Sound transmission loss through metamaterial plate with lateral local resonators in thepresence of external mean flowTing Wang, Meiping Sheng, and Qinghua Qin

Citation: The Journal of the Acoustical Society of America 141, 1161 (2017); doi: 10.1121/1.4976194View online: http://dx.doi.org/10.1121/1.4976194View Table of Contents: http://asa.scitation.org/toc/jas/141/2Published by the Acoustical Society of America

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Vibration damping using a spiral acoustic black holea) a)Portions of this work were presented in “Vibrationdamping using a spiral acoustic black hole,” Proceedings of INTER-NOISE 2016, Hamburg, Germany, 21–24August, 2016.The Journal of the Acoustical Society of America 141, (2017); 10.1121/1.4976687

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Sound transmission loss through metamaterial plate with laterallocal resonators in the presence of external mean flow

Ting Wang,1,a) Meiping Sheng,1 and Qinghua Qin2

1School of Marine Science and Technology, Northwestern Polytechnical University, Youyi Western Road,Number 127, Xi’an, Shaanxi, 710072, People’s Republic of China2College of Engineering and Computer Science, the Australian National University, Canberra,Australian Capital Territory, 2601, Australia

(Received 5 September 2016; revised 5 December 2016; accepted 28 January 2017; publishedonline 28 February 2017)

In the context of sound incident upon a metamaterial plate, explicit formulas for sound transmission

loss (STL) are derived in the presence of external mean flow. Metamaterial plate, consisting of

homogeneous plate and lateral local resonators (LLRs), is homogenized by using effective medium

method to obtain the effective mass density and facilitate the calculation of STL. Results show that

(a) vigorously oscillating LLRs lead to higher STL compared with bare plate, (b) increasing Mach

number of the external mean flow helps obtain higher STL below the coincidence frequency but

decreases STL above the coincidence frequency due to the added mass effect of light fluid loading

and aerodynamic damping effect, (c) the coincidence frequency shifts to higher frequency range for

the refracted effect of the external mean flow. However, effects of the flow on STL within negative

mass density range can be neglected because of the lateral local resonance occurring. Moreover,

hysteretic damping from metamaterial can only smooth the transmission curves by lowering higher

peaks and filling dips. Effects of incident angles on STL are also examined. It is demonstrated that

increasing elevation angle can improve the sound insulation, while the azimuth angle does not.VC 2017 Acoustical Society of America. [http://dx.doi.org/10.1121/1.4976194]

[MRH] Pages: 1161–1169

I. INTRODUCTION

Sound transmission is a classical problem in structural

acoustics. The improvement of insulation capacity of struc-

tures can help facilitate working environment, reduce the

radiation of energy, avoid detectability of submarines by

sonars, and so forth. Much effort has been made to increase

the STL of an object by designing its structures in the form

of double-leaf wall,1 sandwich structures2–4 and attaching

substructures5,6 or inserting air gaps between the walls of the

object.7–10 Analysis of spherical wave transmission loss

through a single-leaf wall was investigated by Yairi et al.11

Explicit formula was derived and its result revealed that

decreased wall impedance deteriorates the insulation perfor-

mance under the spherical wave incidence.

Studies mentioned above are all related to STL through

a static structure. However, many objects are practically

working in moving condition. Within the environment, flow

from the medium may be generated, which may also be a

factor affecting the STL. Study of the effects of external

flow on STL through structures has also been carried out. It

was found that air flow provided modest increases in STL

through a bare plate up to the coincidence frequency,12 and

when a sandwich shell was subject to turbulent boundary

layer fluctuations, the convective flow velocity showed little

influence on the structural response or the interior pressure.13

It was also demonstrated STL through an aeroelastic plate

with orthogonal rib-stiffeners under external mean flow were

improved significantly with the increase of Mach number

over a wide frequency range.14 As external mean flow was

added on one side of a sandwich plate9 or cylindrical shell15

under different lined porous cores, higher STL was yielded

when larger Mach number was applied.

Recently, metamaterials, a kind of artificial structure,

which enables unique properties not existing from basic

structure,16,17 have attracted much attention owing to its neg-

ative effective parameters and band gap properties. Within

band gaps, flexural waves can be efficiently attenuated to

achieve wave suppression and prohibition,18,19 vibration iso-

lation, and sound absorption.20 It is an efficient and elegant

approach for optimizing mechanical constituents by tailoring

the dynamical behaviour of metamaterials with respect to

vibration and acoustic control among others.21 Xiao et al.5

has reported the capability of sound insulation of a metama-

terial with attached resonators with effective medium

method and plane wave expansion. Their results showed that

higher STL can be obtained within the ranges affected by

resonances. Similar work was finished with continuum reso-

nators consisting of rubber and steel attached to a sandwich

plate.22 Meanwhile, STL through a metamaterial plate com-

prising of shunted piezoelectric patches and a thin plate23

was increased and its frequency range was broadened by the

negative capacitance shunting circuits. Moreover, Collet etal.24 proposed an effective approach to minimize the acous-

tic radiation of a semi-active metacomposites by optimizing

the impedance of the shunted circuit. Chronopoulos et al.25

and Antoniadis et al.26 proposed an isolator with negative

a)Also at: Australian National University, Canberra, Australian Capital

Territory, 2601, Australia. Electronic mail: WT323@mail.nwpu.edu.cn

J. Acoust. Soc. Am. 141 (2), February 2017 VC 2017 Acoustical Society of America 11610001-4966/2017/141(2)/1161/9/$30.00

stiffness and embedded it in continuous beams. Drastic

increase for the damping ratio of the flexural waves is

obtained with the structure. Assouar et al.27 applied a plate-

type acoustic metamaterial in an air-borne sound environ-

ment and analysed the sound mitigation performances.

However, when the metamaterial is immersed in compli-

cated environment, which is closer to practical engineering,

STL through metamaterial is rarely mentioned.

As stated in Ref. 18, metamaterial can be applied for

noise filter and wave suppression, and has potential applica-

tion in, for example, building a fuselage, which may be

exposed to external flow stimulus. The investigation on STL

through a metamaterial in the presence of external flow,

therefore, is an interesting and meaningful issue, regardless

the fact that few papers related to STL through metamateri-

als under external mean flow condition have been reported.

The study will be helpful in gaining a basic understanding of

sound insulation under complex acoustic environment. In

this paper, a method to calculate STL through a metamaterial

plate with LLRs attachment is established in the presence of

external mean flow, from which the influence of external

mean flow on the STL through a metamaterial with respect

of structural changes is demonstrated. Furthermore, how the

varying dynamic parameters, e.g., the Mach number and

incident angles, influence the STL is also explored. The

paper is organised as follows. In Sec. II, an effective medium

method28 is adopted to obtain the effective mass density of a

metamaterial with LLR attachments, and an explicit formula

for plane wave propagation through the metamaterial plate

in the presence of external mean flow is derived. In Sec. III,

results and discussions for STL with varying hysteretic

damping, Mach numbers, and incident angles are demon-

strated. Meanwhile, the coincidence frequency of the meta-

material plate is characterized. Finally, conclusions are

presented in Sec. IV.

II. STL CALCULATION

A. Effective parameters of the metamaterial plate viaeffective medium theory

Effective medium theory describes the macroscopic

properties of composite materials via theoretical modelling.

It has been effectively established for the investigation of

elastic metamaterial and its accuracy has been verified

within the subwavelength range.23,28 Via effective medium

method, the effective mechanical parameters of the metama-

terial can be obtained since the lattice constant of the LLR is

much smaller than the flexural wavelength of the plate. And

the related problem can be greatly simplified when the meta-

material plate considered here is treated as a homogeneous

plate with effective mass density.

In this subsection, the equivalent model of a metamate-

rial plate with LLR attachment is built. Figure 1 shows the

whole schematic of the metamaterial plate consisting of a

homogeneous plate with LLRs attached. Figures 1(a) and

1(b) give a typical unit cell of the metamaterial plate in dif-

ferent view angle, with unit length ax along the x direction

and ay along the y direction. The host plate has material

properties of complex Young’s modulus E0 ¼ Eð1þ igpÞ,density q0, Poisson’s ratio �, and thickness h, i ¼

ffiffiffiffiffiffiffi�1p

. The

vertical resonators with mass constant m1 and complex spring

stiffness k01 vibrate in the z direction and the lateral resonators

with mass constant m2 and complex spring stiffness k02 move

in the x direction. The complex spring stiffness includes hys-

teretic damping as k01 ¼ k1ð1þ ig1Þ, k02 ¼ k2 ð1þ ig2Þ. Four-

link-mechanisms are jointed to the host plate with geometri-

cal parameters of vertical distance L and horizontal distance

D. The effective mass density of the metamaterial plate is

analysed based on the thin plate theory and Bloch-Floquet

theory.

The closed form equation for flexural wave propagation

through a thin plate29 is

q0h@2w x;y; tð Þ

@t2¼� ~D

0 @4w x;y; tð Þ@x4

þ2@4w x;y; tð Þ@x2@y2

þ@4w x;y; tð Þ

@y4

�þq x;y; tð Þ; (1)

q 0;0; tð Þ¼�nk01 w 0;0; tð Þ�u1ð Þ�2nk02L

2Dv�u2ð Þ; (2)

in which n ¼ 1=axay and qðx; y; tÞ is the lumped force

applied to this unit. wðx; y; tÞ, u1, u2 are the displacement of

the plate, the vertical mass, and the lateral masses, respec-

tively. ~D0 ¼ E0h3=12ð1� �2Þ is the flexural rigidity. By

using Newton’s second law, the governing equations for m1

and m2 can be expressed as

FIG. 1. (Color online) Effective pro-

cess of the metamaterial plate, (a) front

view, (b) perspective view of the unit

cell of metamaterial plate, (c) metama-

terial plate, (d) equivalent plate with

effective parameters.

1162 J. Acoust. Soc. Am. 141 (2), February 2017 Wang et al.

m1

@2u1

@t2¼ k01 w 0; 0; tð Þ � u1ð Þ þ 2k02

L

2Dv� u2ð Þ; (3)

m2

@2u2

@t2¼ k02 v� u2ð Þ: (4)

Assuming small displacements for the plates and masses, the

displacement relationship of the four-link-mechanism is

obtained,

v ¼ � L

2Du1 � w 0; 0; tð Þð Þ: (5)

Based on the periodicity of the unit cell and the Bloch-

Floquet theorem for the infinite metamaterial plate,

the displacements of the plate and masses can be writ-

ten as

wðx; y; tÞ ¼ Weiðaxþby�xtÞ ¼ w0eiðaxþbyÞ;

u1ðtÞ ¼ U1e�ixt; u2ðtÞ ¼ U2e�ixt; (6)

where a and b are the wavenumbers along x, y directions,

respectively, and a ¼ 2p=k1, b ¼ 2p=k2 with k1 and k2

being the corresponding flexural wavelengths. x is the wave

frequency. W, U1, and U2 are the displacement amplitudes.

Inserting Eqs. (5) and (6) into Eqs. (1)–(4) yields

Qða; bÞ�A ¼ 0½ (7)

with

Q a; bð Þ ¼

q0hx2 � ~D a2 þ b2� �2 � nk01 �

1

2nk02

L

D

� �2

nk01 þ1

2nk02

L

D

� �2

nk02L

D

k01 þ1

2k02

L

D

� �2

m1x2 � k01 �1

2k02

L

D

� �2

�k02L

D

k02L

2D�k02

L

2Dm2x2 � k02

266666664

377777775

(8)

and A ¼ ðW U1 U2ÞT .

From K2 ¼ xffiffiffiffiffiffiffiffiffiffiqef f h

p=ffiffiffiffi~D

p, K ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 þ b2

p, the effective mass density qef f can be obtained as

qef f ¼ q0 þ�2k01k02m1nþ 2k01m1m2nx2 þ k02m1m2nx2c2

h 2k01m2x2 � 2k01k02 þ 2k02m1x2 � 2m1m2x4 þ k02m2x2c2� � : (9)

B. STL in the presence of external mean flow

External mean flow acts as a stimulus when a metamate-

rial is moving. And the effect of the external mean flow on

sound transmission through the metamaterial is too obvious to

be ignored. When external mean flow is added to one side of

the metamaterial with fluid properties of density q1 and sound

speed c1, a harmonic plane pressure with elevation angle u1

and azimuth angle b is incident from the side coupling with

external mean flow to another side containing stationary fluid

properties of density q2 and sound speed c2, as shown in Fig. 2.

The presence of the external mean flow may affect the sound

transmission loss through the metamaterial by adding light fluid

loading on the plate and refracting the transmitted waves angles

with u2 and b respectively. Setting the flow velocity V along

the x direction, the wave equation can be written as

@

@tþ V

@

@x

� �2

pinc þ prefð Þ ¼ c21r2 pinc þ prefð Þ; (10)

where pinc and pref are the pressures of incident and reflected

waves with the expression of

pincðx; y; z; tÞ ¼ pincðx; y; zÞejxt

¼ Pince�iðj1xxþj1yyþj1zzÞeixt; (11)

pref ðx; y; z; tÞ ¼ pref ðx; y; zÞejxt

¼ Pref e�iðj1xxþj1yy�j1zzÞeixt; (12)

with Pinc and Pref being the pressure amplitude of the inci-

dent and reflected sound waves, and

j1x ¼ j1 cos u1 cos b; j1y ¼ j1 cos u1 sin b;

j1z ¼ j1 sin u1: (13)

Substitution of Eqs. (11)–(13) into Eq. (10) yields

j1 ¼ ðx=c1Þð1þM cos u1 cos bÞ�1; (14)

where Mach number M ¼ V=c1. When the pressure is trans-

mitted through the metamaterial plate, the transmission pres-

sure is expressed as

ptrðx; y; z; tÞ ¼ ptrðx; y; zÞeixt ¼ Ptre�iðj2xxþj2yyþj2zzÞeixt:

(15)

Since there is no flow in medium 2, the transmitted wave

propagates referring to the classical wave equation and the

wave numbers are

J. Acoust. Soc. Am. 141 (2), February 2017 Wang et al. 1163

j2x ¼ j2 cos u2 cos b; j2y ¼ j2 cos u2 sin b;

j2z ¼ j2 sin u2; j2 ¼ x=c2; (16)

with u2 representing the refracted angle of the transmitted

waves and c2 representing the sound speed in fluid 2.

With the stimulus of the incident pressure, the flexural

vibration in the metamaterial can be expressed as

wðx; y; tÞ ¼ We�iðjmxxþjmyyÞeixt; (17)

with

jmx ¼ jm cos b; jmy ¼ jm sin b; jm ¼xct; (18)

where ct is the transverse velocity in the metamaterial plate.

The governing equation for the metamaterial plate is

~D0r4wðx; y; tÞ � qef f hx2wðx; y; tÞ

¼ pincðx; y; z; tÞjz¼0 þ pref ðx; y; z; tÞjz¼0

� ptrðx; y; z; tÞjz¼0: (19)

At the boundary of the metamaterial plate, the wavelengths

should be matched in order that the sound waves propagate

normally. Therefore,

j1x ¼ jmx ¼ j2x;

j1y ¼ jmy ¼ j2y: (20)

Inserting Eqs. (13), (14), (16), and (18) into Eq. (20), the

angle of the refraction can be obtained as

u2 ¼ arccosc2 cos u1

c1 1þM cos u1 cos bð Þ

� �: (21)

When the material properties on both sides of the metamate-

rial is the same, i.e., c1 ¼ c2, the incident elevation angle

does not equal to the refracted angle, u1 6¼ u2. It can be

found that the presence of the external mean flow makes the

transmitted wave deviate from the incident wave.

At the interfaces, among these two fluid fields and the

thin metamaterial plate, continuity of normal velocity and

displacement must be satisfied,

� 1

q1

@ pinc x; y; z; tð Þ þ pref x; y; z; tð Þ½ �@z

���z¼0

¼ @

@tþ V

@

@x

� �2

w x; y; tð Þ; (22)

� 1

q2

@ ptr x; y; z; tð Þ½ �@z

���z¼0¼@2w x; y; tð Þ

@t2: (23)

Combining Eqs. (11), (12), (15), (17), and (20), the transmit-

ted wave and reflected wave can be represented by the inci-

dent wave and the deflection of the plate,

Pinc�Pref ¼q1c1 sinu2

q2c2 sinu1 1þMcosu1 cosbð Þ

� Ptr; (24)

Ptr ¼q2c2ixsin u2

W: (25)

Inserting Eqs. (24)–(25) into Eq. (19), the explicit solution

of the displacement coefficient of the plate can be obtained

as

W ¼ 2Pinc

D x;u1; bð Þ ; (26)

where

D x;u1; bð Þ ¼ ~D0

k1 cos u1ð Þ4 � qef f hx2

þ iq1c1xsin u1 1þM cos u1 cos bð Þ þ

iq2c2xsin u2

:

(27)

The oblique sound power transmission coefficient is

s ¼ Ptr

Pinc

��������2

¼ 2ixq2c2

sin u2D x;u1; bð Þ

��������2

(28)

FIG. 2. (Color online) Schematic dia-

gram for STL through an equivalent

plate, (a) perspective view, (b) front

view, (c) vertical view.

1164 J. Acoust. Soc. Am. 141 (2), February 2017 Wang et al.

and

STL¼ 10log10

1

s

� �¼ 20log10

sinu2D x;u1;bð Þ2ixq2c2

��������: (29)

III. RESULTS AND DISCUSSION

Numerical results from the theoretical derivation in Sec.

II for STL through the metamaterial plate are presented in

this section with the parameters described in Table I. Effects

of incident angles, hysteretic damping, and Mach number on

STL are fully investigated. Mechanism of fluid and structure

coupling will also be discussed to give explanation for the

variation of the STL. It is known that coincidence frequency

is an important quantity to characterize the plate, whose

change reflects the degree of difficulty of coupling between

the fluid and the structure. During the calculation, the inves-

tigation is focused on the subwavelength range. Since the

higher resonance of LLRs is frh ¼ 251 Hz, from Table I, the

flexural wavelength of the host plate at the higher resonance

can be obtained as

kp ¼ 2p~D

qhx2rh

!1=4

¼ 0:3429 m (30)

and the ratio of the lattice constant to the flexural wavelength

is

ax=kp ¼ 0:0583� 1: (31)

Hence, the assumption of subwavelength still holds strong in

the study. When above the coincidence frequency, the effect

of the LLRs can be neglected from Fig. 4 as the effective

mass density is much closer to the density of the host plate.

Analysis in this range thus follows classical thin plate theory.

A. The coincidence frequency of the metamaterialplate

Coincidence frequency is an important index for sound

transmission, as when coincidence occurs the structure is the

most susceptible to acoustic excitation and gives rise to a far

more efficient transfer of sound energy from one side to the

other. When the incident angle (elevation angle) is p/6, using

the parameters in Table I, the coincidence frequency of the

plate is fco ¼ 5300Hz, and the ratio a=kpf is equal to 0.27,

which is still small enough to support the subwavelength

assumption. Furthermore, the resonance of the substructure

is far lower than the coincidence frequency. Thus, the effec-

tive medium method still holds in the analysis of the effec-

tive “bare plate” at the coincidence frequency. Using the

effective medium method, in the presence of the external

mean flow, the coincidence frequency of the metamaterial

can be obtained as1

fco ¼c2

2

2p cos2 u2ð Þ

ffiffiffiffiffiffiffiffiffiffiqef f h

D

r� c2

2

2p cos2 u2ð Þ

ffiffiffiffiffiffiffiq0h

D

r: (32)

From Eq. (32), it can be seen that the coincidence frequency

is related to the refracted angle as the properties of the host

plate and the fluid in medium 2 is fixed. Combining Eqs.

(21) and (32), the coincidence frequency changes as the inci-

dent angle and the velocity of the external mean flow vary,

which will be exhibited in the following part.

B. STL through the metamaterial plate

The effective mass density from Eq. (9) is visually

shown in Fig. 3 by using parameters in Table I. It can be

divided into three parts in terms of the frequency. Within the

first part, i.e., from 0 to 120 Hz, the effective mass density

equals the density of the host plate plus the averaging density

of the vertical masses over the unit cell, which is qst ¼ q0

þm1=axay ¼ 1:5q0. While for the second part, from 120 to

305 Hz, LLRs vibrate vigorously, causing significant influ-

ence on the effective mass density. Within part 2, the effec-

tive mass density becomes negative in specific ranges, which

is much different from the classical structure. And how this

negative mass density influences the STL is an interesting

issue. For the third part, above 305 Hz, the ratio qef f=q0

becomes 1, which means that the effect of the attached

TABLE I. Parameters of the metamaterial plate and fluid medium.

Symbol Description Value

Metamaterial plate properties

k1 Vertical spring stiffness 1� 104N=m

m1 Vertical mass 0.005 kg

k2 Lateral spring stiffness 0:5� 104 N=m

m2 Lateral mass 0.002 kg

ax � ay Side length of a unit cell 0:02 m� 0:02 m

L=D Ratio of the four-link mechanism 0.5

q Material density of the host plate 7800 kg=m3

E Young’s modulus of the host plate 2:1� 1011Pa

� Poisson’s ratio of the host plate 0.3

h Thickness of the host plate 0.003 m

Fluid properties

q1 ¼ q2 Fluid density 1:25 kg=m3

c1 ¼ c2 Sound speed in the fluid 343 m=sFIG. 3. (Color online) Effective mass density of the metamaterial plate.

J. Acoust. Soc. Am. 141 (2), February 2017 Wang et al. 1165

substructures is too little to be neglected. In summary, the

variation of the effective mass density of the metamaterial

plate is mainly attributed to the attached substructures.

Therefore, the effects of the substructures on the STL

through the metamaterial should be examined, which is dem-

onstrated in Fig. 4.

Comparisons of the STL through plates with density of

q0, qst, and qef f are presented. During the calculation,

parameters in Table I are used. Damping is zero for both the

plate and the resonators. The incident angles are u1 ¼ p=6

and b ¼ 0.

STL through three different plates is calculated and

shown in Fig. 4. It is noticeable that there are several over-

laps among the three STL curves. For the range from 10 to

120 Hz, STL of the metamaterial plate is almost the same as

that of the bare plate with qst, and about 2 dB higher than

that of the bare plate with q0 because of the static averaged

vertical mass. For the range above 400 Hz, STL through the

metamaterial plate is almost the same as that of the bare

plate with q0, and 2 dB lower than that of bare plate with qst.

From Fig. 3, it is known that in such two ranges, the mass

density of the metamaterial is stable and equals to either qst

or q0, and the overlaps with the two bare plates in density

can be explained by the mass law, i.e., for a bare plate,

obliquely incident mass law

STL hð Þ ¼ 10 log10 1þ xqh cos h2q0c0

� �2 !

� 20 log10

xqh cos h2q0c0

� �: (33)

Equation (33) reveals that higher density leads to higher

STL. Similarly, from Eq. (32), it denotes that higher density

results in higher coincidence frequency. Last but not the

least, for the range from 120 to 400 Hz, vigorous vibrating

LLRs affects the effective density changes dramatically (as

shown in Fig. 3), as well as the STL. Larger STL occurs

when the vibration phases of the resonators are the same as

that of the host plate, while lower STL is the result of the

opposite phases of the resonators and the host plate. From

the analysis, it can be concluded that LLRs have significant

effects on effective mass density, thereby the STL of the

metamaterial plate.

C. Effect of damping on the STL

In practical engineering, damping in structural acoustics

cannot be neglected. And damping always has significant

influences on the response of the structures. Reasonable

damping can smooth the response curves and lower the

response peaks. However, too much damping may deactivate

the effect of the substructures. Although there are several

kinds of damping existing in structure, hysteretic damping

(structural damping) is adopted for the analysis. In this part,

effects of damping from different part of the metamaterial

plate on the STL are examined thoroughly with fixed Mach

number M ¼ 0 and incident angle u1 ¼ p=6 and b ¼ 0.

Figures 5 and 6 show how structural damping of the

resonator and host plate affects STL. Obviously, the most

distinguish influence lies on the peaks and dips on the trans-

mission loss curves. In Fig. 5, it can be seen that the struc-

tural damping of the host plate only influences the

transmission at the coincidence frequency. The bigger the

plate’s damping, the higher the STL at the coincidence fre-

quency. Meanwhile, STL in other frequency ranges does not

change. Similarly, in Fig. 6, only the first and second peaks

and dips on the STL curves are influenced by the hysteretic

damping of the resonators. Comparing the effects of the hys-

teretic damping, it can be found that damping from different

parts of the metamaterial plate provides separate influences

on the STL at different frequencies, which facilitates the

design of the metamaterial when implemented in practical

engineering for sound insulation.

D. Effect of the Mach number on the STL

In the presence of the external mean flow, STL may be

different from that of stable medium immersed metamaterial

FIG. 5. (Color online) STL plotted as a function of an infinite metamaterial

plate immersed in convective fluid for selected structure damping with ele-

vation angle u1 ¼ p=6 and azimuth angle b ¼ 0, resonators’ damping ¼ 0,

Mach number ¼ 0.

FIG. 4. (Color online) Comparison of STL through plate with different mass

density.

1166 J. Acoust. Soc. Am. 141 (2), February 2017 Wang et al.

plate.5 In this subsection, effects of various Mach number

from 0 to 0.8 on the STL of the metamaterial plate are calcu-

lated with other parameters fixed as in Table I. The damping

values for the resonators and host plate are all 0.05, i.e.,

g1 ¼ 0:05, g2 ¼ 0:05, gp ¼ 0:05.

From Fig. 7, there are several peaks and dips on the

transmission curves. The first and second peaks and dips are

due to the resonance of the substructure while the third dip is

the reason of the match of the trace wavelengths of the

acoustic wave and that of the bending wave of the metamate-

rial plate, the so-called coincidence frequency. The peaks

and dips divide the STL curves into three parts. The first part

includes two frequency ranges, i.e., one from 0 to 120 Hz,

the other from 305 Hz to the coincidence frequency. The sec-

ond part contains two peaks and dips and the negative mass

density range, from 120 to 305 Hz. The third part is the fre-

quency range over the incidence frequency. Within those

parts, the trends of the STL differ from each other. As Mach

number gets bigger, several interesting phenomena can be

seen. (1) The coincidence frequency of the metamaterial

plate increases which is consistent with Eq. (32). The

increase is mainly because the external mean flow affects the

transmitted sound wave to refract from its original angle.30

(2) It is obvious that STL is not linear function of the Mach

number since larger increase of Mach number does not lead

to higher increase of the STL. Meanwhile, the STL increases

modestly in the presence of the external mean in the first and

second part of the curves which is similar to the tendency of

bare plate for the density is steady within such ranges, and it

can be found in Refs. 4 and 31. Above the coincidence fre-

quency, the trend of STL shifts decrease as the Mach number

increases. This is the effect of the aerodynamic damping

effect, which can be found in Refs. 32 and 33. (3) For the

range from the first peak to the second dip, including the

negative mass density range, the external mean flow seems

to have little influence on the STL, especially on the negative

slopes. While at the peaks and dips, STL increase dramati-

cally. Comparing the effect of the resonance of the LLRs

and the external mean flow, it can be deduced that the sub-

structures plays a more significant role in the STL curve,

which can be attribute to that the added mass effect of light

fluid loading are negligible in terms of the behaviour of the

substructures. As a result, it is noticeable that effects of

Mach number of the external mean flow are complex and

different within different frequency ranges, and analysis in

above gives general explanation for the variation of the STL,

which can be helpful in designing fuselages and coverages

by utilizing metamaterials.

E. Effect of incident angle on STL

From previous derivation, the explicit formula of the

coincidence frequency and the STL are both related to the

sound incident angle, i.e., the elevation angle and the azi-

muth angle. In this subsection, variations of the transmission

loss through the metamaterial plate are demonstrated in Figs.

8 and 9 for different incident angles, as well as the

FIG. 6. (Color online) STL plotted as a function of an infinite metamaterial

plate immersed in convective fluid for selected resonators’ damping with

elevation angle u1 ¼ p=6 and azimuth angle b ¼ 0, Mach number is 0,

structure damping ¼ 0.

FIG. 7. (Color online) Comparison of STL as a function of frequency of an

infinite metamaterial plate immersed in convective fluid for selected Mach

numbers with elevation angle u1 ¼ p=6 and azimuth angle b ¼ 0, resona-

tors’ damping and structure damping is 0.05.

FIG. 8. (Color online) STL plotted as a function of an infinite metamaterial

plate immersed in convective fluid for selected elevation angle with azimuth

angle b ¼ 0, Mach number ¼ 0.2, structure and resonators’ damping ¼ 0.05.

J. Acoust. Soc. Am. 141 (2), February 2017 Wang et al. 1167

coincidence frequency. In the study, the external mean flow

is added, which is set to 0.2. Meanwhile, hysteretic damping

of the host plate and resonators is 0.05.

In the calculation of the effects of the incident angles on

the STL through the metamaterial plate, either the elevation

angle or the azimuth angle is fixed. Figure 8 reveals that as

the elevation angle increases, the coincidence frequency

moves to higher frequency range, and STL increases up to

the coincidence frequency while decreases above the coinci-

dence frequency. In Fig. 9, the coincidence frequency is a

significant boundary. With the azimuth angle increasing

from 0 to p=3, on the left side of the coincidence frequency,

the STL decreases slightly, while on the right side, on the

contrary, the STL increases. The coincidence frequency

shifts to lower frequency range as the azimuth angle

increases, which mean easier resonance of the metamaterial

plate at large incident azimuth angle.

Comparing Figs. 8 and 9, it can be found that the eleva-

tion angle has more significant influences on the STL, espe-

cially on the coincidence frequency. In the left range of the

coincidence frequency, larger elevation angle leads to higher

STL, however, larger azimuth angle results in slightly lower

STL. In the right range of the coincidence frequency, oppo-

site phenomenon occurs. In summary, higher STL needs

incident sound waves with larger elevation angle and smaller

azimuth angle.

IV. CONCLUSIONS

This paper is concerned with the STL through a meta-

material plate with LLRs in the presence of external mean

flow. Equations for STL are thoroughly obtained with the

help of effective medium theory. Numerical results are

evaluated for a specific set of parameters and reveal that

external mean flow improves STL over a wide range below

the coincidence frequency based on the light fluid loading

on the metamaterial plate, but not in the negative mass

density ranges caused by the rigorous lateral local

resonance. An opposite phenomenon occurs above the

coincidence frequency, owing to the fact that the STL is

influenced by the flow generated from aerodynamic damp-

ing effect. As Mach number increases, the coincidence fre-

quency of the metamaterial shifts to higher frequency

range, attributing to the sound wave refraction by the

external mean flow. Effect of hysteretic damping of the

metamaterial plate only lies in specific frequencies induced

by the LLRs and host plate. Damping can lower the trans-

mission peaks and raise the dips, which provides a practi-

cal solution for the sound insulation. Tendencies of STL

are different with varying incident elevation and azimuth

angles. STL through the metamaterial is always enhanced

within the frequency range below coincidence frequency,

but reduced above the coincidence frequency when

increasing the elevation angle. On the contrary, the azi-

muth angle has inverse effect on the STL in accordance

with that of the elevation angle.

ACKNOWLEDGMENTS

The first author is grateful for the sponsorship from the

China Scholarship Council and Innovation Foundation for

Doctoral Dissertation of Northwestern Polytechnical

University (Grant No. CX201702).

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