Controlling sound transmission with density-near-zero acoustic membrane networkYuan 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 Articles you may be interested in Active control of membrane-type acoustic metamaterial by electric field Appl. Phys. Lett. 106, 091904 (2015); 10.1063/1.4913999 Manipulating acoustic flow by using inhomogeneous anisotropic density-near-zero metamaterials Appl. Phys. Lett. 106, 081912 (2015); 10.1063/1.4914005 Acoustic superlens using membrane-based metamaterials Appl. Phys. Lett. 106, 051901 (2015); 10.1063/1.4907634 Analytical coupled vibroacoustic modeling of membrane-type acoustic metamaterials: Membrane model J. Acoust. Soc. Am. 136, 969 (2014); 10.1121/1.4892870 Decentralized harmonic control of sound radiation and transmission by a plate using a virtual impedanceapproach J. Acoust. Soc. Am. 125, 2978 (2009); 10.1121/1.3106124

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Controlling sound transmission with density-near-zero acoustic membranenetwork

Yuan Gu,1 Ying Cheng,1,2,a) Jingshi Wang,3 and Xiaojun Liu1,2,b)

1Department of Physics, Collaborative Innovation Center of Advanced Microstructures, Nanjing University,Nanjing 210093, China2State Key Laboratory of Acoustics, Chinese Academy of Sciences, Beijing 100190, China3School 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 2015AIP 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

Graci�a-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 Al�u 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 DNZMETAMATERIAL

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.cnb)liuxiaojun@nju.edu.cn

0021-8979/2015/118(2)/024505/6/$30.00 VC 2015 AIP Publishing LLC118, 024505-1

JOURNAL OF APPLIED PHYSICS 118, 024505 (2015)

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basic building block is a cubic compartment of dimension

d¼ 20 mm with exterior rigid boundaries, and filled by the

background medium air. There are four circular openings with

a radius of R¼ 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 Kair¼ 152 kPa, respectively. The dashed

lines in Fig. 1(a) outline the primitive unit cell with a lattice

constant a (¼ffiffiffi2p

d¼ 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 Dr4W þ qmhW¼ Dp. Here, h is the thick of the membrane, D ¼ Eh3=12

ð1� �2Þ 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 0�), the first

resonance of the membrane dynamic system can occur at

f1 ¼ 1:631

R2

ffiffiffiffiffiffiffiffiD

qmh

s: (1)

The dynamic response of the membrane to the force can

be represented by the mechanical impedance Zm ¼Ð Ð

SDpðrÞdS

jx�n.

Here, �n ¼ ð1=SÞÐ 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=2pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimamCam

p) is

equal to the exact frequency of the first resonance f1. Here,

f1 ¼ 1:63ffiffiffiffiffiffiffiffiffiffiffiffiffiffiD=qmh

p=R2¼ 1023 Hz, which can be given by

Eq. (1), leading to the following expressions:11

mam ¼ 1:8830qmh

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 � hÞ=S. Thus, the parame-

ters of the lumped element model in Fig. 1(c) can be given

by ms¼maþmam and Cs¼Cam. According to the LC theory,

the resonance frequency of the system can be expressed as

f ¼ 1

2pffiffiffiffiffiffiffiffiffiffimsCs

p ¼ 1

2pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffima 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� d=Kair, where DSis 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 d¼ 20 mm. (b) Primitive unit cell with a lattice constant of

a¼ 28.28 mm and a membrane radius of R¼ 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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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 Zare 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 sin2k2D�1=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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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 � 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< 600 mm)

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 mm�80 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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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-

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