefficients. We obtain investigated and discussed in Ref. 13. Briefly, point m are connected by springs with a spring constant k . Four GNMDðg ¼ x=xNMD; n ¼ qLÞ 1 1 latt 0 rigid and massless truss members are assembled symmetri- 4 2h21 2 cally as shown to support two sets of microstructures consist- ¼ g ð1 þ h21Þþ ð1 cos nÞ g d21 ing of springs with spring constant, k3, and masses, m3. Point 2h mass m3 is allowed to move only in the y (vertical) direction. þ 21 ð1 cos nÞ¼0; (4) d21 L and D are geometrical parameters shown in Fig. 1(b). For this lattice model, the equations of motion for the where h21 ¼ m2=m1 and d21 ¼ k2=k1 are the non-dimensional jth unit are given by mass ratio and stiffness ratio, respectively. 2 ðjÞ In Ref. 8, an effective elastic solid was developed to d u ðjÞ ðj1Þ ðjþ1Þ m 1 þ k 2u u u represent the original mass-in-mass lattice. It was found that 1 dt2 1 1 1 1  the effective mass density of the equivalent elastic solid is L þ 2k vðj1Þ vðj1Þ frequency-dependent in the form 3 3 1 2D   2 ðjÞ ðjÞ L NMD NMD h21 g q q 1 ; (5) 2k3 v3 v1 ¼ 0; (9) eff ¼ st þ 2 2D 1 þ h21 1 g 2 ðjÞ NMD d v where q ¼ðm1 þ m2Þ=AL is the static mass density in 3 ðjÞ ðjÞ st m3 þ k3 v3 v1 ¼ 0: (10) which A and L are the cross-sectional area and the lattice dt2 spacing, respectively. This model is denoted as “NMD” From geometrical relations based on the assumption of small since it exhibits negative effective mass density in the range displacements, we have x pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 < g ¼ < 1 þ h21: (6) ðjÞ ðjÞ ðjÞ 2 ðjÞ 2 xNMD ðjÞ ð2L Du ÞDu ð2L Du Þ ðDu Þ 0 v ¼ 1 1 þ 1 1 þ 1 4D 16D3 The time-independent for plane harmonic L L DuðjÞ ¼ ðuðjþ1Þ uðjÞÞ: (11) waves is simply given by 2D 1 2D 1 1

2888 J. Acoust. Soc. Am., Vol. 132, No. 4, Pt. 2, October 2012 H. H. Huang and C. T. Sun: Negative phase velocity

Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 50.81.134.248 On: Fri, 11 Sep 2015 23:03:19 In a similar manner, the dispersion relation is obtained as C. Model of double negativity (DN)

GNYMðg ¼ x=xNYM; n ¼ qLÞ Consider a 1D infinite lattice shown in Fig. 2. This latt 0  model is a combination of the two models in Fig. 1 such that 4 2h31 d31l 2 ¼ g 1 þ 1 þ ð1 cos nÞ g mass m1 in the NYM model is replaced by the mass unit in d31 2 NMD. This model is denoted as “DN” as it will be shown to 2h31 possess double negativity. Let the displacements of masses þ ð1 cos nÞ¼0; (12) ðjÞ ðjÞ ðjÞ d31 m1, m2, and m3 at the jth unit be u1 ðtÞ, u2 ðtÞ, v3 ðtÞ, respec- tively. The equations of motion of the system at the jth cell NYM NYM wherepffiffiffiffiffiffiffiffiffiffiffiffih31 ¼ m3=m1, d31 ¼ k3=k1, g ¼ x=x0 , x0 are readily obtained as 2 ¼ k3=m3, and l ¼ðL=DÞ . The relation of the non- dimensional wave reads 2 ðjÞ @ u1 ðj1Þ ðjÞ ðjþ1Þ ðjÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m1 ¼ k1ðu1 u1 Þþk1ðu1 u1 Þ @t2  g ¼ g ðh31d21Þ=ðh21d31Þ: (13) ðjÞ ðjÞ ðj1Þ ðj1Þ L þ k2ðu2 u1 Þ2k3ðv3 v1 Þ If this lattice system is represented by an equivalent elastic  2D solid, the effective Young’s modulus of the solid is found ðjÞ ðjÞ L þ 2k3ðv v Þ ; (17) frequency-dependent in the form13 3 1 2D  d l g2 2 ðjÞ NYM NYM 31 @ u2 ðjÞ ðjÞ Eeff ¼ Est 1 þ ; (14) m ¼ k u u ; (18) 2 g2 1 2 @t2 2 1 2

NYM 2 ðjÞ where E ¼ k L=A is the static Young’s modulus. This @ v ðjÞ ðjÞ st 1 m 3 ¼ k v v : (19) model, denoted as “NYM,” exhibits negative effective 3 @t2 3 1 3 Young’s modulus in the range sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The steady-state solution of the equations of motion (17)– 2 x (19) is determined by letting the displacements be < g ¼ NYM < 1: (15) 2 þ d31l x 0 ðjþnÞ iðnXþnngTÞ ðjþnÞ iðnXþnngTÞ u1 ¼ u^1e ; u2 ¼ u^2e ; and

The wave equation takes the form of Eq. (7), in which the ðjþnÞ iðnXþnngTÞ NYM v3 ¼ v^3e ; (20) effective Young’s modulus, Eeff ¼ Eeff , is given by Eq. (14) and the effective mass density is the static formulation, NYM where, again, n ¼ qL is the non-dimensional wavenumber, q ¼ q ¼ m1=AL. (Note that the masses m3 in this eff st NMD model are assumed to move only in the y direction so that X ¼ x=L and T ¼ x0 t are the non-dimensional para- they contribute zero inertia in the x direction.) The meters in space and time, respectively. Substituting these dispersion relation is given by harmonic displacements in Eqs. (17)–(19) and solving the resulting eigenvalue problem of the coefficients, we obtain NYM NYM NYM 2 2 NYM 2 2 Gelast ðg;nÞ¼qst ðx0 Þ g Eeff n =L ¼ 0: (16) the dispersion relation

 DN 6 d31h21 h21 d31l 4 Glatt ðg; nÞ¼g þ 1 þ þ 21þ ð1 cos nÞþd21 g d21h31 d21 2  2 d31h21 h21 d31h21 d31l 2 d31h21 ð1 þ h21Þ þ 2 þ 1 þ ð1 cos nÞ g þ 2 2 ð1 cos nÞ¼0: (21) d21h31 d21 d21h31 2 d21h31

On the other hand, the equation of motion of the equivalent III. EFFECTIVE DYNAMIC PARAMETERS AND 1D elastic solid is given by Eq. (7). In this formulation DISPERSION RELATION the effective mass density and the effective Young’s modu- For numerical illustrations, we present two cases with DN NMD lus are given by Eq. (5), qeff ¼ qeff ¼ qeff ,andEq.(14), the following assumed material constants: DN NYM Eeff ¼ Eeff ¼ Eeff , respectively. Consequently, the disper- (Case 1) h21 ¼ m2=m1 ¼ 1:5, h31 ¼ m3=m1 ¼ 1, d21 2 sion relation of the representative elastic solid is obtained ¼ k2=k1 ¼ 0:1, d31 ¼ k3=k1 ¼ 1, l ¼ðL=DÞ ¼ 2:78, as and (Case 2) h21 ¼ 2:25, h31 ¼ 0:4, d21 ¼ 0:3, d31 ¼ 0:25, DN DN NMD 2 2 DN 2 2 Gelastðg; nÞ¼qeff ðx0 Þ g Eeff n =L ¼ 0: (22) l ¼ 25.

J. Acoust. Soc. Am., Vol. 132, No. 4, Pt. 2, October 2012 H. H. Huang and C. T. Sun: Negative phase velocity 2889

Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 50.81.134.248 On: Fri, 11 Sep 2015 23:03:19 FIG. 2. (Color online) One-dimensional lattice model exhibiting DN.

As has been mentioned in Sec. II C, the model in Fig. 2 bandgaps, where waves attenuated and stopped, exist due to is denoted as “DN” because for some range of wave frequen- the effect of the local resonance. cies it could exhibit double negativity. This range can be The accuracy of the elastic solid representation can be obtained from the negative modulus and mass density prop- verified by comparing the dispersion curves obtained with the erties and their corresponding spectral regions. The effective lattice and the elastic solid models. In Figs. 5 and 6,thedis- mass density in Eq. (5) and the effective Young’s modulus persion curves of the equivalent elastic solid are obtained in Eq. (14) with respect to the non-dimensional wave fre- using Eqs. (8), (16),and(22) for the NMD, NYM, and DN quency for Case 1 and Case 2 are illustrated in Figs. 3 and 4, models, respectively. Good approximation is observed for the respectively. lower branch for all models. For the higher branch some devi- In Fig. 3 for Case 1, the material constants are selected ation is seen at the high frequency range. The reason is that so that the frequency range of the negative quantity of the the equivalent elastic solid employs an approximate formula- NMD model lies apart from that of the NYM model. In other tion from the original lattice system based on a long wave- words, for the frequency range in-between the two regions length approximation. It is noted that the shaded areas in the with negative quantity (e.g., g approximately ranges in 1.6 figures indicate the negative properties derived from Eqs. (6) and 2.5 in Fig. 3), both the effective mass density and the and (15) for the NMD and NYM model, respectively. For effective Young’s modulus are positive. On the other hand, both cases and for the DN model, the bandgaps closely match in Fig. 4 for Case 2, the material constants are chosen so that the shaded areas of the separate models. the frequency ranges of both negative quantities of the NMD and the NYM models overlap. IV. PHASE VELOCITY AND GROUP VELOCITY Next, the dispersion curves for both cases obtained using the exact formulation [Eqs. (4), (12), and (21) for A. Phase velocity NMD, NYM, and DN models, respectively] of the lattice In the present study, we adopt the definition of the phase model are shown in Figs. 5 and 6, respectively, with solid velocity of a harmonic wave as32–34 lines. In each figure, three columns labeled DN, NMD, and NYM represent the dispersion curves of the double negativ- x vp ¼ : (23) ity model, the negative mass density model, and the negative q Young’s modulus model, respectively. For each diagram, All points of the same phase propagate in space with this velocity. In the dispersion x-q diagram, the phase velocity

FIG. 3. (Color online) Effective dynamic non-dimensional parameters for Case 1 using Eqs. (5) and (14). The bounds of negative quantities are FIG. 4. (Color online) Effective dynamic non-dimensional parameters for described in Eqs. (6) and (15). Case 2. Descriptions as in Fig. 3.

2890 J. Acoust. Soc. Am., Vol. 132, No. 4, Pt. 2, October 2012 H. H. Huang and C. T. Sun: Negative phase velocity

Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 50.81.134.248 On: Fri, 11 Sep 2015 23:03:19 sffiffiffiffiffiffiffiffi x Eeff vp ¼ ¼ 6 : (25) q qeff

It is still not possible to determine the direction of a wave in the acoustic metamaterial from the sign of phase velocity, or the , based on the given dispersion relation, since the real wave vector q could be either positive or nega- tive. We shall proceed to the next subsections for further investigation.

B. Group velocity of DN acoustic metamaterial Of particular interest is to investigate the sign of group velocity of a wave group or packet in an acoustic metamate- rial. This wave group or packet corresponds to a superposi- FIG. 5. (Color online) Dispersion curves for Case 1. DN model, NMD tion of harmonic waves with very similar wave frequencies, model, NYM model; solid lines: mass-spring lattice model from Eqs. (4), (12), and (21) for NMD, NYM, and DN models, respectively; circles: equiv- x, and , q. In the dispersion x-q diagram, the alent elastic solid from Eqs. (8), (16), and (22) for NMD, NYM, and DN group velocity of a wave group or packet is given by the models, respectively. slope of the dispersion curve. By using the implicit differen- tiation, the final expression of the group velocity reads is given by the slope of the straight line connecting the origin  dx @G 1 @G and a certain point on the dispersion curve. Since negative vg ¼ ¼ wave frequency is not in our consideration, it is obvious that dq @x @q * the wave vector, q, points in the direction of phase velocity. 2Eeffq ¼ : (26) For an effective elastic solid of an acoustic metamate- @qeff @Eeff qeff 2 2qeffx þ x rial, the dispersion relation can be obtained by, for instance, @x @x Eeff employing Eq. (7) together with qeff in Eq. (5) and Eeff in Eq. (14), and substituting the harmonic plane-wave solution From the expressions given by Eqs. (5) and (14), it can be into Eq. (7). The dispersion relation reads shown that NMD 2 2 @qeff 2qst k g Gðx; qÞ¼qeffx Eeffq ¼ 0: (24) ¼ 0 (27) NMD 2 2 @x x0 ð1 g Þ From the dispersion relation, one can immediately conclude that in order for the elastic medium to have wave propaga- and

tion without attenuation (i.e., a real solution for q), the me- NYM @Eeff Est d31l g dium must have either both positive qeff and Eeff or both ¼ 0; (28) @x xNYM g2 2 negative qeff and Eeff. 0 ð 1Þ The phase velocity is thus given by where k ¼ h21=ð1 þ h21Þ. The term ½ð@Eeff=@xÞðqeff= 2 EeffÞx in the denominator in Eq. (26) is always positive if qeff and Eeff are of the same sign. Substituting Eqs. (5) and (26) in the remainder of the denominator of Eq. (26), 2 2qeffx þð@qeff=@xÞx , we obtain  @q 2q x þ eff x2 eff @x 2qNMDx ¼ st ½ð1 kÞð1 g2Þ2 þ k0: (29) ð1 g2Þ2

Since 0 k 1 is always true, this inequality holds for all frequencies. It is, therefore, proved that the denominator of Eq. (26) is positive for identical signs of qeff and Eeff. Obviously, from Eq. (26), group velocity, vg, is propor- tional to Eeffq. The sign of group velocity is, hence, affected by the sign of the effective Young’s modulus, Eeff, and the wavenumber, q. From Sec. IV A, in order for an elastic me- FIG. 6. (Color online) Dispersion curves for Case 2. Solid lines: mass- spring lattice model; Circles: elastic solid model; detailed description as in dium to have wave propagation without attenuation, the me- Fig. 5. dium must possess either both positive qeff and Eeff or both

J. Acoust. Soc. Am., Vol. 132, No. 4, Pt. 2, October 2012 H. H. Huang and C. T. Sun: Negative phase velocity 2891

Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 50.81.134.248 On: Fri, 11 Sep 2015 23:03:19 negative qeff and Eeff.Ifqeff and Eeff of the medium are both positive, group velocity, vg, is obviously positive for a posi- tive wavenumber. On the other hand, if qeff and Eeff are simultaneously negative, waves, therefore, propagate in the medium with opposite signs of phase velocity, vp, and group velocity, vg.

C. Negative phase velocity For waves in a lossless elastic medium the energy veloc- ity is identical to the group velocity.35 In view of positive energy flow for a lossless elastic medium, the group velocity is always positive. Once again, we exclude the negative group-velocity media resulting in fast-light phenomena dis- cussed in, for instance, Refs. 26–29. That negative group- velocity phenomenon is beyond the scope of the present FIG. 7. (Color online) Snapshots of sinusoidal wave propagation in NMD study. finite DN model with excitation frequency g ¼ x=x0 ¼ 0:346 and NMD For the acoustic metamaterial with frequency-dependent x0 ¼ 115:47 rad=s (in the double-positivity region; ts ¼ time step; T NMD ¼ x0 t ¼ ts DT with DT ¼ 0:462). effective mass density, qeff, and Young’s modulus, Eeff,if qeff and Eeff are both positive, a positive group velocity results in a positive wavenumber. On the other hand, ifq the time steps hereafter. Section 1 (ts ¼ 1–4), for instance, is eff the initial state and section 3 (ts ¼ 101–104) is considered and Eeff are negative simultaneously, in order to have a posi- close to steady state. tive group velocity, wavenumber must be negative. This means that in the double negativity frequency region, the Selection of the time increment, DT , for visualization of simulation results is nontrivial. An improper selection of sign of phase velocity, vp, in Eq. (25) must be negative. the increments may, on one hand, produce incorrect simula- tion results, or, on the other hand, simply cause expensive V. DYNAMIC CHARACTERISTICS OF WAVE computation. Here we provide a simple guide for the balance PROPAGATION BY SIMULATION of accuracy and efficiency. The definition of the time incre- A. Harmonic waves ment reads To investigate how a group of waves of essentially the t DT ¼ xNMD ¼ xNMD Dt; (30) same propagates with a negative phase velocity, 0 ts 0 we perform numerical simulations of transient wave propa- gations. One of the reasons the transient wave propagation where Dt is the time step in seconds. Substituting the non- NMD rather than steady-state wave propagation is studied is dimensional wave frequency g ¼ x=x0 in Eq. (30) and because the direction of disturbances, hence the direction of using the relationship of and period of group velocity, can be ensured. time we obtain Dynamic simulations are implemented by use of the ABAQUS Explicit solver (Dassault Syste`mes, Ve`lizy- Villacoublay, France). In each simulation, a lattice model shown in Fig. 2 is constructed with a sinusoidal displace- ment, UðtÞ¼U0 sinðxtÞ HðtÞ, applied at the left end of the lattice. The material properties of Case 2 listed in Sec. III NMD together with x0 ¼ 115:47 rad=s are used. If the lattice system is long enough, the reflected wave does not appear in the window of numerical results. Figures 7 and 8 show the snapshots of the displacement fields in space with two different wave frequencies, NMD g ¼ x=x0 ¼ 0.346 and 1.559, respectively. U denotes the normalized of the displacements with respect to U0. In each figure, a typical propagation mode is demon- strated. Three consecutive sections of time, T ¼ ts DT with the non-dimensional time increment DT ¼ 0:462, show the behavior of wave propagation. For now, let us assume that the time increment DT is sufficiently small in capturing the dynamic behavior of the model. We shall FIG. 8. (Color online) Snapshots of sinusoidal wave propagation in come to this point immediately in the following paragraph. NMD NMD finite DN model with excitation frequency g ¼ x=x0 ¼ 1:559 and Note that, again, T ¼ x t is the non-dimensional NMD 0 x0 ¼ 115:47 rad=s (in the double-negativity region; ts ¼ time step; NMD parameter in time domain and t is in seconds. “ts” denotes T ¼ x0 t ¼ ts DT with DT ¼ 0:462).

2892 J. Acoust. Soc. Am., Vol. 132, No. 4, Pt. 2, October 2012 H. H. Huang and C. T. Sun: Negative phase velocity

Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 50.81.134.248 On: Fri, 11 Sep 2015 23:03:19 2p Dt DT ¼ : (31) g Tp

For a harmonic wave, Dt Tp=8 should be sufficient in cap- turing the dynamic behavior of the models in the present study. For Case 2, for instance, this inequality Dt Tp=8 and the wave frequency, g ¼ 1:559, together with Eq. (31), result in DT 0:504. Clearly, the time increments chosen for both cases are satisfactory. In Fig. 7, given the first section from ts ¼ 1–4, the prop- agation direction of the front of the dynamic disturbance is seen pointing from left to right, indicating the direction of the energy flow. However, due to the nature of transient wave propagation, the forerunner of the dynamic disturbance may contain frequencies other than the excitation frequency, x. Further time steps, in the middle and bottom windows of Fig. 7, need to be carried out in order to make sure the energy direction of the sinusoidal wave with exactly the ex- citation frequency, x. In the second section, ts ¼ 51–54, it is seen that the transient part of the wave already propagates to the right side of the spatial window, leaving the transition to steady state at the left part of the window. From the third section, ts ¼ 101–104, a more complete configuration approaching the steady-state wave propagation can be observed. At this stage, the propagation direction of the FIG. 9. (Color online) Two typical wave packets in time and frequency do- wave phase is observed pointing from left to right. NMD main with (a) frequency band approximately from g ¼ x=x0 ¼ 0:22 to Figure 8 shows the profile of wave propagation with a g ¼ 0:47 with the central frequency at g ¼ 0:346, and (b) frequency band NMD frequency in the region of double negativity. Noticeable dif- approximately from g ¼ x=x0 ¼ 1:10 to g ¼ 2:10 with the central fre- ference to Fig. 7 is the negative propagation direction of the quency at g ¼ 1:559. wave phase with respect to positive energy flow. In the second and third windows, the phase direction is clearly seen pointing quency is located at the positive effective mass density and from right to left during the sequence of time evolution. positive effective elastic modulus region as shown in Fig. 4. The wave packet propagates without much distortion since most of the spectral content travel with approximately the same B. Wave packets speed, which is also evident in view of the dispersion curves Group velocity is the velocity of the of a wave shown in Fig. 6. The energy flow direction is the moving direc- group or packet. For a wave packet corresponding to a super- tion of the wave packet, which is clearly from left to right. position of harmonic waves with a finite frequency band, the Note that a wave packet contains not just a single frequency, propagation behavior in a dispersive media like DN acoustic metamaterial should be of particular interest. One great advantage of considering the wave packet is that the fre- quency content of the packet can be controlled and the energy propagation direction can be easily identified as the propagation direction of the wave packet. To this end, wave packets propagating in the metamate- rial with double negativity are simulated in a similar fashion. Two wave packets in time as well as in frequency domain used for the simulation are illustrated in Fig. 9. In Fig. 9(a), the wave packet possesses a frequency band approximately NMD from g ¼ x=x0 ¼ 0:22 to g ¼ 0:47 (in the double posi- tivity region) with the central frequency at g ¼ 0:346, while in Fig. 9(b), the wave packet possesses a frequency band NMD approximately from g ¼ x=x0 ¼ 1:10 to g ¼ 2:10 (in the double negativity region) with the central frequency at g ¼ 1:559. Each packet is sent from the left end of the model. Same material properties are chosen so that the dis- FIG. 10. (Color online) Snapshots of wave-packet propagation in finite DN model. Wave packet consists of frequency band approximately from persion relation of the model is shown in Fig. 6. NMD g ¼ x=x0 ¼ 0:22 to g ¼ 0:47 with the central frequency at g ¼ 0:346 Figure 10 shows a few snapshots of wave propagation of NMD and x0 ¼ 115:47 rad=s (in the double-positivity region; ts ¼ time step; NMD the wave packet shown in Fig. 9(a), of which the central fre- T ¼ x0 t ¼ ts DT with DT ¼ 0:462).

J. Acoust. Soc. Am., Vol. 132, No. 4, Pt. 2, October 2012 H. H. Huang and C. T. Sun: Negative phase velocity 2893

Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 50.81.134.248 On: Fri, 11 Sep 2015 23:03:19 the slope of the dispersion curve is negative. For example, let us refer to the dispersion diagram in Fig. 12 for illustration. The dispersion curves are reproduced from Fig. 6,Case2in Sec. V, for the DN model. Usually the first quadrant of the diagram is considered where x and q are both positive. For the first lowest propagation mode, both phase velocity and group velocity, according to the definition in Sec. IV,areposi- tive. However, for the second lowest propagation mode in the first quadrant, the phase velocity is positive while the group velocity is negative. The misleading and confusing conclusion of the “negative group velocity” may have come from the fact that energy velocity is not directly provided by the disper- sion curves in the first quadrant. In fact, the second branch of the dispersion curve should be plotted in the second quad- rant where x is positive and q is negative. By so doing, the FIG. 11. (Color online) Snapshots of wave-packet propagation in the group velocity is positive while the phase velocity is, hence, DN model. Wave packet consists of frequency band approximately from NMD negative. g ¼ x=x0 ¼ 1:10 to g ¼ 2:10 with the central frequency at g ¼ 1:559 NMD and x0 ¼ 115:47 rad=s (in the double-negativity region; ts ¼ time step; NMD T ¼ x0 t ¼ ts DT with DT ¼ 0:462). VI. CONCLUSION but a finite frequency band. It is therefore difficult to identify the phase velocity of the propagation of the wave packet. We have proposed a 1D mechanical model representing Next, consider the case when the frequency components an acoustic metamaterial. The system can be modeled with an of the wave packet are selected in the frequency region of equivalent elastic solid which exhibits simultaneously nega- double negativity. A wave packet of Fig. 9(b) is generated tive effective mass density and effective Young’s modulus in for simulation. The frequency band of this wave packet is a certain frequency range. In this double negativity frequency inside the double negativity region shown in Fig. 4. The range, harmonic waves propagate without attenuation. How- wave packet is sent from the left end and is clearly seen to ever, the phase velocity becomes opposite to the energy flow propagate in the positive direction (left to right) from the top direction. In view of positive energy direction, it is the back- and bottom windows of Fig. 11. Negative phase direction of ward wave that propagates with negative phase velocity. the packet is expected and can be verified from the middle Transient wave simulation results not only agree well with the window of Fig. 11. analytical prediction, but also clearly illustrate the unusual wave phenomenon.

C. Discussion ACKNOWLEDGMENT The region of the dispersion curve with negative slope This work was supported by an AFOSR Grant No. means a spectral range where the phase and group velocities FA9550-10-1-0061. Dr. Les Lee was the program manager. are opposite in direction. Some researchers25,30,36–38 have termed the backward-wave phenomenon along with “negative 1V. G. Veselago, “The electrodynamics of substances with simultaneously group velocity” simply because, perhaps, in the x-q diagram negative values of e and l,” Sov. Phys. Usp. 10, 509–514 (1968). 2Z. Liu, X. Zhang, Y. Mao, Y. Y. Zhu, Z. Yang, C. T. Chan, and P. Sheng, “Locally resonant sonic materials,” Science 289, 1734–1736 (2000). 3Z. Liu, C. T. Chan, and P. Sheng, “Analytic model of phononic with local ,” Phys. Rev. B 71, 014103 (2005). 4G. W. Milton and J. R. Willis, “On modifications of Newton’s second law and linear continuum elastodynamics,” Proc. R. Soc. London, Ser. A 463, 855–880 (2007). 5S. Yao, X. Zhou, and G. Hu, “Experimental study on negative mass effec- tive mass in a 1D mass-spring system,” New J. Phys. 10, 043020 (2008). 6H. H. Huang, C. T. Sun, and G. L. Huang, “On the negative effective mass density in acoustic metamaterials,” Int. J. Eng. Sci. 47, 610–617 (2009). 7H. H. Huang and C. T. Sun, “Wave attenuation mechanism in an acoustic metamaterial with negative effective mass density,” New J. Phys. 11, 013003 (2009). 8H. H. Huang and C. T. Sun, “Locally resonant acoustic metamaterials with 2D anisotropic effective mass density,” Philos. Mag. 91, 981–996 (2011). 9S. H. Lee, C. M. Park, Y. M. Seo, Z. G. Wang, and C. K. Kim, “Acoustic metamaterial with negative density,” Phys. Lett. A 373, 4464–4469 (2009). 10M. T. Islam and G. Newaz, “Metamaterial with mass-stem array in acous- tic cavity,” Appl. Phys. Lett. 100, 011904 (2012). 11N. Fang, D. Xi, J. Xu, M. Ambati, W. Srituravanich, C. Sun, and FIG. 12. (Color online) Schematic of group velocity and phase velocity in X. Zhang, “Ultrasonic metamaterials with negative modulus,” Nature x-q diagram. Mater. 5, 452–456 (2006).

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