This document describes a design for a density-near-zero (DNZ) acoustic membrane network that can control sound transmission. The network is made of circular membranes arranged in a square lattice inside a waveguide. The unit cell is modeled as an equivalent lumped-circuit of inductors and capacitors. Simulations show the effective mass density approaches zero near 987 Hz, the resonance frequency predicted by the circuit model, demonstrating DNZ behavior. Further simulations then examine how the DNZ membrane network can achieve applications like cloaking, high transmission through sharp corners, and wave splitting.

1. Controlling sound transmission with density-near-zero acoustic membrane network
Yuan Gu, Ying Cheng, Jingshi Wang, and Xiaojun Liu
Citation: Journal of Applied Physics 118, 024505 (2015); doi: 10.1063/1.4922669
View online: http://dx.doi.org/10.1063/1.4922669
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/118/2?ver=pdfcov
Published by the AIP Publishing
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2. Controlling sound transmission with density-near-zero acoustic membrane
network
Yuan Gu,1
Ying Cheng,1,2,a)
Jingshi Wang,3
and Xiaojun Liu1,2,b)
1
Department of Physics, Collaborative Innovation Center of Advanced Microstructures, Nanjing University,
Nanjing 210093, China
2
State Key Laboratory of Acoustics, Chinese Academy of Sciences, Beijing 100190, China
3
School of Electronics and Information, Nantong University, Nantong 226019, China
(Received 5 March 2015; accepted 6 June 2015; published online 14 July 2015)
We demonstrate a design of two-dimensional density-near-zero (DNZ) membrane structure to
control sound transmission. The membrane structure is theoretically modeled as a network of
inductors and capacitors, and the retrieved effective mass density is confirmed to be close to zero at
the resonance frequency. This scheme proposes a convenient way to construct the unit cell for
achieving DNZ at the designed frequency. Further simulations clearly demonstrate that the
membrane-network has the ability to control sound transmission such as achieving cloaking,
high transmission through sharp corners, and high-efficient wave splitting. Different from
the phononic-crystal-based DNZ materials, the compact DNZ membrane-network is in deep
subwavelength scale and provides a strong candidate for acoustic functional devices. VC 2015
AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4922669]
I. INTRODUCTION
Zero-index metamaterials (ZIMs) denote the artificial
structures in which the effective dynamic constitutive param-
eters approach zero in a certain frequency region. The initial
proposal and the recent realization in electromagnetics (EM)
permit the phase velocity and the wavelength to be infinitely
large so that the EM waves do not undergo any spatial
phase change.1–7
This unique characteristic induces unusual
EM responses, such as cloaking through ZIMs waveguide
embedded with rectangular dielectric defects,2,3
perfect
transmission in waveguides with sharp bends and corners,4–6
and equivalent perfect magnetic conductor.7
Inspired by the EM ZIMs, acoustic ZIMs have attracted
much attention recently due to their rich physics and practi-
cal applications. The principle of e-near-zero EM metamate-
rials equals valid for that of density-near-zero (DNZ)
acoustic metamaterials. The ability of the near-zero mass
density to possess unusual acoustic response was conceptu-
ally demonstrated by Wei et al.8
Various prototypes have
been further investigated to provide physical realizations.9–18
Gracia-Salgado designed a two-dimensional (2D) DNZ
metamaterials with periodical distributions of structured cy-
lindrical scatterers embedded in a waveguide.14
Liang and Li
achieved low-loss near-zero density using coiling up space
with curled perforations.15
Liu et al. observed that both the
effective mass density and the inverse of bulk modulus
vanish simultaneously in phononic crystals with the Dirac
cones.16
Fleury and Alu studied the anomalous sound
transmission and the uniform energy squeezing through
ultranarrow acoustic channels filled with zero-density meta-
materials.17
However, the requirements of extreme low
velocity, large dimension comparable with the working
wavelength, or complex geometric structures in the inclu-
sions of the prototypes limit their practical applications.
In this paper, based on the one-dimensional (1D) acous-
tic ZIM with periodic clamped membranes and open chan-
nels,11
we design a simple 2D membrane-network structure
behaving as DNZ metamaterials for airborne sound. The
lumped-circuit (LC) model of the metamaterial unit cell
exhibits a significant resonance. Further parameter retrievals
find that the effective mass density approaches zero at the
frequency near the resonance. By using both full-wave simu-
lations and lumped-circuit simulations, the unique transmis-
sion property of the metamaterial slab at the DNZ frequency
is demonstrated, in which the phase velocity and the wave-
length approach infinity. We further investigate the extraor-
dinary sound transmission induced by the proposed DNZ
metamaterials such as efficient cloaking effect, perfect trans-
mission in waveguides with sharp bends, and wave splitter.
Different from the previous schemes, no extreme parameters
are required and the unit cell of membrane-network DNZ
metamaterial is in deep subwavelength scale. The micro-
structure of the membrane network is comparatively simple,
and its lump-circuit model provides a convenient way to
construct the unit cell for achieving DNZ at the designed fre-
quency. The near-zero effective density has been ascribed to
the first order of the air-membrane resonance, and hence the
disappearance of density because the restoring force from
the membrane adds a negative term to the effective mass.9,10
II. DESCRIPTION OF THE PROPOSED DNZ
METAMATERIAL
The 2D DNZ acoustic metamaterial is composed of
circular membranes array fixed inside a 2D waveguide
arranged in square lattice network, as shown in Fig. 1(a). The
a)
chengying@nju.edu.cn
b)
liuxiaojun@nju.edu.cn
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JOURNAL OF APPLIED PHYSICS 118, 024505 (2015)
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3. basic building block is a cubic compartment of dimension
d ¼ 20mm with exterior rigid boundaries, and filled by the
background medium air. There are four circular openings with
a radius of R ¼ 9 mm on the steel boundaries of the block in
xz- and yz-planes, which contain identical 0.125 mm-thick
low-density polyethylene membranes. The mass density (qm),
Young’s modulus (E), and Poisson’s ratio () of the mem-
brane are 1420 kg/m3
, 2.5 GPa, and 0.34, respectively.
The mass density and the bulk modulus of air are qair
¼ 1.39 kg/m3
and Kair ¼ 152 kPa, respectively. The dashed
lines in Fig. 1(a) outline the primitive unit cell with a lattice
constant a (¼
ffiffiffi
2
p
d ¼ 28.28 mm), which is depicted in Fig. 1(b).
Note that the lattice constant should be small enough compared
to the wavelength k.
First, we convert the physical unit cell to its equivalent
acoustic circuit representation [Fig. 1(c)] and predict the
DNZ metamaterials’ working frequency. The membrane is
modeled as a thin plate with a transverse displacement W(r),
which satisfies the flexural wave’s equation Dr4
W þ qmhW
¼ Dp. Here, h is the thick of the membrane, D ¼ Eh3
=12
ð1 À 2
Þ represents its flexural rigidity, and Dp is the sub-
traction of the sound pressure on two faces of the membrane.
Under the assumption that the pressure distribution over the
plate is uniform, which is verified in the case of plane-wave
incidence and small displacements, and considering only axi-
ally symmetrical modes and the fixed constraint at the exte-
rior boundary (½Wðr ¼ RÞ ¼ 0; dW=drjr¼R ¼ 0Š), the first
resonance of the membrane dynamic system can occur at
f1 ¼ 1:63
1
R2
ffiffiffiffiffiffiffiffi
D
qmh
s
: (1)
The dynamic response of the membrane to the force can
be represented by the mechanical impedance Zm ¼
Ð Ð
S
DpðrÞdS
jxn
.
Here, n ¼ ð1=SÞ
Ð Ð
S nðrÞdS represents the mean transverse
displacement over the plate surface, where S is the cross sec-
tion area of the unit. If the pressure distribution over the plate
is uniform, the acoustic impedance can be expressed as
Zam ¼ Zm=S2
. Thus, the acoustic model of the membrane can
be described with a series impedance Zam, which is equivalent
to a series resonant circuits comprised of acoustic mass mam
and compliance Cam. The values of mam and Cam are calcu-
lated such that the impedance is equal to Zam and the reso-
nance frequency of the circuit unit (f ¼ 1=2p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
mamCam
p
) is
equal to the exact frequency of the first resonance f1. Here,
f1 ¼ 1:63
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
D=qmh
p
=R2
¼ 1023 Hz, which can be given by
Eq. (1), leading to the following expressions:11
mam ¼ 1:8830
qmh
pR2
; Cam ¼
pR6
196:51D
: (2)
Although the membranes lead to a discontinuity of the
acoustic pressure field, the acoustic velocity is continuous
across the membrane.
Further, taking the air-membrane interaction into con-
sideration, the air section can be described by an acoustic
mass ma and a compliance Ca. As the lattice constant is far
less than the wavelength of sound wave, Ca is small enough
to be neglected and ma ¼ qairðd À hÞ=S. Thus, the parame-
ters of the lumped element model in Fig. 1(c) can be given
by ms ¼ ma þ mam and Cs ¼ Cam. According to the LC theory,
the resonance frequency of the system can be expressed as
f ¼
1
2p
ffiffiffiffiffiffiffiffiffiffi
msCs
p ¼
1
2p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ma þ mamð ÞCam
p ¼ 987 Hz: (3)
This resonance frequency is the theoretical working fre-
quency, where the effective density of the metamaterial
could turn to zero. This resonance frequency can be con-
trolled by adjusting the basic setup of the unit cell (see
Eq. (3)), which provides a feasible way to tune the DNZ
phenomenon. Furthermore, the lumped circuit model of air
can be expressed as a regular right-handed acoustic transmis-
sion line, and then the acoustic mass and the acoustic
compliance of the lumped element model can be calculated
according to the density (qair) and the bulk modulus (Kair) of
air such as mR ¼ qaird=DS and CR ¼ DS Â d=Kair, where DS
is the area of the cross section.11
Thus, the sound transmis-
sion in both the membrane network and the background air
can be simulated by using the periodic arrangement of basic
circuit units.
In order to verify the near-zero density of the metamate-
rial, a rigorous retrieving method is used to obtain its
effective parameters.19–23
In the retrieving process, the
reflectance and the transmittance are first numerically calcu-
lated through the pressure fields under a plane wave incident
on the membrane-network metamaterials by COMSOL.
Figure 2(a) shows the schematic setup which is used to
retrieve the effective acoustic parameters. The thickness of
the metamaterial is set to be one layer (20 mm) in order to
FIG. 1. (a) Composition pattern of the membranes network that constructs the DNZ metamaterials. The network is made of circular membranes array fixed
inside a 2D waveguide in square lattice. The basic unit is a cubic compartment of a dimension d ¼ 20 mm. (b) Primitive unit cell with a lattice constant of
a ¼ 28.28 mm and a membrane radius of R ¼ 9 mm, outlined by the dashed lines in Fig. 1(a). (c) The equivalent circuit model of the unit cell. The resonance
frequency of the system can be determined according to the LC theory.
024505-2 Gu et al. J. Appl. Phys. 118, 024505 (2015)
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4. solve the problem of the branch number selection.19
The
reflectance and the transmittance are shown in Fig. 2(b). It is
observed that the transmittance comes to a peak value at
1020 Hz, which is near the resonance frequency of the struc-
ture. Then, the effective refractive index n and impedance Z
are retrieved as a function of the incident frequency. At last,
the effective q and 1/K are calculated from n and Z.20–23
Figure 2(c) depicts the retrieved q (solid line), Z (dashed
line), and 1/K (dashed-dotted line). Note that all the parame-
ters here are normalized to the background medium air. The
inset figure amplifies the near-zero region of the effective
mass density q from 987 Hz to 1000 Hz. Here, three remarks
should be noted: (1) q is close to zero near 987 Hz, and the
DNZ frequency is identical to the working frequency pre-
dicted by Eq. (3) in the circuit model; (2) q and 1/K undergo
a relatively gradual increment in the frequency region,
where efficient DNZ effect exists; and (3) As we can see
from Fig. 2(b), at the frequency where q is exact zero, a big
impedance mismatch is occurred, which may cause a strong
reflectance.
In order to further investigate the effect of impedance
mismatch, sound transmissions at multiple frequencies
around 987 Hz are calculated and a trade-off effect is
observed. First, deflecting from the zero-density frequency
leads to more phase variation due to the limited effective
sound speed. For example, it exhibits a phase difference of
0.08 rad for the sound wave between the output and input
surfaces at 990 Hz. However, deflecting from the zero-
density frequency also results in an improved impedance
match, which can significantly increase the transmittance
(e.g., 0.92 at 990 Hz). Therefore, the efficient DNZ effects
can be achieved by choosing moderate frequency region. In
the following simulations, we have chosen 990 Hz as the
near-zero frequency, instead of the exact zero-density
frequency (987 Hz). At 990 Hz, the contradiction between
the impedance mismatch and the zero-density effect is bal-
anced, e.g., the metamaterials keep a strong zero-density
phenomenon and the relative impedance is also compara-
tively matched so that the transmittance is relatively high.
For further demonstration, we have considered the model of
the sound transmission through a uniform layer whose pa-
rameters are equal to the DNZ metamaterial at 990 Hz. Then
the transmittance of sound pressure can be expressed as
tp ¼ 2
½4 cos2k2DþðZ1=Z2þZ2=Z1Þ2
sin2
k2DŠ1=2, where D (0.02 m) and k2
represent the thickness and the wave vector of interface
layer, respectively. At the DNZ frequency of 990 Hz, the im-
pedance Z2/Z1 is 0.2029, and the mass density q2/q1 is
0.0278, and then k2 (¼2pf/c2) is 2.575. Thus, tp is 0.99 at
990 Hz. This behavior means that the transmittance between
the background air and the membrane network could be very
large at the density-near-zero frequency due to the small
parameter k2D although the impedance was not perfectly
matched. The results are consistent with the retrieving results
shown in Fig. 2. We have also calculated the transmittance
and the effective parameters for longer membrane network
(from 2 to 15 unit cells). Even if several peaks of the trans-
mittance appear due to the coupling resonances of the mem-
branes,24
the DNZ frequency band is equivalent to that
shown in Fig. 2. Thus, the effective parameters retrieved
numerically verify the theoretically predicted DNZ band,
which further convinces that the near-zero effective density
originates from the resonance of the coupled air and the
membranes.
III. RESULTS AND DISCUSSION
In order to present the unique transmission property of
the metamaterial slab at the DNZ frequency, numerical
FIG. 2. (a) One-layer membrane struc-
ture used to retrieve the effective
acoustic parameters. Incident plane
waves irradiate vertically on the inter-
face between the metamaterial slab
and the air. (b) The transmittance and
the reflectance of the structure
retrieved from the incident, reflected,
and transmitted waves. The transmit-
tance reaches to a peak value at
1020 Hz. (c) The effective mass den-
sity q (solid line), the acoustic imped-
ance Z (dashed line), and the inverse of
bulk modulus 1/K (dashed-dotted line)
of the metamaterials. All the parame-
ters are normalized to the background
medium air. q approaches zero near
987 Hz, and the DNZ frequency is con-
sistent with the theoretical prediction.
024505-3 Gu et al. J. Appl. Phys. 118, 024505 (2015)
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5. experiments are performed through three-dimensional (3D)
full-wave acoustic simulations as well as LC simulations. In
acoustic simulations, the metamaterial slab is constructed by
periodically repeating the unit cell in Fig. 1(b) in the plane
parallel to the membranes, and the transport properties of the
planar array are similar to that of a single unit cell.25
In LC
simulations, the metamaterials can be represented by the 2D
array of the unit circuit structures as shown in Fig. 1(c), and
the background medium is modeled as transmission lines
represented by a series of acoustic impedances. The sound
transmissions can be calculated by using SPICE. Figure 3(a)
depicts the schematics of the simulation domain setup in
which the DNZ slab is made of 6 Â 12 unit cells. A plane
harmonic acoustic wave p ¼ P0eik0x
transmits along the
waveguide from left to right, vertically onto the interface
between the metamaterial slab and the air, where P0 is the
amplitude of the incident wave and k0 is the wave number in
the air. The external interfaces of the domain are radiation
boundaries which absorb the outgoing waves without reflec-
tion. Considering the rigidness of steel, the internal bounda-
ries are treated as hard boundaries. The pressure inside the
structure increases due to the action of the membrane
tension, which provides a restoring force for the air column
in the cubic building block. This restoring force adds a nega-
tive term to the effective mass, which makes the effective
mass density of the structure turn to zero at the resonance
frequency. Figure 3(b) shows the acoustic experimental
result of the normalized pressure field in the waveguide
when the metamaterial slab is inserted at the DNZ frequency
by using the COMSOL multi-physics finite element package,
whereas Fig. 3(c) shows the circuit experimental result under
the same circumstance.
It is found that the transmitted wave reserves its plane
wavefront without obvious loss or scattering, as shown in
Figs. 3(b) and 3(c). It has been demonstrated that the pressure
transmission in DNZ metamaterials should be a quasi-static
process with an infinitely large phase velocity (wavelength).
Thus, the spatial phase of the transmitted wave should keep
constant. In order to guarantee a strong transmittance, the
chosen DNZ frequency is not at the exact zero-density fre-
quency, which makes a small phase change in DNZ metama-
terials. The phase distribution of the pressure field along
x-axis in the waveguide is shown in Fig. 3(d), in which the
solid lines (open circles) represent the results of full-wave
(lumped-circuit) simulations, and that of the incident wave
(dashed line) is also shown for comparison. It is observed
that the phase of the sound pressure keeps almost constant
before and after the slabs when the frequency is selected at
the DNZ frequency. We can also observe that the phase
change between the incident wave and the transmitted wave
is very small, indicating the efficient DNZ character. Note
that the phase of the total sound pressure and the incident
sound waves are different in the input region (x 600mm)
because of the reflected waves caused by the slight imped-
ance mismatch between the air and the DNZ metamaterials.
Thus, the phase distortion originates from the superposition
of the incident and the reflected waves. In addition, we
also investigate the cases when acoustic waves illuminate
obliquely on the surface and when the interface of the struc-
ture is along the C-M direction, and the results confirm the
efficient DNZ effects in both cases. Therefore, the DNZ
acoustic membrane network can work well along C-X and
C-M directions, indicating the high isotropy.
DNZ metamaterials can induce many fascinating phe-
nomena such as cloaking, perfect transmission through sharp
corners, and power splitting. Here, we first demonstrate the
cloaking of a rectangular obstacle by membrane-network
DNZ metamaterials. Figures 4(a) and 4(c) show the pressure
fields in a waveguide with a vertically inserted steel obstacle
in the middle of the waveguides, which are respectively
attained by using full-wave acoustic simulations and
lumped-circuit simulations. The thickness of the waveguide
is 160 mm, and the obstacle is a rectangular bar of a dimen-
sion of 20 mmÂ80 mm (¼d  4d). The boundaries of the
steel obstacle are treated as ideal hard boundaries due to the
huge impedance-mismatch between the air and the steel.
The incident plane wave is scattered severely because the
length of the obstacle is comparable with the wavelength.
It is found that the scattered field shows complex and disor-
ganized patterns in Figs. 4(a) and 4(c). If we encircle the
FIG. 3. The transmission property of the metamaterial slab at the DNZ
frequency. (a) The schematic setup of the simulation. The slab is made of 6
 15 unit cells. (b) Full-wave acoustic simulation results and (c) lumped-
circuit simulation results of the pressure field in the waveguide when the
incident plane waves irradiate vertically on the interface between the meta-
material slab and the air. (d) The phase distribution of the pressure field along
x-axis. Note only a minor phase change between the incident wave and the
transmitted wave. (b)–(d) share the same x-coordinate for clear comparison.
024505-4 Gu et al. J. Appl. Phys. 118, 024505 (2015)
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6. obstacle with the membrane-network DNZ metamaterial in a
proper thickness, the wavefront preserves its original plane
pattern after it passes through the metamaterial slab and the
rigid obstacle, as shown in Figs. 4(b) and 4(d). The plane
wave can transmit as if the steel obstacle does not exist.
Therefore, the membrane-network DNZ metamaterials have
the efficient cloaking effect, which is different from the
cloaking based on transformation acoustics since the param-
eters of the metamaterials are irrelevant to the shape of the
cloaking.
The high sound transmission is further realized in wave-
guides with sharp bends and corners by using the extraordi-
nary transmission property of the DNZ metamaterial. Figure
5(a) depicts the pressure field in an L-shaped waveguide
without the proposed DNZ metamaterials. The white arrows
indicate the direction of the incident sound waves, and the
width of the waveguide is 200 mm. It is obvious that the
transmitted waves in the perpendicular waveguides undergo
a very severe scattering and cannot keep its original plane
wavefront. With the membrane-network DNZ metamaterial
in the corner, the sound waves are transmitted without
scattering and hence the wavefront reserves the plane pattern
along the perpendicular direction, as shown in Fig. 5(b).
Similar results have been experimentally demonstrated in
EM waves by using e-near-zero photonic crystals and in
acoustic waves by using near-zero-index phononic crystals.
In these designs, the dimension of unit cell for the photonic/
phononic crystal should be comparable with the wave-
length,26,27
whereas the unit of membrane-network DNZ
metamaterial in this work is in deep subwavelength scale.
Finally, the application of DNZ metamaterial is demon-
strated in a waveguide splitter, which can divide the energy
of input acoustic signals into multiple outputs with a perfect
power distribution regardless of the shape of the metamate-
rial or the orientations of the waves. Figure 5(d) shows the
pressure field of acoustic waveguide splitter which has one
input port with the thickness of 160 mm in the left-side and
two output ports with the thickness of 80 mm in the right-
side. Both of the two output waves keep the plane wavefront.
Figure 5(c) depicts the sound pressure field in the same
waveguides without the membrane-network DNZ metamate-
rial. It is found that the transmitted sound energy is weak in
the output ports due to the severe backward scattering at the
corners. The efficient transmission of energy is found to
increase 80% in the case of the DNZ metamaterial by com-
paring Figs. 5(c) and 5(d).
IV. CONCLUSION
We have designed a 2D DNZ metamaterial constructed
by the membrane-network structures. Based on the lumped-
circuit model, the theoretical predicted DNZ property is con-
firmed by the retrieved effective parameters, which provides
a convenient way to design the unit cell to realize the DNZ
effects. Both full-wave and lumped-circuit simulations have
demonstrated the unique transmission properties of the DNZ
metamaterial such as cloaking, high transmission through
sharp corners, and wave splitter.
ACKNOWLEDGMENTS
This work was supported by the National Basic Research
Program of China (No. 2012CB921504), SRFDP (No.
20130091130004), and NSFC (Nos. 11474162, 11274171,
11274099, and 11204145).
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FIG. 4. Acoustic cloaking effect by the DNZ metamaterials. (a) The scat-
tered pressure field caused by the rectangular steel obstacles through full-
wave acoustic simulation. (b) The pressure field in the waveguides with the
DNZ metamaterials. The transmitted wave maintains the plane wavefront as
if the obstacle cannot exist. The efficient cloaking effect is confirmed by
the corresponding lumped-circuit simulation results (c) and (d). The DNZ
metamaterial slab is indicated by the black grids.
FIG. 5. Normalized pressure fields of the transmissions through the (a) and
(b) perpendicular bend and (c) and (d) power divider. Left: without the DNZ
metamaterials; right: with the DNZ metamaterials indicated by the grids.
The white arrows denote the directions of the sound propagation.
024505-5 Gu et al. J. Appl. Phys. 118, 024505 (2015)
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