Anomalous wave propagation in a one-dimensional acoustic metamaterial having simultaneously negative mass density 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 acoustic metamaterial that exhibits simultaneously negative mass density and negative Young’s modulus was proposed. Wave propagation was studied in the frequency range of double negativity. In view of positive energy flow, it was found that the phase velocity 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 wave packet. 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 refraction 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 waves in anomalous dis- sustain waves with group velocity 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- metamaterials 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 elastic modulus in a spectral metamaterials possessing, for instance, negative effective band. The local resonance 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 negative refraction, 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 dispersion 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 equations of motion for the jth unit cell are given by It also distinguishes the present metamaterials from the 2 ðjÞ d u1 ðjÞ ðjÀ1Þ ðjþ1Þ ðjÞ ðjÞ m1 2 þk1 2u1 Àu1 Àu1 þk2 u1 Àu2 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 À u ¼ 0; (2) Àqeffx 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 stress-strain relation. The harmonic wave solution for the equivalent elastic ðjþnÞ iðnXÃþnnÀgTÃÞ inXà uc ¼ u~ce : (3) solid is given by u^ ¼ ue~ . Hence, the dispersion relation is obtained as In Eq. (3), n ¼ qL is the non-dimensional wavenumber, g NMD NMD NMD NMD NMD 2 2 NMD 2 2 ¼ px=ffiffiffiffiffiffiffiffiffiffiffiffix0 is the non-dimensional wave frequency, x0 Gelast ðg; nÞ¼qeff ðx0 Þ g À Est n =L ¼ 0: (8) à ¼ k2=m2 is the local resonance frequency, X ¼ x=L is the non-dimensional spatial parameter with the lattice spac- à NMD B. Model of negative Young’s modulus (NYM) ing L,andT ¼ x0 Á 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 resonators shown in Fig. 1(b). This model was value problem of the coefficients. We obtain investigated and discussed in Ref. 13. Briefly, point masses 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 À 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 À 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 ðjÀ1Þ ðjþ1Þ m 1 þ k 2u À u À 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 vðjÀ1Þ À vðjÀ1Þ frequency-dependent in the form 3 3 1 2D 2 ðjÞ ðjÞ L NMD NMD h21 g q q 1 ; (5) À 2k3 v3 À v1 ¼ 0; (9) eff ¼ st þ 2 2D 1 þ h21 1 À 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 À 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 À Du ÞDu ð2L À Du Þ ðDu Þ 0 v ¼ 1 1 þ 1 1 þÁÁ 1 4D 16D3 The time-independent wave equation for plane harmonic L L À DuðjÞ ¼ ðuðjþ1Þ À 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 À 1 þ 1 þ ð1 À 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 À cos nÞ¼0; (12) ðjÞ ðjÞ ðjÞ d31 m1, m2, and m3 at the jth unit be u1 ðtÞ, u2 ðtÞ, v3 ðtÞ, respec- tively.