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Vol. 52, No. 2
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2004
REFLECTION/TRANSMISSION OF PLANE SH WAVE
THROUGH A SELF-REINFORCED ELASTIC LAYER
BETWEEN TWO HALF-SPACES
Sushil CHAUDHARY1, Ved P. KAUSHIK1 and Sushil K. TOMAR2
1
Department of Mathematics, Kurukshetra University, Kurukshetra -136119, India
2
Department of Mathematics, Panjab University, Chandigarh -160014, India
e-mail: [email protected]
Abstract
The reflection and transmission of plane SH wave through a self-reinforced elastic layer sandwiched between two homogeneous visco-elastic solid
half-spaces has been studied. The expressions for reflection and transmission
coefficients are obtained. These coefficients are computed against the angle of
incidence, normalized wave number and the reinforced parameters. The results
obtained have been presented graphically.
The problem studied by Borcherdt has been reduced as a special case of
our problem. It is found that reflection and transmission coefficients depend on
the angle of incidence, normalized wave number and the reinforcement parameters of the sandwiched layer.
Key words: SH wave, visco-elastic half-space, self-reinforced elastic layer,
reflection coefficient, transmission coefficient.
1. INTRODUCTION
The knowledge of elastic wave propagation through the earth is helpful in investigating the internal structure of the earth and exploration of valuable minerals buried. The
seismic signals propagating through the earth medium have to travel through various
types of minerals present in the earth in the form of layers. Doubtless to say that the
220
S. CHAUDHARY et al.
elastic properties of these layered materials do affect the propagation of the seismic
waves. Moreover, the discontinuities present in the earth between the layers produce
the reflection and refraction phenomena.
The subject of propagation, reflection and refraction of the elastic waves from the
boundary surfaces is very important and researchers have investigated several problems by assuming different models. The propagation and reflection of general plane
SH wave in homogeneous visco-elastic media has been studied, e.g., by Cooper
(1967), Schoenberg (1971), Borcherdt (1977; 1982). Buchan (1971a; b) studied the
propagation of plane SH waves in a linear visco-elastic solid. The problem of transmission and reflection of inhomogeneous plane SH waves at an interface between homogeneous and inhomogeneous visco-elastic half-spaces has been studied by Kaushik
and Chopra (1980). Kaushik and Chopra (1983) studied the reflection and transmission of general SH waves through n-layered visco-elastic media. Biswas (1967) studied the transmission of SH waves through a visco-elastic layer embedded between two
elastic half-spaces and concluded that the effect of imaginary components of elastic
properties on the transmission of waves increases as the frequency increases. Shaw
and Bugl (1968) studied the transmission of time harmonic plane P and SV waves
in a layered infinite linear visco-elastic medium. The generation and propagation of
SH-type waves due to stress discontinuity in a linear visco-elastic layered medium was
studied by Pal and Kumar (1995). Pipkin and Rogers (1971) developed the plane
strain theory of finite deformation for fiber-reinforced materials. Belfield et al. (1983)
gave a linear model for such a material, which was later used by Verma and Rana
(1983) for studying the problem of rotation of a circular cylindrical tube reinforced by
fibers lying along helices. Verma (1986) examined the influence of an external magnetic field on the propagation of purely transverse waves polarized parallel to the
plane faces in a homogeneous, initially unstressed, infinitely conducting, selfreinforced elastic flat plate extending to infinity. Verma et al. (1988) studied the problem of magneto-elastic transverse surface waves in self-reinforced elastic solids.
Green (1991) studied the problem of reflection and transmission phenomenon of transient stress waves in fiber composite laminates. Chattopadhyay and Chaudhury (1995)
studied the propagation of magneto-elastic shear waves in an infinite self-reinforced
plate. The problem of surface waves in fiber-reinforced anisotropic elastic media was
studied by Sengupta and Nath (2001).
It is well known in the literature that the earth medium exhibits various types of
elastic properties. The hard and/or soft rocks, which may be self-reinforced in nature,
might be present beneath the earth surface. These rocks when come in the way of
seismic waves do affect their propagation and such seismic signals are always influenced by the elastic properties of the media through which they travel. Keeping in
view this type of geophysical situation, we analyze the problem of reflection and refraction of plane SH waves using the model of a self-reinforced elastic layer sandwiched between two homogeneous visco-elastic half-spaces.
REFLECTION/TRANSMISSION OF PLANE SH WAVE
221
2. FORMULATION OF THE PROBLEM AND BASIC EQUATIONS
Consider a layer MII having width d of a perfectly conducting self-reinforced linearly
elastic solid between two homogeneous visco-elastic half-spaces MI and MIII. Let
Z = 0 be the interface between
MI and MII and Z = −d between
MII and MIII. The Z-axis of the rectangular coordinate system is
pointing into the lower half-space
MI. A plane SH wave is assumed
to be incident at an angle θ1. The
geometry of the problem is shown
in Fig. 1. We shall denote the density and complex rigidity, respectively, by ρ1, µ1 in MI and by ρ3, µ3
in MIII. We postulate that an incident SH wave will give rise to: (i)
a reflected SH wave at an angle θ1
in MI, (ii) a refracted SH wave at
an angle θ2 in the layer MII , and
Fig. 1. Geometry of the problem.
(iii) the refracted SH wave in MII
which will again reflect the same
wave from the interface Z = −d
along with a transmitted SH wave at an angle θ3 in the half-space MIII.
The equations of motion in a self-reinforced perfectly conducting medium MII
having only the electromagnetic force, that is, Lorentz force J × B, are given by
τ ij , j + (J × B )i = ρ
∂ 2 ui
,
∂t 2
(i, j = 1, 2, 3) ,
(1)
where τij are the Cartesian components of the stress tensor, J is the electric current,
B is the magnetic induction, ui are the components of displacement vector u, ρ is the
mass density and comma denotes the partial derivative with respect to space coordinates.
J and B are given by the well-known Maxwell’s fundamental equations:
∇×H = J ,
∇×E = −
∂B
,
∂t
∇ ⋅B = 0 ,
∂u
⎡
⎤
J = σ ⎢E +
× B⎥ .
∂
t
⎣
⎦
B = µe H ,
(2)
(3)
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S. CHAUDHARY et al.
Also, the Maxwell’s stress tensor is given by
(τ ij ) M = µe ( H i h j + H j hi − H k hk δ ij ) ,
(4)
where E is the induced electric field, H = (Hx, Hy, Hz) is the total applied and induced
magnetic field, µe is the magnetic permeability, σ is the electrical conductivity, and h
is the change in the basic magnetic field. Following Belfield et al. (1983), the stressstrain relation in medium MII, in which reinforcement is to make the material locally
transversely isotropic, whose preferred direction is that of unit vector a, is given by
τ ij = λ ekk δ ij + 2 µT eij + α (ak am ekm δ ij + ekk ai a j )
+ 2( µ L − µT )( ai ak ekj + a j ak eki ) + β ak am ekm ai a j ,
(5)
where λ, µT, µL, α and β are the material constants with the dimension of stress, and
other symbols have their usual meanings. Using eqs. (2) and (3) we can write
⎡∂ H
⎛ ∂u
⎞⎤
∇ 2 H = σµe ⎢
− ∇×⎜
× H ⎟⎥ .
∂
∂
t
t
⎝
⎠⎦
⎣
(6)
For the motion of SH wave, we take u = (0, V2, 0) and since we are considering a
two-dimensional problem in X−Z plane, we have ∂ ∂Y ≡ 0 . Thus, from eq. (6) we get
∂H x
1 2
=
∇ Hx ,
∂t
σµe
(7a)
∂H y
∂V2 ⎞
∂V2 ⎞
∂ ⎛
∂ ⎛
1 2
=
∇ Hy +
+
H
H
,
σµe
∂t
∂X ⎜⎝ x ∂t ⎟⎠
∂Z ⎜⎝ z ∂t ⎟⎠
(7b)
∂H z
1 2
=
∇ Hz .
σµe
∂t
(7c)
For perfect conductor, i.e., when σ → ∞, we have from eqs. (7a, b, c)
∂H x
∂H z
=
= 0,
∂t
∂t
(8a)
∂H y
∂
=
∂t
∂X
(8b)
∂V2 ⎞
∂V2 ⎞
∂ ⎛
⎛
⎜ H x ∂t ⎟ + ∂Z ⎜ H z ∂t ⎟ .
⎝
⎠
⎝
⎠
It is clear from eq. (8a) that there is no perturbation in Hx and Hz. Therefore, we may
take small perturbation h2 in Hy, so that
H x = H 01 ,
H y = H 02 + h2 ,
H z = H 03 .
Taking
H 01 = H 0 cos φ ,
H 02 = 0 ,
H 03 = H 0 sin φ ,
(9)
REFLECTION/TRANSMISSION OF PLANE SH WAVE
223
where H0 = ⎪H0 ⎢ and φ is the angle at which the wave crosses the magnetic field.
Hence,
H = ( H 0 cos φ , h2 , H 0 sin φ ) .
(10)
Using eq. (10) into eq. (8b), and integrating with respect to t we get
∂V
⎛ ∂V
⎞
h2 = H 0 ⎜ 2 cos φ + 2 sin φ ⎟ .
X
Z
∂
∂
⎝
⎠
(11)
Since the initial value of h2 is zero, we get from eq. (2)
1
⎡
⎤
J × B = µe ⎢( H ⋅∇ ) H − ∇H 2 ⎥ .
2
⎣
⎦
(12)
Using eq. (10) into eq. (12), we get
(J × B )1 = (J × B)3 = 0 ,
(13a)
∂h
⎛ ∂h
⎞
(J × B )2 = µe H 0 ⎜ 2 cos φ + 2 sin φ ⎟ .
∂Z
⎝ ∂X
⎠
(13b)
Equation (13b) with the aid of eq. (11) yields
⎛ ∂ 2V2
⎞
∂ 2V2
∂ 2V2
2
φ
φ
+
+
(J × B )2 = µe H 02 ⎜
cos
sin
2
sin 2 φ ⎟ .
2
2
∂X ∂Z
∂Z
⎝ ∂X
⎠
(14)
Using a = (a1, 0, a3) into eq. (5), the non-zero stress components are obtained as
τ 23 = µT
∂V2
∂V ⎞
⎛ ∂V
+ ( µ L − µT ) ⎜ a32 2 + a1a3 2 ⎟ ,
∂Z
∂
∂X ⎠
Z
⎝
(15a)
τ 21 = µT
∂V2
∂V ⎞
⎛ ∂V
+ ( µ L − µT ) ⎜ a12 2 + a1a3 2 ⎟ .
∂X
∂Z ⎠
⎝ ∂X
(15b)
Using expressions given by eqs. (14) to (15b) into (1), we obtain
∂ 2V 2
⎡⎣ µ T + µ e H 02 sin 2 φ + a32 ( µ L − µ T ) ⎤⎦
∂Z 2
+ ⎡⎣ 2a1a3 ( µ L − µT ) + µ e H 02 sin 2φ ⎤⎦
∂ 2V2
∂X ∂Z
+ ⎡⎣ µT + µe H 02 cos 2 φ + a12 ( µ L − µT ) ⎤⎦
∂ 2V2
∂ 2V2
ρ
=
.
∂X 2
∂t 2
(16)
For time harmonic wave propagating in positive X-direction, the solution of eq. (16)
may be taken as
224
S. CHAUDHARY et al.
⎧
S ⎞ ⎤
S ⎞ ⎤⎫
⎡ ⎛
⎡
⎛
V2 = ⎨ A2 exp ⎢ik ⎜ γ 2 −
Z ⎥ + B2 exp ⎢ −ik ⎜ γ 2 +
Z exp ⎣⎡i (ω t − kX ) ⎦⎤ ,
⎟
2P ⎠ ⎦
2 P ⎟⎠ ⎥⎦ ⎭⎬
⎝
⎣ ⎝
⎣
⎩
(17)
where A2 and B2 are unknown and
γ2 =
⎞
1 ⎛ ρω 2
S2
+
−Q⎟ ,
⎜
2
2
P⎝ k
4P
⎠
P = µT + µe H 02 sin 2 φ + a32 ( µ L − µT ) ,
S = 2a1a3 ( µ L − µT ) + µe H 02 sin 2φ ,
(18a)
Q = µT + µ e H 02 cos 2 φ + a12 ( µ L − µT ) .
(18b)
Using eq. (17) into eq. (15a), the stress components are
⎡ ⎧ ⎛
⎡ ⎛
⎡
S ⎞
S ⎞ ⎤
S ⎞
S ⎞ ⎤⎫
⎛
⎛
τ 23 = ⎢ P ⎨ik ⎜ γ 2 −
A2 exp ⎢ik ⎜ γ 2 −
Z ⎥ − ik ⎜ γ 2 +
B2 exp ⎢ −ik ⎜ γ 2 +
Z
⎟
⎟
⎟
2P ⎠
2P ⎠ ⎦
2P ⎠
2 P ⎟⎠ ⎦⎥ ⎭⎬
⎝
⎝
⎢⎣ ⎩ ⎝
⎣ ⎝
⎣
−
⎡ ⎛
⎡ ⎛
ikS ⎧
S ⎞ ⎤
S ⎞ ⎤ ⎫⎤
M
A exp ⎢ik ⎜ γ 2 −
⎟ Z ⎥ + B2 exp ⎢ −ik ⎜ γ 2 + 2 P ⎟ Z ⎥ ⎬ ⎥ exp [i(ω t − kX )] − (τ 23 ) , (19)
2 ⎩⎨ 2
2
P
⎠ ⎦
⎠ ⎦ ⎭ ⎥⎦
⎣ ⎝
⎣ ⎝
∂V
⎛ ∂V2
⎞
cos φ + 2 sin φ ⎟ .
∂Z
⎝ ∂X
⎠
where (τ 23 ) M = µ e H 02 sin φ ⎜
Following Kaushik and Chopra (1982), the equation of motion governing
SH-type of motion in Mi (i = I, III) is given by
∇ 2 Vi + k β2 i Vi = 0 ,
(20)
where k β2 = ρi ω 2 µi and the complex displacement vector in medium MI and MIII is
i
given by
Vi = ( 0, Viy , 0 ) exp ( iω t ) .
(21)
The general solution of eq. (20) is of the form
Vi = ∑ Dij exp ( − A ij ⋅ r ) exp ⎡⎣i (ω t − Pij ⋅ r ) ⎤⎦ yˆ ,
2
(22)
j =1
where Dij are the complex amplitude, r is the position vector and ŷ is a unit vector in
the Y-direction. The propagation and attenuation vectors Pij and Aij are defined, respectively, by
Pij = ki R xˆ + (−1) j d β R zˆ ,
Aij = −ki1 xˆ + (−1) j +1 d β1 ẑ ,
(23)
where ki = kR + ik1 is the complex wave number, ẑ is a unit vector in the Z-direction,
subscripts R and I denote the real and imaginary parts and d β2i = k β2i − k i2 . The vectors Pij and Aij satisfy
225
REFLECTION/TRANSMISSION OF PLANE SH WAVE
Pij ⋅ Pij − A ij ⋅ A ij = Re k β2i ,
1
Pij ⋅ A ij = Pij ⋅ A ij cos γ ij = − I m k β2i .
(24)
2
From eqs. (24), we can write
Pij
⎧ ⎡
⎛
⎪1 ⎢
2
= ⎨ ⎢ Re k βi + ⎜ Re k β2i
⎜⎜
⎪2 ⎢
⎝
⎩ ⎣
A ij
⎧
⎡
⎛
⎪ 1⎢
= ⎨ − ⎢ Re k β2i + ⎜ Re k β2i
⎜⎜
⎪ 2⎢
⎝
⎣
⎩
(
(
)
2
)
+
2
(
I m k β2i
)
⎞
⎟
⎟⎟
⎠
2
cos γ ij
2
(I
+
2
m k βi
)
2
cos 2 γ ij
12
⎞
⎟
⎟⎟
⎠
12
⎤⎫
⎥⎪
⎥⎬
⎥⎦ ⎪
⎭
12
,
(25)
⎤⎫
⎥⎪
⎥⎬ ,
⎥⎦ ⎪
⎭
(26)
12
where γij is the angle between the propagation vector Pij and attenuation vector Aij.
3. SPECIFICATION OF THE PROBLEM
Since the SH wave is made incident at the interface Z = 0 obliquely in the medium MI,
the displacement in MI is given by
V1 = ⎡⎣ D11 exp ( − A11 ⋅ r ) exp ( i(ω t − P11 ⋅ r ) )
+ D12 exp ( − A12 ⋅ r ) exp ( i(ω t − P12 ⋅ r ) ) ⎤⎦ yˆ ,
(27)
where D11 is the amplitude of incident SH wave and D12 is that of reflected SH wave.
In medium MII , the displacement and stresses are given by eqs. (17) and (18b). In
the half-space MIII , the displacement is given by
V3 = D31 exp ( − A13 ⋅ r ) exp ( i(ω t − P13 ⋅ r ) ) yˆ ,
(28)
where D31 is the complex amplitude of the transmitted wave.
The reflection and transmission coefficients of SH wave will be determined by
the boundary conditions to be satisfied at the interfaces. These conditions are the continuity of displacement and stresses at the interfaces Z = −d and Z = 0. Mathematically, these boundary conditions are
V3 = V2 ,
⎡ ∂V3 ⎤
⎢ µ3 ∂Z ⎥ = [τ 23 ] II
⎣
⎦ III
at
Z = −d ,
(29a)
V1 = V2 ,
⎡ ∂V1 ⎤
⎢ µ1 ∂Z ⎥ = [τ 23 ] II
⎣
⎦I
at
Z = 0.
(29b)
Also, Snell’s law is given by
c =
β1
sin θ1
=
β2
sin θ 2
=
β3
sin θ 3
,
(30)
226
S. CHAUDHARY et al.
where β 22 = µT ρ and β1, β3 are the complex shear wave velocities in the visco-elastic
half-spaces. Following Borcherdt (1977), the quantities k1 and k2 obey the following
relation:
(31)
k = P11 sin θ1 − i A11 sin(θ1 − γ 1 ) = k1 = k 2 ,
where P11 and A11 are the propagation and the attenuation vectors in MI, and θ1 and γ1
are the angle of incidence and the angle between P11 and A11, respectively.
Using eqs. (17), (18b), (27) and (28) into the above boundary conditions given by
(29) and Snell’s law (30), we shall obtain the four simultaneous equations. From these
equations, one can obtain the following formulae for reflection and transmission coefficients:
R = ∆1 ∆ ,
where R = D12 D11 ,
T = ∆2 ∆ ,
(32)
T = D31 D11 ,
⎧⎪⎛
µ3 d β 3 ⎞⎛
k d2 ⎞
⎡
S ⎞ ⎤
⎛
∆ = ⎨⎜ d1 −
d
⎟⎜1 −
⎟ exp ⎢ −ik ⎜ γ 2 −
k
d
P ⎟⎠ ⎥⎦
2
µ
⎝
1 β1 ⎠
⎣
⎠⎝
⎩⎪⎝
µ3 d β3 ⎞ ⎛ k d1
⎞
⎛
⎡ ⎛
S ⎞ ⎤ ⎫⎪
+ ⎜ d2 +
+ 1⎟ exp ⎢ik ⎜ γ 2 +
d exp ⎡⎣ −id β3 d ⎤⎦ ,
⎜
⎟
k ⎠⎝ µ1 d β 1 ⎠
2 P ⎟⎠ ⎥⎦⎪⎬
⎣ ⎝
⎝
⎭
(33a)
⎧⎪⎛
µ3 d β 3 ⎞⎛
k d2 ⎞
S ⎞ ⎤
⎡
⎛
∆1 = ⎨⎜ d1 −
d
⎟⎜ 1 +
⎟ exp ⎢ −ik ⎜ γ 2 −
µ
k
d
P ⎟⎠ ⎥⎦
2
⎝
1 β1 ⎠
⎣
⎠⎝
⎩⎪⎝
µ3 d β3 ⎞ ⎛ k d1
⎞
⎛
S ⎞ ⎤⎪⎫
⎡ ⎛
+ ⎜ d2 +
⎟ ⎜ µ d − 1⎟ exp ⎢ik ⎜ γ 2 + 2 P ⎟ d ⎥ ⎬ exp ⎡⎣ −id β3 d ⎤⎦ ,
k
⎠ ⎦⎪⎭
⎣ ⎝
⎝
⎠⎝ 1 β 1 ⎠
⎛ S ⎞
∆ 2 = 2 ( d1 + d 2 ) exp ⎜ ik d ⎟ .
⎝ P ⎠
(33b)
(33c)
4. SPECIAL CASE
When the thickness of the layer is zero, the problem reduces to the problem of reflection and refraction of SH wave at a plane interface between two homogeneous viscoelastic half-spaces in welded contact. Thus, putting d = 0 into eqs. (32), the following
expressions for reflection and transmission coefficients are obtained:
R =
µ3 d β3 − µ1d β1
,
µ1d β1 + µ3 d β3
T =
2 µ1d β1
.
µ1d β1 + µ3 d β3
(34)
REFLECTION/TRANSMISSION OF PLANE SH WAVE
227
These are the same formulae for the reflection and transmission coefficients as obtained earlier by Borcherdt (1977) for the relevant problem.
5. NUMERICAL CALCULATION
For the purpose of numerical calculations, we introduce the following non-dimensional parameters:
Q1−1 =
µ1I
,
µ1R
Q3−1 =
µ3 I
,
µ3 R
ξ =
µ1R
,
µ3 R
where µ1 = µ1R +µ1I , µ3 = µ3R +µ3I , so that β iR2 = µiR ρi , (i = 1, 3), and Q1−1 and Q3−1
are the loss parameters in the medium MI and MIII , respectively.
The numerical computations have been performed using the following values of
non-dimensional material parameters: Q1 = 45, Q3 = 30, µ L µT = 2 , ρ3 ρ1 = 0.7,
ρ2 ρ1 = 0.9, µ1R µT = 1.6 , µ3 R µT = 0.5, c β1R = 0.5, c β3 R = 0.8 . With these values we have computed the absolute values of reflection and transmission coefficients.
Fig. 2. Variation of reflection
coefficient R versus angle of incidence θ. Curves 1−4 when kd = 0.1,
0.2, 0.3, 0.4, respectively; φ = 30o,
εH = 0.5, a1 = 0.4, a3 = 0.925.
Figure 2 shows the variation of reflection coefficient with angle of incidence θ1
(hereafter θ) for different values of kd. We notice that as kd takes the values 0.1, 0.2,
0.3, 0.4, the reflection coefficient increases for all values of angle of incidence, except
228
S. CHAUDHARY et al.
for few values of θ near the normal incidence. Near the normal incidence, the reflection coefficient fluctuates and the magnitude of fluctuation increases with increasing
value of kd. Also, in the neighbourhood of θ = 70o, the reflection coefficient is almost
independent of kd.
Figure 3 shows the variation of transmission coefficient with angle of incidence
for different values of kd. Note that at θ = 1o, 72o and 90o, the transmission coefficient
does not depend on kd. However, at θ = 1o and 90o, the transmission coefficient is
nearly zero. Note that the value of transmission coefficient has a jump near the normal
incidence; thereafter, as the angle of incidence increases, the value of transmission
coefficient increases. At θ = 72o, the transmission coefficient attains equal value for
different values of kd and thereafter it goes on decreasing towards zero as θ approaches 90o. This also shows that at grazing incidence, no transmitted wave take
place.
Fig. 3. Variation of transmission coefficient T versus
angle of incidence θ. Curves
1−4 when kd = 0.1, 0.2, 0.3,
0.4, respec-tively; φ = 30o,
εH = 0.5, a1 = 0.4, a3 = 0.925.
Figure 4 shows the variation of reflection coefficient with normalized wave
number kd for different values of φ when θ = 45o. It is clear that for kd = 0.001, the
reflection coefficient is independent of φ and has minimum value. It increases monotonically with the increase of kd. Also as φ increases, the reflection coefficient increases for all values of kd when θ = 45o.
REFLECTION/TRANSMISSION OF PLANE SH WAVE
229
Fig. 4. Variation of reflection
coefficient R versus normalized wave number kd when
o
θ = 45 , εH = 0.5. Curves 1−4
o
o
o
o
when φ = 0 , 30 , 60 , 90 ,
respectively.
Fig. 5. Variation of transmission coefficient T versus
normalized wave number kd
o
when θ = 45 , εH = 0.5.
o
Curves 1−4 when φ = 0 ,
o
o
o
30 , 60 , 90 , respectively.
230
S. CHAUDHARY et al.
Figure 5 shows the variation of transmission coefficient with normalized wave
number kd for different values of φ when θ = 45o. It can be observed that the transmission coefficient decreases monotonically with increasing kd and its value approaches
zero as kd approaches 10. However, the transmission coefficient is different for different values of kd at different values of φ.
Figure 6 shows the variation of reflection coefficient with angle φ for different
values of θ. It is clear from figure that at φ = 0o, the reflection coefficient increases as
θ increases through the values θ = 30o, 45o and 90o. For θ = 30o and 45o, the value of
reflection coefficient starts with 0.30 and 0.71, respectively, and then there is slight
decrease in its value with increasing φ. It is also noticed that when θ = 90o, the reflection coefficient first increases and then there is a sudden fall at φ = 22o. Thereafter, it
decreases slowly as φ increases.
Figure 7 shows the variation of transmission coefficient with φ for different values of angle of incidence. When θ = 30o, 45o and 90o, the value of transmission coefficient remains in the neighbourhood of value 0.53, 0.76 and 0.0, respectively, for all
values of φ running through 0o to 90o. This shows that the transmission coefficient is
not much influenced by the parameter φ.
Figure 8 shows the variation of reflection coefficient with angle of incidence for
different values of ∈H. We note that the reflection coefficient is influenced most by ∈H
at grazing incidence and its value becomes larger and larger as ∈H takes higher and
higher values.
Fig. 6. Variation of reflection coefficient R
o
versus angle φ. Curves 1, 2, 3 when θ = 30 ,
o
o
45 , 90 , respectively, and kd = 0.2, εH =
0.5, a1 = 0.4.
Fig. 7. Variation of transmission coefficient T versus angle φ. Curves 1, 2, 3
o
o
o
when θ = 30 , 45 , 90 , respectively, and
kd = 0.2, εH = 0.5, a1 = 0.4.
REFLECTION/TRANSMISSION OF PLANE SH WAVE
231
Fig. 8. Variation of reflection coefficient R versus angle of incidence θ. Curves 1−4 when εH =
o
0.05, 0.1, 0.5, 0.9, respectively, and kd = 0.2, φ = 30 , a1 = 0.4.
Fig. 9. Variation of transmission coefficient T versus angle of incidence θ. Curves 1−4 when
o
εH = 0.05, 0.1, 0.5, 0.9, respectively, and kd = 0.2, φ = 30 , a1 = 0.4.
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S. CHAUDHARY et al.
Figure 9 shows the variation of transmission coefficient with angle of incidence
for different values of ∈H. Here this coefficient increases with increasing value of ∈H.
Fig. 10. Variation of reflection
coefficients R versus εH. Curves
o
o
1, 2, 3 when (θ, φ) = (45 , 30 ),
o
o
o
o
(30 , 60 ), (45 , 45 ), respectively.
Fig. 11. Variation of transmission
coefficient T versus εH. Curves
o
o
1, 2, 3 when (θ, φ) = (45 , 30 ),
o
o
o
o
(30 , 60 ), (45 , 45 ), respectively.
REFLECTION/TRANSMISSION OF PLANE SH WAVE
233
Its value is found to be independent of ∈H at grazing incidence, where transmission
coefficient vanishes showing that no transmitted wave takes place there.
Figures 10 and 11 shown the variation of reflection and transmission coefficient
with ∈H for different sets of values of θ and φ. For each set of values, the reflection
coefficient shows reverse behaviour than that of transmission coefficient as ∈H goes
through the values 0.0 to 0.2.
6. REMARKS AND CONCLUSIONS
The phenomena of reflection and refraction of seismic plane waves in the layered
earth model has been of great importance in geophysics for exploration of various materials beneath the earth’s surface and to understand the earthquake processes. Studies
of the past earthquakes equipped with other geophysical data are helpful to find out
the location and size of the future earthquake. These estimates are also helpful to
earthquake engineers for developing earthquake-resistant structures. Reflection and
refraction seismology enables geophysicists to determine gross earth structures, specifically, the crustal and upper mantle structures. Moreover, the resolution obtainable
from reflection and refraction seismology is an important method being used by almost all oil exploration companies to map oil reservoirs. Seismic reflection/refraction
profiling and wide-angle reflection techniques are of great concern to geophysicists in
determining the seismic structure of the oceanic crust. The problem studied here will
be helpful in modelling the construction of the reinforced bedding of an earthquake
resistant design and dams to control the transmission of earthquake energy to the upper surface in order to avoid the destruction.
In conclusion, a mathematical study of the problem of SH wave propagation
through a self-reinforced layer sandwiched between two homogeneous visco-elastic
half-spaces is made. Theoretical results indicate that reflection and transmission coefficients depend upon the reinforcement parameters of the sandwiched layer, frequency
as well as on the incidence angle. The numerical study reveals that the reflection and
transmission coefficients exhibit the reverse behaviour with the increasing values of
reinforcement parameters of the sandwiched layer and the normalized wave number. It
is found that near the normal incidence both the reflection and transmission coefficients are slightly influenced at small values of kd and ∈H. On the other hand, both the
amplitudes ratios are greatly influenced by the increasing values of kd and ∈H .
A c k n o w l e d g e m e n t. One of the authors (Sushil Chaudhary) is grateful
to Kurukshetra University, Kurukshetra, for financial support in the form of University Research Scholarship for the completion of this work.
234
S. CHAUDHARY et al.
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Received 16 October 2002
Received corrected version 16 October 2003
Accepted 6 November 2003