SONIC COMPOSITES AS NOISE BARRIERS

Abstract

We address an alternative road traffic noise barrier as a sonic composite consisted of an array of acoustic scatterers embedded in air. Acoustic scatterers are piezoceramic hollow spheres of functionally graded materials-the Reddy graded hollow spheres. Multiple (Bragg) scattering lead to a selective sound attenuation in the frequency bands called band gaps or stop bands for certain spacing and size of the scatterers. Size variations of the band gaps are discussed in this paper by taking into account the intrinsic acoustic properties of the scatterers. We reveal that the noise barrier simulation in the context of the road traffic noise confirm the Bragg band gaps existence and the predicted multiple resonances at frequencies below the first Bragg band gap.

Full text

Ro. J. Techn. Sci.  Appl. Mechanics, Vol. 65, N 2, P. 122135, Bucharest, 2020
SONIC COMPOSITES AS NOISE BARRIERS
LIGIA MUNTEANU, VETURIA CHIROIU, CIPRIAN DRAGNE, VALERIA MOȘNEGUȚU,
NICOLETA NEDELCU, IULIAN GIRIP, CRISTIAN RUGINĂ
Abstract. We address an alternative road traffic noise barrier as a sonic composite
consisted of an array of acoustic scatterers embedded in air. Acoustic scatterers are
piezoceramic hollow spheres of functionally graded materials - the Reddy graded
hollow spheres. Multiple (Bragg) scattering lead to a selective sound attenuation in
the frequency bands called band gaps or stop bands for certain spacing and size of the
scatterers. Size variations of the band gaps are discussed in this paper by taking into
account the intrinsic acoustic properties of the scatterers. We reveal that the noise
barrier simulation in the context of the road traffic noise confirm the Bragg band gaps
existence and the predicted multiple resonances at frequencies below the first Bragg
band gap.
Key words: Sonic composite, band-gaps, Multiple resonances, Noise barrier.
1. INTRODUCTION
In the past few years, the accurate predictions of the interactions of acoustic
waves with periodic structures show the potential of sonic composites to work as
noise barriers in the road traffic noise [1‒6]. According to the Bragg’s theory, the
sonic composites can generate large band-gaps from multiple reflected waves with
large acoustic impedance ratios between the scatterers and the matrix, respectively
[7‒9]. The sonic composites are composed of scatterers embedded into air. The
scatterers are local resonators which scater, diffuse or disperse energy, such as
spheres, rods or cylinders. The sonic composite is the sonic version of the photonic
crystal being architectured such that the sound is not allowed to propagate in
certain full band-gaps due to complete reflections. Existence of the band-gaps
makes this material the main candidate for applications as the acoustic filters [10],
acoustic barriers [11] or wave guides [12, 13].
Many unique properties of the sonic composites come from the generation of
large band-gaps at different frequencies inverse proportional to the central distance
between two scatterers [14]. The band-gap generation mechanism means the
completely reflected waves in the frequency range where all partial band-gaps
overlap. This can be explained by evanescent waves [15] which lead to localized
modes withe no real wave number [16, 17].
Institute of Solid Mechanics, Romanian Academy, Bucharest

Sonic composites as noise barriers
123
In the present paper, we investigate an alternative noise barrier as a 3D sonic
composite with scatterers made from functionally graded materials with radial
polarization [18‒20].
Architectural acoustics deals with the sound quality of closed and open
spaces [28]. Sound perception in the free field is different from halls because
reflections on the walls are missing. In the free field, only the direct sound comes
to the listener. When the space is closed, the sound gives rise to a multitude of
reflections that decrease in time and space.
Unwanted noise is perceived as a dangerous factor for human health [21] and an
environmental stressor in the road traffic especially during the night time in the
urban areas [22, 23]. The extra low frequency band gaps in the sonic composites
can be exploited in the noise barrier located above the ground surface as suggested
in [3, 24] where cylindrical PVC embedded in air or arrangements of triangular,
square or elliptical rigid scatterers (square lattice) were used for scatterers.
The efficiency of a barrier along the highways depends on the moving traffic
and the vehicle type and velocity [28]. According to STAS 9783/1‒94 Standard
Acoustics in Constructions [29], the normalized traffic noise spectrum measured in
A-weighted decibel (dBA) lies between 100 Hz to 5 KHz, with the main noise
energy centred at 1 kHz. In addition, the normalized traffic noise spectrum does not
take into accounts the temporal effects of the moving traffic. Efficiency of a road
traffic noise barrier is measured by the Insertion Loss (IL) expressed in Decibel
[dB] as
10
20log
dir
tr
p
IL
p
, (1)
where
dir
p
and
tr
p
denote the pressure obtain without and with the barrier,
respectively.
Conventional attenuating of the traffic noise knows two main mechanisms highly
frequency dependent. At the lower end of the frequency spectrum of interest
(below 1 kHz) the ground effect is the most important mechanism. At the
frequencies greater than 2 kHz, the noise can be attenuated by absorbing of the noise
by leaf. In both cases, it was necessary to have planted vegetation belt to obtain a
little attenuation.
Tree plantations arranged periodically can also attenuate the noise at low
frequencies.
High attenuations at low frequencies (<500 Hz) may occur by devices of
destructive interference of the scattered waves. The research of the sonic
composites barriers exploits the properties of the scatterers which scater, diffuse or
disperse energy. The scatterers can be spheres, rods or cylinders. The noise barriers
made of the sonic composites offer an alternative mechanism of absorption, in
addition to multiple scattering waves in the periodic structure. The influence of

Ligia Munteanu, Veturia Chiroiu, Ciprian Dragne, Valeria Mosnegutu, (...)
124
wind generated by the noise also affects the efficiency of the noise barrier. It was
found that the adverse influence is absent up to wind speeds of 30 m/s [3].
The bandgap generation requires a large contrast in the density and velocity
of sound between the scatterer and the matrix material. We show this in the
following.
The acoustic impedance of a material
Z
, and the wave velocity
v
, are
expressed as
Z v E
   
,
K
v

, (2)
where

,
E
and
K
are the density, the Young’s elasticity modulus and bulk
modulus of the material, respectively.
In a sonic crystal with square lattice, the fundamental Bragg resonance
frequencies in the lattice main directions are separated by a factor of
2
as [3]
,
2
Bragg x
c
f
l
, ,
2 2
Bragg y
c
f
l
, (3)
where
c
is the speed of sound (344 m/s in air at room temperature 20ºC).
On the other hand, the hexagonal lattice the Bragg resonances are separated
by
3
which result in a wider bandgap for a higher acoustic impedance mismatch
between the matrix and the scatterer [3].
The nature of the matrix is the difference between the sonic and phononic
crystals. If the matrix is solid then the term phononic crystal is used for the
artificial crystal. The phononic crystals exhibit both longitudinal and transverse
shear waves, but in contrast, the sonic crystals are independent of the transverse
waves. The scatterers are made of solid materials and the matrix is air to obtain
high acoustic impedance contrast between them.
2. THE SONIC COMPOSITE
The sonic composite discussed in this paper contains an array of 144
piezoceramic hollow spheres of functionally graded materials with diameter
a
and
embedded in air. This panel is displayed in Fig.1. For simplicity, we consider all
spheres are tangent to each other. The length of the panel is
L
, the width is
d
, and
its thickness is
e a

. The coordinate system
1 2 3
Ox x x
is introduced with origin in
the middle plane of the plate, the axis
1
Ox
in-plane normal to the layers and the
axis
3
Ox
out-plane normal to the plate.
Absorbing boundary conditions in the
1
x
-direction at 1
0
x

and 1
x l

have the role to avoid the unphysical reflections. The transducer send the plane

Sonic composites as noise barriers
125
monochromatic waves in the
1
x
-direction and the receiver measures the
displacements at both sides of the plate at
1
x b

and
1
x l b
 
, respectively. The
ratio of the displacement at the receiver and the input transducer, respectively
measures the sound attenuation.
Fig. 1 – The sonic composite consisted of an array of acoustic scatterers embedded in air.
Fig. 2 – Artificial perfectly matched layers at the boundaries of the sonic composite; a) location of
the layers; b) pressure map at 1.2 kHz computed by FEM.

Ligia Munteanu, Veturia Chiroiu, Ciprian Dragne, Valeria Mosnegutu, (...)
126
Fig. 3 – Insertion Loss with respect to the frequency computed by FEM.
The Sommerfeld condition of radiation means the absence of all reflection
from the boundaries. To obtain an unbounded medium an artificial perfectly
matched layers may to be introduced at the boundaries of the panel. In these layers,
the wave equation contains a damping term which attenuates proportionally with
distance in the direction perpendicular to the interface with the physical domain.
Location of the perfectly matched layers at the boundaries of the panel is shown in
Fig. 2a.
The pressure map computed by FEM for 1.2 kHz is presented in Fig. 2b. We
see that the wave incident upon the layers is absorbed only in the outgoing
direction, while the tangential waves to the interface between the layers and the
physical domain remain unaffected. Important is that the wave incident upon these
layers from environment do not reflect at the interface.
Fig. 3 shows the variation of the Insertion Loss with respect to frequency.
The panel is predicted to produce the highest amplitude Bragg band gap at 1.3 kHz.
This suggests that much of the incoming wave front is blocked by the larger
surface of the scatterer. Also, the Bragg band gap is predicted to contain two peaks
instead of one.
The elastic, piezoelectric and dielectric constants depend on the radial
coordinate
r
. The following notations are introduced:
ij

the stress tensor,

the
electric potential,
i
D
the electric displacement vector,
ij
C
the elastic constants,
66 11 12
( ) / 2
C C C  ,
ij
f
the piezoelectric constants
ij
f
,
ij

the dielectric
constants,
, ,
i r
  
, and

the density of the material.

Sonic composites as noise barriers
127
The constitutive equations for the piezoelectric hollow sphere in the
spherical coordinate system
( , , )
r
 
are given by [25‒27]
11 12 13 31
rr r
r C S C S C S f r
  
     
,
12 11 13 31 ,
rr r
r C S C S C S f r
  
     
,
13 13 33 33 ,
rr rr r
r C S C S C S f r
 
     
,
44 15 ,
2
r r
r C S f
  
   
,
44 15 ,
2 csc
r r
r C S f
  
   
,
66
2
r C S
 
  ,
15 11 ,
2r
rD C S
  
   
,
15 11 ,
2 csc
r
rD f S
  
   
,
31 31 33 33 ,
r rr r
rD f S f S f S r
 
     
.
(4)
The strain components
ij

are related to the displacement components
i
u
,
, ,
i r
  
by
,
rr r r
r ru
  , ,
r
r u u
  
  
,
,
csc cot
r
r u u u
   
     
,
, ,
2r r r
r u ru u
   
   
, , ,
2 csc
r r r
r u ru u
   
    
,
, ,
2 csc cot
r u u u
     
     
.
(5)
The electrostatic charge is described as
, , ,
( ) csc ( sin ) csc ( ) 0
r r r
r rD rD rD rD
   
      
. (6)
To simplify the motion equations, the Chen functions
F
,
G
and
w
, and the
stress functions
1

and
2

are introduced [25‒27]
, ,
csc
u F G
  
    ,
, ,
csc
u F G
  
   , r
u w

,
1, 2,
csc
r
r
  
     
,
1, 2,
csc
r
r
  
    
.
(7)

Ligia Munteanu, Veturia Chiroiu, Ciprian Dragne, Valeria Mosnegutu, (...)
128
Consequently, the motion equations can be written as two independent sets
of equations
,r
rA MA
, ,r
rB PB
, (8)
2
[ , , , , , ]
T
rr r
B r G w rD
   
,
2
2 2
66
2
1
44
2 ( 2)
1
C r
M
t
C
 

     
 

 
 
 
, (9)
where
2 2
2 2
2 2
cot csc
  
     

 
. The matrix
P
has the components:
11
2 1
P
  
,
2
12
P
 
,
2
13 1
P k
 
,
2
2
14 1
2
2P k r
t

   

,
15 25 64
2 2
P P P
    
, 21
P
 
, 22
2
P
 
,
2
2 2
23 2 66
2
2P k C r
t

    

,
24 1
P k
 
,
1
32 44
P C

,
33 34 55
1
P P P
   
, 1
36 44 15
P C f

, 1
41 33
P
  
,
2
43
P
 
,
44
2
P
  
, 1
45 33
P f

  ,
2 2
52 44 15
P C f

 
,
2
56 3
P k
 
,
1
61 33
P f

  ,
2
63
P
 
, 1
65 33
P C

  ,
(10)
with
2
33 33 33
C f
    , 1
13 33 31 33
( )
C f f

     ,
1
13 33 33 31
( )
C f C f

    ,
1 13 31 11 12
2( ) ( )
k C f C C      ,
2 1 66
0.5
k k C
  ,
2 1
3 11 15 44
k f C

   .
(11)
The functionally graded material is described by the Reddy law [18‒20]
(1 )
p z
M M M
 
    
, (12)
where
0

is the ratio of the inner and outer radii of the hollow sphere,

is the
gradient index,
p
M
and
z
M
are material constants of two materials, namely PZT-
4 and ZnO. The case
0
 
corresponds to a homogeneous PZT-4 hollow sphere
and
  
, to a homogeneous ZnO hollow sphere.
Sharp periodic boundary conditions in displacement and traction are
considered at the interfaces between the hollow spheres and air.

Sonic composites as noise barriers
129
The sets of equations (8) imply two independent classes of free vibrations.
The first class does not involve the piezoelectric or dielectric parameters, and
corresponds to the isotropic elastic sphere. The second class depends on the
piezoelectric and the dielectric parameters. When the gradient index

increases,
the natural frequencies increase for all modes.
The equations (8)‒(12) are solved by the cnoidal method [8, 9].
3. NOISE BARRIER
The research objective of the paper is to use the sonic composite shown in
Fig. 1 to design an outdoor noise barrier. Multiple independent resonance bandgaps
below the first Bragg bandgap (due to the periodicity of the scatterers) between 400
and 1600 Hz are needed for the traffic noise.
The noise barrier is a 1

144 array of hollow piezoelectric ceramic hollow
spheres of diameter
0.11m
a

embedded in air. The length of the plate is
1.76m
L

the width is
0.99m
d

, and its thickness is
0.11m
e a
 
. The
thickness of the hollow sphere is 0.00275m and 0
0.3
  .
The numerical results are carried out for the following constants [2]:
for PZT-4
10 2
11
13.9 10 N/m
C  ,
10 2
12
7.8 10 N/m
C  ,
10 2
13
7.4 10 N/m
C  ,
10 2
33
11.5 10 N/m
C  ,
10 2
44
2.56 10 N/m
C  ,
2
15
12.7C/m
f,
2
31
5.2C/m
f  ,
2
33
15.1C/m
f, 11
11
650 10 F/m

   , 11
33
560 10 F/m

   ,
3
7500kg/m
  ;
for ZnO
10 2
11
20.97 10 N/m
C  ,
10 2
12
12.11 10 N/m
C  ,
10 2
13
10.51 10 N/m
C  ,
10 2
33
21.09 10 N/m
C  ,
10 2
44
4.25 10 N/m
C  ,
2
15
0.59C/m
f  ,
2
31
0.61C/m
f  ,
2
33
1.14C/m
f, 11
11
7.38 10 F/m

   ,
11
33
7.83 10 F/m

   ,
3
5676kg/m
  ,
0.5
 
;
and for air
3
1.2kg/m
air
  and speed of sound
1
344 ms

.

Ligia Munteanu, Veturia Chiroiu, Ciprian Dragne, Valeria Mosnegutu, (...)
130
Fig. 4 – a) Plan view of the noise barrier; b) side view of the noise barrier.
Fig. 5 – Direct and the transmitted signals variation with respect to time.
We assume that the receiver microphone is located at 50 mm from the
opposite face of the source. The source and receiver are located at 1.2 m above the
ground. The loud speaker is placed between 1.5 m and 1.63 m away from the
barrier such that the source-receiver axis is normal to the barrier orientation Fig. 4.
The input signal is simulated by removing any signals with frequency
beyond the 10 kHz.
Fig. 5 shows the time signals corresponding to the direct and transmitted
fields of signals through the barrier.
Fig.6 plots the dispersion curve including the first partial band-gaps for the
sonic composite. The reduced units for the frequency are
0
/ 2
a c
 
, with
0
c
the
speed of sound in air.

Sonic composites as noise barriers
131
Fig. 6 – Linear dispersion for panel.
The guided waves are accompanied by evanescent waves which extend to
the periodic array of the scatterers surrounding the wave-guide. Using the
Joannopoulus theory of the bad-gap structure [14], Fig. 7 presents the band
structure with the evanescent modes with exponential decay. The central grey
region is the full band-gap ranged between 8.02 kHz and 8.72 kHz, given by the
real part of the wave vector constrained in the first Brillouin zone for each
frequency. The left region shows the imaginary part of the wave vector for
longitudinal direction frequency, while the right region is the imaginary part of the
wave vector for transverse direction frequency. The red lines represent the
imaginary part of the wave vector of the evanescent modes inside the bad-gap.
A full band-gap can be obtained by adding band-gaps for both longitudinal
and transverse waves in the same frequency region.
Fig. 7 – Band structure for the panel in the case of Reddy law.

Ligia Munteanu, Veturia Chiroiu, Ciprian Dragne, Valeria Mosnegutu, (...)
132
Fig. 8 – The input and coupled waves for panel in the case of Reddy law.
The influence of the diameter size of the hollow sphere is investigated for
three values of
a
, i.e. 0.12m, 0.09 m and 0.08 m with similar wall thickness of
0.00275m and 0
0.3
  .
Figs. 9‒11 show the Insertion Loss spectra of the panel for three sizes of
diameter 0.12 m, 0.09 m and 0.08 m respectively. We observed that by increasing
the size of diameter, the resonance appears to lower frequency, namely 1.8 kHz,
1.5kHz and 1.1 kHz respectively.
Fig. 9 – Insertion loss spectra of the panel for a diameter 0.12 m.

Sonic composites as noise barriers
133
Fig. 10 – Insertion loss spectra of the panel for a diameter 0.09 m.
Fig. 11 – Insertion loss spectra of the panel for a diameter 0.08 m.
4. CONCLUSIONS
An alternative noise barrier for the road traffic is reported in this paper using
an array of piezoceramic hollow spheres of functionally graded materials - the
Reddy graded hollow spheres. The sonic composites exhibit a selective noise
attenuation in the band gaps related to the size of the scatterers. The enhancing the

Ligia Munteanu, Veturia Chiroiu, Ciprian Dragne, Valeria Mosnegutu, (...)
134
band gaps in the low frequency domain is investigated the intrinsic acoustic
properties of the scatterers. We observed that by increasing the size of diameter,
the resonance appears to lower frequency, namely 1.8 kHz, 1.5kHz and 1.1 kHz
respectively. The results confirmed the existence of the Bragg band gaps for noise
barriers and the predicted multiple resonances at frequencies below the first Bragg
band gap.
Acknowledgements. This work was supported by the Romanian Academy. The support is
gratefully acknowledged.
Received on July 24, 2020
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