Download Wave reflection and refraction in triclinic crystalline media

1
Wave reflection and refraction in triclinic crystalline media
A.Chattopadhyay*
Geomathematics Group, University of Kaiserslautern,
Erwin-Schrödinger-Straße, Postfach 3049,
67663 Kaiserslautern, Germany
* Permanent address: Department of Applied Mathematics,
Indian School of Mines, Dhanbad-826004, Jharkhand, India
Email: [email protected] , Fax : 00913262202380
Abstract:
In this paper, the reflection and refraction of a plane wave at an interface between
.two half-spaces composed of triclinic crystalline material is considered. It is shown that
due to incidence of a plane wave three types of waves namely quasi-P (qP), quasi-SV
(qSV) and quasi-SH (qSH) will be generated governed by the propagation condition
involving the acoustic tensor. A simple procedure has been presented for the calculation
of all the three phase velocities of the quasi waves. It has been considered that the
direction of particle motion is neither parallel nor perpendicular to the direction of
propagation. Relations are established between directions of motion and propagation,
respectively. The expressions for reflection and refraction coefficients of qP, qSV and
qSH waves are obtained. Numerical results of reflection and refraction coefficients are
presented for different types of anisotropic media and for different types of incident
waves. Graphical representation have been made for incident qP waves and for incident
qSV and qSH waves numerical data are presented in two tables.
Key words: Reflection, refraction, incident wave, triclinic medium, quasi-P,quasi-SV,
quasi-SH
1. Introduction
The study of reflection and refraction phenomena of elastic waves is of
considerable interest in the field of Seismology, in particular seismic prospecting as the
Earth’s surface might be supposed of consist of different layers having different material
properties. The elastic properties of a crystalline material depend on the internal structure
of the material.
Effect of earthquake on artificial structures is of prime importance to engineers
and architects. During an earthquake and similar disturbances a structure is excited into a
more or less violent, with resulting oscillatory stresses, which depend both upon the
ground vibration and physical properties of the structure. So, wave propagation in
anisotropic medium plays a very important role in civil engineering and geophysics.
The propagation of body waves and surface waves in anisotropic media is
fundamentally different from their propagation in isotropic media. In seismology
anisotropy manifests itself most straightforwardly by a variation of the phase speed of
seismic waves with their direction of propagation. A material displaying velocity
anisotropy must have its effective elastic constants arranged in some form of crystalline
symmetry. Cramplin [1977] has pointed out that the behaviour of both body and surface
waves in anisotropic structures differs from that in isotropic structures, and variation of
2
velocity with direction is only one of the anomalies which may occur. Within an
anisotropic material three body waves propagate in any direction, having different and
varying velocity and different and varying polarization. In highly anisotropic medium the
P, SV and SH are coupled. This coupling introduces polarization anomalies which may
be used to investigate anisotropy within the earth.
The problem of reflection and refraction of elastic waves have been discussed by
several authors. Without going into the details of the research work in this field we
mention the papers by Knott [1899], Gutenberg [1944], Achenbach [1976], Keith and
Crampin [1977, 1977a, 1977b], Tolstoy [1982], Norris [1983], Pal and Chattopadhyay
[1984], Auld [1990], Ogden and Sotirropoulos [1997,1998], Chattopadhyay and
Rogerson [2001].
Crampin and Taylor [1971] studied surface wave propagation in examples of
unlayered and multilayered anisotropic media, which is examined numerically with a
program using as extension of the Thompson-Haskell matrix formulation. They studied
some examples of surface wave propagation in anisotropic media to interpret a possible
geophysical structure. Crampin [1975] showed that the surface waves have distinct
particle motion when propagating in a structure having a layer of anisotropic material
with certain symmetry relations.
In this paper we have studied the reflection and refraction of a plane wave at the
interface of of two triclinic crystalline media. Relations have been established between
directions of motion and propagation, respectively. Reflection and refraction coefficients
due to incident qP,qSV and qSH waves have been computed for different types of
anisotropic media. It has been observed that triclinic media plays a significant role in case
of reflection and refraction.
2. Formulation of the problem
Consider a homogeneous triclinic medium having twenty one elastic constants.
We assume ui = ui ( x2 , x3 , t ) , i=1,2,3.
(1)
The stress-strain relations are
τ 11 = C11e11 + C12 e22 + C13e33 + C14 e23 + C15 e13 + C16 e12 ,
τ 22 = C12 e11 + C 22 e22 + C23e33 + C 24 e23 + C25 e13 + C 26 e12 ,
τ 33 = C13e11 + C 23e22 + C33e33 + C34 e23 + C35 e13 + C36 e12 ,
(2a)
τ 23 = C14 e11 + C 24 e22 + C34 e33 + C 44 e23 + C 45 e13 + C 46 e12 ,
τ 13 = C15 e11 + C 25 e22 + C35 e33 + C45 e23 + C55 e13 + C56 e12 ,
τ 12 = C16 e11 + C26 e22 + C36 e33 + C46 e23 + C56 e13 + C66 e12
where
Cij = C ji , 2eij = (ui , j + u j ,i ) and u i (i=1,2,3) are the displacement components.
The equations of motion without body forces are
τ ij , j = ρui ,i=1,2,3.
(2b)
The following nonvanishing equations of motion are obtained after using equations (1)
and (2)
3
∂ 2u1
∂ 2u1
∂ 2u1
∂ 2u 2
∂ 2u 2
∂ 2u 2
C
C
C
+
2
+
)
+
{
+
(
C
+
C
)
+
}
C
26
46
25
45
66
56
∂x32
∂x 2 ∂x3
∂x22
∂x32
∂x 2 ∂x3
∂x22
(C55
∂ 2 u3
∂ 2 u3
∂ 2 u3
∂ 2u1
ρ
+
+
+
=
,
C
C
C
(
)
}
46
36
45
∂x32
∂x2∂x3
∂x22
∂t 2
+ {C35
(3)
∂ 2u1
∂ 2u1
∂ 2u1
∂ 2u 2
∂ 2u 2
∂ 2u 2
C
C
C
C
+
(
+
)
+
}
+
{
+
2
C
+
}
C
22
24
44
26
25
46
∂x32
∂x2∂x3
∂x22
∂x32
∂x2∂x3
∂x22
{C45
+ {C34
∂ 2 u3
∂ 2 u3
∂ 2 u3
∂ 2u 2
ρ
+
+
+
=
,
C
C
C
(
)
}
24
23
44
∂x32
∂x2∂x3
∂x22
∂t 2
(4)
∂ 2u1
∂ 2u1
∂ 2u1
+
(
+
)
+
}
C
C
C
46
45
36
∂x32
∂x2∂x3
∂x22
{C35
+ {C34
∂ 2u 2
∂ 2u 2
∂ 2u 2
+
(
+
)
+
}
C
C
C
24
23
44
∂x32
∂x 2 ∂x3
∂x22
∂ 2u3
∂ 2u3
∂ 2u3
∂ 2 u3
+
+
=
.
(5)
ρ
C
C
2
}
44
34
∂x32
∂x2∂x3
∂x22
∂t 2
Let p (0, p 2( n ) , p3( n ) ) denote the unit propagation vector, cn is the phase velocity and k n is
the wavenumber of plane waves propagating in the x2 x3 -plane.
We consider plane wave solution of equations (3) to (5) as
 u1( n ) 
 d1( n ) 




 u 2( n )  = An  d 2( n )  exp(iηn )
(6)
 (n) 
 ( n) 
 d3 
 u3 




where
d (d1( n ) , d 2( n ) , d 3( n ) ) is the unit displacement vector and
+ {C33
η n = k n ( x2 p2( n ) + x3 p3( n ) − cn t ) .
Inserting the expressions of (6) into the equations (3) to (5), we have
( S − c 2 )d1( n ) + Td 2( n ) + Pd 3( n ) = 0 ,
Td
(n)
1
+ (Q − c )d
2
(n)
2
+ Rd
(n)
3
Pd + Rd + (W − c )d
where
Cij
ρc 2
c 2 = n , Cij =
,
C44
C44
(n)
1
(n)
2
2
( n)
3
(7)
(8)
= 0,
(9)
=0
(10)
S = C55 p32 + 2C56 p2 p3 + C66 p22 ,
T = C 45 p32 + (C 46 + C 25 ) p2 p3 + C26 p 22 ,
P = C35 p32 + (C36 + C 45 ) p 2 p3 + C 46 p 22 ,
Q = C 44 p32 + 2C 24 p2 p3 + C 22 p22 ,
R = C34 p32 + (C 23 + C 44 ) p2 p3 + C 24 p22 ,
4
W = C33 p32 + 2C34 p2 p3 + C44 p22 .
From equations (8),(9) and (10), we obtain
d 3( n ) T 2 − (Q − c 2 )( S − c 2 )
=
,
d1( n )
P(Q − c 2 ) − RT
d 2( n ) R( S − c 2 ) − PT
=
d1( n ) P(Q − c 2 ) − RT
and
d 3( n ) T 2 − (Q − c 2 )( S − c 2 )
.
=
d 2( n )
R( S − c 2 ) − PT
(11)
(12)
(13)
(14)
The equations (12) to (14) may be used to calculate d in terms of p .
Eliminating d1( n ) , d 2( n ) , d 3( n ) from (8),(9) and (10), we have
c 6 + a1c 4 + a2 c 2 + a3 = 0
where
a1 = −( S + Q + W ) ,
(15)
a2 = QS + WS + QW − R 2 − T 2 − P 2 ,
a3 = −( SQW − SR 2 − WT 2 + 2 PTR − P 2Q) .
(16)
Solving the equation (15), we will obtain the phase velocities of quasi-P(qP),quasiSV(qSV) and quasi-SH(qSH) as
ϕ a
ρc L2 = −2r cos( ) − 1 ,
(17)
3
3
ϕ a
2
ρcSV
(18)
= 2r cos(60 0 + ) − 1 ,
3
3
ϕ a
2
ρcSH
= 2r cos(60 0 − ) − 1
(19)
3
3
where
2a 3 a a
3a − a12
2 q = 1 − 1 2 + a3 , 3 p = 2
,
27
3
3
q
r = − p , ϕ = cos −1 ( 3 ) .
(20)
r
In isotropic case
C11 = C22 = C33 = λ + 2 µ ,
C12 = C13 = C 23 = λ ,
C 44 = C55 = C66 = µ
(21)
and all other elastic constants are zero.
Substituting (21) in equations (17),(18) and (19) and after simplification, we obtain the
following compressional velocity ( c L ) and the repeated roots ( cSV and cSH ) for shear
velocity as
λ + 2µ 2
µ
2
cL2 =
, cSV = cSH
= .
(22)
ρ
ρ
5
We solved the equation (15) and obtained three real roots of c 2 . The largest root is
assigned to the phase velocity of qP waves, the second largest is the phase velocity of
qSV waves and the lowest root for the phase velocity of qSH waves. The phase velocities
of quasi-transverse waves (qSV and qSH) will not be identical in case of triclinic
medium. The result was tested with different sets of data as mentioned in section 4. If any
geophysical evidence exists that the qSH wave velocity is more than qSV wave velocity
then the nature of the graphs of the reflected qSV and reflected qSH of this paper are to
be interchanged. This method of solution for calculating the velocities of all the three
quasi-waves is most general and will be helpful to identify the phase velocities for
different types of anisotropy.
3. Solution of the problem
Consider a triclinic crystalline medium. The x3 -axis is taken along the free
surface and x2 -axis is vertically downward. Plane wave is incident at the free boundary
x2 = 0 . Incident qP or qSV or qSH waves will generate reflected qP, reflected qSV,
reflected qSH waves and also refracted qP, refracted qSV, refracted qSH waves. It is also
clear from the equations (3),(4) and (5) that all the displacement components are coupled.
Let n=0,1,2,3,4,5,6 be assumed for incident wave, reflected qP, qSV, qSH and refracted
qP, qSV, qSH waves respectively.
In the plane x2 = 0 , the displacements and stresses of incident and reflected waves are
represented by
u (jn ) = An d (j n ) exp(iη n ) , j=1,2,3.
τ 12( n ) = P1n ik n An exp(iη n ) ,
τ 22( n ) = Qn ik n An exp(iηn ) ,
( n)
τ 23
= Rn ik n An exp(iηn )
where
P1n = C 26 p2( n ) d 2( n ) + C36 p3( n ) d 3( n ) + C46 {d 2( n ) p3( n ) + d 3( n ) p2( n ) }
+ C56 d1( n ) p3( n ) + C66 d1( n ) p2( n ) ,
(23)
(24)
Qn = C 22 p2( n ) d 2( n ) + C23 p3( n ) d 3( n ) + C24 {d 2( n ) p3( n ) + d 3( n ) p2( n ) }
+ C 25 d1( n ) p3( n ) + C 26 d1( n ) p2( n ) ,
Rn = C 24 p d
( n)
2
(n)
2
(n)
45 1
+C d
+ C34 p d
(n)
3
p
(n)
3
(n)
3
+C d
(n)
46 1
+ C44 {d
p
(n)
2
(25)
(n)
2
p
(n)
3
+d
(n)
3
( n)
2
p }
,
η n = k n ( x3 p3( n ) − cn t )
and n=0,1,2,3,4,5,6.
(26)
For n=4,5,6 the elastic constants Cij to be replaced by Cij/ and accordingly equations (23)
to (26) will be changed for the refracted waves in the upper half-space.
For incident plane waves
p2( 0) = − cos θ 0 , p3( 0) = sin θ 0 , c0 = c I .
For reflected qP waves
6
p2(1) = cos θ 1 , p3(1) = sin θ 1 , c1 = c L1 .
For reflected qSV waves
p2( 2) = cos θ 2 , p3( 2) = sin θ 2 , c2 = cT .
For reflected qSH waves
p2(3) = cos θ 3 , p3( 3) = sin θ 3 , c3 = cT 1
For refracted qP waves
p2( 4) = − cos θ 4 , p3( 4 ) = sin θ 4 , c1 = c L/ .
For refracted qSV waves
p2(5) = − cos θ 5 , p3(5) = sin θ 5 , c2 = cT/ .
For refracted qSH waves
p2( 6) = − cos θ 6 , p3( 6) = sin θ 6 , c3 = cT/ 1
(27)
where cI , cL1 , cT , cT 1 , , c L/ 1 , cT/ and cT/ 1 are the phase velocities of incident plane wave,
reflected qP, reflected qSV, reflected qSH waves, refracted qP, refracted qSV and
refracted qSH waves respectively.
The boundary conditions at x2 = 0 are
u1( 0) + u1(1) + u1( 2) + u1( 3) = u1( 4) + u1(5) + u1( 6) ,
u
u
(0)
2
(0)
3
+u
+u
(1)
2
(1)
3
+u
+u
( 2)
2
( 2)
3
+u
+u
( 3)
2
( 3)
3
=u
=u
( 4)
2
( 4)
3
+u
+u
(5)
2
(5)
3
(28)
+u ,
+u ,
(6)
2
(6)
3
(29)
(30)
τ 12( 0) + τ 12(1) + τ 12( 2) + τ 12(3) = τ 12( 4) + τ 12( 5) + τ 12( 6) ,
( 0)
(1)
( 2)
( 3)
( 4)
( 5)
(6)
τ 22
+ τ 22
+ τ 22
+ τ 22
= τ 22
+ τ 22
+ τ 22
,
( 0)
(1)
( 2)
( 3)
( 4)
( 5)
(6)
τ 23 + τ 23 + τ 23 + τ 23 = τ 23 + τ 23 + τ 23 .
Using the boundary conditions and the equations (23) to (26), we obtain,
A0 d1( 0 ) exp{ik 0 ( x3 p3( 0 ) − c I t )} + A1d1(1) exp{ik1 ( x3 p3(1) − c L1t )}
(31)
(32)
(33)
+ A2 d1( 2) exp{ik 2 ( x3 p3( 2) − cT t )} + A3 d1( 3) exp{ik3 ( x3 p3(3) − cT 1t )}
= A4 d1( 4) exp{ik 4 ( x3 p3( 4) − c L/ 1t )} + A5 d1(5) exp{ik 5 ( x3 p3( 5) − cT/ t )}
P10 A0 k 0 exp{ik 0 ( x3 p3( 0) − cI t )} + P11 A1k1 exp{ik1 ( x3 p3(1) − c L1t )}
+ A6 d1( 6 ) exp{ik 6 ( x3 p3( 0 ) − cT/ 1t )}
(34)
A0 d 2( 0 ) exp{ik 0 ( x3 p3( 0 ) − c I t )} + A1d 2(1) exp{ik1 ( x3 p3(1) − c L1t )}
+ A2 d 2( 2) exp{ik 2 ( x3 p3( 2) − cT t )} + A3 d 2( 3) exp{ik3 ( x3 p3(3) − cT 1t )}
= A4 d 2( 4) exp{ik 4 ( x3 p3( 4) − c L/ 1t )} + A5 d 2(5) exp{ik 5 ( x3 p3( 5) − cT/ t )}
+ A6 d 2( 6 ) exp{ik 6 ( x3 p3( 0 ) − cT/ 1t )}
A0 d
(0)
3
exp{ik 0 ( x3 p
(0)
3
(35)
− c I t )} + A d
(1)
1 3
exp{ik1 ( x3 p
(1)
3
− c L1t )}
+ A2 d 3( 2) exp{ik 2 ( x3 p3( 2) − cT t )} + A3 d 3( 3) exp{ik3 ( x3 p3(3) − cT 1t )}
= A4 d 3( 4) exp{ik 4 ( x3 p3( 4) − c L/ 1t )} + A5 d 3(5) exp{ik 5 ( x3 p3( 5) − cT/ t )}
+ A6 d 3( 6 ) exp{ik 6 ( x3 p3( 0 ) − cT/ 1t )}
P10 A0 k 0 exp{ik 0 ( x3 p3( 0) − cI t )} + P11 A1k1 exp{ik1 ( x3 p3(1) − c L1t )}
(36)
7
+ P12 A2 k 2 exp{ik 2 ( x3 p3( 2 ) − cT t )} + P13 A3 k 3 exp{ik 3 ( x3 p3(3) − cT 1t )}
= P14 A4 k 4 exp{ik 4 ( x3 p3( 4 ) − c L/ 1t )} + P15 A5 k5 exp{ik5 ( x3 p3( 5) − cT/ t )}
+ P16 A6 k 6 exp{ik 6 ( x3 p3( 6) − cT/ 1t )} = 0 ,
(37)
Q0 A0 k 0 exp{ik 0 ( x3 p3( 0 ) − c I t )} + Q1 A1k1 exp{ik1 ( x3 p3(1) − cL1t )}
+ Q2 A2 k 2 exp{ik 2 ( x3 p3( 2) − cT t )} + Q3 A3 k3 exp{ik3 ( x3 p3( 3) − cT 1t )}
= Q4 A4 k 4 exp{ik 4 ( x3 p3( 4) − c L/ 1t )} + Q5 A5 k 5 exp{ik5 ( x3 p3(5) − cT/ t )}
+ Q6 A6 k 6 exp{ik 6 ( x3 p3( 6) − cT/ 1t )} = 0 ,
(38)
R0 A0 k 0 exp{ik 0 ( x3 p3( 0) − c I t )} + R1 A1k1 exp{ik1 ( x3 p3(1) − c L1t )}
+ R2 A2 k 2 exp{ik 2 ( x3 p3( 2 ) − cT t )} + R3 A3 k 3 exp{ik3 ( x3 p3(3) − cT 1t )}
= R4 A4 k 4 exp{ik 4 ( x3 p3( 4 ) − c L/ 1t )} + R5 A5 k5 exp{ik 5 ( x3 p3( 5) − cT/ t )}
+ R6 A6 k 6 exp{ik 6 ( x3 p3( 6) − cT/ 1t )} = 0 ,
The above equations are valid for all values of x3 and t. Therefore, we have
k 0 ( x3 sin θ 0 − c I t ) = k1 ( x3 sin θ1 − c L1t ) = k 2 ( x3 sin θ 2 − cT t ) = k3 ( x3 sin θ 3 − cT 1t )
(39)
= k 4 ( x3 sin θ 4 − c L/ 1t ) = k 5 ( x3 sin θ 5 − cT/ t ) = k 6 ( x3 sin θ 6 − cT/ 1t )
(40)
which gives
k 0 c I = k1c L1 = k 2 cT = k3cT 1 = k 4 c L/ 1 = k 5 cT/ = k 6 cT/ 1 = k ,
(41)
and
k 0 sin θ 0 = k1 sin θ1 = k 2 sin θ 2 = k 3 sin θ 3 = k 4 sin θ 4 = k5 sin θ 5 = k 6 sin θ 6 = ω
(42)
where k and ω are apparent wave number and circular frequency respectively. The
A
A A A A A
amplitude ratios of qP,qSV and qSH are denoted by 1 , 2 , 3 , 4 , 5 ,and 6 .
A0 A0 A0 A0 A0
A0
Solving the equations (34)-(39), the reflection and refraction coefficients of qP, qSV and
qSH may be obtained as
A1 D1 A2 D2 A3 D3 A4 D4 A5 D5 A6 D6
=
,
=
,
=
,
=
,
=
,
=
(43)
A0 D0 A0 D0 A0 D0 A0 D0 A0 D0 A0 D0
where
a1
a2
a3
− a4
− a5
− a6
D0 =
b1
b2
b3
− b4
− b5
− b6
c1
c2
c3
− c4
− c5
− c6
e1
e2
e3
− e4
− e5
− e6
f1
f2
f3
− f4
− f5
− f6
g1
g2
g3
− g4
− g5
− g6
,
8
D1 =
D2 =
D3 =
D4 =
D5 =
−1
a2
a3
− a4
− a5
− a6
−1
b2
b3
− b4
− b5
− b6
−1
c2
c3
− c4
− c5
− c6
−1
e2
e3
− e4
− e5
− e6
−1
f2
f3
− f4
− f5
− f6
−1
g2
g3
− g4
− g5
− g6
a1
−1
a3
− a4
− a5
− a6
b1
−1
b3
− b4
− b5
− b6
c1
−1
c3
− c4
− c5
− c6
e1
−1
e3
− e4
− e5
− e6
f1
−1
f3
− f4
− f5
− f6
g1
−1
g3
− g4
− g5
− g6
a1
a2
−1
− a4
− a5
− a6
b1
b2
−1
− b4
− b5
− b6
c1
c2
−1
− c4
− c5
− c6
e1
e2
−1
− e4
− e5
− e6
f1
f2
−1
− f4
− f5
− f6
g1
g2
−1
− g4
− g5
− g6
a1
a2
a3
−1
− a5
− a6
b1
b2
b3
−1
− b5
− b6
c1
c2
c3
−1
− c5
− c6
e1
e2
e3
−1
− e5
− e6
f1
f2
f3
−1
− f5
− f6
g1
g2
g3
−1
− g5
− g6
a1
a2
a3
− a4
−1
− a6
b1
b2
b3
− b4
−1
− b6
c1
c2
c3
− c4
−1
− c6
e1
e2
e3
− e4
−1
− e6
f1
f2
f3
− f4
−1
− f6
g1
g2
g3
− g4
−1
− g6
,
,
,
,
,
9
D6 =
a1
a2
a3
− a4
− a5
−1
b1
b2
b3
− b4
− b5
−1
c1
c2
c3
− c4
− c5
−1
e1
e2
e3
− e4
− e5
−1
f1
f2
f3
− f4
− f5
−1
g1
g2
g3
− g4
− g5
−1
and
ai =
d1(i )
,
d1( 0)
bi =
d 2(i )
,
d 2( 0)
ci =
d 3(i )
,
d 3( 0)
,i=1,2…,6
P1i ki
Q k
R k
, f i = i i , g i = i i , i=1,2,3,…6
P10 k 0
Q0 k 0
R0 k 0
R
Q
P
P1i = 1i , Qi = i , Ri = i , i=0,1,2,3,…6.
C 44
C 44
C 44
ei =
(39)
4. Numerical Calculations and Discussions
Numerical calculations were performed for incident qP, qSV and qSH waves with
different types of anisotropic data. We have considered eight hypothetical data in case of
Data-1 and twelve hypothetical data in case of Data-3 to get the effect of 21 elastic
constants.
The following cases have been considered:
Data-1: The 13 elastic constants for the case of AT-cut quartz are (Tiersten[1969])
C11 = 86.74GPa, C 22 = 129.77GPa, C33 = 102.83GPa ,
C12 = −8.25GPa, C13 = 27.15GPa, C14 = −3.66GPa,
C 23 = −7.42GPa, C 24 = 5.7GPa, C34 = 9.92GPa,
C 44 = 38.61GPa, C55 = 68.81GPa, C66 = 29.01GPa, C56 = 2.53GPa ,
ρ = 2.649 gm / cm 3 .
To test the effect of triclinic structures, we have considered the following hypothetical
values of the constants:
C15 = C16 = C25 = C26 = C35 = C36 = C45 = C46 = 7.5 GPa.
Data-2: The 13 elastic constants of Data-1 case and
C15 = C16 = C 25 = C 26 = C35 = C36 = C45 = C 46 = 0.5 GPa
Data-3: The 9 elastic constants for Rochelle salt (Auld [1990]) are
C11 = 28.0GPa, C 22 = 41.4GPa, C33 = 39.4GPa,
C 44 = 6.66GPa, C55 = 2.85GPa, C66 = 9.6GPa,
C12 = 17.4GPa, C13 = 15.0GPa, C 23 = 19.7GPa.
ρ / = 2.7 gm / cm 3 .
We have considered alongwith the set of Data-3, the following hypothetical data
C14 = C34 = C 24 = C56 = 0.5 GPa
10
and
C15 = C16 = C 25 = C 26 = C35 = C36 = C45 = C 46 = 0.5 Gpa.
Curve-1 has been drawn considering the values of the upper layer as Data-1 and for the
lower layer as Data 3. Curve-2 has been drawn with the set of vaules for the upper layer
as Data-3 and the lower layer as Data-2.
Figures 1 to 6 have been drawn for incident qP waves. It has been observed that the
existence of the angles for amplitude ratios of reflected and refracted waves are upto
680 . Curves 1 and 2 have some jumps at certain angles for each diagrams from 1 to 6.
Due to want of practical data the actual behaviour cannot be presented but idea can be
made by considering these hypothetical data about the behaviour of reflection and
refraction of waves in a triclinic media which is highly anisotropic in nature.
Table-1
The reflection coefficients for incident qSV waves for the set of data of curve-1 as
mentioned above:
θ in
degrees
0
10
20
30
40
50
60
70
A1 / A0
A2 / A0
A3 / A0
A4 / A0
A5 / A0
A6 / A0
0.5011
0.8549
-0.13409
0.3268
0.5807
-0.9412
1.1948
1.5806
-0.7285
-.7938
-2.4629
-3.0822
-3.3594
3.1518
-2.7522
-2.5766
-0.7182
-0.9305
1.7378
2.4645
2.4821
-2.2028
-1.9326
-1.8422
0.1916
0.0236
-0.5539
-0.4448
2.4825
-2.6552
1.5351
1.3135
0.0303
-0.0082
-0.0201
0.0258
0.1714
-0.2639
0.2678
0.2564
-0.1682
0.0416
0.6569
0.4792
-2.6587
2.7961
-1.47136
-.9914
Table-2
The reflection coefficients for incident qSH waves for the set of data of curve-1 as
mentioned above:
θ in
degrees
0
10
20
30
40
50
60
70
A1 / A0
A2 / A0
A3 / A0
A4 / A0
A5 / A0
A6 / A0
-0.5187
1.9077
0.4372
0.1879
0.5246
0.7984
0.6717
-2.3998
-1.2574
0.5953
-1.7627
-2.8943
-3.2924
-2.7937
-1.8051
2.6543
0.9662
-3.2210
0.7549
2.3863
2.7141
2.2873
-1.7599
-0.0552
0.6328
1.3696
1.0272
0.5420
3.5593
2.6008
1.2689
-0.6534
0.0136
0.0425
0.0758
0.0573
0.1596
0.2034
0.1439
-0.3901
-0.4530
-1.2106
-0.9497
-0.4573
-3.6311
-2.5651
-0.9705
1.3588
11
Acknowledgement: The work was completed while I was visiting the University of
Kaiserslautern, Department of Geomathematics as Visiting Professor. The author is
grateful to Professor (Dr.) W. Freeden for providing DAAD fellowship and all the
facilities for conducting research. I am also grateful to Dr. V.Michel for various
discussion about the research work and also for all kind of help during my stay at
Kaiserslautern. This award is very gratefully acknowledged.
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12