Advances in Materials Physics and Chemistry, 2011, 1, 86-90
doi:10.4236/ampc.2011.13014 Published Online December 2011 (http://www.SciRP.org/journal/ampc)
Band Gaps and Single Scattering of Phononic Crystal*
Xiaoyi Huang1#, Jingcui Peng2, Huanyou Wang1, Gui Jin1
1Department of Physics, Xiangnan University, Chenzhou, China
2Department of Applied Physics, Hunan University, Changsha, China
Received August 28, 2011; revised September 29, 2011; accepted October 11, 2011
Abstract
A method is introduced to study the transmission and scattering properties of acoustic waves in two-dimen-
sional phononic band gap (PBG) materials. First, it is used to calculate the transmission coefficients of PBG
samples. Second, the transmitted power is calculated based on the far field approach. We have also calcu-
lated the scattering cross section, the results indicate that phononic band gap appear in frequency regions
between two well separated resonance states.
Keywords: Phononic Crystal, Band Gap, Single Scattering
1. Introduction
The acoustic properties of a locally homogeneous and
isotropic composite material is characterized by a set of
parameters varying in space: mass densityρ, Lamé coef-
ficients λ, and μ. In this paper we focus on the composite
materials, which consist of homogeneous particles dis-
tributed periodically in a host medium. They are charac-
terized by different mass densities and Lamé coefficients.
When identical particles are distributed periodically in a
host medium, the composite material may be referred to
as a phononic crystal. Recently the propagation of elastic
or acoustic waves (EL or AC wave) in a phononic crystal
has received much renewed attention [1-9]. These new
materials can be of real interest since a large contrast
between the elastic parameters is allowed. For example,
systems composed of very soft rubber [10] are more
likely to obtain the low-frequency gaps with a structure
of small dimension. This can lead to promising applica-
tions such as a low-frequency vibration/noise devices
such as lenses and acoustic interferometers [11]. On the
other hand, more sophisticated combinations such as
fluids infiltrated in a drilled solid [5] or solid-solid sys-
tems [7] have been demonstrated to produce a full pho-
nonic band gap for ultrasounds. Phononic crystal make
possibility of the achievement of complete frequency
band gaps that are useful to prohibit specific vibrations in
accurate technologies such as transducers and sonar.
In the plane-wave expansion method, the finite differ-
ence time domain and the multiple-scattering theory are
commonly used in order to study the elastic response of
phononic crystal [12-15]. In this work, in order to study
the propagation of acoustic waves in phononic crystal,
we consider a two-dimensional periodic system consist-
ing of finite cylinders of circular cross section. The sys-
tem is periodic in the x-y plane and within it there is a
translational invariance in the direction (z) parallel to the
cylinders. The intersection of the cylinders with a trans-
verse plane makes a square lattice. We treated finite PBG
samples as scattering objects in open geometry, The ra-
diation boundary condition was naturally imposed. Con-
sidering the far-field approach, we have independently
adopted this method to study the transmission and scat-
tering properties of finite PBG samples. In the case of
transmission, a generalized transmission coefficient can
be defined in terms of the far-field total scattering am-
plitude, from the total scattering amplitude we can re-
trieve the dispersion relations and the decay length inside
a gap. By adopted this method, the incident field, scat-
tered field and the total scattering amplitude become
very simple form, the calculating can be extremely sim-
plified. We explicitly demonstrate that this method can
produce transmission results that are in excellent quanti-
tative agreements with the available experimental data.
2. Model and Formula
The displacement vector


,tUr in a homogeneous
elastic medium of mass density ρ and Lamé coefficients
λ, μ satisfies the following equation:
*Supported by the Key Project of Education Department of Hunan
Province (09A086), Science and Technology Project of Chenzhou.

X. Y. HUANG ET AL.87



2
20
t
  UUU
it 

(1)
In the case of a harmonic elastic wave with angular
frequency ω, we have
 
,Re expt  

Ur Ur, (2)
and Equation (1) was reduced to the following time-in-
dependent form



2
2    uu0u. (3)
Defining
 ulmn, (4)
where

l, (5)

 m

Z

Z
, (6)

 n, (7)
where is the unit vector along the z-axis. Z

, χ and
ψ are the displacement potential functions of longitudinal
and two transverse waves respectively. The displacement
potential function of the incident longitudinal waves can
be expanded in terms of the cylindrical Bessel Function
[16]
  
exp exp
n
inczn lr
jk zjJkrjn





, (8)
where

12
22
lrl z
kkk is the radial component of the
incident wave vector, Jn is the Bessel function of the first
kind of order n, kz is the z-axis component of the incident
wave vector; kl is the longitudinal wave numbers, r is the
normal distance of the field spot away from z-axis; and θ
is the angle of direction.
The displacement potential functions of the longitudi-
nal and transverse scattered waves can also be expanded:
  
exp exp
sczn nlr
n
jkzA Hkrjn





, (9)



exp exp
sczn nlr
n
jkzBHkrjn





, (10)
  
exp exp
sczn nlr
n
jk zC Hkrjn





, (11)
where Hn is the Hankel function. Using the same method
we can expand the displacement potential functions of
the incident transverse waves in terms of the cylindrical
Bessel functions. Therefore the displacement potential
functions of the incident transverse waves inside the cyl-
inders are expanded as:
 




0
0
exp
exp
inznnlrn n
n
nn trnn
jkzA HkrD
BJ k rEjn










 (12)
 




0
0
exp
exp
inznnlrn n
n
nn trnn
jkzA JkrD
BJ k rEjn









 

 (13)



0
exp exp
inznnlrn n
n
jk zC JkrFjn








(14)
where nn
A
F

are coefficients and

12
22
trt z
kkk .
In the following we consider a sample of the two-di-
mensional periodic arrays system. The sample was made
of d-radius rods with lattice constant a. The position of
the rod with index j corresponds to

,

j
jj
r

r. What
are around this rod are incident waves involving external
sources and scattered waves from other rods. The total
field around this rod is inc scatt
uu u

.
The coefficients nn
A
F

are defined depending on
the boundary conditions.
In the light of the continuity of the displacements,
there are

:,,
inc scin
iii
rd rdrd
uuu ir 



z , (15)
Due to the continuity of the stresses, there exists:

:,,
inc scin
iii
rd rdrd
pppir 



z
z
, (16)
where

,: ,,
iijj
j
pnijr  

and




222
22,,:
ijt ijttijll
l
cuccui jlrz   
,,.
where σij are the stress tensor elements and uij are the
strain tensor elements that result from the components of
the displacement vector. The superscripts inc, sc, in, de-
note the incident, the scattered and the inner field respec-
tively.
In the far field, when ,

1
lt
kr kr






exp
scatt s
uf  rikrr.
The total scattering amplitude of the longitudinal
waves from Equations (9)-(11) is
 
2exp
π
nN
sn
n
l
fjA
kjn




. (17)
For acoustic wave transmission, a slit with width w
along the y direction is put between a source and the
sample. Acoustic waves propagate along x direction. In
this case, the incident field can be obtained from the
Kirchoff integral formula [17]:
 
2
01
2
,d
4
w
l
incl l
w
kx
uxy yHkriHkr
r


  



 







, (18)
where

2
2
rxyy


, in the case of far field,

X. Y. HUANG ET AL. 88
when , Equation (18) becomes rw
 

0
ua

exp
inc l
ikrr  r with
 
π42
0
2sin2 sin
ecos2
2πsin
l
il
l
kw
k
ak





. (19)
The vector of energy flux density is:
 
Im
8πl
cuu
k

 

 

S
rrr. (20)
Therefore the far-field energy flux has the form
 


0
,8π
s
ca fr
rS
r

r








Sr . (21)
So we define a transmission coefficient as the ratio of
transmission energy flux to that of the incident wave at
0

.
Therefore:

 
2 2
π4
2
0
02π
11e
0
si
s
l
f
T
akw
 0f. (22)
According to Equation (17) and the definition of scat-
tering cross section, the dimensionless scattering cross
sections of the longitudinal and transverse scattered
waves have the form:
2
2
1
0
1
1
ˆt
lnn
n
ll
c
a
kd c











2
n
b


(23)
2
22
1
0
1
1
ˆl
tnn
n
tt
cab
kd c











n





(24)
where ˆl

and ˆt

are the scattering cross sections for
longitudinal and transverse incident wave respectively, kl
and kt are the longitudinal and transverse wave numbers
for the host, cl1 and ct1 are the longitudinal and transverse
wave velocities for the host, d is the diameter of cylinder,
an and bn are the longitudinal and transverse scattered
wave coefficients.
For elastic media, there is a reasonable amount of cal-
culations for infinite systems [18-20]. However, systems
are finite and there are boundaries. Therefore under a
proper choice of parameters, states sliding and propagat-
ing along the surface and localized in the normal to the
surface, i.e., should appear, these are analogous to elec-
tronic surface in crystals [21] and to those calculated for
photonic systems [22]. According to M. Torres et al. that
surface state solutions are consubstantial with finite sys-
tems and exist for sonic propagation in finite elastic me-
dia. They deal with several realizations of structures for
ultrasonic propagation in elastic media to observe such
surface state modes and localization phenomena in linear
and point defects [3].
3. Numerical Results
In this paper, the finite-sized PBG sample used in the
calculation consisted of 6 rows along the x axis, 36 col-
umn rods with steel rods arranged in air host as square
with the lattice constant a, with filling fraction of f = 0.55,
rod radius mains = 0.35a, and a room of temperature
25˚C. The mass density of phononic crystal is ρ = 7800
g/cm3, longitudinal wave velocity cl = 5940 m/s, trans-
verse wave velocity ct = 3220 m/s [23], width of slit w =
3.5a, and is placed at a distance of l = 2.1a. From Equa-
tion (22) we have calculated the transmission coefficient
and total transmitted power as

π
2
0
2
T
Psd

 .
Figure 1 shows the calculated results of transmission
coefficient and total transmitted power, in dimensionless
frequency region (1.75 to 2.25). Acoustic wave propaga-
tion is inhibited forming frequency band gaps. The
transmission coefficient curve is finite and total trans-
mitted power becomes zero. The results are in excellent
agreement with previous results from Ref. [1].
Taking T ≈ 0 in Equation (22) which gives




0
s
f
afwa inside the bands


0
s
f show
large variations in frequency. This is related to the phase
shift of the scattered waves, if we assuming 1T






 

0
002sin2exp
s
fa ii  2 and




02sin2
s

f
af  wa
, where Φ is the
phase difference between outgoing and incoming waves.
Φ changes rapidly near band edges. The derivative of Φ(f)
gives information on group velocity vg. At band edge,
dΦ/df diverges and vg approaches zero. Therefore from
fs(0) we are able to extract the effective elastic constant
for frequencies inside a band and the decay length for
frequencies inside a gap.
(a) (b)
6
5
4
3
2
1
0
Figure 1. Acoustic band structure and transmission coeffi-
cients and total transmitted power for a square array of
rigid stainless steel cylinders in air host. The filling fraction
is f = 0.55. (a) The band structures reproduced from Ref.
[1]; (b) Solid curves: transmission coefficients. Dashed
curves: total transmitted power.

X. Y. HUANG ET AL.89
From Equation (23) and Equation (24) we have calcu-
lated the dimensionless scattering cross sections.
Figure 2 shows the calculated results of dimensionless
scattering cross sections, The gap appears in 1.9 ≤ kld(ktd)
≤ 2.8. The arrows denote the position of band gap, Fig-
ure 2(c) shows one full band gap. These results agree
with Figure 1.
Here, we try to connect the appearance of a gap and
other characteristics of the band structure in a periodic
system consisting of cylinder inclusions in a homogene-
ous matrix with the form of the cross section from a sin-
gle inclusion. This connection determines to what extent
single scattering is an important factor in determining
some characteristic features in the band structure, and
how it can be used to predict the possible existence of
gaps. For cylinder inclusions in a host material, the exis-
tence of full gaps has been connected to the following
picture: There are two channels for propagation. One is
mainly using the host material and the other is employing
the resonance states. Coherent jumping from resonance
state creates this second channel. In analogy, with the
linear combination of atomic orbital (LCAO, otherwise
called tight binding approximation) in the electronic
band structure.
In attempting this extension of the LCAO approach to
AC, one should keep in mind some important differences
between the two cases. Resonances are not true eigenstates,
rigorously localized inside and around each scattering as
the atomic-like orbital. On the other hand, because ω2
corresponds to the case where the electronic energy is
higher than the maximum of the potential, there is an addi-
tional the host material. It means that resonant states for
AC are states embedded in the continuum. This is an as-
pect of the problem not encountered in the electronic case.
4. Conclusions
In this work, we have investigated theoretically the
(a) (b) (c)
Figure 2. Dimensionless scattering cross sections for steel
(a), rigid (b) cylinder embedded in air host. Panel (c): re-
sults by subtracting the amplitudes of (a) and (b). Solid
curves: dimensionless scattering cross sections for longitu-
dinal wave. Dashed curves: dimensionless scattering cross
sections for transverse wave.
propagation of acoustic waves in a binary 2D phononic
crystal constituted of a square array of parallel, circular,
steel cylinder in air resin matrix. We have limited the
wave propagation to the plane perpendicular to the cyl-
inders. The numerical calculations prove unambiguously
the existence of absolute stop band independent of the
direction of propagation of the acoustic waves. Besides
the band gaps, one can establish some qualitative and
even semiquantitative correspondences between the ex-
perimental and theoretical transmission spectra inside the
pass bands. However, a more quantitative comparison
would need to repeat such experiments with other sam-
ples (for instance to check the possibility of defects dur-
ing the sample preparation, different thicknesses of the
samples, etc.), in this respect, an analysis of the eigen-
vectors associated with the different modes would be
also helpful for an understanding of the details of the
experimental transmission spectra.
We extended the far field approach and presented
transmissive and scattering properties of acoustic waves
in finite-sized phononic band gap (PBG) material. This
method make the calculating can be extremely simplified.
We found that full band gap is created between well
separated resonance states in which one can’t achieve
coherent jumping from a resonance state to a neighbor-
ing resonance states in analogy with the linear combina-
tion of atomic orbits in the electronic band structure; On
the other hand, such that the propagation along the host
material is inhibited. This results in full band gap ap-
pearing.
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