1 s2.0-s0022460 x17307472-main

Spectro-spatial analysis of wave packet propagation in
nonlinear 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, China
b
Department of Mechanical and Aerospace Engineering, University of Missouri, Columbia, MO 65211, USA
c
Institute of Engineering Mechanics, Beijing Jiaotong University, Beijing, China
d
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 2017
Received in revised form 9 October 2017
Accepted 13 October 2017
Available online 31 October 2017
Keywords:
Nonlinear acoustic metamaterial
Spectro-spatial analysis
Direction-biased waveguide
a b s t r a c t
The objective of this work is to analyze wave packet propagation in weakly nonlinear
acoustic metamaterials and reveal the interior nonlinear wave mechanism through
spectro-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 transmitted
waves and also verified by the perturbation method (the L-P method). We found that the
spectro-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 the
optical wave modes in the nonlinear metamaterial are sensitive to the parameters of the
nonlinear constitutive relation. Specifically, a significant frequency shift phenomenon is
found in the middle-wavelength region of the optical wave branch, which makes this
frequency region behave like a band gap for transient waves. This special frequency shift is
then used to design a direction-biased waveguide device, and its efficiency is shown by
numerical 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 structures
and materials, acoustic metamaterials (AMs) are a very promising candidate [1e11]. AMs are, broadly speaking, materials
with 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 to
their engineered microstructural configurations. Inasmuch as the underlying physics involved in realizing the unusual
behavior 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 for
engineering structural applications may thus be viewed as a new design philosophy with a unified focus on tailoring their
form, function and performance using microstructural configurations across length-scales [12].
* Corresponding author. Department of Mechanical and Aerospace Engineering, University of Missouri, Columbia, MO 65211, USA.
E-mail address: huangg@missouri.edu (G.L. Huang).
Contents lists available at ScienceDirect
Journal of Sound and Vibration
journal homepage: www.elsevier.com/locate/jsvi
https://doi.org/10.1016/j.jsv.2017.10.023
0022-460X/© 2017 Elsevier Ltd. All rights reserved.
Journal of Sound and Vibration 413 (2018) 250e269

The combination of nonlinearity into material or boundary periodicity may produce many new wave phenomena
including the gap solitons [13], tunable band gaps [14], envelope and dark solitons [15,16] to name a few. Nonlinear periodic
materials/structures have attracted considerable interest in recent years due to the fantastic wave phenomena in them. For
example, Narisetti et al. [17] developed a perturbation analysis-based approach to obtain approximate solutions for the
dispersion behavior of various one-dimensional cubically nonlinear periodic chains. Manktelow et al. [18] used a multiple
scales technique to analyze wave-wave interactions in a one-dimensional monoatomic mass-spring chain with cubic
nonlinearity and found that the interaction of two waves generates different amplitude- and frequency-dependent dispersion
branches, 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 well
as other interesting phenomena, such as the propagation of nanoptera [21], a strongly localized solitary wave followed by a
small amplitude oscillatory tail, can be achieved/observed. The generation of such oscillatory tail was further discussed in
detail 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 bright
and 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-mass
chain with Hertzian nonlinear local resonators.
Most of studies on nonlinear periodic structures concentrate on dispersion characteristics or propagation of solitary waves
in 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 better
understanding of the solitons or discrete breathers in short-wavelength region needs more considerations. Thus, Ganesh and
Gonella [27] provided an elegant method to study the spectro-spatial wave features of nonlinear periodic chains, which could
present both the local wave properties (such as existence of solitary waves) and the global wave features (such as the
dispersion 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 oscillator
microstructure. The LindstedtePoincare (L-P) perturbation method [17] is first adopted to obtain the approximate dispersion
relation and displacement solutions. The displacement and velocity fields are then set as initial conditions to numerically
simulate 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 to
elucidate 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 transient
wave packet with initial frequency band located in this ‘gap’, an intensive frequency shift phenomenon can be observed
during the propagation, and the frequency components of the wave packet are transferred out of the ‘gap’ region, making this
region behave like a band gap. We should emphasize that such ‘gap’ works only for transient wave signal but not for
monochrome harmonic wave. Such kind of frequency transformation is totally different from the traditional frequency shift in
nonlinear phononic crystals. Finally, by employing such special but strong frequency shift phenomenon, a wave device for
direction-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 a mass-in-mass chain, as shown in Fig.1. Each unit-cell consists of a
rigid mass, m1, a nonlinear spring whose stiffness is governed by two parameters e a linear stiffness, k1 and a nonlinear
parameter, G, the cubic nonlinear parameter, and an interior oscillator consisting of a mass, m2, and a linear spring, k2. The
length of a cell is a. For the weak nonlinearity, we introduce a small parameter ε so that the restoring force fr in the nonlinear
spring 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) 250e269 251

fr ¼ k1d þ 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 À unÀ1 À unþ1Þ þ εG
h
ðun À unÀ1Þ3
þ ðun À unþ1Þ3
i
þ k2ðun À vnÞ ¼ 0 (2a)
m2 €vn þ k2ðvn À unÞ ¼ 0 (2b)
where un and vn indicate the displacements of the n-th outer and inner masses, respectively, and the superscripted dot
denotes time derivation. Eq. (2) can be normalized with respect to the linear spring constant k1, mass m1, and can be
expressed as
k2
m2u2
d
€un þ ð2un À unÀ1 À unþ1Þ þ εG
h
ðun À unÀ1Þ3
þ ðun À unþ1Þ3
i
1
u2
d
€vn þ ðvn À unÞ ¼ 0 (3b)
where G ¼ G=k1, ud ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
k2=m2
p
, m2 ¼ m2=m1 and k2 ¼ k2=k1.
The LindstedtePoincare (L-P) perturbation method [17] is adopted to obtain the dispersion relation of the system. By
defining the dimensionless time t ¼ ut, the normalized equations of wave motion (3) can be rewritten as
k2U2
m2
v2
un
vt2
þ ð2un À unÀ1 À unþ1Þ þ εG
h
ðun À unÀ1Þ3
þ ðun À unþ1Þ3
i
U2v2
vn
vt2
þ ðvn À unÞ ¼ 0 (4b)
where U ≡
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u2=u2
d
q
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 u
ð0Þ
n þ εu
ð1Þ
n vn ¼ v
ð0Þ
n þ εv
ð1Þ
n (5)
Specifically, by substituting Eq. (5) into Eq. (4) and separating the coefficients of ε0; ε1, we obtain the governing equations
as
k2U2
0
m2
d2
u
ðaÞ
n
dt2
þ

2u
ðaÞ
n À u
ðaÞ
nÀ1 À u
ðaÞ
nþ1

þ k2

u
ðaÞ
n À v
ðaÞ
n

¼ Ma (6a)
U2
0
v2
v
ðaÞ
n
vt2
þ

v
ðaÞ
n À u
ðaÞ
n

¼ Na (6b)
in which the subscript a ¼ 0; 1 and
M0 ¼ 0
N0 ¼ 0
(7a)
M1 ¼ À2
k2
m2
U0U1
d2
u
ð0Þ
j
dt2
À G

u
ð0Þ
n À u
ðaÞ
nÀ1
3
þ

u
ð0Þ
n À u
ðaÞ
nþ1
3
!
N1 ¼ À2U0U1
d2
v
ð0Þ
n
dt2
(7b)
The lowest order recovers the governing equation of the corresponding linear AM and the higher order expansions
manifest as a heterogeneous form of the linear system with the nonlinear terms acting as the forcing function. The frequency
correction 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) 250e269252

u
ð0Þ
n ¼
A0
2
einkeit þ c:c (8a)
v
ð0Þ
n ¼
B0
2
einkeit þ c:c (8b)
where i ¼
ffiffiffiffiffiffiffi
À1
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 have
n
1 À U2
0
h
À k2U2
0 þ 2m2ð1 À cos kÞ
i
À k2m2U2
0
o
A0 ¼ 0 (9a)
B0 ¼ Ku0
A0 (9b)
where Ku0
¼ 1
1ÀU2
0
. For nontrivial solutions, the dispersion relation of the linear wave can be determined as

1 À U2
0
h
À k2U2
0 þ 2m2ð1 À cos kÞ
i
À k2m2U2
0 ¼ 0 (10)
It can be proved that Eq. (10) yields two positive real roots, the smaller one corresponds to the acoustic branch of the
system 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
U2
0
d2
dt2
þ 1
!
k2
m2
U2
0
d2
u
ð1Þ
n
dt2
þ

2u
ð1Þ
n À u
ð1Þ
nÀ1 À u
ð1Þ
nþ1

#
þ k2U2
0
d2
dt2
u
ð1Þ
n ¼ c1einkeit þ c2e3inke3it þ c:c
(11a)

U2
0
d2
dt2
þ 1

v
ð1Þ
n ¼ U0U1KuA0einkeit þ u
ð1Þ
n (11b)
where
c1 ¼

1 À U2
0

k2
m2
U0U1A0 À
3
2
Gð1 À cos kÞ2
A2
0A*
0
#
þ k2U0U1Ku0
A0 (12a)
c2 ¼
1
2
G k2 À 9
k2
m2
U2
0
!

2 cos3
k þ 3 cos2
k À 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 À cos kÞ2
jA0j2
2U0

k2 þ m2k2K2
u0
(13)
Finally, for a weakly nonlinear mass-in-mass chain, the reconstituted dispersion relation is then given as
Uac ¼ U þ ε
3Gð1 À cos kÞ2
jA0j2
2U0ac

k2 þ m2k2K2
u0ac
(14a)
Uop ¼ U0op þ ε
3Gð1 À cos kÞ2
jA0j2
2U0op

k2 þ m2k2K2
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 as m2 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 the
analytical solution, a full-scale transient analysis of finite mass-in-mass chains is also conducted to obtain the numerical

dispersive curves in the nonlinear mass-in-mass system (see the next section), which is also plotted in Fig. 2. The way in
which 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 weakly
nonlinear 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 wave
branch, in which the unique frequency shift is actually observed for the propagating wave packet with the frequency band in
the range of 1:78 U 2:12, which cannot be captured by using the L-P perturbation method. That is understandable because
although the L-P method is capable of predicting the dispersion relation of plane waves, it fails to capture the evolution of
wave packets in nonlinear AMs. In the following section, we will show that a wave packet in this frequency range propagating
in the nonlinear chain with local motions will evolve into pieces of wave packets: A high-frequency solitary wave and some
low-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 transient
wave analysis [27]. Consider a finite mass-in-mass system with 500 unit cells, as shown in Fig. 3. The perfectively matched
layer (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 n with 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 Hann modulated Ncy-cycle burst to ensure the
excitation 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 ¼
A0
2

Hðn À 1Þ À H n À 1 À Ncy

2p
k
!'
1 À cos

nk
Ncy
!
sinðnkÞ
v0ðnÞ ¼ Ku0
u0ðnÞ
(16a)
_u0ðnÞ ¼
A0
2


2p
k
!'
UðnÞ
_v0ðnÞ ¼ Ku0
_u0ðnÞ
(16b)
Fig. 2. The dispersion curves predicted by analytical and numerical methods.

where the initial velocity profile has been chosen to suppress the left-going waves, HðxÞ is the Heaviside function, and
function UðnÞ is
UðnÞ ¼
udU
Ncy
sin

nk
Ncy

sinðnkÞ À udU 1 À cos

nk
Ncy
!
cosðnkÞ (17)
It should be noted that Eq. (16) describes the wave packet, where the factor
A0
2


2p
k
!'
1 À cos

nk
Ncy
!
determines the envelope of the wave packet, while sinðnkÞ corresponds
to the carrier wave with Ncy being the number of cycles. In the numerical simulations, we have taken Ncy ¼ 7 in Sections 3 and
4, and Ncy ¼ 60 in Section 5 for different purposes. It should be noted that the energy of the wave packet will be concentrated
around 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 of
the useful features of the pulse and the continuous sinusoidal tone. It can therefore be particularly effective when the
application involves pulse and sinewave (harmonic) testing.
The input signal is prescribed in the form of initial displacement and velocity profile
uðx½nŠ; 0Þ ¼ u0ðx½nŠÞ vðx½nŠ; 0Þ ¼ v0ðx½nŠÞ
_uðx½nŠ; 0Þ ¼ _u0ðx½nŠÞ _vðx½nŠ; 0Þ ¼ _v0ðx½nŠÞ
(18)
The numerical integration routine of MATLAB, ODE45, is used to integrate the nonlinear system. The 2D-FFT (Fast Fourier
Transform) of the time-space response is evaluated to numerically determine the dispersion relation. The wavenumber and
frequency corresponding to the maximum spectral amplitude of the 2D-FFT are determined by sweeping wave frequencies at
the 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 the
acoustic 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 wave
is 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. In
Fig. 4, the vertical coordinate is the normalized displacement u=A0. From Fig. 4(a)e(c), it is observed that, for very small
wavenumbers 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 and
k ¼ 7p=9, dispersion-associated distortion is observed in the linear system in the form of stretching of the wave packet and
decrease of amplitude. At the same time, it can be noticed that stretching of the wave profile with a significantly different
distribution of amplitude is observed with the increase of the magnitude of nonlinearity. The generated wave packet in the
nonlinear mass-in-mass chain exhibits two distinctive features: A low-amplitude distributed feature and a high-amplitude
localized feature. The high-amplitude localized wave is a solitary wave resulting from the interplay between nonlinearity
and dispersion [27].
The optical-mode wave packet at the end of simulation is plotted in Fig. 5 for different values of nonlinearity and different
wavenumbers. Similarly, it is found that, for very small wavenumber k ¼ p=9, the wave packet is not dispersive and almost
unaffected by nonlinearity. However, for optical-mode wave packet with wavenumber k ¼ p=2, the behavior of the wave
packet is totally different from that of acoustic mode with the same wavenumber, since its distortion becomes more obvious
Fig. 3. The finite nonlinear mass-in-mass system with PML on both ends.

in the form of stretching of the wave packet and reduction of amplitude as the nonlinearity is increased. Furthermore, we
observe 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 wave
packets 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 wave
packet with wavenumber k ¼ 7p=9, propagating wave in nonlinear chain has two features, the ﬁrst one is high-amplitude
localized wave and the other one is a low-amplitude distributed wave. Furthermore, the amplitude of the initial wave
packet is preserved as the wave travels through the nonlinear AM, i.e., the maximum amplitude of the wave packet at the end
of simulation is almost equal to the maximum amplitude of the initial wave packet. This conservation of maximum amplitude
is 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, the
distribution of spatial stretch of wave packet is non-smooth, from which we can expect that the distribution of spectral
amplitude is non-smooth. In the following section, the distribution of spectral amplitude is investigated by monitoring the
short-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 the
wave packet in space and time. Only the space-dependent wavenumber/frequency characteristics are monitored since the
amplitude 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 for measurement of the wave distortion
[27]. In the study, we focus on the spatiotemporal analysis of optical-mode wave propagating in the linear and nonlinear
systems.
Fig. 6 shows the spatial spectrograms of the optical-mode wave in different nonlinear metamaterial systems for two
distinct wavenumbers k ¼ p=2 and k ¼ 7p=9 in the Brillouin zone. For the wavenumber k ¼ p=9, the spatial spectrograms are
not presented since it can be deduced directly from Fig. 5(B-c) that the initial band of frequencies is preserved in both linear
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.

and nonlinear AMs. The region between the two white dashed lines denotes Hann window. In Fig. 6(a), the wavenumber
content 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 the
simulation 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 second
packet are in the Hann window and are also dispersive. Finally, the last wave packet is nondispersive according to the energy
density distribution, localized in a region higher than the Hann window. In the following section, the 2D-FFT simulation
results 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 solitary
wave. We can further observe that most of the energy density is located outside the Hann window, indicating a strong
frequency shift of the initial wave packet. Such evolution of the wave packet into pieces of wave packets accompanied with a
strong 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. As
also shown in Fig. 6(def), spectral amplitude is distributed over the length of the distorted wave in the linear metamaterial
system 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 shift
the frequency content of the signal, enhance the distortion of the frequency content or even divide the initial signal into
pieces of wave profiles by generating extra frequency bands. As the wavenumber is increased, the localization of spectral
content is more prominent and the difference in the distribution of spectral amplitude between linear and nonlinear systems
is 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 of
maximum spectral amplitude were determined from the wave field of each numerical simulation. In contrast, one could
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) nonlinear
chain with εGjA0j2
¼ 0:03; (c) nonlinear chain with εGjA0j2
¼ 0:06.

monitor the complete spectrum in the neighborhood of the carrier frequencyewavenumber pair with the objective of
studying the profile of the entire spectral content of the wave. Fig. 7 shows the contour lines of the spectral amplitude
function U(x, U) of optical-mode wave for different values of nonlinearity in two points of the Brillouin zone as k ¼ p=2 and
k ¼ 7p=9.
Fig. 7 indicates that the evolutions of spectral contours in the linear and nonlinear AMs are significantly different. For
example 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 contour
profile above the initial frequency band, a quasi-linear contour profile in the initial band, and a nonlinear contour profile
below the initial band. From the comparison of Fig. 7(b) with Fig. 6(b) we can conclude that: 1) the lowest wave packet in
Fig. 6(b) is dispersive because its spectral contour profile is nonlinear; 2) the wave packet in the Hann window (i.e. the initial
frequency band) is weakly dispersive; 3) the wave packet above the Hann window is a solitary wave since its spectral contour
profile is linear, which means it has a constant group velocity and consequently can propagate with an unchanged waveform.
Fig. 6. Spectrograms of the optical mode wave packets in space at the beginning and end of simulation for different wavenumbers and different magnitudes of
nonlinearity. (aec): k ¼ p=2; (def): k ¼ 7p=9.

The comparison of Fig. 7(c) with Fig. 6(c) also illustrates the existence of a solitary wave along with multiple dispersive wave
packets. Furthermore, most of the wave energy density is confined within the solitary wave as well as the dispersive wave
packets with frequency components below the initial band. This observation phenomenologically illustrates the mechanism
of the formation of pseudo-gap. A wave packet with frequency in the pseudo gap will evolve into several pieces of wave
packets, including a solitary wave and multiple dispersive wave packets. These wave packets with frequency components
distributed out of the initial frequency band possess most of the wave energy. Thus, within the Hann window the energy of
the received wave packet at the end of the simulation is much weaker than the initial wave energy, making the wave packet
look 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 wave
energy 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 is
observed, and there are also two distinctive spectral features in the nonlinear system, one is a dominant linear contour profile
while the other is a nonlinear contour profile. The linear contour profile represents solitary wave, while the nonlinear contour
leads to distortion.
The spectral contours of the nonlinear chain in Fig. 7 are qualitatively different from the nature of the global dispersion
curve observed in Fig. 2. In order to enunciate the global dispersive characteristic of the nonlinear system, the dispersion
curves 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
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.

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 not
exactly 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 is
increased, the evolution of spectral contours in the nonlinear AMs becomes significantly different from the prediction by the
L-P method. Two distinct features of the spectral contours are observed: strong linear contour profiles and a much weaker but
nonlinear contour profile. The nonlinear contour profile is consistent with the prediction from the L-P method for linear
system, and strong linear profiles standing for solitary waves are caused by the interplay between the nonlinearity and
dispersion. It should be pointed out that the maximum spectral amplitude points in each linear profile locate in the dispersion
curve 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 in
prediction 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 than
the long wave region. The multiple scales method [18,28] has been employed to theoretically confirm that solitary waves are
indeed 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 structures
have 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 by
Fleury et al. [35] to break the wave transmission reciprocity among the three acoustic inputs and outputs of a circulator, which
consists 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 is
possible to construct a direction-biased waveguide to estimate the magnitude of nonreciprocal wave. In the study, the
acoustic diode is designed and constructed by combining a finite nonlinear AM system with a finite linear AM system. Fig. 9
illustrates schematics of the wave rectification mechanism by using the nonlinear AM system. The acoustic diode is designed
so 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 solid
curve 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.)

signal is a transient wave packet with frequency band in the pseudo gap of the nonlinear chain (or the band gap of the linear
chain). If such an excitation signal is applied on the left side (nonlinear chain) of the system, significant frequency shift occurs
and 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 linear
system. 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 with
150 cells, as shown in Fig. 10. In order to reduce the impedance mismatch between the linear and nonlinear system, the outer
masses mL1 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 chain
overlaps the band gap of the linear chain. Therefore, the frequency range of the band gap for the linear system and the pseudo
gap for the nonlinear chain is 1:929 U 2:074, and the frequency range of the band gap for the nonlinear system is
0:874 U 1:414. The excitation signal is selected as
uex ¼
1
2
A0 HðtÞ À H

t À
2p
udU
Ncy
!
1 À cos

udUt
Ncy
!
sinðudUtÞ (19)
Fig. 9. Conceptual diagrams of unidirectional acoustic device. Forward configuration: driving on the nonlinear AM side, the significant frequency shift are
generated 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 cells
linked 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.

It should be mentioned that the location and width of the conversion region are determined by the system parameters, the
amplitude of nonlinear parameter εGjA0j2
and the cycle number Ncy ¼ 60 of the excitation bursts. The propagation time of the
signal in the system is set as tsimulate ¼ 2 s, which is long enough for the main 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 be
seen, 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 and
output displacement, as shown in Fig. 12, reveal that the excitation frequency components are tailored when the wave is
propagated from left to right. As the wave passes first through the nonlinear system, it permits propagation due to the shifting
of the optical mode to lower and higher frequencies as predicted before. Therefore, after traversing the nonlinear system, the
propagated wave has a predominant frequency content out of the bandgap frequency range of the linear system and the
resultant output at the right end of the device retains the high wave amplitude while its frequency content is shifted
significantly as evidenced from its spectrum in Fig. 12 (a). In contrast, the wave cannot be propagated from the right to left in
the 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 the
n-th outer masses of the input and output signals are determined by
Pn ¼ À
h
ðun À unÀ1Þ þ εGðun À unÀ1Þ3
i vun
vt
for nonlinear chain
Pn ¼ ÀkL1ðun À unÀ1Þ
vun
vt
for linear chain
(20)
and the wave transmission ratio can be defined as
Fig. 11. The input signal and the output signal received on the other end of the acoustic diode. (a) Forward configuration; (b) Reverse configuration.

a ¼
Z TLast
0
Pres dt
Z TLast
0
Pin dt
(21)
where Pin represents the input energy flux density and Pres is the flux density of the respond signal received at the other
end of the device, TLast is the ending time of simulation. It can be found that for the wave from left to right, the wave
transmission ratio is as high as aþ ¼ 0:585, whilst for the reverse direction, the transmission ratio is as low as
aÀ ¼ 1:178 Â 10À5. The asymmetric ratio s ¼ aþ=aÀ is an important parameter in quantifying the transmission asymmetry
of an acoustic diode. The device designed in Ref. [30] possesses an asymmetric ratio sz104 with a time-averaged energy
transmission coefficient of ~0.35%. In Ref. [33], the asymmetry ratio of the reported device is sz104, with a weak energy
transmission coefficient of $ 10À3. A high asymmetric ratio of sz106 was experimentally obtained in Ref. [31] with a
wide 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 lies
close to 1. The acoustic diode proposed here exhibits a large asymmetric ratio of sz5 Â 105, together with a high energy
transmission 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 for
a certain narrow frequency range. As a result, its robustness in terms of working frequency is not satisfactory. However, by
broadening the width 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 packet
with 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.

be induced. Then, if we make the band gap range of the linear chain in Fig.10 overlap the range of the pseudo-gap, an acoustic
diode with a broad operating frequency range, large asymmetric ratio, and relatively high forward transmission ratio can be
achieved. 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 studied
using approximate analytical models and numerical simulations. The spatial and spectral features of wave propagation in
each case have been monitored to study the interplay between nonlinear, dispersion and local resonance mechanisms in
wave packet propagation. The analysis is based on full-scale transient analysis of the finite system, from which dispersion
curves are generated from the transmitted waves, which is also verified by the perturbation methods. The evolution of
spatial and spectral features is monitored using spatial-spectrogram analysis and 2D FFT simulations, and the interplay of
dispersive and nonlinear and local resonance mechanisms in the process of waveform distortion is evaluated. In general, it
can be concluded that nonlinearity gives rise to nondispersive features in wave propagation. It is also found that the optical
wave modes propagating in the nonlinear metamaterial are sensitive to the parameters of the nonlinear constitutive
relation and significant wave shift phenomena is found. Specifically, a strong frequency shift phenomenon is observed for a
transient wave packet with frequency located in the pseudo gap. Such wave packet evolution into pieces of wave packets
with a strong frequency shift is a distinctive characteristic of wave packet propagation in nonlinear chains with local
resonance. Finally, the feasibility of the nonlinear acoustic metamaterials for direction-biased waveguides is numerically
demonstrated and good behavior of the acoustic diode is observed i.e., the asymmetric coefficient is sz5 Â 105, with good
energy transmission of forward configuration, as large as 58:5%, much more effective than most of the existing designed
acoustic rectifiers.
Acknowledgements
The work is supported by the Air Force Office of Scientific Research under Grant No. AF 9550-15-1-0016 with Program
Manager Dr. Byung-Lip (Les) Lee and the National Natural Science Foundation of China (Nos. 11532001, 11532001 and
11621062). 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, the
solution of the wave equation (2) can be assumed as

unðtÞ ¼ u0
ðxn; ~t; fnÞ þ cu1
ðxn; ~t; fnÞ þ c2
u2
ðxn; ~t; fnÞ þ /
¼
X∞
p¼0
cp
u
p
n;n
vnðtÞ ¼ v0
ðxn; ~t; fnÞ þ cv1
ðxn; ~t; fnÞ þ c2
v2
ðxn; ~t; fnÞ þ /
¼
X∞
p¼0
cp
vp
n;n
(B-1)
where ε is a small quantity defined in the main text, and hence c ¼
ffiffiffi
ε
p
is also a small quantity, u
p
m;n ¼ upðxm; ~t; fnÞ,
v
p
m;n ¼ vpðxm; ~t; fnÞ and
xn ¼ cðna À ltÞ; ~t ¼ c2
t; fn ¼ nqa À 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 as
8

:
m1
v2
vt2
þ k2
!
u
p
n;n þ k1

2u
p
n;n À u
p
n;nþ1 À u
p
n;nÀ1

À k2v
p
n;n ¼ M
p
n
m2
v2
vt2
þ k2
!
v
p
n;n À k2u
p
n;n ¼ N
p
n
ðp ¼ 0; 1; 2Þ (B-3)
where
M0
n ¼ 0; N0
n ¼ 0
M1
n ¼ 2m1l
v
vxn
v
vt
u0
n;n þ k1a
v
vxn

u0
n;nþ1 À u0
n;nÀ1

N1
n ¼ 2m2l
v
vxn
v
vt
v0
n;n
(B-4)
and
M2
n ¼ m1 2l
v
vxn
v
vt
u1
n;n À l2 v2
vx2
n
u0
n;n À 2
v
v~t
v
vt
u0
n;n
!
ÀG

u0
n;n À u0
n;nþ1
3
þ

u0
n;n À u0
n;nÀ1
!3
þk1
1
2
a2 v2
vx2
n

u0
n;nþ1 þ u0
n;nÀ1

þ a
v
vxn

u1
n;nþ1 À u1
n;nÀ1

!
N2
n ¼ m2 2l
v
vxn
v
vt
v1
n;n À l2 v2
vx2
n
v0
n;n À 2
v
v~t
v
vt
v0
n;n
!
(B-5)
Decoupling u
p
n;n from v
p
n;n, we obtain from Eq. (B-3)
8

:
uÀ2
d
v2
vt2
þ 1
!
mÀ1
2 uÀ2
d
v2
vt2
þ 1
!
up
n;n þ k
À1
2 uÀ2
d
v2
vt2
þ 1
!

2up
n;n À up
n;nþ1 À up
n;nÀ1

À up
n;n
¼ uÀ2
d
v2
vt2
þ 1
!
kÀ1
2 Mp
n þ kÀ1
2 Np
n
uÀ2
d
v2
vt2
þ 1
!
v
p
n;n ¼ kÀ1
2 N
p
n þ u
p
n;n
(B-6)
When p ¼ 0

8

:
uÀ2
d
v2
vt2
þ 1
!
mÀ1
2 uÀ2
d
v2
vt2
þ 1
!
u0
j;j þ k
À1
2 uÀ2
d
v2
vt2
þ 1
!

2u0
n;n À u0
n;nþ1 À u0
n;nÀ1

À u0
n;n ¼ 0
uÀ2
d
v2
vt2
þ 1
!
v0
n;n ¼ u0
n;n
(B-7)
The solution to Eq. (B-7)1 can be obtained as
u0
n;n ¼ Aðxn; ~tÞeifn þ c:c;
fn ¼ nk À ut;
(B-8)
where u satisﬁes
mÀ1
2 u4
À
h
mÀ1
2 þ 1

þ 2k
À1
2 ð1 À cosðkÞÞ
i
u2
du2
þ 2k
À1
2 ð1 À cosðkÞÞu4
d ¼ 0 (B-9)
In view of Eq. (B-8), we obtain the solution to Eq. (B-7)2 as
v0
n;n ¼
1
1 À u2
.
u2
d
Aðxn; ~tÞeifn þ c:c; (B-10)
Substituting Eqs. (B-8) and (B-10) into (B-5) (with p ¼ 1), we have
M1
n;n ¼ À2im1ul
v
vxn
Aeifn þ 2ik1asinðkÞ
v
vxn
Aeifn þ c:c
N1
n;n ¼
À2im2ul
1 À u2
.
u2
d
v
vxn
Aeifn þ c:c
(B-11)
Substituting of Eq. (B-11) into Eq. (B-6) yields
8

:
uÀ2
d
v2
vt2
þ 1
!
mÀ1
2 uÀ2
d
v2
vt2
þ 1
!
u1
n;n þ k
À1
2 uÀ2
d
v2
vt2
þ 1
!

2u1
n;n À u1
n;nþ1 À u1
n;nÀ1

À u1
n;n
¼ À2i

1 À uÀ2
d u2

2
6
4mÀ1
2 uÀ2
d ul À k
À1
2 asinðkÞ þ
uÀ2
d ul

1 À u2
.
u2
d
2
3
7
5
v
vxn
Aeifn þ c:c
uÀ2
d
v2
vt2
þ 1
!
v1
n;n ¼
À2iuuÀ2
d l
1 À u2
.
u2
d
v
vxn
Aeifn þ u1
n;n þ c:c
(B-12)
Hence the solvability equation is
l ¼
m2ud

1 À U2
2
sinðkÞa
k2U

1 À U2
2
þ m2k2U
! ¼
vu
vq
(B-13)
Then u1
n;n could be set zero, i.e. u1
n;n ¼ 0, and the solution of v1
n;n is
v1
n;n ¼
À2iuuÀ2
d
l

1 À u2
.
u2
d
2
v
vxn
Aeifn (B-14)
Substitution of Eqs. (B-8), (B-10), (B-11) and (B-14) into Eq. (B-5) yields

M2
n ¼
(
2im1u
v
v~t
A þ

k1a2
cos k À m1l2
v2
vx2
n
A þ 6Gf4 cosðkÞ À cosð2kÞ À 3gA2
A
)
eifn
þ2½3 cos k À 3 cosð2kÞ þ cosð3kÞ À 1ŠGA3
e3ifn þ c:c
N2
n ¼ m2
0
B
@
À4u2
uÀ2
d l2

1 À u2
.
u2
d
2
v2
vx2
n
Aeifn À
l2
1 À u2
.
u2
d
v2
vx2
n
Aeifn þ
2iu
1 À u2
.
u2
d
v
v~t
Aeifn
1
C
A þ c:c
(B-15)
By inserting this equation into Eq. (B-5) (with p ¼ 2), we obtain
uÀ2
d
v2
vt2
þ 1
!
m1
v2
vt2
þ k2
!
u2
n;n þ k1 uÀ2
d
v2
vt2
þ 1
!

2u2
n;n À u2
n;nþ1 À u2
n;nÀ1

À k2u2
n;n
¼
2
6
4

k
À1
2 a2
cos k À mÀ1
2 uÀ2
d l2

1 À uÀ2
d u2

À

3u2
uÀ2
d þ 1

uÀ2
d l2

1 À u2
.
u2
d
2
3
7
5
v2
vx2
n
Aeifn
þ

2imÀ1
2 uÀ2
d u

1 À uÀ2
d u2

þ
2iuuÀ2
d
1 À u2
.
u2
d
#
v
v~t
Aeifn
þ6

1 À uÀ2
d u2
G
k2
f4 cosðkÞ À cosð2kÞ À 3gA2
Aeifn
þ2

1 À 9uÀ2
d u2

½3 cosðkÞ À 3 cosð2kÞ þ cosð3kÞ À 1Š
G
k2
A3
e3ifn
(B-16)
The solvability condition of this equation is the nonlinear Schr€odinger equation (NSE)
i
v
v~t
A þ
P
2
v2
vx2
n
A þ QA2
A ¼ 0 (B-17)
where
P ¼ À
k2

1 À U2
3
þ m2k2

1 þ 3U2
!
l2
þ m2a2

1 À U2
3
cosðkÞu2
d
k2Uud

1 À U2
3
þ m2k2Uud

1 À U2
¼
v2
u
vq2
Q ¼ À6
Gud
k2
m2UÀ1
½1 À cosðkÞŠ2

1 À U2
2
,

1 À U2
2
þ m2
!
(B-18)
The solution of Eq. (B-18) depends on PQ [16]

PQ 00envelope solitons
PQ 00dark solitons
(B-19)
When PQ 0, Eq. (B-18) admits a solution of the form
A ¼ fsech
ffiffiffiffiffiffi
Q
2P
r
fcðx À ltÞ
#
eic2Q
2
f2
t
(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

qðx; tÞ ¼ f
(
1 À m2
sech2
ffiffiffiffiffiffiffiffi
ÀQ
2P
r
mfcðx À ltÞ
#)1=2
(B-22)
and
j ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
À
m2
À 1
Á
Q
2P
s
fcðx À ltÞ
þtanÀ1
(
m
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 À m2
p tanh
ffiffiffiffiffiffiffiffi
ÀQ
2P
r
mfεðx À ltÞ
#)
À c2Q
2

3 À m2

f2
t
(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 envelope
soliton solutions because PQ is positive. In the contrast, in the range of 0 k 1:63, the system has no envelope soliton
solutions because of negative PQ but could have dark solitons. An envelope soliton can be induced by lunching a wave packet.
However, the dark soliton cannot be induced by lunching a wave packet with finite wavelengths. In the study, only envelope
solitons whose wavenumber in the range of 1:63 k p can be induced in the nonlinear mass-in-mass chain by lunching a
wave 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.

[15] N. Nadkarni, C. Daraio, D.M. Kochmann, Dynamics of periodic mechanical structures containing bistable elastic elements: from elastic to solitary wave
propagation, 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.

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