Journal of Sound and Vibration 413 (2018) 250e269

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Journal of Sound and Vibration

journal homepage: www.elsevier .com/locate/ jsvi

Spectro-spatial analysis of wave packet propagation innonlinear acoustic metamaterials

W.J. Zhou a, b, X.P. Li b, Y.S. Wang c, W.Q. Chen a, d, G.L. Huang b, c, *

a Department of Engineering Mechanics, Zhejiang University, Yuquan Campus, Hangzhou 310027, Chinab Department of Mechanical and Aerospace Engineering, University of Missouri, Columbia, MO 65211, USAc Institute of Engineering Mechanics, Beijing Jiaotong University, Beijing, Chinad Key Laboratory of Soft Machines and Smart Devices of Zhejiang Province, Zhejiang University, Yuquan Campus, Hangzhou 310027,China

a r t i c l e i n f o

Article history:Received 24 February 2017Received in revised form 9 October 2017Accepted 13 October 2017

Keywords:Nonlinear acoustic metamaterialSpectro-spatial analysisDirection-biased waveguide

* Corresponding author. Department of MechanicE-mail address: huangg@missouri.edu (G.L. Hua

https://doi.org/10.1016/j.jsv.2017.10.0230022-460X/© 2017 Elsevier Ltd. All rights reserved.

a b s t r a c t

The objective of this work is to analyze wave packet propagation in weakly nonlinearacoustic metamaterials and reveal the interior nonlinear wave mechanism throughspectro-spatial analysis. The spectro-spatial analysis is based on full-scale transient anal-ysis of the finite system, by which dispersion curves are generated from the transmittedwaves and also verified by the perturbation method (the L-P method). We found that thespectro-spatial analysis can provide detailed information about the solitary wave in short-wavelength region which cannot be captured by the L-P method. It is also found that theoptical wave modes in the nonlinear metamaterial are sensitive to the parameters of thenonlinear constitutive relation. Specifically, a significant frequency shift phenomenon isfound in the middle-wavelength region of the optical wave branch, which makes thisfrequency region behave like a band gap for transient waves. This special frequency shift isthen used to design a direction-biased waveguide device, and its efficiency is shown bynumerical simulations.

© 2017 Elsevier Ltd. All rights reserved.

1. Introduction

Among the many avenues of current research interest in the quest for enhancing the properties of engineering structuresand materials, acoustic metamaterials (AMs) are a very promising candidate [1e11]. AMs are, broadly speaking, materialswith man-made “microstructures” that exhibit unusual elastic wave propagation behavior not found in natural materials.They derive their advantageous dynamic properties not just due to the constituent material compositions but more so due totheir engineered microstructural configurations. Inasmuch as the underlying physics involved in realizing the unusualbehavior displayed by AMs is well-understood and straightforward, the physical consequences resulting from their engi-neered microstructural configurations are nonetheless advantageous and without precedent. The development of AMs forengineering structural applications may thus be viewed as a new design philosophy with a unified focus on tailoring theirform, function and performance using microstructural configurations across length-scales [12].

al and Aerospace Engineering, University of Missouri, Columbia, MO 65211, USA.ng).

W.J. Zhou et al. / Journal of Sound and Vibration 413 (2018) 250e269 251

The combination of nonlinearity into material or boundary periodicity may produce many new wave phenomenaincluding the gap solitons [13], tunable band gaps [14], envelope and dark solitons [15,16] to name a few. Nonlinear periodicmaterials/structures have attracted considerable interest in recent years due to the fantastic wave phenomena in them. Forexample, Narisetti et al. [17] developed a perturbation analysis-based approach to obtain approximate solutions for thedispersion behavior of various one-dimensional cubically nonlinear periodic chains. Manktelow et al. [18] used a multiplescales technique to analyze wave-wave interactions in a one-dimensional monoatomic mass-spring chain with cubicnonlinearity and found that the interaction of twowaves generates different amplitude- and frequency-dependent dispersionbranches, in contrast to a single amplitude-dependent branch when only a single wave is present. If the microstructure-induced local resonance is further taken into consideration, low-frequency amplitude-dependent band-gap [19,20] as wellas other interesting phenomena, such as the propagation of nanoptera [21], a strongly localized solitary wave followed by asmall amplitude oscillatory tail, can be achieved/observed. The generation of such oscillatory tail was further discussed indetail in Refs. [22,23]. On the other hand, based on recently proposed locally resonant granular system with/without pre-compression, Liu et al. [24,25] theoretically investigated the weakly/strong nonlinear waves and the existence of brightand dark breathers were proved strictly. Recently, Wallen et al. [26] investigated the discrete breathers in a granular meta-material composed of a monolayer of spheres on an elastic half-space by modelling the complex system as a mass-in-masschain with Hertzian nonlinear local resonators.

Most of studies on nonlinear periodic structures concentrate on dispersion characteristics or propagation of solitary wavesin long-wavelength domain. The dispersion relation can provide an estimate of nonlinear effects on the cut off frequency,however, other important characteristics of wave propagation are not well documented [27e29]; on the other hand, a betterunderstanding of the solitons or discrete breathers in short-wavelength region needs more considerations. Thus, Ganesh andGonella [27] provided an elegant method to study the spectro-spatial wave features of nonlinear periodic chains, which couldpresent both the local wave properties (such as existence of solitary waves) and the global wave features (such as thedispersion relation and the existence of solitary waves in the first Brillouin zone).

In the present paper, the nonlinear AM is modeled by the simple nonlinear mass-in-mass chain with linear oscillatormicrostructure. The LindstedtePoincar�e (L-P) perturbation method [17] is first adopted to obtain the approximate dispersionrelation and displacement solutions. The displacement and velocity fields are then set as initial conditions to numericallysimulate the propagation process of a transient wave packet by using the numerical integration routine of MATLAB, ODE45.After that, signal processing techniques such as short-term Fourier transform and wavenumber filtering are utilized toelucidate the relation between topological (spaceetime domain) and dispersion (wavenumberefrequency domain) features.Simulation results show that there is a gap-like region located in the middle domain of the optical branch. For a transientwave packet with initial frequency band located in this ‘gap’, an intensive frequency shift phenomenon can be observedduring the propagation, and the frequency components of the wave packet are transferred out of the ‘gap’ region, making thisregion behave like a band gap. We should emphasize that such ‘gap’ works only for transient wave signal but not formonochrome harmonic wave. Such kind of frequency transformation is totally different from the traditional frequency shift innonlinear phononic crystals. Finally, by employing such special but strong frequency shift phenomenon, a wave device fordirection-biased waveguide is designed and its high efficiency is shown by numerical simulations.

2. Spectral wave analysis of nonlinear acoustic metamaterials

The nonlinear acoustic metamaterial is represented by amass-in-mass chain, as shown in Fig. 1. Each unit-cell consists of arigid mass, m1, a nonlinear spring whose stiffness is governed by two parameters e a linear stiffness, k1 and a nonlinearparameter, G, the cubic nonlinear parameter, and an interior oscillator consisting of a mass, m2, and a linear spring, k2. Thelength of a cell is a. For the weak nonlinearity, we introduce a small parameter ε so that the restoring force fr in the nonlinearspring can be expressed as

Fig. 1. The mass-in-mass system with nonlinear inner stiffness.

W.J. Zhou et al. / Journal of Sound and Vibration 413 (2018) 250e269252

fr ¼ k1dþ εGd3 (1)

where d is the displacement of the spring.The equations of motion for the mass-in-mass system in the absence of external forces can be written as

m1 €un þ k1ð2un � un�1 � unþ1Þ þ εGhðun � un�1Þ3 þ ðun � unþ1Þ3

iþ k2ðun � vnÞ ¼ 0 (2a)

m2€vn þ k2ðvn � unÞ ¼ 0 (2b)

where un and vn indicate the displacements of the n-th outer and inner masses, respectively, and the superscripted dotdenotes time derivation. Eq. (2) can be normalized with respect to the linear spring constant k1, mass m1, and can beexpressed as

k2m2u

2d

€un þ ð2un � un�1 � unþ1Þ þ εGhðun � un�1Þ3 þ ðun � unþ1Þ3

iþ k2ðun � vnÞ ¼ 0 (3a)

1u2d

€vn þ ðvn � unÞ ¼ 0 (3b)

where G ¼ G=k1, ud ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffik2=m2

p, m2 ¼ m2=m1 and k2 ¼ k2=k1.

The LindstedtePoincar�e (L-P) perturbation method [17] is adopted to obtain the dispersion relation of the system. Bydefining the dimensionless time t ¼ ut, the normalized equations of wave motion (3) can be rewritten as

k2U2

m2

v2unvt2

þ ð2un � un�1 � unþ1Þ þ εGhðun � un�1Þ3 þ ðun � unþ1Þ3

iþ k2ðun � vnÞ ¼ 0 (4a)

2v2vn

Uvt2

þ ðvn � unÞ ¼ 0 (4b)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 2

q

where U ≡ u =ud is the dimensionless frequency. The perturbation expansion of U, as well as the displacements of the

outer and inner masses can be truncated to the first order as:

U ¼ U0 þ εU1 un ¼ uð0Þn þ εuð1Þn vn ¼ vð0Þn þ εv

ð1Þn (5)

Specifically, by substituting Eq. (5) into Eq. (4) and separating the coefficients of ε0; ε1, we obtain the governing equations

as

k2U20

m2

d2uðaÞn

dt2þ�2uðaÞn � uðaÞn�1 � uðaÞnþ1

�þ k2

�uðaÞn � v

ðaÞn

�¼ Ma (6a)

v2vðaÞ � �

U20

n

vt2þ v

ðaÞn � uðaÞn ¼ Na (6b)

in which the subscript a ¼ 0; 1 and

M0 ¼ 0N0 ¼ 0 (7a)

d2uð0Þ �� � � � �

M1 ¼ �2

k2m2

U0U1j

dt2� G uð0Þn � uðaÞn�1

3 þ uð0Þn � uðaÞnþ1

3

N1 ¼ �2U0U1d2vð0Þn

dt2

(7b)

The lowest order recovers the governing equation of the corresponding linear AM and the higher order expansionsmanifest as a heterogeneous form of the linear systemwith the nonlinear terms acting as the forcing function. The frequencycorrection for each order of nonlinearity is then obtained by enforcing the requirement of non-secular solutions.

The harmonic solution for the linear system can be assumed as

W.J. Zhou et al. / Journal of Sound and Vibration 413 (2018) 250e269 253

uð0Þn ¼ A0

2einkeit þ c:c (8a)

ð0Þ B0 ink it

vn ¼2

e e þ c:c (8b)

where i ¼ffiffiffiffiffiffiffi�1

p, k ¼ qa is the dimensionless wavenumber, q is the wavenumber, c.c denotes the complex conjugate, A0 and B0

are the wave amplitudes of the outer and inner masses, respectively. By substituting Eq. (8) into Eq. (6), we haven�1� U2

0

�h� k2U

20 þ 2m2ð1� cos kÞ

i� k2m2U

20

oA0 ¼ 0 (9a)

B0 ¼ Ku0A0 (9b)

where Ku0 ¼ 12. For nontrivial solutions, the dispersion relation of the linear wave can be determined as

1�U0�1� U2

0

�h� k2U

20 þ 2m2ð1� cos kÞ

i� k2m2U

20 ¼ 0 (10)

It can be proved that Eq. (10) yields two positive real roots, the smaller one corresponds to the acoustic branch of thesystem and denoted as U0ac, while the larger one to the optical branch denoted as U0op.

The governing equations for a ¼ 1 can be rewritten as

U20d2

dt2þ 1

!"k2m2

U20

d2uð1Þn

dt2þ�2uð1Þn � uð1Þn�1 � uð1Þnþ1

�#

þ k2U20d2

dt2uð1Þn ¼ c1e

inkeit þ c2e3inke3it þ c:c

(11a)

�2 d2

�ð1Þ ink it ð1Þ

U0 dt2

þ 1 vn ¼ U0U1KuA0e e þ un (11b)

where

c1 ¼�1� U2

0

�" k2m2

U0U1A0 �32Gð1� cos kÞ2A2

0A*0

#þ k2U0U1Ku0A0 (12a)

1

k!� �

c2 ¼2G k2 � 9 2

m2U20 2 cos3 kþ 3 cos2 k� 1 A3

0 (12b)

In Eq. (12), the superimposed asterisk denotes complex conjugate. The forcing term with spatial form eink on the right-hand side of Eq. (11a) is secular and must be eliminated. Hence, by setting c1 ¼ 0, we obtain

U1 ¼ 3Gð1� cos kÞ2jA0j2

2U0

�k2 þm2k2K2

u0

� (13)

Finally, for a weakly nonlinear mass-in-mass chain, the reconstituted dispersion relation is then given as

Uac ¼ Uþ ε

3Gð1� cos kÞ2jA0j2

2U0ac

�k2 þm2k2K2

u0ac

� (14a)

3Gð1� cos kÞ2jA j2

Uop ¼ U0op þ ε

0

2U0op

�k2 þm2k2K2

u0op

� (14b)

Fig. 2 shows the dispersion curves in the nonlinear mass-in-mass system predicted by the L-P method. In the figure, the

normalized parameters of the mass-in-mass system is selected asm2 ¼ 1; k2 ¼ 1; ud ¼ 103rad=s and the parameters of the

nonlinearity being εGjA0j2 ¼ 0 and εGjA0j2 ¼ 0:06, respectively, to represent the linear and nonlinear AMs. To validate theanalytical solution, a full-scale transient analysis of finite mass-in-mass chains is also conducted to obtain the numerical

Fig. 2. The dispersion curves predicted by analytical and numerical methods.

W.J. Zhou et al. / Journal of Sound and Vibration 413 (2018) 250e269254

dispersive curves in the nonlinear mass-in-mass system (see the next section), which is also plotted in Fig. 2. The way inwhich the numerical dispersion curves shown are created is provided in a flow chart given in Appendix A.

It can be clearly noticed from Fig. 2 that the L-P method can accurately predict the dispersion relation of the weaklynonlinear mass-in-mass system, including both acoustic and optical wave modes. Comparing with the acoustic wave mode,the optical mode is much more sensitive to the stiffness nonlinearity, especially for the higher wavenumber. Particularly,there is a frequency “gap” (which will be named as pseudo gap in the following) in the middle of the optical-mode wavebranch, in which the unique frequency shift is actually observed for the propagating wave packet with the frequency band inthe range of 1:78<U<2:12, which cannot be captured by using the L-P perturbation method. That is understandable becausealthough the L-P method is capable of predicting the dispersion relation of plane waves, it fails to capture the evolution ofwave packets in nonlinear AMs. In the following section, wewill show that awave packet in this frequency range propagatingin the nonlinear chain with local motions will evolve into pieces of wave packets: A high-frequency solitary wave and somelow-frequency wave packets. This distinct wave packet evolution phenomenon is the underlying mechanism for the for-mation of the pseudo-gap.

3. Spatial wave analysis of nonlinear acoustic metamaterials

To address the problem, the spatial characteristics of wave packet propagation in nonlinear AMs are studied by a transientwave analysis [27]. Consider a finite mass-in-mass system with 500 unit cells, as shown in Fig. 3. The perfectively matchedlayer (PML) [17] is adopted on both ends of the chain. The PML is composed of a gradually varied damped (linear viscous)chain, so that the incoming wave will be efficiently absorbed and dissipated while minimizing wave reflection on each end.The PML damping profile is chosen as [17]

CðnÞ ¼ Cmax

n

Npml

!3

(15)

where CðnÞ is the damping coefficient at the cell nwith n starting from 1 at the beginning of the PML and ending at Npml, Cmax

represents the maximum damping coefficient on the PML. The input signal is a Hannmodulated Ncy-cycle burst to ensure theexcitation of a minimal band of frequencies centered at a prescribed carrier frequency and enable the signals with quasi-monofrequency content. The functions of the initial displacement and velocity are

u0ðnÞ ¼A0

2

Hðn� 1Þ �H

�n� 1� Ncy

�2pk

���1� cos

�nkNcy

��sinðnkÞ

v0ðnÞ ¼ Ku0u0ðnÞ(16a)

A0 � �

2p��

_u0ðnÞ ¼ 2Hðn� 1Þ �H n� 1� Ncy

kUðnÞ

_v0ðnÞ ¼ Ku0_u0ðnÞ

(16b)

Fig. 3. The finite nonlinear mass-in-mass system with PML on both ends.

W.J. Zhou et al. / Journal of Sound and Vibration 413 (2018) 250e269 255

where the initial velocity profile has been chosen to suppress the left-going waves, HðxÞ is the Heaviside function, andfunction UðnÞ is

UðnÞ ¼ udUNcy

sin�nkNcy

�sinðnkÞ � udU

�1� cos

�nkNcy

��cosðnkÞ (17)

It should be noted that Eq. (16) describes the wave packet, where the factor

A02

Hðn� 1Þ � H

�n� 1� Ncy

�2pk

���1� cos

�nkNcy

��determines the envelope of the wave packet, while sinðnkÞ corresponds

to the carrier wavewithNcy being the number of cycles. In the numerical simulations, we have takenNcy ¼ 7 in Sections 3 and4, and Ncy ¼ 60 in Section 5 for different purposes. It should be noted that the energy of the wave packet will be concentratedaround the central frequency, while Ncy controls the bandwidth of the energy concentration. The more cycles in the function,the narrower the energy spectrum becomes around the central frequency. The wave packet (or tone burst) combines some ofthe useful features of the pulse and the continuous sinusoidal tone. It can therefore be particularly effective when theapplication involves pulse and sinewave (harmonic) testing.

The input signal is prescribed in the form of initial displacement and velocity profile

uðx½n�;0Þ ¼ u0ðx½n�Þ vðx½n�;0Þ ¼ v0ðx½n�Þ_uðx½n�;0Þ ¼ _u0ðx½n�Þ _vðx½n�;0Þ ¼ _v0ðx½n�Þ (18)

The numerical integration routine of MATLAB, ODE45, is used to integrate the nonlinear system. The 2D-FFT (Fast FourierTransform) of the time-space response is evaluated to numerically determine the dispersion relation. The wavenumber andfrequency corresponding to the maximum spectral amplitude of the 2D-FFT are determined by sweeping wave frequencies atthe desired range to numerically determine the dispersive curve, as plotted in Fig. 2.

3.1. Spatial characteristics of wave motion

To monitor the wave spatial evolution of the input signal, consider the transient wave propagation in the finite mass-in-mass system, shown in Fig. 3. The initial condition is described by Eqs. (16) and (18), with carrier cycle Ncy ¼ 7. For both theacoustic mode and the optical mode, three initial wavenumbers are chosen as k ¼ p=9, k ¼ p=2 and k ¼ 7p=9. In the simu-lation, the propagation time of acoustic-mode wave is set to be Tsimulate ¼ 2 s and the propagation time of optical-mode waveis chosen to be Tsimulate ¼ 0:8 s.

The acoustic-mode wave packet at the end of simulation is plotted in Fig. 4 for different nonlinearities (nonlinear pa-

rameters being εGjA0j2 ¼ 0 (linear system), εGjA0j2 ¼ 0:03 and εGjA0j2 ¼ 0:06, respectively) and different wavenumbers. InFig. 4, the vertical coordinate is the normalized displacement u=A0. From Fig. 4(a)e(c), it is observed that, for very smallwavenumbers k ¼ p=9 , the wave packet is unaffected by dispersion and travels through the system without any distortion.The effect of nonlinearity on the wave packet is also negligible in this region. As the wavenumber is increased to k ¼ p=2 andk ¼ 7p=9, dispersion-associated distortion is observed in the linear system in the form of stretching of the wave packet anddecrease of amplitude. At the same time, it can be noticed that stretching of the wave profile with a significantly differentdistribution of amplitude is observed with the increase of the magnitude of nonlinearity. The generated wave packet in thenonlinear mass-in-mass chain exhibits two distinctive features: A low-amplitude distributed feature and a high-amplitudelocalized feature. The high-amplitude localized wave is a solitary wave resulting from the interplay between nonlinearityand dispersion [27].

The optical-mode wave packet at the end of simulation is plotted in Fig. 5 for different values of nonlinearity and differentwavenumbers. Similarly, it is found that, for very small wavenumber k ¼ p=9, the wave packet is not dispersive and almostunaffected by nonlinearity. However, for optical-mode wave packet with wavenumber k ¼ p=2, the behavior of the wavepacket is totally different from that of acoustic mode with the same wavenumber, since its distortion becomes more obvious

Fig. 4. Spatial profile of the acoustic mode wave packet at the end of simulation for different values of nonlinearity. (a) linear chain with εGjA0j2 ¼ 0; (b)nonlinear chain with εGjA0j2 ¼ 0:03; (c) nonlinear chain with εGjA0j2 ¼ 0:06.

W.J. Zhou et al. / Journal of Sound and Vibration 413 (2018) 250e269256

in the form of stretching of the wave packet and reduction of amplitude as the nonlinearity is increased. Furthermore, weobserve that the initial wave packet now evolves into pieces of wave packets, among which there is a relatively high-amplitude localized component which is a solitary wave which will be shown in the following section. These wavepackets have different group velocities, implying that they may possess different frequency components.

In summary, for the acoustic-mode wave packet with wavenumber k ¼ p=2 or k ¼ 7p=9, as well as the optical-mode wavepacket with wavenumber k ¼ 7p=9, propagating wave in nonlinear chain has two features, the first one is high-amplitudelocalized wave and the other one is a low-amplitude distributed wave. Furthermore, the amplitude of the initial wavepacket is preserved as the wave travels through the nonlinear AM, i.e., the maximum amplitude of the wave packet at the endof simulation is almost equal to the maximum amplitude of the initial wave packet. This conservation of maximum amplitudeis attributed to the interplay between nonlinearity and dispersion, leading to the formation of solitary waves. This phe-nomenon will be studied in-depth in the following sections. It is also noticed that, for nonlinear wave propagation, thedistribution of spatial stretch of wave packet is non-smooth, from which we can expect that the distribution of spectralamplitude is non-smooth. In the following section, the distribution of spectral amplitude is investigated by monitoring theshort-space characteristics of the spatiotemporal evolution of wave motion.

3.2. Short-space spectral characteristics of wave motion

The Short Term Fourier Transform (STFT) method is then utilized to study the evolution of spectral characteristics of thewave packet in space and time. Only the space-dependent wavenumber/frequency characteristics are monitored since theamplitude of super-harmonics generated in the nonlinear system is negligible. A Hann window is used to contain the short-space components and its length is chosen to be equal to the length of the initial burst formeasurement of thewave distortion[27]. In the study, we focus on the spatiotemporal analysis of optical-mode wave propagating in the linear and nonlinearsystems.

Fig. 6 shows the spatial spectrograms of the optical-mode wave in different nonlinear metamaterial systems for twodistinct wavenumbers k ¼ p=2 and k ¼ 7p=9 in the Brillouin zone. For the wavenumber k ¼ p=9, the spatial spectrograms arenot presented since it can be deduced directly from Fig. 5(B-c) that the initial band of frequencies is preserved in both linear

Fig. 5. Spatial profile of the optical mode wave packet at the end of simulation for different values of nonlinearity. (a) linear chain with εGjA0j2 ¼ 0; (b) nonlinearchain with εGjA0j2 ¼ 0:03; (c) nonlinear chain with εGjA0 j2 ¼ 0:06.

W.J. Zhou et al. / Journal of Sound and Vibration 413 (2018) 250e269 257

and nonlinear AMs. The region between the two white dashed lines denotes Hann window. In Fig. 6(a), the wavenumbercontent of the wave packet at the end of the simulation is a little distorted compared with that of the initial wave packet,which means the wave packet is slightly more dispersive. In Fig. 6(b) and (c), the wavenumber content at the end of thesimulation is distorted severely. Particularly, the spectrograms in Fig. 6(b) clearly show the existence of three wave packets.The first packet has low-wavenumber/frequency components and is dispersive. The wavenumber components of the secondpacket are in the Hannwindow and are also dispersive. Finally, the last wave packet is nondispersive according to the energydensity distribution, localized in a region higher than the Hann window. In the following section, the 2D-FFT simulationresults will show that the third wave packet is a solitary wave. We can also observe multiple wave packets from the spec-trograms in Fig. 6(c), including some dispersive wave packets and a nondispersive localized wave packet, which is a solitarywave. We can further observe that most of the energy density is located outside the Hann window, indicating a strongfrequency shift of the initial wave packet. Such evolution of the wave packet into pieces of wave packets accompanied with astrong frequency shift is the direct cause of a pseudo-gap. For the case of k ¼ 7p=9 (see Fig. 6(def)), extra-frequency/wavenumber content out of the Hann window is also generated, although the main frequency content is not shifted. Asalso shown in Fig. 6(def), spectral amplitude is distributed over the length of the distorted wave in the linear metamaterialsystem whereas localization of the initial frequency content is observed in nonlinear system.

The spectrograms in Fig. 6 have shown that the interplay of dispersion, nonlinearity and local motions may localize or shiftthe frequency content of the signal, enhance the distortion of the frequency content or even divide the initial signal intopieces of wave profiles by generating extra frequency bands. As the wavenumber is increased, the localization of spectralcontent is more prominent and the difference in the distribution of spectral amplitude between linear and nonlinear systemsis clearly demarcated.

3.3. Spectral characteristics of wave motion

In order to numerically reconstruct the dispersion curves, the wavenumber and frequency corresponding to the point ofmaximum spectral amplitude were determined from the wave field of each numerical simulation. In contrast, one could

Fig. 6. Spectrograms of the optical mode wave packets in space at the beginning and end of simulation for different wavenumbers and different magnitudes ofnonlinearity. (aec): k ¼ p=2; (def): k ¼ 7p=9.

W.J. Zhou et al. / Journal of Sound and Vibration 413 (2018) 250e269258

monitor the complete spectrum in the neighborhood of the carrier frequencyewavenumber pair with the objective ofstudying the profile of the entire spectral content of the wave. Fig. 7 shows the contour lines of the spectral amplitudefunction U(x, U) of optical-mode wave for different values of nonlinearity in two points of the Brillouin zone as k ¼ p=2 andk ¼ 7p=9.

Fig. 7 indicates that the evolutions of spectral contours in the linear and nonlinear AMs are significantly different. Forexample of k ¼ p=2 (see Figs. (B-c)), the spectral contour lines in linear system (Fig. 7(a)) present as a quasi-linear profile,which indicates that the wave propagation is nearly undistorted. However, the spectral contours of this wave packet prop-

agating in the nonlinear chain with εGjA0j2 ¼ 0:03 (Fig. 7(b)) have three distinct spectral features: A dominant linear contourprofile above the initial frequency band, a quasi-linear contour profile in the initial band, and a nonlinear contour profilebelow the initial band. From the comparison of Fig. 7(b) with Fig. 6(b) we can conclude that: 1) the lowest wave packet inFig. 6(b) is dispersive because its spectral contour profile is nonlinear; 2) the wave packet in the Hannwindow (i.e. the initialfrequency band) is weakly dispersive; 3) the wave packet above the Hannwindow is a solitary wave since its spectral contourprofile is linear, which means it has a constant group velocity and consequently can propagate with an unchanged waveform.

Fig. 7. 2-D spectral contours of the optical mode wave packets for different points in the first Brillouin zone. (aec): k ¼ p=2; (def): k ¼ 7p=9.

W.J. Zhou et al. / Journal of Sound and Vibration 413 (2018) 250e269 259

The comparison of Fig. 7(c) with Fig. 6(c) also illustrates the existence of a solitary wave along with multiple dispersive wavepackets. Furthermore, most of the wave energy density is confined within the solitary wave as well as the dispersive wavepackets with frequency components below the initial band. This observation phenomenologically illustrates the mechanismof the formation of pseudo-gap. A wave packet with frequency in the pseudo gap will evolve into several pieces of wavepackets, including a solitary wave and multiple dispersive wave packets. These wave packets with frequency componentsdistributed out of the initial frequency band possess most of the wave energy. Thus, within the Hann window the energy ofthe received wave packet at the end of the simulation is much weaker than the initial wave energy, making the wave packetlook like being attenuated. We should emphasize here that such a pseudo-gap mechanism is not a real band gap, because the‘missing’ energy of the input wave packet is not dissipated or captured by the local resonance. Instead, most of the waveenergy is transferred out of the initial wavenumber/frequency range.

For the short wavelength optical-mode wave k ¼ 7p=9 (Fig. 7(def)), the nonlinear contour profile in linear chain isobserved, and there are also two distinctive spectral features in the nonlinear system, one is a dominant linear contour profilewhile the other is a nonlinear contour profile. The linear contour profile represents solitary wave, while the nonlinear contourleads to distortion.

The spectral contours of the nonlinear chain in Fig. 7 are qualitatively different from the nature of the global dispersioncurve observed in Fig. 2. In order to enunciate the global dispersive characteristic of the nonlinear system, the dispersioncurves are regenerated with the complete spectral content instead of using the maximum spectral amplitude, i.e., the su-perposition of multiple bursts is performed to reconstruct the global dispersion curve.

The global dispersion curves of the optical wave modes for different magnitudes of nonlinearity are shown in Fig. 8

(nonlinear parameters being εGjA0j2 ¼ 0 and εGjA0j2 ¼ 0:06, respectively). For the comparison, the dispersion curves pre-dicted by the L-P method is also plotted in the figure. Fig. 8(a) depicts the case of the linear system. It is observed that the

W.J. Zhou et al. / Journal of Sound and Vibration 413 (2018) 250e269260

reconstructed global dispersion curve from the consecutive spectral contours almost overlap with the linear dispersion curve

predicted by the L-P method. For the nonlinear chain with εGjA0j2 ¼ 0:06 (Fig. 8(b)), consecutive spectral contours do notexactly overlap to form a single curve. It indicates that there is no difference between the spectral contours and the pre-dictions by the L-P method in nonlinear wave propagation in the long wavelength limit. However, as the wavenumber isincreased, the evolution of spectral contours in the nonlinear AMs becomes significantly different from the prediction by theL-Pmethod. Two distinct features of the spectral contours are observed: strong linear contour profiles and amuchweaker butnonlinear contour profile. The nonlinear contour profile is consistent with the prediction from the L-P method for linearsystem, and strong linear profiles standing for solitary waves are caused by the interplay between the nonlinearity anddispersion. It should be pointed out that themaximum spectral amplitude points in each linear profile locate in the dispersioncurve of nonlinear AM chain predicted by L-P method, which in turn is used in Fig. 2 to indicate the validity of L-P method inprediction of spectral features of nonlinear AM chain.

In Fig. 8(b) as well as Fig. 7(b), (c), (e), and (f), we observe that solitary waves only exist in the short wave region rather thanthe long wave region. The multiple scales method [18,28] has been employed to theoretically confirm that solitary waves areindeed located in the short wave region and the induced solitary wave is an envelope soliton for the wave number k>1:63,and the detailed derivation can be found in Appendix B.

4. Direction-biased waveguide

Direction-biased wave guide devices have attracted numerous interest these years [30,31]. Nonlinearities in structureshave been utilized to construct waveguides having such direction-sensitive propagation behavior. For example, Ma et al. [32]used a cubically nonlinear oscillator attached to a linear periodic lattice by taking advantage of the up-conversion of fre-quencies due to the generation of higher harmonics to create an acoustic rectifier. Liang et al. numerically [33] and experi-mentally [34] demonstrated an acoustic diode formed by coupling a linear super lattice with a strongly nonlinear medium.The Zeeman effect, which is commonly applied to achieve nonreciprocal electromagnetic propagation, was introduced byFleury et al. [35] to break thewave transmission reciprocity among the three acoustic inputs and outputs of a circulator, whichconsists of a resonant ring cavity biased by a circulating fluid.

With the knowledge of wave frequency shift in the optical-branch of dispersion curve in the nonlinear system, it ispossible to construct a direction-biased waveguide to estimate the magnitude of nonreciprocal wave. In the study, theacoustic diode is designed and constructed by combining a finite nonlinear AM system with a finite linear AM system. Fig. 9illustrates schematics of the wave rectification mechanism by using the nonlinear AM system. The acoustic diode is designedso that the frequency edges of the pseudo gap of the nonlinear AM overlap those of the band gap in linear AM. The excitation

Fig. 8. Complete spectral contours of optical mode wave in a linear and nonlinear chain. (a) linear chain; (b) nonlinear chain with εGjA0j2 ¼ 0:06; The red solidcurve represents the analytical dispersion curve of nonlinear chain, while the green dashed curve is the analytical dispersion curve of linear chain. (For inter-pretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 9. Conceptual diagrams of unidirectional acoustic device. Forward configuration: driving on the nonlinear AM side, the significant frequency shift aregenerated and thus wave transmit through system. Reverse configuration: driving at the linear chain, the band gap filters out vibrations at frequencies in the gap.

Fig. 10. Schematics of the unidirectional acoustic device. (a), (b): Schematics of the AM system used in the experiments, composed of 350 mass-in-mass cellslinked by nonlinear spring and 150 mass-in-mass cells linked by linear spring. The band gap of the linear chain overlap the frequency band of input signal.

W.J. Zhou et al. / Journal of Sound and Vibration 413 (2018) 250e269 261

signal is a transient wave packet with frequency band in the pseudo gap of the nonlinear chain (or the band gap of the linearchain). If such an excitation signal is applied on the left side (nonlinear chain) of the system, significant frequency shift occursand therefore the transformed wave can propagate in the system. However, the signal excited on the right side (linear chain)of the system cannot propagate because the frequency band of the signal is within the band gap frequency range of the linearsystem. In this way, a unidirectional transmission of wave packet is achieved.

Consider a device composed of a nonlinear mass-in-mass chain with 350 unit-cells and a linear mass-in-mass chain with150 cells, as shown in Fig. 10. In order to reduce the impedance mismatch between the linear and nonlinear system, the outermassesmL1 in linear chain are assumed to be equal to those (m1) in nonlinear chain. The parameters of the nonlinear system

are m2 ¼ 1; k2 ¼ 1; εGjA0j2 ¼ 0:06, and the parameters of the linear system are

mL2 ≡ mL2=m1 ¼ 0:0397; kR2 ¼ kR2=k1 ¼ 0:1643 and kR1 ≡ kR1=k1 ¼ 1:2939, so that the pseudo gap of the nonlinear chainoverlaps the band gap of the linear chain. Therefore, the frequency range of the band gap for the linear system and the pseudogap for the nonlinear chain is 1:929<U<2:074, and the frequency range of the band gap for the nonlinear system is0:874<U<1:414. The excitation signal is selected as

uex ¼ 12A0

�HðtÞ � H

�t � 2p

udUNcy

���1� cos

�udUtNcy

��sinðudUtÞ (19)

Fig. 11. The input signal and the output signal received on the other end of the acoustic diode. (a) Forward configuration; (b) Reverse configuration.

W.J. Zhou et al. / Journal of Sound and Vibration 413 (2018) 250e269262

It should bementioned that the location andwidth of the conversion region are determined by the system parameters, the

amplitude of nonlinear parameter εGjA0j2 and the cycle numberNcy ¼ 60 of the excitation bursts. The propagation time of thesignal in the system is set as tsimulate ¼ 2 s, which is long enough for themain passable signal to propagate through the system.

The input and output displacement histories for the left to right and right to left propagation is shown in Fig. 11. As can beseen, the output for an input signal with central frequency U ¼ 2 for left to right preserves large amounts of wave energy,however, the total wave attenuation can be observed for right to left wave propagation. The frequency spectra of the input andoutput displacement, as shown in Fig. 12, reveal that the excitation frequency components are tailored when the wave ispropagated from left to right. As thewave passes first through the nonlinear system, it permits propagation due to the shiftingof the optical mode to lower and higher frequencies as predicted before. Therefore, after traversing the nonlinear system, thepropagated wave has a predominant frequency content out of the bandgap frequency range of the linear system and theresultant output at the right end of the device retains the high wave amplitude while its frequency content is shiftedsignificantly as evidenced from its spectrum in Fig. 12 (a). In contrast, the wave cannot be propagated from the right to left inthe device, which acts as direction-based waveguide for mechanical waves.

Finally, to evaluate the efficiency of the device for the nonreciprocal wave, the normalized energy flux density Pn of then-th outer masses of the input and output signals are determined by

Pn ¼ �hðun � un�1Þ þ εGðun � un�1Þ3

i vunvt

for nonlinear chain

Pn ¼ �kL1ðun � un�1Þvunvt

for linear chain(20)

and the wave transmission ratio can be defined as

W.J. Zhou et al. / Journal of Sound and Vibration 413 (2018) 250e269 263

a ¼

Z TLast

0Pres dtZ TLast

0Pin dt

(21)

where Pin represents the input energy flux density and Pres is the flux density of the respond signal received at the otherend of the device, TLast is the ending time of simulation. It can be found that for the wave from left to right, the wavetransmission ratio is as high as aþ ¼ 0:585, whilst for the reverse direction, the transmission ratio is as low asa� ¼ 1:178� 10�5. The asymmetric ratio s ¼ aþ=a� is an important parameter in quantifying the transmission asymmetryof an acoustic diode. The device designed in Ref. [30] possesses an asymmetric ratio sz104 with a time-averaged energytransmission coefficient of ~0.35%. In Ref. [33], the asymmetry ratio of the reported device is sz104, with a weak energytransmission coefficient of � 10�3. A high asymmetric ratio of sz106 was experimentally obtained in Ref. [31] with awide operating band (10 kHz) and an audible effect. Among the three ports of the compact circulator designed in Ref. [35],an isolation of 40 dB between two ports is obtained, while the transmission value in the other propagation direction liesclose to 1. The acoustic diode proposed here exhibits a large asymmetric ratio of sz5� 105, together with a high energytransmission of 58:5%.

Since our acoustic diode is designed such that it has a large asymmetry ratio along with a high forward transmission ratio,the band gap of the linear chain exactly overlaps the frequency range of the input signal. Thus, it has a high efficiency only fora certain narrow frequency range. As a result, its robustness in terms of working frequency is not satisfactory. However, bybroadening thewidth of the band gap range of the linear chain, we immediately improve the robustness of the acoustic diode,but at the cost of reducing the forward transmission ratio. In the previous section, we have shown that when a wave packetwith central wavenumber/frequency range located in the pseudo-gap is lunched in the nonlinear chain, a relatively high-amplitude solitary wave as well as multiple wave packets with frequency components located outside the pseudo-gap will

Fig. 12. The FFT plots of the input and output signal, (a) Forward configuration; (b) Reverse configuration.

W.J. Zhou et al. / Journal of Sound and Vibration 413 (2018) 250e269264

be induced. Then, if wemake the band gap range of the linear chain in Fig. 10 overlap the range of the pseudo-gap, an acousticdiode with a broad operating frequency range, large asymmetric ratio, and relatively high forward transmission ratio can beachieved. Thus, understanding the mechanism of the pseudo-gap can offer valuable guidance for the design of high-performance acoustic diodes.

5. Conclusion

Wave packet propagation of nonlinear acoustic metamaterials with nonlinear stiffness in the microstructure is studiedusing approximate analytical models and numerical simulations. The spatial and spectral features of wave propagation ineach case have been monitored to study the interplay between nonlinear, dispersion and local resonance mechanisms inwave packet propagation. The analysis is based on full-scale transient analysis of the finite system, from which dispersioncurves are generated from the transmitted waves, which is also verified by the perturbation methods. The evolution ofspatial and spectral features is monitored using spatial-spectrogram analysis and 2D FFT simulations, and the interplay ofdispersive and nonlinear and local resonance mechanisms in the process of waveform distortion is evaluated. In general, itcan be concluded that nonlinearity gives rise to nondispersive features in wave propagation. It is also found that the opticalwave modes propagating in the nonlinear metamaterial are sensitive to the parameters of the nonlinear constitutiverelation and significant wave shift phenomena is found. Specifically, a strong frequency shift phenomenon is observed for atransient wave packet with frequency located in the pseudo gap. Such wave packet evolution into pieces of wave packetswith a strong frequency shift is a distinctive characteristic of wave packet propagation in nonlinear chains with localresonance. Finally, the feasibility of the nonlinear acoustic metamaterials for direction-biased waveguides is numericallydemonstrated and good behavior of the acoustic diode is observed i.e., the asymmetric coefficient is sz5� 105, with goodenergy transmission of forward configuration, as large as 58:5%, much more effective than most of the existing designedacoustic rectifiers.

Acknowledgements

The work is supported by the Air Force Office of Scientific Research under Grant No. AF 9550-15-1-0016 with ProgramManager Dr. Byung-Lip (Les) Lee and the National Natural Science Foundation of China (Nos. 11532001, 11532001 and11621062). Partial support from the Fundamental Research Funds for the Central Universities is also acknowledged.

Appendix A. A flow chart to determine dispersion curve

The flow chart to numerically determine the dispersion curves is shown in Fig. A1.

Fig. A1. The flow chart to obtain the dispersion curves in Fig. 2.

Appendix B. Derivation of the formation of solitary wave

The multiple scales method [18,28] is adopted to obtain the solitary wave solutions in Eq. (2). Based on this method, thesolution of the wave equation (2) can be assumed as

W.J. Zhou et al. / Journal of Sound and Vibration 413 (2018) 250e269 265

unðtÞ ¼ u0ðxn; ~t;fnÞ þ cu1ðxn; ~t;fnÞ þ c2u2ðxn; ~t;fnÞ þ/

¼X∞p¼0

cpupn;n

vnðtÞ ¼ v0ðxn; ~t;fnÞ þ cv1ðxn; ~t;fnÞ þ c2v2ðxn; ~t;fnÞ þ/

¼X∞p¼0

cpvpn;n

(B-1)

ffiffiffip p p

where ε is a small quantity defined in the main text, and hence c ¼ ε is also a small quantity, um;n ¼ u ðxm; ~t;fnÞ,vpm;n ¼ vpðxm; ~t;fnÞ and

xn ¼ cðna� ltÞ; ~t ¼ c2t; fn ¼ nqa� ut (B-2)

where q is the wavenumber, u is the frequency and l is a parameter to be determined. By substituting of Eq. (B-1) into Eq. (2),and equating the coefficients of c0; c1 and c2 separately, we obtain the governing equations as8>>>>>>><

>>>>>>>:

m1

v2

vt2þ k2

!upn;n þ k1

�2upn;n � upn;nþ1 � upn;n�1

�� k2v

pn;n ¼ Mp

n

m2

v2

vt2þ k2

!vpn;n � k2u

pn;n ¼ Np

n

ðp ¼ 0;1;2Þ (B-3)

where

M0n ¼ 0; N0

n ¼ 0

M1n ¼ 2m1l

v

vxn

v

vtu0n;n þ k1a

v

vxn

�u0n;nþ1 � u0n;n�1

�

N1n ¼ 2m2l

v

vxn

v

vtv0n;n

(B-4)

and

M2n ¼ m1

2l

v

vxn

v

vtu1n;n � l2

v2

vx2nu0n;n � 2

v

v~t

v

vtu0n;n

!

�G

��u0n;n � u0n;nþ1

�3 þ �u0n;n � u0n;n�1

��3

þk1

12a2

v2

vx2n

�u0n;nþ1 þ u0n;n�1

�þ a

v

vxn

�u1n;nþ1 � u1n;n�1

�!

N2n ¼ m2

2l

v

vxn

v

vtv1n;n � l2

v2

vx2nv0n;n � 2

v

v~t

v

vtv0n;n

!(B-5)

Decoupling upn;n from vpn;n, we obtain from Eq. (B-3)

8>>>>>>>>>>><>>>>>>>>>>>:

u�2d

v2

vt2þ 1

! m�1

2 u�2d

v2

vt2þ 1

!upn;n þ k

�12

u�2d

v2

vt2þ 1

!�2upn;n � upn;nþ1 � upn;n�1

�� upn;n

¼ u�2d

v2

vt2þ 1

!k�12 Mp

n þ k�12 Np

n

u�2d

v2

vt2þ 1

!vpn;n ¼ k�1

2 Npn þ upn;n

(B-6)

When p ¼ 0

W.J. Zhou et al. / Journal of Sound and Vibration 413 (2018) 250e269266

8>>>>><>>>>>:

u�2d

v2

vt2þ 1

! m�1

2 u�2d

v2

vt2þ 1

!u0j;j þ k

�12

u�2d

v2

vt2þ 1

!�2u0n;n � u0n;nþ1 � u0n;n�1

�� u0n;n ¼ 0

u�2d

v2

vt2þ 1

!v0n;n ¼ u0n;n

(B-7)

The solution to Eq. (B-7)1 can be obtained as

u0n;n ¼ Aðxn; ~tÞeifn þ c:c;

fn ¼ nk� ut;(B-8)

where u satisfies

m�12 u4 �

h�m�1

2 þ 1�þ 2k

�12 ð1� cosðkÞÞ

iu2du

2 þ 2k�12 ð1� cosðkÞÞu4

d ¼ 0 (B-9)

In view of Eq. (B-8), we obtain the solution to Eq. (B-7)2 as

v0n;n ¼ 1

1� u2.u2d

Aðxn; ~tÞeifn þ c:c; (B-10)

Substituting Eqs. (B-8) and (B-10) into (B-5) (with p ¼ 1), we have

M1n;n ¼ �2im1ul

v

vxnAeifn þ 2ik1asinðkÞ

v

vxnAeifn þ c:c

N1n;n ¼ �2im2ul

1� u2.u2d

v

vxnAeifn þ c:c

(B-11)

Substituting of Eq. (B-11) into Eq. (B-6) yields8>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>:

u�2d

v2

vt2þ 1

! m�1

2 u�2d

v2

vt2þ 1

!u1n;n þ k

�12

u�2d

v2

vt2þ 1

!�2u1n;n � u1n;nþ1 � u1n;n�1

�� u1n;n

¼ �2i�1� u�2

d u2�264m�1

2 u�2d ul� k

�12 asinðkÞ þ u�2

d ul�1� u2

.u2d

�2375 v

vxnAeifn þ c:c

u�2d

v2

vt2þ 1

!v1n;n ¼ �2iuu�2

d l

1� u2.u2d

v

vxnAeifn þ u1n;n þ c:c

(B-12)

Hence the solvability equation is

l ¼m2ud

�1� U2

�2sinðkÞa�

k2U�1� U2

�2 þm2k2U� ¼ vu

vq(B-13)

Then u1n;n could be set zero, i.e. u1n;n ¼ 0, and the solution of v1n;n is

v1n;n ¼ �2iuu�2d l�

1� u2.u2d

�2 v

vxnAeifn (B-14)

Substitution of Eqs. (B-8), (B-10), (B-11) and (B-14) into Eq. (B-5) yields

W.J. Zhou et al. / Journal of Sound and Vibration 413 (2018) 250e269 267

M2n ¼

(2im1u

v

v~tAþ

�k1a

2 cos k�m1l2� v2

vx2nAþ 6Gf4 cosðkÞ � cosð2kÞ � 3gA2A

)eifn

þ2½3 cos k� 3 cosð2kÞ þ cosð3kÞ � 1�GA3e3ifn þ c:c

N2n ¼ m2

0B@ �4u2u�2

d l2�1� u2

.u2d

�2 v2

vx2nAeifn � l2

1� u2.u2d

v2

vx2nAeifn þ 2iu

1� u2.u2d

v

v~tAeifn

1CAþ c:c

(B-15)

By inserting this equation into Eq. (B-5) (with p ¼ 2), we obtain u�2d

v2

vt2þ 1

! m1

v2

vt2þ k2

!u2n;n þ k1

u�2d

v2

vt2þ 1

!�2u2n;n � u2n;nþ1 � u2n;n�1

�� k2u

2n;n

¼

264�k�1

2 a2 cos k�m�12 u�2

d l2��

1� u�2d u2

���3u2u�2

d þ 1�u�2d l2�

1� u2.u2d

�2375 v2

vx2nAeifn

þ"2im�1

2 u�2d u

�1� u�2

d u2�þ 2iuu�2

d

1� u2.u2d

#v

v~tAeifn

þ6�1� u�2

d u2� G

k2f4 cosðkÞ � cosð2kÞ � 3gA2Aeifn

þ2�1� 9u�2

d u2�½3 cosðkÞ � 3 cosð2kÞ þ cosð3kÞ � 1� G

k2A3e3ifn

(B-16)

The solvability condition of this equation is the nonlinear Schr€odinger equation (NSE)

iv

v~tAþ P

2v2

vx2nAþ QA2A ¼ 0 (B-17)

where

P ¼ �

�k2�1� U2

�3 þm2k2�1þ 3U2

��l2 þm2a

2�1� U2

�3cosðkÞu2

d

k2Uud

�1� U2

�3 þm2k2Uud

�1� U2

� ¼ v2u

vq2

Q ¼ �6Gud

k2m2U

�1½1� cosðkÞ�2�1� U2

�2,��1� U2

�2 þm2

� (B-18)

The solution of Eq. (B-18) depends on PQ [16]PQ >00envelope solitonsPQ <00dark solitons

(B-19)

When PQ >0, Eq. (B-18) admits a solution of the form

A ¼ fsech

" ffiffiffiffiffiffiQ2P

rfcðx� ltÞ

#eic

2Q2f

2t (B-20)

where f is the amplitude of the envelope.When PQ <0, Eq. (B-18) admits a solution of the form

A0 ¼ qðx; tÞeijt (B-21)

where

W.J. Zhou et al. / Journal of Sound and Vibration 413 (2018) 250e269268

qðx; tÞ ¼ f

(1�m2sech2

" ffiffiffiffiffiffiffiffi�Q2P

rmfcðx� ltÞ

#)1=2

(B-22)

and

j ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�m2 � 1

�Q

2P

sfcðx� ltÞ

þtan�1

(mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1�m2p tanh

" ffiffiffiffiffiffiffiffi�Q2P

rmfεðx� ltÞ

#)� c2

Q2

�3�m2

�f2t

(B-23)

where m is a parameter that controls the depth of the modulation of the amplitude 0 � m � 1. The center amplitude is

maximum at m ¼ 0, decreases with the increasing m, and vanishes at m ¼ 1 [16].

Fig. B1. PQ as a function of the dimensionless wavenumber k.

Fig. B1 shows the value of PQ as a function of k in the optical branch. In the range of 1:63< k � p, the system has envelopesoliton solutions because PQ is positive. In the contrast, in the range of 0 � k<1:63, the system has no envelope solitonsolutions because of negative PQ but could have dark solitons. An envelope soliton can be induced by lunching awave packet.However, the dark soliton cannot be induced by lunching a wave packet with finite wavelengths. In the study, only envelopesolitons whose wavenumber in the range of 1:63< k � p can be induced in the nonlinear mass-in-mass chain by lunching awave packet, and this explains why solitary waves observed in Fig. 8 only exist in the short wave region.

References

[1] Z. Liu, X. Zhang, Y. Mao, Y. Zhu, Z. Yang, C.T. Chan, P. Sheng, Locally resonant sonic materials, Science 289 (5485) (2000) 1734e1736.[2] J. Li, C.T. Chan, Double-negative acoustic metamaterial, Phys. Rev. E 70 (5) (2004) 055602.[3] N. Fang, D. Xi, J. Xu, M. Ambati, W. Srituravanich, C. Sun, X. Zhang, Ultrasonic metamaterials with negative modulus, Nat. Mater. 5 (6) (2006) 452e456.[4] Y. Ding, Z. Liu, C. Qiu, J. Shi, Metamaterial with simultaneously negative bulk modulus and mass density, Phys. Rev. Lett. 99 (9) (2007) 093904.[5] Y. Cheng, J. Xu, X. Liu, One-dimensional structured ultrasonic metamaterials with simultaneously negative dynamic density and modulus, Phys. Rev. B

77 (4) (2008) 045134.[6] C.T. Chan, J. Li, K.H. Fung, On extending the concept of double negativity to acoustic waves, J. Zhejiang Univ. Sci. A 7 (1) (2006) 24e28.[7] H.H. Huang, C.T. Sun, G.L. Huang, On the negative effective mass density in acoustic metamaterials, Int. J. Eng. Sci. 47 (4) (2009) 610e617.[8] R. Zhu, G.L. Huang, H.H. Huang, C.T. Sun, Experimental and numerical study of guided wave propagation in a thin metamaterial plate, Phys. Lett. A 375

(30) (2011) 2863e2867.[9] M.R. Haberman, M.D. Guild, Acoustic metamaterials, Phys. Today 3 (12) (2016) 31e39.

[10] S.A. Cummer, J. Christensen, A. Alù, Controlling sound with acoustic metamaterials, Nat. Rev. Mater. 1 (2016) 16001.[11] G. Ma, P. Sheng, Acoustic metamaterials: from local resonances to broad horizons, Sci. Adv. 2 (2) (2016), e1501595.[12] Y. Chen, H. Liu, M. Reilly, H. Bae, M. Yu, Enhanced acoustic sensing through wave compression and pressure amplification in anisotropic metamaterials,

Nat. Commun. 5 (2014) 5247.[13] Y.S. Kivshar, N. Flytzanis, Gap solitons in diatomic lattices, Phys. Rev. A 46 (12) (1992) 7972e7978.[14] J.M. Manimala, C.T. Sun, Numerical investigation of amplitude-dependent dynamic response in acoustic metamaterials with nonlinear oscillators, J.

Acoust. Soc. Am. 139 (6) (2016) 3365e3372.

W.J. Zhou et al. / Journal of Sound and Vibration 413 (2018) 250e269 269

[15] N. Nadkarni, C. Daraio, D.M. Kochmann, Dynamics of periodic mechanical structures containing bistable elastic elements: from elastic to solitary wavepropagation, Phys. Rev. E 90 (2) (2014) 023204.

[16] M. Remoissenet, Waves Called Solitons: Concepts and Experiments, Springer-Verlag, Berlin, 1999.[17] R.K. Narisetti, M.J. Leamy, M. Ruzzene, A perturbation approach for predicting wave propagation in one-dimensional nonlinear periodic structures, J.

Vib. Acoust. 132 (3) (2010) 031001.[18] K. Manktelow, M.J. Leamy, M. Ruzzene, Multiple scales analysis of waveewave interactions in a cubically nonlinear monoatomic chain, Nonlinear Dyn.

63 (2011) 193e203.[19] R. Khajehtourian, M.I. Hussein, Dispersion characteristics of a nonlinear elastic metamaterial, AIP Adv. 4 (12) (2014) 124308.[20] L. Bonanomi, G. Theocharis, C. Daraio, Wave propagation in granular chains with local resonances, Phys. Rev. E 91 (3) (2015) 033208.[21] E. Kim, F. Li, C. Chong, G. Theocharis, J. Yang, P.G. Kevrekidis, Highly nonlinear wave propagation in elastic woodpile periodic structures, Phys. Rev. Lett.

114 (11) (2015) 118002.[22] H. Xu, P.G. Kevrekidis, A. Stefanov, Traveling waves and their tails in locally resonant granular systems, J. Phys. A 48 (19) (2015) 195204.[23] P.G. Kevrekidis, A.G. Stefanov, H. Xu, Traveling waves for the mass in mass model of granular chains, Lett. Math. Phys. 106 (2016) 1067e1088.[24] L. Liu, G. James, P. Kevrekidis, A. Vainchtein, Strongly nonlinear waves in locally resonant granular chains, Nonlinearity 29 (2016) 3496.[25] L. Liu, G. James, P. Kevrekidis, A. Vainchtein, Breathers in a locally resonant granular chain with precompression, Phys. D Nonlinear Phenom. 331 (2016)

27e47.[26] S.P. Wallen, J. Lee, D. Mei, C. Chong, P.G. Kevrekids, N. Boechler, Discrete breathers in a mass-in-mass chain with hertzian local resonators, Phys. Rev. E

95 (2017) 022904.[27] R. Ganesh, S. Gonella, Spectro-spatial wave features as detectors and classifiers of nonlinearity in periodic chains, Wave Motion 50 (4) (2013) 821e835.[28] G. Huang, Soliton excitations in one-dimensional diatomic lattices, Phys. Rev. B 51 (18) (1995) 12347e12360.[29] V. Tournat, V.E. Gusev, B. Castagnede, Self-demodulation of elastic waves in a one-dimensional granular chain, Phys. Rev. E 70 (5) (2004) 056603.[30] N. Boechler, G. Theocharis, C. Daraio, Bifurcation-based acoustic switching and rectification, Nat. Mater. 10 (9) (2011) 665e668.[31] T. Devaux, V. Tournat, O. Richoux, V. Pagneux, Asymmetric acoustic propagation of wave packets via the self-demodulation effect, Phys. Rev. Lett. 115

(23) (2015) 234301.[32] C. Ma, R.G. Parker, B.B. Yellen, Optimization of an acoustic rectifier for uni-directional wave propagation in periodic massespring lattices, J. Sound Vib.

332 (20) (2013) 4876e4894.[33] B. Liang, B. Yuan, J. Cheng, Acoustic diode: rectification of acoustic energy flux in one-dimensional systems, Phys. Rev. Lett. 103 (10) (2009) 104301.[34] B. Liang, X.S. Guo, J. Tu, D. Zhang, J.C. Cheng, An acoustic rectifier, Nat. Mater. 9 (12) (2010) 989e992.[35] R. Fleury, D.L. Sounas, C.F. Sieck, et al., Sound isolation and giant linear nonreciprocity in a compact acoustic circulator, Science 343 (6170) (2014)

516e519.