Reflection and Transmission of Plane Waves from a Fluid-Porous Piezoelectric
Solid Interface
Anil K. Vashishth* and Vishakha Gupta
Department of Mathematics
Kurukshetra University
Kurukshetra -136 119 INDIA
Email: anil_vashishth@yahoo.co.in
1
ABSTRACT
The reflection and transmission of plane waves from a fluid-porous piezoelectric solid
interface is studied. The porous piezoelectric solid, having 6mm symmetry, is supposed
to be filled with viscous fluid. The expressions for amplitude ratios and energy ratios
corresponding to reflected wave and transmitted waves are derived analytically. The
Christoffel equation of leaky wave propagating along the surface of porous piezoelectric
solid is derived. The effects of angle of incidence, frequency, porosity, piezoelectric
interaction and anisotropy on the reflected and transmitted energy ratios are studied
numerically for a particular model BaTiO3 . The porous piezoelectric solid half space is
assumed to be loaded with water. The effects of porosity and frequency on the leaky
wave velocity are also studied.
PACS numbers: 43.20.Gp, 43.38.Fx
2
I. INTRODUCTION
The theory of electro-acoustic waves in piezoelectric solids poses numerous challenging
problems that attract wide attention. Much of the interest in the subject of electro-acoustic
waves is directed at the applications in the areas of signal processing, transduction and
frequency control, where transmission and reflection of acoustic energy at boundary
surfaces play an important role. A number of problems, which are related to phenomena
of reflection and refraction of plane waves in piezoelectric materials, can be found in the
texts1-3. In the analysis of the reflection of plane electro-acoustic waves at the boundary
of a piezoelectric half space, four partial waves are required in the reflection field so that
the mechanical and electrical boundary conditions can be satisfied4. Recently, reflection
studies5-7 at the boundaries of piezoelectric media have also been carried out.
Knowledge and control of the reflection and transmission of ultrasonic waves at the
boundary between piezoelectric materials and water are important problems in designing
acoustic transducers, which in most of the cases are piezoelectric ceramics, for use in
under water imaging8. Noorbehesht and Wade9 studied the reflection and transmission of
plane elastic waves at the boundary between piezoelectric material and water. Nayfeh and
Chien10-11 derived the analytical expressions for the reflected and transmitted amplitude
ratios for the fluid-loaded piezoelectric plate and fluid-loaded piezoelectric half-space in
order to study influence of piezoelectricity on such waves.
3
Wave propagation and reflection-transmission phenomena in viscous fluid in the
presence of electric field have been dealt separately by many authors12-14. The electrohydrodynamic instability of a plane layer of dielectric fluid which is in hydrostatic
equilibrium between two semi-infinite conducting fluids with surface charges in porous
media was studied and dispersion relation was derived by Sayed15.
Piezoelectric crystals find many applications in ultrasonic devices, such as resonators in
electromechanical filters, sensors and ultrasonic delay lines in surface acoustic wave
(SAW) devices. One of the most important problems in designing SAW devices is the
observation and investigation of the properties of surface waves such as Rayleigh waves,
leaky waves etc. The relative efficiency of excitation of piezoelectric surface waves can
be judged from the piezoelectric surface velocities under different electrical boundary
conditions16.
Different
laminated
structures17-19
were
analyzed
for
dispersion
characteristics of leaky surface acoustic waves.
Porous piezoelectric materials offer lower acoustic impedance, higher hydrostatic
coefficients, higher piezoelectric sensitivity, lower density and stiffness than monolithic
piezo-ceramics. The interest in porous piezo-ceramics has grown rapidly in the recent
years with the demands from new fields of applications. Porous piezoelectric ceramics
have shown their advantages on dense ceramics in many possible applications20-22. Use of
piezoelectric effect in porous piezoelectric ceramic offers an original method for studying
the coupling between the electric, mechanical, permeability and piezoelectric properties
of porous systems. The effects of dynamic fluid compressibility and permeability on oil
4
filled porous piezoelectric ceramics23 were studied with PZT hydrophones. The bone
tissue can accurately be characterized by a porous saturated piezoelectric model in which
piezoelectric effect, porosity and the relative pore fluid motion are found as interrelated
phenomena24. Different theoretical models25-26 were developed to study the porous
piezoelectric ceramics and 0-3/3-3 connectivity piezoelectric composites. Gomez et al.27
made an experimental study on wave propagation in porous piezoelectric materials.
Different experimental studies28-31, related to the manufacturing, synthesis and
characterization of porous piezoelectric materials, have been presented. Gupta and
Venkatesh32 developed a finite element model to study the effects of porosity on the
electromechanical responses of porous piezoelectric materials. A micromechanics based
method was developed to evaluate the performance of 1-3 piezoelectric composites with
a porous non-piezoelectric matrix33. A survey of literature reveals that although a lot of
experimental work has been done in the field of porous piezoelectric materials but
theoretical work is much less, in comparison. Recently, Vashishth and Gupta34 derived
the constitutive equations for porous piezoelectric materials using Biot theory and electric
enthalpy density function. Wave propagation in porous piezoelectric materials, having
6mm symmetry, has been studied analytically by Vashishth and Gupta35. The effects of
porosity, frequency and direction of propagation on the phase velocity, attenuation and
polarization were studied therein.
In this paper, the Christoffel equation for plane harmonic waves propagating in porous
piezoelectric materials in a plane is derived in Section II. Porous piezoelectric solid is
considered of the type 6mm and is supposed to be filled with a viscous fluid. Next, the
5
reflection and transmission of waves from fluid- porous piezoelectric solid interface is
studied in Section III. The characteristic equation of leaky waves is obtained in Section
IV. Finally, the effects of angle of incidence, frequency, porosity, anisotropy and
piezoelectric interaction on the reflected and transmitted energy ratios are observed
numerically for a particular model BaTiO3 in Section V. The variation of leaky wave
velocity with the frequency is also studied.
II. CHRISTOFFEL EQUATION
Based on Biot theory for porous materials, the details of the constitutive equations for
porous piezoelectric materials are given in the paper34 and these equations are given as
 ij  cijkl kl  mij *  ekij Ek   kij Ek* ,
 *  mij ij  R *   i Ei  ei* Ei* ,
Di  eikl  kl   i *  ij E j  Aij E *j ,
Di*   ikl kl  ei* *  Aij E j  ij* E*j .
(1)
where σ( ij ) / σ* ( *ij ) are the stress tensors acting on the solid/ fluid phase of porous
aggregate. D( Di ), E( Ei ) / D* ( Di* ), E* ( Ei* ) are the electric displacement and electric field
vectors for the solid/ fluid phase of porous bulk material respectively. ε( ij ) / ε* (  *ij ) are
the strain tensors for the solid/fluid phase respectively. eijk , ij / ek* , ij* are the piezoelectric
and dielectric constants for the solid/ fluid phase respectively. mij ;  ijk ,  k ; Aij are the
material parameters which take into account the elastic; piezoelectric; dielectric coupling
6
between the two phases of porous aggregate. cijkl are the elastic coefficients for the solid
phase of porous aggregate. The elastic constant R measures the pressure to be exerted on
fluid to push its unit volume into the porous matrix.
For an infinitesimal deformation, the elastic strain components  ij and  * are related to the
components of mechanical displacements u (ui ) and U (Ui ) respectively as
1
2
 ij  (ui , j  u j , i ),  *  U i , i .
The
electric
field
vectors
(2)
E and E* are
related
to
the
electric
potentials
 and  * respectively as
E   ,
E*   * .
(3)
The equations of motion in x1  x3 plane, in the absence of body forces and free charge
density, are34
.σ = ρ11 u + ρ12 U + b ( u - U* ) ,
.σ* = ρ12 u + ρ 22 U  b ( u - U* ) ,
.D = 0 ,
.D* = 0 ,
(i, j  1,3) .
(4)
Here ρ11 ( ij11 ), ρ12 ( ij12 ) and ρ22 ( ij22 ) are dynamical coefficients which depend upon the
porosity ( f ), density of porous aggregate (  ), pore fluid density (  * ) and the inertial
coupling parameters. The dissipation tensor b(bij ) steers the effect of wave frequency
(  ), fluid viscosity (  ), solid-matrix permeability   ij ) and the porosity.
The dissipation tensor b 36 is
7
b  f 2  χ -1 , for low frequency waves where 0 

fc
 0.15 .
(5)
For high frequency waves,  is replaced by  F ( ) , where F ( ) is a complex function
of frequency and is given by
F ( ) 
1
 tanh(  )
,
3 (1  tanh(  )  )
(6)
and   6  0 * /  f .
where  0 is the norm of the permeability matrix. Following Biot theory37, the complex
function F ( ) , in term of non-dimensional parameter
 8 
8
tanh 

fc
fc 
1

F ( ) 
3
 8 
8
1  tanh 

fc
 fc 





, can be written as
fc
,
where f c is the Biot characteristic frequency.
Consider a harmonic plane wave propagating in x1  x3 plane at a given angular
frequency (  ), the associated physical quantities can be expressed as
1
(uk , U k ,  ,  * )  ( Bk , Fk , G, H ) exp ( ( x1  q x3  t )) ,
c
( k =1, 3),
(7)
where q is unknown slowness parameter. c is the apparent phase velocity given by
c  v sin  ,
(8)
where v is the phase velocity of wave propagating in x1  x3 plane, along a direction
making an angle  with x3 axis. ( B1 , B3 , F1 , F3 , G, H ) are the amplitudes associated with
the harmonic waves.
8
The equation (4), along with the equations (1) and (7), reduces to a system
ΓS  0,
(9)
where S  [ B1 , B3 , F1 , F3 , G, H ] T
and Γ is a symmetric matrix whose elements are listed in the Appendix A. This system
is consistent if
det(Γ)  0 .
(10)
This leads to
T1q10  T2 q8  T3q 6  T4 q 4  T5 q 2  T6  0 .
(11)
The coefficients T j ( j  1, 2,.., 6) are given in the Appendix B. q1 , q3 , q5 , q7 and q9
correspond to the roots whose imaginary part are positive and q2 , q4 , q6 , q8 and q10 to
those whose imaginary part are negative. Here, q1 , q3 correspond to the electric potential
component wave modes and q5 , q7 and q9 correspond to the propagating quasi P1 mode,
quasi S1 mode and quasi P2 mode. For each qi (i  1, 2,..,10) , the corresponding
amplitude ratios are defined as
Wi 
B3i
F
F
G
H
, U i  1i , Wi *  3i ,  i  i , *i  i .
B1i
B1i
B1i
B1i
B1i
(12)
These can be written in terms of eigen solutions as
Wi 
c(62 )qi
c(61 )qi
, Ui 
c(63 )qi
c(61 ) qi
, Wi* 
c(64 ) qi
c(61 ) qi
, i 
c(65 ) qi
c(61 ) qi
, *i 
c(66 ) qi
c(61 ) qi
,
(13)
where c(ij ) qi denotes the cofactor of ij corresponding to the eigen value qi .
9
The amplitudes ( B1 , B3 , F1 , F3 , G, H ) of the plane harmonic waves decrease as these
waves progress in porous piezoelectric medium. The amplitudes of the plane waves
propagating in porous piezoelectric solid also depend on the frequency.
The formal solution for the mechanical displacement and electrical potential becomes
(u1 , u3 , U1 , U 3 ,  ,  * ) 
10
 (1,W ,U
i 1
i
*
i
, Wi* ,  i , *i ) B1i exp (i (
x1
 qi x3  t )) .
c
(14)
III. REFLECTION AND TRANSMISSION COEFFICIENTS
A. AMPLITUDE RATIOS
Let us consider a porous piezoelectric half-space, having 6mm symmetry, loaded with
elastic fluid half space (FHS). The porous piezoelectric half-space (PPHS) occupies a
region x3  0 and the fluid half-space occupies a region x3  0 . A plane wave, making an
angle  with x3 axis, becomes incident at the interface. This wave results in one
reflected wave in FHS and five transmitted modes in PPHS. The transmitted wave modes
are represented by quasi P1 , quasi S1 and quasi P2 and the other two modes, represented
by PE1 and PE2 , corresponding to electric potential wave modes. The formal solution for
the mechanical displacements, electrical potentials, stress components and electric
displacements, in porous piezoelectric half space, are
(u1 , u3 ,U1 ,U 3 ,  ,  * )   (1, Wi , U i* , Wi* , i , *i ) B1i exp (i (
i
x1
 qi x3  t )) ,
c
and
10
( 31 ,  33 ,  * , D3 , D3* )  i  ( D1i , D3i , D4i , D5i , D6i ) B1i exp ( i (
i
x1
 qi x3  t )), (i  1,3,5, 7,9)
c
(15)
where D1i , D3i , D4i , D5i and D6i are given in the Appendix C.
The displacements and normal stress in the FHS can be written as
(u1f , u3f ) 
 33f  i
 (1,W
p 1,2

p 1,2
f
p
) U pf exp (i  (
 f cU pf exp (i  (
x1
 ( 1) p 1 q f x3  t )) ,
c
x1
 (1) p 1 q f x3  t )) ,
c
(16)
where
W1 f  q f c, W2f  q f c, q f 
1 c2
1 .
c c 2f
Here c f is the longitudinal incident wave velocity in fluid medium. p  1 for the incident
wave and p  2 for the reflected wave.
The boundary conditions at the interface x3  0 are
(a) Mechanical Boundary Conditions
(i)  33   *   33f ,
(ii)  *  f  33f ,
(iii) 13  0,
(iv) (1  f ) u3  f U 3  u3f .
(17)
Here, dot represents the differentiation with respect to time.
(b) Electrical (Free case) Boundary Conditions
(v) D3  0 ,
11
(vi) D3*  0 .
(18)
The equations (15)-(18), result into a non-homogeneous system
AX B ,
(19)
where
X  [ B11 , B13 , B15 , B17 , B19 ,U 2f ]T , B  [0,0,0,  f c , f  f c, q f c]T U1f and elements of matrix
A are given in the Appendix C. On solving system (19), the transmitted and reflected
amplitude ratios are obtained as
B1i 2  f c( dtr  f dtr 1 )
,

U1f
A'  Y  Y1
and
U 2f
A'  Y  Y1
.

U1f A'  Y  Y1
(20)
where expressions for A' , Y , Y1 and dtr (r  1, 2,..,10) are given in the Appendix C.
B. ENERGY RATIOS
Distribution of energy between different reflected and transmitted waves is considered
across a surface element of unit area at the interface x3  0 . Following Ikeda38 and
Vashishth and Gupta34, the normal acoustic flux P in a porous piezoelectric solid is
P  ( 31 u1   33 u3   U 3  D3   D3*  * ) .
(21)
The average energy flux of incident and reflected waves are
12
2
1
 P I   2 q f  f c 2 U1f ,
2
and
2
1
 P R    2 q f  f c 2 U 2f .
2
(22)
The average energy flux of transmitted waves are derived as
1
2
 Ps    2 Re( D1s  D3 sWs  D4 sWs*  D5 s  i  D6 s *s ) B1s , ( s  1, 2,..,5) .
2
(23)
The energy ratios of the reflected and transmitted waves are defined as
ER 
 PR 
,
 PI 
Es 
 Ps 
,
 PI 
( s  1, 2,..,5).
(24)
Following Vashisth and Gogna39, the interaction energy ratios, which account for
interaction between stress/electric potential field and mechanical/ electric displacement
field of different transmitted waves, are described as
Est 
 Pst    Pts 
,
 PI 
(25)
where
1
 Pst    2 Re[ D1s B1s B1t  D3s B1s B1tWt  D4 s B1s B1tWt *  D5 s B1s B1t t  D6 s B1s B1t *t ],
2
( s, t  1, 2,..,5and s  t ).
(26)
The energy is conserved if
5
E
s 1
s
 Eint  ER  1 ,
(27)
13
where Eint 
5
E
s , t 1
s t
s t
st
is the resultant interaction energy between the transmitted waves.
IV LEAKY WAVE EQUATION
The expressions for reflection and transmission amplitude ratio coefficients contain, as a
by product, the characteristic equation for leaky wave propagating along the surface of
porous piezoelectric half space. The characteristic equation of leaky wave, obtained by
equating denominator of reflection coefficient to zero, is given by
A  Y  Y1  0 .
(28)
The equation (28) leads to
T1' c3  T2' c 2  T3' c  T4'  0 .
(29)
The coefficients Tl ' (l  1, 2,3, 4) are listed in the Appendix D. Out of three roots of
Equation (29), the root having positive real part and negative imaginary part corresponds
to the leaky wave velocity. The leaky wave velocity is denoted by vL . The equation (28)
reduces to the characteristic equation A  0 when  f  0 i.e. the characteristic equation
of Rayleigh waves in porous piezoelectric solid half space.
V NUMERICAL DISCUSSION
The energy ratios for reflected and transmitted waves are calculated for a particular
model BaTiO3 . The porous piezoelectric half space is loaded with water. The elastic,
14
dielectric, piezoelectric and dynamical coefficients, followed from Gupta and
Vankatesh33 and Stoll and Kan40, are listed in the Table I and Table II.
The energy ratios of reflected wave, transmitted waves and interaction energy
coefficients are computed using the equations (24) - (27). The Fig. 1 shows the variation
of energy ratios with the angle of incidence (  ) of compressional wave propagating in
the fluid medium at frequency=1MHz. The energy ratios corresponding to reflected and
transmitted waves are represented by ER and Es ( s  1, 2,..,5) respectively. The total
interaction energy ratio between the transmitted waves is denoted by Eint . It is observed
that
before
  13 ,
all
the
transmitted
wave
modes,
namely
PE1 , PE2 ,quasi P1 ,quasi S1 and quasi P2 , propagate in the PPHS and after   13
the
transmitted quasi P1 wave mode is no longer excited. Between   13 and   31 , two
modes i.e. quasi S1 and quasi P2 propagate in PPHS and after   31 , only quasi P2 mode
is excited. The energy carried out by the quasi S1 and quasi P2 modes increase after the
critical angle of transmitted quasi P1 mode.
The energy carried out by the electric
potential components PE1 and PE2 is very small. The contribution of interaction energy
ratio between the transmitted waves is almost negligible except at   33 and   45 . In
Fig. 1, we observe a dip in the amplitude of the reflection coefficient at a polar angle
corresponding to a single quasi P2 wave mode excitation in PPHS medium. At this angle
of incidence there is almost mode conversion from the low impedance fluid medium to
high impedance solid medium. For a compressional wave, incident from the fluid
medium, there are three critical angles 1  13 ,  2  31 and 3  41 corresponding to
15
transmitted P1 wave, S1 wave and P2 wave respectively in porous piezoelectric half space.
Beyond the third critical angle i.e.   41 , the incident wave is totally reflected and
energy reflection coefficient equals unity.
In the case of perfect elastic medium i.e. for a lossless medium, transmitted wave decays
with distance from the interface for supercritical incidence ( Krebes41 ). However, for a
medium with loss, it is possible to have transmitted waves whose amplitude grows with
distance from the interface for some angles of incidence beyond critical angle as in
viscoelasticity, critical angles for transmitted wave are isolated (Stoll and Kan40,
Krebes41). For anelastic reflection-refraction problems, contrasts in anelastic absorption
at a boundary give rise to inhomogeneous waves for all angles of incidence as opposed to
the elastic reflection-refraction problems where inhomogeneous wave exist only beyond
critical angles (Borcherdt et al.42). In the present paper, the dissipation due to the viscous
coupling between the fluid phase and the solid phase motions is considered which is
small, in general, in comparison to the viscoelastic dissipation of the skeleton frame
(Rasolofosaon43, Vashishth and Sharma44).
The major portion of incident energy is reflected back which signifies the fact that the
transmitting medium is much denser. The results are in agreement with the law of
conservation of energy. There is no null in the reflection coefficient which reveals the
effect of anisotropy (Ankan et al.45 ).
16
Figure 1 Variation of Reflected and Transmitted energy ratios with the angle of incidence (  ); (i)
Reflected wave, (ii) Transmitted PE1, (iii) Transmitted PE2, (iv) Transmitted
(vi) Transmitted
P1 , (v) Transmitted S1 ,
P2 , (vii) Interaction energy;   1MHz , f  0.2 .
17
Figure 2 Variation of Reflected and Transmitted energy ratios with the angle of incidence (  ); (i)
Reflected wave, (ii) Transmitted PE1, (iii) Transmitted PE2, (iv) Transmitted
(vi) Transmitted
P1 , (v) Transmitted S1 ,
P2 , (vii) Interaction energy;   10MHz , f  0.2 .
18
Figure 3 Variation of Reflected and Transmitted energy ratios with the angle of incidence (  ); (i)
Reflected wave, (ii) Transmitted PE1, (iii) Transmitted PE2, (iv) Transmitted
(vi) Transmitted
P1 , (v) Transmitted S1 ,
P2 , (vii) Interaction energy;   100MHz , f  0.2 .
Figs. 2 and 3 depict the variation of energy ratios with the angle of incidence at frequency
=10MHz and 100MHz respectively. Comparison of Figs. 1-3 shows that the kinks in the
energy ratios corresponding to P1 and S1 waves beyond their corresponding critical angle
disappear as the frequency increases. It is observed that at frequency =100 MHz, the
P1 and S1 waves become evanescent beyond their respective critical angles. The energy
ratio corresponding to PE1 and PE2 modes decreases with the increase in frequency.
19
The effects of porosity on the variation of reflected and transmitted energy ratios with the
angle of incidence are depicted in Figs. 4 and 5. Fig. 4 shows that the energy carried out
by the reflected wave decreases as the porosity increases. The porosity does not affect the
value of critical angles. It is observed that the energy ratio corresponding to P2 wave
increases with increase in porosity while that corresponding to other transmitted modes in
porous piezoelectric solid half space decreases (Fig. 5). The resultant interaction energy
ratio also increases with porosity (Fig. 5). It is also observed that the energy flux once
again appears to occur in quasi P1 , S1 and P2 wave modes transmitted into PPHS after the
critical angle as the porosity increases.
Figure 4 Variation of Reflected energy ratios with the angle of incidence (  ) for different porosity;
  1MHz .
20
Figure 5 Variation of Transmitted energy ratios with the angle of incidence (  ) for different
porosity; (i) Transmitted energy
P1 , (ii) Transmitted energy S1 , (iii) Transmitted energy P2 , (iv)
Transmitted energy PE1, (v) Transmitted energy PE2, (vi) Interaction energy;
  1MHz .
Next we will observe the effects of piezoelectricity and anisotropy on the variation of
reflected energy ratio with the angle of incidence. The elastic and other coefficients for
transversely isotropic and isotropic non-piezoelectric porous material are listed in the
Table III and IV respectively.
The results are computed for three data sets:
Set1 = The values of the elastic, piezoelectric, dielectric and dynamic coefficients
as given in the Table I and Table II for porous piezoelectric materials;
represented by solid curves.
21
Set2 = The values of the elastic and dynamic coefficients as given in the Table III
and Table II for non-piezoelectric porous materials; represented by dotted
curves.
Set3 = The values of the elastic and dynamic coefficients as given in the Table IV
for isotropic porous materials; represented by dashed curves.
It is observed that, in the absence of piezo-electric interaction and anisotropy, a sharp dip
occurs at   28 which corresponds to the excitation of surface mode (see dashed curve,
Fig. 6(i)). A minimum in the reflection coefficient at   28 also signifies the fact that
for such an angle of incidence, a large amount of incident energy is transmitted into the
porous piezoelectric solid. In the absence of piezoelectricity, there are four modes in the
transmitting medium instead of six, as expected theoretically also. Comparison of solid
and dotted curves reveals that the critical angle corresponding to S1 wave is shifted due
to piezoelectric interaction while critical angle corresponding to P1 and P2 waves remains
unaffected (Fig. 6 (ii-iv)). In the absence of piezoelectric effect, transmission coefficients
of P1 and S1 phase change towards lower side.
22
Figure 6 Variation of Reflected and Transmitted energy ratios with the angle of incidence (  ); (i)
Reflected wave, (ii) Transmitted
P1 , (iii) Transmitted S1 , (iv) Transmitted P2 ;   1MHz ,
f  0.2 .
The effect of porosity on the variation of leaky wave velocity ( vL ) with the frequency
(  ) is depicted in the Fig. 7. The leaky wave velocity decreases with increase in
frequency and increases with increase in porosity.
Figure 7 Variation of Leaky wave velocity ( vL ) with frequency (  ) for different porosity;
  50
.
23
VI CONCLUSION
The reflection and transmission of waves from an interface separating the fluid half space
and porous piezoelectric half space is studied in the present paper. The problem studied
in present paper is of significant practical interest, since reflection and transmission of
ultrasonic waves at the boundary between porous piezoelectric materials and water is an
important problem in designing acoustic transducers for use in underwater imaging. The
analytical expressions for reflected and transmitted amplitude and energy ratios are
derived. The incident wave is totally reflected and energy reflection coefficient equals
unity after   41 . The resultant interaction energy flux and energy flux corresponding to
electric potential wave modes are less significant in comparison to those of propagating
quasi P1 , quasi S1 and quasi P2 modes. The results are in agreement with the law of
conservation of energy. It is also observed that the mode conversion occurs at a waterporous piezoelectric medium interface. There is no null in the reflection coefficient as
expected in case of transversely isotropy. The change in porosity and frequency does not
affect significantly the value of critical angle. The critical angle corresponding to quasi
S1 wave is shifted when piezoelectric effect is neglected. The dispersion equation of
leaky waves in transversely isotropic porous piezoelectric solid is also obtained from the
expressions of amplitude ratios. The leaky wave velocity decreases with frequency while
increases with porosity.
Acknowledgement
24
The second author is grateful to University Grant Commission (Grant No. f.no.8(42)/2010 (MRP/NRCB)) for providing financial support for this work.
Appendix A
11  c11 / c 2  c44 q 2  1111 ; 12  (c13  c44 )q / c ; 13  m11 / c 2  1112 ; 14  m11q / c ;
11
15  (e15  e31 ) q / c ; 16  ( 15   31 ) q / c ;  22  c44 / c 2  c33 q 2  33
; 23  m33q / c ;
12
 24  m33q 2  33
;  25  e15 / c 2  e33q 2 ;  26   15 / c 2   33 q 2 ; 33  R / c 2  1122 ;
34  R q / c ; 35   3 q / c ; 36  e3* q / c ;  44  Rq 2  3322 ;  45   3 q 2 ;  46  e3* q 2 ;
55  11 / c 2  33 q 2 ; 56   A11 / c 2  A33 q 2 ;  66  11* / c 2  33* q 2 .
where
ρ11  ρ11  (i /  ) b; ρ12  ρ11  (i /  ) b; ρ22  ρ22  (i / ) b .
Appendix B
T1  1 y1  5 y10  9 y17  13 y26 ,
T2   2 y1  1 y2   6 y10  5 y11  10 y17  9 y18  14 y26  13 y27 ,
T3  3 y1   2 y2  1 y3   7 y10   6 y11  11 y17  10 y18   9 y19  15 y26  14 y27  13 y28 ,
T4   4 y1  3 y2   2 y3  8 y10  7 y11  12 y17  11 y18  10 y19  16 y26  15 y27  14 y28 ,
T5   4 y2  3 y3  12 y18  11 y19  8 y11  16 y27  15 y28 ,
T6   4 y3  12 y19  16 y28 .
where
25
1  p1 y12  p4 y14  p7 y15 ;  2  p2 y12  p1 y13  p5 y14  p8 y15  p7 y16 ;
3  p3 y12  p2 y13  p6 y14  p9 y15  p8 y16 ;  4  p3 y13  p9 y16 ; 5  p1 y4  p4 y6  p7 y8 ;
 6  p2 y4  p1 y5  p5 y6  p4 y7  p8 y8  p7 y9 ;
 7  p3 y4  p2 y5  p6 y6  p5 y7  p9 y8  p8 y9 ; 8  p3 y5  p6 y7  p9 y9 ;
9  r1 y4  r4 y6  r7 y8 ; 10  r2 y4  r1 y5  r5 y6  r4 y7  r8 y8  r7 y9 ;
11  r3 y4  r2 y5  r6 y6  r5 y7  r9 y8  r8 y9 ; 12  r3 y5  r6 y7  r9 y9 ; 13  s1 y4  s4 y6  s7 y8
14  s2 y4  s1 y5  s5 y6  s4 y7  s8 y8  s7 y9 ; 15  s3 y4  s2 y5  s6 y6  s5 y7  s9 y8  s8 y9
16  s3 y5  s6 y7  s9 y9 .
where
p1  y22 y33  y24 y31 ; p2  y22 y34  y23 y33  y25 y31  y24 y32 ; p3  y23 y34  y25 y32 ;
p4  y20 y33  y24 y29 ; p5  y20 y34  y21 y33  y24 y30  y25 y29 ; p6  y21 y34  y25 y30 ;
p7  y20 y31  y22 y29 ; p8  y20 y32  y21 y31  y23 y29  y22 y30 ; p9  y21 y32  y23 y30 .
r1  y14 y33  y15 y31 ; r2  y14 y34  y16 y31  y15 y32 ; r3   y16 y32 ; r4  y12 y33  y15 y29 ;
r5  y12 y34  y13 y33  y16 y29  y15 y30 ; r6  y13 y34  y16 y30 ; r7  y12 y31  y14 y29 ;
r8  y12 y32  y14 y30  y13 y31 ; r9  y13 y32 .
s1  y14 y24  y15 y22 ; s2  y14 y25  y16 y22  y15 y23 ; s3   y16 y23 ; s4  y12 y24  y15 y20
s5  y12 y25  y13 y24  y16 y20  y15 y21 ; s6  y13 y25  y16 y21 ; s7  y12 y22  y14 y20 ;
s8  y12 y23  y14 y21  y13 y22 ; s9  y13 y23 .
where
y1  e33 x1   33 x6 ; y2  (e15 x1   15 x6 ) / c 2  (c44  c13 ) / c  e33 x2   33 x7 ;
11
y3  (e15 x2   15 x7 ) / c 2 ; y4  e33 x3   33 x8  c33 ; y5  (e15 x3   15 x8  c44 ) / c 2  33
;
26
y6  e33 x4   33 x9  m33 / c ; y7  (e15 x4   15 x9 ) / c 2 ; y8  e33 x5   33 x10  m33 ;
12
y9  (e15 x5   15 x10 ) / c 2  33
; y10   3 x1  e3* x6 ; y11  m11 / c   3 x2  e3* x7 ;
12
y12  m33   3 x3  e3* x8 ; y13   33
; y14  R / c   3 x4  e3* x9 ; y15  R   3 x5  e3* x10 ;
y16   3322 ; y17   x133  x6 A33 ; y18  ( x111  x6 A11 ) / c 2  (e15  e31 ) / c  33 x2  A33 x7 ;
y19  ( x211  x7 A11 ) / c 2 ; y20  e33  x333  x8 A33 ; y21  (e15  x311  x8 A11 ) / c 2 ;
y22   3 / c  A33 x9  33 x4 ; y23  ( x411  x9 A11 ) / c 2 ; y24   x10 A33  x533   3 ;
y25  ( x511  x10 A11 ) / c 2 ; y26   x1 A33  x633* ;
y27  ( x1 A11  x611* ) / c 2  ( 15   31 ) / c  x2 A33  x733* ; y28  ( x2 A11  x711* ) / c 2 ;
y29   33  x3 A33  x833* ; y30  ( 15  x3 A11  x811* ) / c 2 ; y31  e3* / c  33* x9  A33 x4 ;
y32  ( x4 A11  x911* ) / c 2 ; y33   x1033*  x5 A33  e3* ; y34  ( x5 A11  x1011* ) / c 2 .
where
x1  c44 d c ;
x2  (c11 / c2  1111 ) d c  (m11 / c2  1112 ) c /  3  (m11 / c 2  1112 )(e15  e31 ) d c /  3 ;
x3  (m33 (e15  e31 )d  m33 ) /  3  (c13  c44 ) d
x4  (m11 / c2  1112 ) d c  ( R / c2  1122 ) c /  3  ( R / c2  1122 )(e15  e31 ) d c /  3 ;
x5  ( R(e15  e31 )d  R) /  3  m11 d ; x6  c44  3 d c / e3* ;
x7  (c11 / c2  1111 ) d c  3 / e3*  (m11 / c2  1112 )(e15  e31 ) d c / e3* ;
x8  (c13  c44 ) d  3 / e3*  m33 (e15  e31 ) d / e3* ;
x9  (m11 / c2  1112 ) d c  3 / e3*  ( R / c2  1122 )(e15  e31 ) d c / e3* ;
x10  m11 d  3 / e3*  R (e15  e31 ) d / e3* .
27
d  e3* /(e3* (e15  e31 )   3 ( 15   31 )) .
Appendix C
D1i  c55 qi  (c55Wi  e15 i   15*i ) / c ,
D3i  (c33 Wi  m33 Wi *  e33 i   33*i ) qi  (c31  m33 U i* ) / c ,,
D4i  (m33 Wi  RWi*   3 i  e3* *i ) qi  (m11  RUi* ) / c ,
D5i  (e33 Wi   3 Wi *  33  i  a33 *i ) qi  (e31   3 U i* ) / c ,
D6i  ( 33 Wi  e3* Wi *  a33  i  33* *i ) qi  ( 31  e3* U i* ) / c .
D53
 D51
 D
D63
61

 D11
D13

A   D31  D41 D33  D43
 D41
D43

 (1  f )W1 (1  f )W3
  fW *
 fW3*
1

D55
D65
D15
D35  D45
D45
(1  f )W5
D57
D67
D17
D37  D47
D47
(1  f )W7
D59
D69
D19
D39  D49
D49
(1  f )W9
 fW5*
 fW7*
 fW9*





 f c 
 f  f c


qf c 

0
0
0
A'  ( D31  D41 ) dt1  ( D33  D43 ) dt3  ( D35  D45 ) dt5  ( D37  D47 ) dt7  ( D39  D49 ) dt9
Y  [((1  f ) W1  f W1* ) dt1  ((1  f ) W3  f W3* ) dt3  ((1  f )W5  f W5* ) dt5
 ((1  f ) W7  f W7* ) dt7  ((1  f ) W9  f W9* ) dt9 ]  f / q f
Y1  [((1  f ) W1  f W1* ) dt2  ((1  f ) W3  f W3* ) dt4  ((1  f ) W5  f W5* ) dt6
 ((1  f ) W7  f W7* ) dt8  ((1  f ) W9  f W9* ) dt10 ] f  f / q f
dt1  det([ D53 D55 D57 D59 ; D63 D65 D67 D69 ; D13 D15 D17 D19 ; D43 D45 D47 D49 ]) ;
28
dt3  det([ D51 D55 D57 D59 ; D61 D65 D67 D69 ; D11 D15 D17 D19 ; D41 D45 D47 D49 ]) ;
dt5  det([ D51 D53 D57 D59 ; D61 D63 D67 D69 ; D11 D13 D17 D19 ; D41 D43 D47 D49 ]) ;
dt7  det([ D51 D53 D55 D59 ; D61 D63 D65 D69 ; D11 D13 D15 D19 ; D41 D43 D45 D49 ]) ;
dt9  det([ D51 D53 D55 D57 ; D61 D63 D65 D67 ; D11 D13 D15 D17 ; D41 D43 D45 D47 ]) ;
dt2  det([ D53 D55 D57 D59 ; D63 D65 D67 D69 ; D13 D15 D17 D19 ; D43  D33 D45  D35 D47  D37 D49  D39 ])
dt4  det([ D51 D55 D57 D59 ; D61 D65 D67 D69 ; D11 D15 D17 D19 ; D41  D31 D45  D35 D47  D37 D49  D39 ])
dt6  det([ D51 D53 D57 D59 ; D61 D63 D67 D69 ; D11 D13 D17 D19 ; D41  D31 D43  D33 D47  D37 D49  D39 ])
dt8  det([ D51 D53 D55 D59 ; D61 D63 D65 D69 ; D11 D13 D15 D19 ; D41  D31 D43  D33 D45  D35 D49  D39 ])
dt10  det([ D51 D53 D55 D57 ; D61 D63 D65 D67 ; D11 D13 D15 D17 ; D41  D31 D43  D33 D45  D35 D47  D37 ])
Appendix D
T1'  s1' ( s5' s9'  s6' s8' )  s2' ( s4' s9'  s6' s7' )  s3' ( s4' s8'  s7' s5' );
T2'  s1' ( s5' t9'  s9' t5'  s8' t6'  s6' t8' )  s2' (s4' t9'  s9' t 4'  s7' t6'  s6' t7' )  s3' (s4' t8'  s8' t 4'  s7' t5'  s5' t 7' )
 t1' ( s5' s9'  s8' s6' )  t2' ( s4' s9'  s7' s6' )  t3' ( s4' s8'  s7' s5' );
T3'  t1' ( s5' t9'  s9' t5'  s8' t6'  s6' t8' )  t 2' (s4' t9'  s9' t 4'  s7' t6'  s6' t 7' )  t3' (s4' t8'  s8' t 4'  s7' t 5'  s5' t 7' )
 s1' (t5' t9'  t8' t6' )  s2' (t4' t9'  t7' t6' )  s3' (t4' t8'  t7' t5' );
T4'  t1' (t5' t9'  t6' t8' )  t2' (t4' t9'  t6' t7' )  t3' (t4' t8'  t7' t5' );
where
s1'  p1' p6'  p2' p5' ; s2'  p1' p7'  p3' p5' ; s3'  p1' p8'  p4' p5' ; s4'  p1' p10'  p2' p9' ;
s5'  p1' p11'  p3' p9' ; s6'  p1' p12'  p4' p9' ; s7'  p1' p14'  p2' p13' ; s8'  p1' p15'  p3' p13' ;
s9'  p1' p16'  p4' p13' ; t1'  p1' r2'  p2' r1' ; t2'  p1' r3'  p3' r1' ; t3'  p1' r4'  p4' r1' ; t4'  p1' r6'  p2' r5' ;
t5'  p1' r7'  p3' r5' ; t6'  p1' r8'  p4' r5' ; t7'  p1' r10'  p2' r9' ; t8'  p1' r11'  p3' r9' ; t9'  p1' r12'  p4' r9' .
where
29
r1'  D51 y2'  D53 y1' ; r2'  D51 y3'  D55 y1' ; r3'  D51 y4'  D57 y1' ; r4'  D51 y5'  D59 y1' ;
r5'  D51 ( f y7'  (1  f ) y12' )  D53 ( f y6'  (1  f ) y11' );
r6'  D51 ( f y8'  (1  f ) y13' )  D55 ( f y6'  (1  f ) y11' );
r7'  D51 ( f y9'  (1  f ) y14' )  D57 ( f y6'  (1  f ) y1' 1 );
r8'  D51 ( f y0'  (1  f ) y15' )  D59 ( f y6'  (1  f ) y11' ); r9'  ( D51 y12'  D53 y11' ) q f ;
r10'  ( D51 y13'  D55 y11' ) q f ; r11'  ( D51 y14'  D57 y11' ) q f ; r12'  ( D51 y15'  D59 y11' )q f .
p1'  D51 D63  D53 D61 ; p2'  D51 D65  D55 D61 ; p3'  D51D67  D57 D61 ; p4'  D51D69  D59 D61 ;
p5'  D51 x2'  D53 x1' ; p6'  D51 x3'  D55 x1' ; p7'  D51 x4'  D57 x1' ; p8'  D51 x5'  D59 x1' ;
p9'  D51 ( f x7'  (1  f ) x12' )  D53 ( f x6'  (1  f ) x11' );
p10'  D51 ( f x8'  (1  f ) x13' )  D55 ( f x6'  (1  f ) x11' );
p11'  D51 ( f x9'  (1  f ) x14' )  D57 ( f x6'  (1  f ) x11' );
p12'  D51 ( f x10'  (1  f ) x15' )  D59 ( f x6'  (1  f ) x11' );
p13'  D51 ( f  f x17'  q f x12' )  D53 ( f  f x16'  q f x11' );
p14'  D51 ( f  f x18'  q f x13' )  D55 ( f  f x16'  q f x11' );
p15'  D51 ( f  f x19'  q f x14' )  D57 ( f  f x16'  q f x11' );
'
p16'  D51 ( f  f x20
 q f x15' )  D59 ( f  f x16'  q f x11' ).
where
x1'  c55 q1 ; x2'  c55 q3 ; x3'  c55 q5 ; x4'  c55 q7 ; x5'  c55 q9 ,
30
x6'  (c33 W1  m33 W1*  e33 1   33 1* ) q1 ; x7'  (c33 W3  m33 W3*  e33  3   33 *3 ) q3 ;
x8'  (c33 W5  m33 W5*  e33  5   33 *5 ) q5 ; x9'  (c33 W7  m33 W7*  e33  7   33 *7 ) q7 ;
x10'  (c33 W9  m33 W9*  e33  9   33 *9 ) q9 ; x11'  (m33 W1  RW1*   3 1  e3* 1* ) q1 ;
x12'  (m33 W3  RW3*   3  3  e3* *3 ) q3 ; x13'  (m33 W5  RW5*   3  5  e3* *5 ) q5 ;
x14'  (m33 W7  RW7*   3  7  e3* *7 ) q7 ; x15'  (m33 W9  RW9*   3  9  e3* *9 ) q9 ;
x16'  (1  f )W1  fW1* ; x17'  (1  f )W3  fW3* ; x18'  (1  f )W5  fW5* ; x19'  (1  f )W7  fW7* ;
'
x20
 (1  f )W9  fW9* .
y1'  (c55 W1  e15 1   15 1* ); y2'  (c55 W3  e15  3   15 *3 ); y3'  (c55 W5  e15  5   15 *5 );
y4'  (c55 W7  e15  7   15 *7 ); y5'  (c55 W9  e15  9   15 *9 ); y6'  (c31  m33 U1* );
y7'  (c31  m33 U 3* ); y8'  (c31  m33 U 5* ); y9'  (c31  m33 U 7* ); y10'  (c31  m33 U 9* );
y11'  (m11  RU1* ); y12'  (m11  RU 3* ); y13'  (m11  RU 5* ); y14'  (m11  RU 7* );
y15'  (m11  RU 9* ).
References
1
V.Z. Parton and B.A. Kudryavtsev, “Electromagnetoelasticity, Piezoelectrics and
electrically Conductive Solids”, Gordan and Breach, New York (1988), 503 p.
2
B.A. Auld, “Acoustic Field and waves in solids”, Wiley Interscience, New York (1973),
Vol. II, 423 p.
3
C. Galassi, M. Dinescu, K. Uchino, K. Sayer, M. (Eds.), “Piezoelectric materials:
Advances in Science, Technology and applications, Vol.76, Springer (2000), 420 p.
4
A.G. Every and V.I. Neiman, “Reflection of electroacoustic waves in piezoelectric
solids: Mode conversion into four bulk waves”, J. Appl. Phys. 71(12), 6018-6024 (1992).
31
5
S.G. Joshi, B.D. Zaitsev and I.E. Kuznetsova, “Reflection of normal acoustic waves in
thin plates from a grating with a periodically distributed mechanical load”, Acoustical
Physics 48 (2), 162-165 (2002).
6
S.I. Burkov, B.P Sorokin, D.A. Glushkov and K.S. Aleksandrov, “Theory and computer
simulation of the reflection and refraction of bulk acoustic waves in piezoelectrics under
the action of external electric field”, Crystallography Reports 50 (6), 986-993 (2005).
7
A.N. Abd-alla and F.A. Alsheikh, “Reflection and refraction of plane quasi-longitudinal
waves at an interface of two piezoelectric media under initial stresses”, Arch. Appl.
Mech. 79(9), 843-857 (2009).
8
B. Noorbehesht and G. Wade, “Spatial frequency characteristics of opto-acoustic
transducers”, in Acoustic Imaging, edited by K. Wang (Plenum, New York, 1980), Vol
9, pp. 139-154.
9
B. Noorbehesht and G. Wade, “Reflection and transmission of plane acoustic waves at
the boundary between piezoelectric materials and water”, J. Acoust. Soc. Am. 67(6),
1947-1953 (1980).
10
A. D. Nayfeh and H.T. Chien, “The influence of piezoelectricity on free and reflected
waves from fluid-loaded anisotropic plates, J. Acoust. Soc. Am. 91(3), 1250-1261 (1992).
32
11
A.D. Nayfeh and H.T. Chien, “Wave propagation interaction with free and fluid loaded
piezoelectric substrates”, J. Acoust. Soc. Am. 91(6), 3126-3135 (1992).
J.A. Shercliff, “Anisotropic surface waves under a vertical magnetic force”, J. Fluid
12
Mech. 38 (2), 353-364 (1969).
13
M. Deschamps, B. Poiree and O. Poncelet, “Energy velocity of complex harmonic
waves in viscous fluids”, Wave Motion 25(1), 51-60 (1997).
14
H. Tang, C. Luo and X. Zhao, “Sonic resonance in a sandwiched electrorheological
panel”, J. Appl. Phys. 98, 016103-1-016103-3 (2005).
15
M. F. El-Sayed, “Electrohydrodynamic instability of dielectric fluid layer between two
semi infinite identical conducting fluids in porous medium”, Physics A: Statistical
Mechanics and its Applications 367, 25-41 (2006).
16
A.A. Oliner, “Acoustic surface waves”, Springer Verlag, Berlin-Heidelberg, New York,
1978, 325 p.
17
V.I. Grigor and Y.V. Gulyaev, “Propagation of fast leaky waves in a system of metal
electrodes on a lithium tetraborate crystal”, Acoustic Physics 46(4), 405-410 (2000).
33
18
G. Bu, D. Ciplys, M.S. Shur, G. Namkoong, W.A. Doolittle and W.D. Hunt, “Leaky
surface acoustic waves in Z-LiNbO3 substrates with epitaxial AIN overlays”, Appl. Phys.
Letters 85(15), 3313-3315 (2004).
19
A.N. Darinskii and M. Weihnacht, “Acoustic waves guided by a fluid layer on a
piezoelectric substrate”, J. Appl. Phys. 104(054904), 1-9 (2008).
20
T. Arai, K. Ayusawa, H. Sato, T. Miyata, K. Kawamura and K. Kobayashi, “Properties
of hydrophone with porous piezoelectric ceramics”, Japanese Journal of Applied Physics
30, 2253-2255 (1991).
21
T. Hayashi, K. Sugihara and K. Okazaki, “Processing of porous 3-3 PZT ceramics using
capsule-free O2 H.P”, Japanese Journal of Applied Physics 30, 2243-2246 (1991).
22
K. Mizumura, Y. Kurihara, H. Ohashi, S. Kumamoto and K. Okuno, “Porous
piezoelectric ceramic transducer”, Japanese Journal of Applied Physics 30, 2271-2273
(1991).
23
O. Lacour, M. Lagier and D. Sornette, “Effect of dynamic fluid compressibility and
permeability on porous piezoelectric ceramics”, J. Acoust. Soc. Am. 96(6), 3548-3557
(1994).
34
24
Y.A. Avdeev and S.A. Regirer, “Mathematical model of bone tissue as a porou-elastic
piezoelectric material”, Mechanics of Composite material 15(5), 566-569 (2004).
25
T.E. Gomez and F. Montero, “New constitutive relations for piezoelectric composites
and porous piezoelectric ceramics”, J. Acoust. Soc. Am. 100(5), 3104-3114 (1996).
26
T.E. Gomez and F. Montero, “Characterization of porous piezoelectric ceramics: The
length expander case”, J. Acoust. Soc. Am. 102(6), 3507-3515 (1997).
27
T.E. Gomez, A.J. Mulholland, G. Hayward and J. Gomatam, “Wave propagation in 0-
3/3-3 connectivity composites with complex microstructure”, Ultrasonics 38, 897-907
(2000).
28
E. Roncari, C. Galassi, F. Craciun, C. Capiani and A. Piancastelli, “A microstructural
study of porous piezoelectric ceramics obtained by different methods”, J. European
Ceramic Soc. 21, 409-417 (2001).
C.R. Bowen, A. Perry, A.C.F. Lewis and H. Kara, “Processing and properties of porous
29
piezoelectric materials with high hydrostatic figures of merit”, J. European Ceramic Soc.
24, 541-545 (2004).
35
30
B.P. Kumar, H.H. Kumar and D.K. Kharat, “Study on pore forming agents in
processing of porous piezoceramics”, J. Materials Sci.: Materials in Electronics 16, 681686 (2005).
C. Galassi, “Processing of porous ceramics: Piezoelectric materials”, J. European
31
Ceramic Soc. 26, 2951-2958 (2006).
R.K. Gupta and T.A. Venkatesh, “Electromechanical response of porous piezoelectric
32
materials”, Acta Materialia 54, 4063-4078 (2006).
C.N. Della and D. Shu, “The performance of 1-3 piezoelectric composites with a porous
33
non-piezoelectric matrix”, Acta Materialia 56, 754-761 (2008).
34
A.K. Vashishth and V. Gupta, “Vibrations of porous piezoelectric ceramic plates”,
Journal of Sound and Vibrations 325, 781-797 (2009).
35
A.K. Vashishth and V. Gupta, “Wave propagation in transversely isotropic porous
piezoelectric materials”, Int. J. Solids and Structures 46, 3620-3632 (2009).
36
G.S. Sammelmann, “Biot - Stoll model of the high frequency reflection coefficient of
sea ice”, J. Acoust. Soc. Am. 94 (1), 371-385 (1993).
36
37
M.A. Biot, “Theory of propagation of elastic waves in a fluid-saturated porous solid. II
Higher frequency range”, J. Acoust. Soc. Am. 28 (2), 179-191 (1956).
38
T. Ikeda, “Fundamentals of piezoelectricity”, Oxford Science Publications, Oxford
University Press (1996), pp. 65-66.
39
A.K. Vashisth and M.L. Gogna, “Effect of loose boundaries on wave propagation-
reflection and refraction of plane waves at an interface between viscoelastic and
poroviscoelastic solids”, J. Phys. Earth 44, 173-192 (1996).
R.D. Stoll and T.K. Kan, “Reflection of acoustic waves at a water-sediment interface”.
40
J. Acoust. Soc. Am. 70, 149-156 (1981).
41
E.S. Krebes, “The viscoelastic reflection/Transmission problem: Two special cases”,
Bulletin of the Seismological Society of America 73 (6), 1673-1683 (1983).
R.D. Borcherdt, G. Glassmoyer, L. Wennerberg, “Influence of welded boundaries in
42
anelastic media on energy flow, and characteristics of P, S-I and S-II waves:
Observational evidence for inhomogeneous body waves in low-loss solids”, Journal of
Geophysical Research 91 (B11), 11503-11518, 1986.
37
P.N.J. Rasolofosaon, “Plane acoustic waves in linear viscoelastic porous media:
43
Energy, particle displacement, and physical interpretation”, J. Acoust. Soc. Am. 89 (4),
1532-1550, 1991.
44
A.K. Vashishth and M.D. Sharma, “Reflection and refraction of acoustic waves at
poroelastic ocean bed”, Earth Planet Space 61, 675-687, 2009.
45
O. Ankan, E. Telatar and A. Atalar, “Reflection coefficient null of acoustic waves at a
liquid-anisotropic solid interface”, J. Acoust. Soc. Am. 85 (1), 1-10 (1989).
TABLE I: Elastic constants, Piezoelectric constants and Dielectric constants of
BaTiO3 crystal.
Elastic constants
Piezoelectric
( GPa )
( C m2 )
( nC Vm )
c11  150.4
e15  11.4
c12  65.63
e31  4.32
11  10.8
 33  13.1
c13  65.94
e33  17.4
c33  145.5
m11  8.8
 15  4.56
 31  1.728
 33  6.96
11*  11.8
m33  5.2
e3*  3.6
c44  43.86
R  20
constants Dielectric constants
 33*  13.9
A11  12.8
A33  15.1
 3  7.5
38
TABLE II: Dynamical coefficients, Permeability tensor and other parameters:
Dynamical Coefficients
Permeability tensor
Parameters
( Kg / m3 )
( m2 )
1111 = 3762
11 = 2.5  10 10
 ( Kg / m3 ) = 5700
11
 33
= 3876
 33 = 4.5  10 10
 * ( Kg / m3 ) =1000
1112 = -855
 f ( Kg / m3 ) = 1000
12
33
= -741
 ( Ns / m 2) = 1 10 3
1122 = 3648
3322 = 3762
 ( MHz )  1
f  0.2
39
c f (m / s ) =1500
TABLE III Elastic and other coefficients for non-piezoelectric porous material
e15  e31  e33   15   31   33  e3*   3  11  33  11*  33*  A11  A33  0
Rest of the values are same as in the TABLE I.
40
TABLE IV Elastic and dynamic coefficients for isotropic porous material
c13  c12 , c33  c11 , c66  c44 , m33  m11;
11
11
12
12
33
 11
, 33
 11
, 3322  1122 ; 33  11
Rests of the values are same as given in the TABLE II and
TABLE III.
41
Figures Captions
Figure 1 Variation of Reflected and Transmitted energy ratios with the angle of incidence (  ); (i)
Reflected wave, (ii) Transmitted PE1, (iii) Transmitted PE2, (iv) Transmitted
(vi) Transmitted
P1 , (v) Transmitted S1 ,
P2 , (vii) Interaction energy;   1MHz , f  0.2 .
Figure 2 Variation of Reflected and Transmitted energy ratios with the angle of incidence (  ); (i)
Reflected wave, (ii) Transmitted PE1, (iii) Transmitted PE2, (iv) Transmitted
(vi) Transmitted
P1 , (v) Transmitted S1 ,
P2 , (vii) Interaction energy;   10MHz , f  0.2 .
42
Figure 3 Variation of Reflected and Transmitted energy ratios with the angle of incidence (  ); (i)
Reflected wave, (ii) Transmitted PE1, (iii) Transmitted PE2, (iv) Transmitted
(vi) Transmitted
P1 , (v) Transmitted S1 ,
P2 , (vii) Interaction energy;   100MHz , f  0.2 .
Figure 4 Variation of Reflected energy ratios with the angle of incidence (  ) for different porosity;
  1MHz .
Figure 5 Variation of Transmitted energy ratios with the angle of incidence (  ) for different
porosity; (i) Transmitted energy
P1 , (ii) Transmitted energy S1 , (iii) Transmitted energy P2 , (iv)
Transmitted energy PE1, (v) Transmitted energy PE2, (vi) Interaction energy;
  1MHz .
Figure 6 Variation of Reflected and Transmitted energy ratios with the angle of incidence (  ); (i)
Reflected wave, (ii) Transmitted
P1 , (iii) Transmitted S1 , (iv) Transmitted P2 ;   1MHz ,
f  0.2 .
Figure 7 Variation of Leaky wave velocity ( vL ) with frequency (  ) for different porosity;
  50
.
43