Applied Mathematics, 2011, 2, 145-154
doi:10.4236/am.2011.22017 Published Online February 2011 (http://www.SciRP.org/journal/am)
Copyright © 2011 SciRes. AM
145
G-Type Seismic Wave in Magnetoelastic Monoclinic Layer
Amares Chattopadhyay, Shishir Gupta, Abhishek K. Singh, Sanjeev A. Sahu
Department of Ap pl i e d Mat hematics, Indian School of Mines, Dhanbad, India
E-mail: amares.c@gmail.com
Received July 15, 201 0; revised October 28, 2010; accepted November 2, 2010
Abstract
This paper deals with the study of propagation of G type waves along the plane surface at the interface of
two different types of media. The upper medium is taken as monoclinic magnetoelastic layer whereas the
lower half-space is inhomogeneous isotropic. Dispersion equation and condition for maximum energy flow
near the surface are obtained in compact form. The dispersion equation is in assertion with the classical
Love-type wave equation for the isotropic case. Effect of magnetic field and inhomogeneity on phase veloc-
ity and variation of group velocity with scaled wave number has been depicted by means of graphs. It is ob-
served that inhomogenetity decreases phase velocity and the magnetic field has the favouring effect. A com-
parative study for the case of isotropic layer and monoclinic layer over the same isotropic inhomogeneous
half space has been made through graphs.
Keywords: G-Waves, Magnetoelastic, Dispersion Equation, Monoclinic, Transform Technique
1. Introduction
Horizontally polarized surface wave of shear type is
known as G-type wave after B. Gutenberg [1-2], who
established the existence of a low velocity layer in the
earth mantle. It has been studied by researchers that such
waves propagate with a group velocity of 4.4 km/s [3-5]
and since the group velocity of Love waves over the pe-
riod range from about 40 to 300 s is same, so on other
hand Love waves with long periods (60 to 300 s) may
also be recognized as G-waves. These waves are fol-
lowed by dispersed Love waves, especially for continen-
tal paths and they exhibit a transient pulse-like character
in record. A sequence of G-waves may be observed after
a large earthquake. An outstanding case of well devel-
oped G-type waves was provided by the earthquake in
Peru (January, 1960). Bath and Arroyo [6] presented the
result obtained from this earthquake, especially with re-
gard to absorption and velocity dispersion of G-waves.
Aki [7] discussed the generation and propagation of G-
waves from the Niigata earthquake of June 16, 1964.
Some other notable works in this field are done by Jef-
freys [8], Bhattacharya [9], Chattopadhyay [10], Haskell
[11] and others. Possibility of generation of G-waves in
different medium has been investigated by different au-
thors. Mal [12] studied the generation of G-waves taking
the medium to be isotropic. The wave propagation in
crystalline media plays a very interesting role in geophy-
sics and also in ultrasonic and signal processing. Chat-
topadhyay and Keshri [13] presented the low velocity
layer by assuming the law of va riation in the lower semi-
infinite anisotropic medium of two different media of
monoclinic symmetry. Recently, Chattopadhyay et al. [14]
discussed the dispersion of G-type waves in low velocity
layer. The variation for the half-space in elastic constants
and density reduces the equation of motion into Hill’s
equation with periodic coefficients which is solved by
the method given by Valeev [15]. Valeev considered a
certain class of system of linear differential equations
with periodic coefficients which have the property that,
by means of Laplace transformation, they may be con-
verted to a system of linear difference equations, which
in turn may be solved by the method of infinite determi-
nants. This method of solving Hill’s differential equation
has been successfully employed by Bhattacharya [9], Mal
[12] and Chattopadhyay [10].
Theoretical and experimental studies regarding to bet-
ter understanding of the real earth has led a need for more
realistic representation of the medium thro ugh which sei-
smic waves propagate. The pr op agatio n of seis mic wave s
is affected by the elastic properties of the layered materi-
als. Moreover, the materials of the layer might be mag-
netoelastic in nature and th e interplay of electromagnetic
field with the motion of a deformable solid has its impor-
tance in various fields of science and technology.
In the present paper we study of the propagation of G
A. CHATTOPADHYAY ET AL.
Copyright © 2011 SciRes. AM
146
type waves along the plane surface at the interface of two
different types of media. The upper medium is taken as
monoclinic magnetoelastic layer whereas the lower half-
space is inhomogeneous isotropic. Keeping terms up to
first order, the Laplace transform of the displacement is
obtained. Dispersion equation and condition for maxi-
mum energy flow near the surface are obtained in com-
pact form. The dispersion equation is in assertion with
the classical Love-type wave equation for the isotropic
case. Effect of magnetic field and inhomogeneity on
phase velocity and variation of group velocity with scaled
wave number has been depicted by means of graphs. It is
observed that inhomogenetity decreases phase velocity
and the magnetic field has the favouring effect. A com-
parative study for the case of isotropic layer and mono-
clinic layer over the same isotropic inhomogeneous half
space has bee n made through graphs.
The formulation part and solution part of the problem
has been dealt in Section 2. In the same section disper-
sion equation, the condition for maximum energy flow
near the surface and the expression for group velocity
have been obtained. Special cases for the dispersion
equation obtained in Section 2 are considered in Section
3. Section 4 deals with the numerical calculation and gra-
phical illustratio n for the problem. Finally, Section 5 con-
cludes the study.
2. Formulation and Solution of the Problem
We consider a monoclinic magnetoelastic layer of thick-
ness H lying over an inhomogeneous isotropic half-space.
The variation for half-space is taken in following manner


20
20
1cos
1cos
s
y
s
y
 
 


(1)
where
is small positive con stant and
s
is real depth
parameter. The axes Z and Y are taken horizontally and
vertically downwards respectively (Figure 1).
At first, we deduce the equation governing the propa-
gation of shear wave in monoclinic magnetoelastic crus-
tal layer.
The strain-displacement relations for monoclinic me-
dium are
123 4
56
,,, ,
,
uvwwv
SSS S
x
yzyz
uw vu
SS
zx xy



 
 
 
(2)
where ,,uvw are displacement components in the di-
rection x, y, z respectively, and

1, 2,, 6
i
Si are the
strain components.
Also, the stress-strain relation for a rotated y-cut quartz
Monoclinic
Magnetoela s t ic medi um
Z
Y
H
O
66 56 55
,,,CCC
20
1cos
s
y
 

20
1cos
s
y
 

Inhomogeneous
Isotropic half-space
Figure 1. Geometry of the problem.
plate, which exhibits monoclinic symmetry with x being
the diagonal axis are
111 1122133144
212122 223324 4
313 1232333344
414124 234344 4
555 556 6
656 5666
,
,
,
,
,
TCSCSCSCS
TCSCSCSCS
TCSCSCSCS
TCSCSCSCS
TCSCS
TCSCS


 



(3)
where
1, 2,, 6
i
Ti are the stress components and
1, 2,, 6
ij ji
CCi are the elastic constants.
Equations governing the propagation of small elastic
disturbances in a perfectly conducting monoclinic me-
dium having electromagnetic force JB (the Lorentz
force, J being the electric current density and B be-
ing the magnetic induction vector) as the only body for-
ces are



2
65
12
2
624 2
2
53
42
,
,
x
y
z
TT
Tu
JB
xyz t
TTT v
JB
xyz t
TT
Tw
JB
xyz t









(4)
where
is the density o f the layer.
For SH wave propagating in the z-direction and caus-
ing displacement in the x-direction only, we shall assume
that

,, ,0and0.uuyztvw x
 
(5)
Using Equations (2) and (5), the stress-strain relation
(3) becomes
A. CHATTOPADHYAY ET AL.
Copyright © 2011 SciRes. AM
147
1234
55556
65666
0,
,
.
TTTT
uu
TCC
zy
uu
TC C
zy







(6)
Using Equation (6) in Equation (4), the only non-vani-
shing equati on we h ave

222 2
6656 55
222
2.
x
uuu u
CCCJB
yz
yzt
 



(7)
The well known Maxwell’s equations governing the
electromagnetic field are
0, ,
with ,
e
t
t






B
BE HJ
u
BHJ EB
 
(8)
where E is the induced electric field, J is the current
density vector and magnetic field H includes both pri-
mary and induced magnetic fields. e
and
are the
induced permeability and conduction coefficient respec-
tively.
The linearized Maxwell’s stress tensor

0x
M
ij
due to
the magnetic field is given by


0x
M
ijeijj ikkij
HhHh Hh

.
Let

123
,, ,,,
H
HH uvwHu
and

123
,,
i
hhhh
where i
h is the change in the magnetic field. In writing
the above equations, we have neglected the displacement
current. From Equation (8), we get
2.
ett

 

 





Hu
HH (9)
In component form, Equation (9) can be written as
23
2
11
2
22
2
33
1,
1,
1.
e
e
e
uu
HH
Htt
H
tyz
HH
t
HH
t









 



(10)
For perfectly conducting medium i.e.
, the
Equation (10) beco mes
3
20,
H
H
tt

 (11)
and
23
1.
uu
HH
Htt
ty z







 
(12)
It is clear from Equation (11) that there is no perturba-
tion in 2
H
and 3
H
, however from Equation (12) there
may be perturbation in 1
H
. Therefore, taking small per-
turbation, say 1
h in 1
H
, we have 1011
H
Hh
,
202
H
H
and 303
H
H
, where

01 0203
,,HHH are
components of the initial magnetic field 0
H.
We can write
000
0,sin ,cosHH
H, where
00
HH and
is the angle at which the wave
crosses the m agneti c fi el d. Th us we have
10 0
,sin,coshH H
H (13)
We shall take initial value of 1
h as 10h
. Using
Equation (13) in Equation (12), we get
00
1
sin cos
.
uu
HH
htt
ty z








 
(14)
Integrating with respect to t, we get
10 0
sincos .
uu
hH H
yz


(15)
Considering

2
2
H

 

 HHHH
and
Equation (8), we get

2.
2
eH

 



JBH H (16)
In the component form Equation (16) can be written as

222
22 2
022
0and
sinsin 2cos.
yz
e
x
uuu
Hyz
yz
 



 




JB JB
JB
(17)
Using Equations (7) and (17), we find the equation of
motion for the magnetoelastic monoclinic medium in the
form
2222
6656 55
222
2.
uuuu
MMM
yz
yzt




(18)
where
2
66 66
2
55
5566 66
56
5666 66
1sin,
cos ,
cos sin
H
H
H
MC m
C
MC m
C
C
MC m
C









(19)
where 2
0
66
e
H
H
mC
is monoclinic-magnetoelastic cou-
pling parameter.
A. CHATTOPADHYAY ET AL.
Copyright © 2011 SciRes. AM
148
Hence, the equation of motion for the propagation of
shear wave in monoclinic magnetoelastic crustal layer is
2222
6656 55
222
2.
uuuu
MMM
yz
yzt




(20)
Now we consi der


11
,, ik zct
uyztUye
(21)
where k is wave number and c is wave velocity.
Substituting Equation (21) in Equation (20), we get

222
11
665655 1
220
dU dU
MikMkcMU
dy
dy

. (22)
Solution of Equation (22 ) is
 
2
1cos sin
y
Uy eAPyBPy
 (23)
where
2
2
2
5655 56
6666 6666
2,
ikMM kkM
P
MMMM






and ,
A
B
are constants.
Since the upper surface is stress free therefore,
11
6656 0
uu
MM
yz


at
y
H , gives
  
2
1 1666656
sin cos sin
2
y
UybeMPyH MPPyH ikMPyH


 




(24)
In lower inhomogeneous half-space the displacement
2,,uyzt satisfy the differential equation
 
2
222
000
2
1cos1cos 1cos
uuu
sysy sy
zzyy t
 









 (25)
we considered

22
,, ik zct
uyzt Uye
(26) Using (25) and (26), the equation of motion for lower
inhomogeneous medium may be written as
 
 
22
22 2
0222
22
22
0
2
22 2222
00 22
222
2
00
2222
0.
2222
isy
isy
dUdU dU
is
ekcUykUy dy
dy dy
dU dU
is
kckU yekcUykU ydy
dy










 




(27)
This is Hill’s different ial equation, which we will so lve
by the method given by Valeev [15]. We apply Laplace transform with respect to y, i.e. we multiply (27) by
p
y
e and integrate with respect to y from 0 to
, we get

 

 

2
222
022
22
2
0
0
2
22 2
022
22
2
0
0
222 2
0
22
20
0
2222
2222
0.
pisy
pisy
py
dU dU
is
ekcUykUy dy
dy
dy
dU dU
is
ekcUykUy dy
dy
dy
dU
ekckUydy
dy









 





 





(28)
Now the boundary conditions are
12
112
6656 2
11
66 56
(i)at 0
(ii)at 0
(iii)0 at.
uu y
uuu
MM y
yzy
uu
MM yH
yz


 




(29)
From (i) of Equation (29) we get

66
21 5666
0sincos
2
M
UbikM PHMPPH

 


(30)
Now we consi der

2
0
0
y
dU
qdy



and

(0)
20
1.


Then (ii) of Equation (29 ) gives
A. CHATTOPADHYAY ET AL.
Copyright © 2011 SciRes. AM
149


66
166 5666
0
2
66 66
566656 5666
0sincos
22
cos sinsincos
22
M
b
qMikM PHPMPH
MM
PikMPH PMPHikMikMPH PMPH




 
 







 
  

 
 


(31)
Now defining the Laplace transform of

2
Uy
as
 
2
0
py
F
peUydy
. (32)
Using the boundary conditions we obtain the following system of equations
 
 


2
22 2
0
0
2
22 2
0
0
22 12
222 2
222 2
is
kckpispis Fpis
is
kckpispis Fpis
pwFpp










 
(33)
where
 
122
22 2
0
0
10,10
and 1
Vq
wk c

  




(34)
To find
F
p from Equation (33), we replace p by
pism and divide throughout by

n
ism ,
0m
.
We then obtain the following infinite system of linear
algebraic equations in the quantities
F
pism,
0, 1, 2,m
 








 





 

2
22 2
0
0
22
2
22 2
0
0
12
111
222 2
111
222 2
n
n
n
n
is
ismkc kpismpismFpism
ismpismwFp ism
is
ismkckpis mpis mFpis m
ismp ism



 




 



(35)
where p may be c onside red as a para meter in th e coef-
ficients. It should be noted that in order not to have to
consider the special case 0m
separately, we include
Equation (33) in (35) by agreeing to regard

1
n
ism
when 0m. Solving the system of difference Equation
(35), we obtain
F
p as the ratio of two infinite de-
terminants, viz.

1
2
Fp
(36)
where
 

 

 

 
 

 

2212
22 222 2
0 0
0 0
112
2 2
22
12
0
22 22
22 22
0
nn
nn
ispis wispis
kc kkc k
P
is is
p isp isp isp is
is pisis pisw

 

 


 

 




 



  
 
 
 
  
A. CHATTOPADHYAY ET AL.
Copyright © 2011 SciRes. AM
150
and
 


  
  

222222
0
0
22 222 2
0 0
22
0 0
22 2
2
22 222
0
0
0
2222
22 22
22 22
02222
nn
nn
is
ispiswisk ckpp
kc kkc k
pw
is is
p isp isp isp is
is
isk ckppispisw


 

 




 



 




  



 


  
 
 
 

The first approximation of Equation (36) is



12
12
2222 022
2
11pBB
p
FP pw pwpw


 
 (37)
where

66
1 15666
sin cos
2
M
BbikMPHPMPH


 




and
66 66
2 16656665666
66
56 5666
sincoscos sin
22 2
sincos .
2
MM
Bb MikMPHPMPHPikMPHPMPH
M
ikMikMPH PMPH

 

 
 

 
 

 

 
 

 







The second approximation of Equation (36) is

3
4
Fp
(38)
where
 

 

 

 
 

 

2212
22 222 2
0 0
0 0
312
2 2
22
12
0
22 22
22 22
0
nn
nn
ispiswisp is
kc kkc k
P
is is
p isp ispisp is
is pisis pisw

 

 


 






 



and
 


   
  

222222
0
0
22 222 2
0 0
22
0 0
42 2
2
22 222
0
0
0
2222
22 22
22 22
02222
nn
nn
is
ispiswisk ckpp
kc kkc k
pw
is is
p isp isp isp is
is
isk ckppispisw


 

 




 



 


 

 



 


A. CHATTOPADHYAY ET AL.
Copyright © 2011 SciRes. AM
151
Neglecting the terms containing 2
an d hi gher powers , we g e t


 








 
22
22222
0
312 0
22
22
12
22
2222
0
12 0
222 2
222 2
nis
sppisw kckpispis
ppis wpis w
is
p ispiswkckpispis
 
 

 


  


and



22
222 2 2
4.
n
s
pwpiswpisw
Hence Equation (38) becomes
 




 




2
222 2
0
12
22 2
2220
2
2222
0
2
222 0
00
2
00
2
Vpisq
p
F
Pkckpisispis
pw pwpis w
Vpisq kckp isispis
pwpis w


 


 




(39)
Then

2
Uy will be obtained from the following in-
version formula:
 
21
2
ipy
i
Uy Fpedp


(40)
The residues 123
,,RRR at the poles pw
,
pwis, pwis are given respectively by
 
 
2
2
122
22
22
1
22 22
00
1
24
00 0
2
22
44
qwV w
Rwsw
qwV V
D
s
w
w
s
wsw




 













(41)
 




2
21
2
00
42
iswy
qwV Dwisw
i
Re
swwis

(42)
and
 




2
21
3
00
42
iswy
qwV Dwisw
Re
swwis


(43)
where
222
0
10
Dkck

Equations (41), (42) and (43) show the conditions for
a large amount of energy to be confined near the surface
are

2000wV q (44)
22
20ws (45)
and

2
000qwV. (46)
Equations (44) and (46) give

2
00qwV (47)
which finally gives the d ispersion relation as

0
66
1
tan .
w
PH PM
 (48)
We will consider only the positive sig n for further dis-
cussion.
Now Equation (45) gives

22
0
0
2
2
kck s

and hence we get the expression for group velocity as

222
2.
2
k
Gkks

(49)
It follows from Equation (49) that 2
G
, i.e. the
group velocity is less than the shear wav e velocity in the
upper mantle.
3. Particular cases
Case 1
When 0
th e Equation (48) reduces to
0
66
tan w
PH PM
(50)
which is the dispersion relation for the case when mono-
clinic magnetoelastic layer lying over an isotropic half-
space.
A. CHATTOPADHYAY ET AL.
Copyright © 2011 SciRes. AM
152
Case 2
When 55661 56
,0CCC
  the Equation (48) re-
duces to




2
1
2
22
02
2
1
2
12
tan 1
1sin
(1) 1
1sin1
1sin
H
HH
c
kH
c
c

 
 






(51)
where 2
0
1
e
H
H
.
Case 3
When 5566 1,CC
 560,C
0
H
m the Equa-
tion (48) reduces to



22
202
222
111
11
tan 11
c
c
kH c








(52)
which is the result obtained by Chatto padhyay et al. [14]
for isotropic layer lying over an inhomogeneous isotropic
half space.
Case 4
When 5566 1,CC
 56 0,C
0
H
m and 0
the Equation (48) reduces to
2
02
22
22
1
12
1
1
tan 1
1
c
c
kH c



 









(53)
which is the usual dispersion equation for Love wave
with 12
c
 (Chattopadhyay [10]).
4. Numerical Examples and Discussion
For the case of monoclinic magnetoelastic layer lying
over a non homogeneous isotropic half-space, we select
the following data:
1) For monoclinic magnetoelastic layer (Tiersten [16])
92 92
55 66
92 3
56
94.010 N/m,93.010 N/m,
11.010N/m,7450Kg/m .
CC
C
 
 
2) For non homogeneous isotropic half-space (Gub-
bins [17])
92 3
00
78.410N/m,3535Kg/m .


Moreover, the following data are used
0, 0.8;0,0.2,0.4
H
m

Figure 2 and Figure 3 represent the variation in di-
mensionless phase velocity 1
c
against dimensionless
wave number kH in monoclinic magnetoelastic layer,
for the case when magnetic field is absent and present
respectively. By the comparative study of these two
graphs we can conclude that presence of magnetic field
increases the phase velocity, whereas increment in
non-homogeneity parameter
decreases the phase
velocity. Figure 3 and Figure 4 represent the variation
in dimensionless phase velocity 1
c
against dimen-
sionless wave number kH in isotropic magnetoelastic
Figure 2. Variation of dimensionless phase velocity against
dimensionless wave number in an monoclinic magnetoelas-
tic layer over an inhomogeneous isotropic semi-infinite me-
dium when 0.0
H
m.
Figure 3. Variation of dimensionless phase velocity against
dimensionless wave number in an monoclinic magnetoelas-
tic layer over an inhomogeneous isotropic semi-infinite me-
dium when 0.8
H
m.
A. CHATTOPADHYAY ET AL.
Copyright © 2011 SciRes. AM
153
Figure 4. Variation of dimensionless phase velocity against
dimensionless wave number in an isotropic magnetoelastic
layer over an inhomogeneous isotropic semi-infinite me-
dium when 0.0
H
.
layer, for the case when magnetic field is absent and
present respectively. The presence of magnetic field in-
creases the phase velocity, whereas increase in the non-
homogeneity parameter
decreases the phase velocity.
It is observed from Figures 2-4 that presence of mono-
clinic medium favours more to the phase velocity as
compared to simply isotropic one. The magnetic field and
inhomogeneity parameter show similar type of tendency
for both the isotropic and monoclinic medium.
Figure 6 shows the variation in dimensionless group
velocity 2
G
with respect to scaled wave number ks.
This graph explains that group velocity increases with
scaled wave number and approaches to shear wave ve-
locity asymptotically. Keeping in the mind the depend-
ence of group velocity G on wave number k and depth
parameter s, surface plot of group velocity against vary-
ing k and
s
has been drawn in Figure 7.
For the case of magnetoelastic isotropic layer lying
over a non homogeneous isotropic half-space, we select
the following data:
3) For isotropic magnetoelastic layer (Gubbins [17])
92 3
11
63.410N/m,3364Kg/m .

 
4) For non homogeneous isotropic half-space (Gub-
bins [17])
92 3
00
78.410N/m,3535Kg/m .


Moreover, the following data are used
2
0
1
0,0.8;0,0.2,0.4
e
H
H

 
Figure 5. Variation of dimensionless phase velocity against
dimensionless wave number in an isotropic magnetoelastic
layer over an inhomogeneous isotropic semi-infinite me-
dium when 0.8
H
.
Figure 6. Variation of dimensionless group velocity
G
against scaled wave number ks.
5. Conclusions
Dispersion equation for the propagation of G-type seis-
mic wave in monoclinic magnetoelastic layer lying over
an inhomogeneous isotropic half space is obtained, using
the transform technique and Valeev’s method [15]. Con-
dition for maximum energy to be confined near the sur-
face and expression for group velocity are found. Phase
and group velocity curves against wave number are plot-
ted taking variation in inhomogeneity parameter. It is ob-
served that the presence of non-homogeneity decreases
the phase velocity whereas the presence of magnetic field
increases the phase velocity, in both the cases when there
A. CHATTOPADHYAY ET AL.
Copyright © 2011 SciRes. AM
154
Figure 7. Variation of group velocity

G with respect to
parameter k and s.
is isotropic magnetoelastic layer or monoclinic magneto-
elastic layer. It is also observed from the comparative
study that presence of monoclinic medium favours more
to the phase velocity as compared to simply isotropic one.
Group velocity is found lower than the shear wave ve-
locity in the upper mantle. The present study has its ap-
plication especially to the problem of waves and vibra-
tions where the wave signals have to travel through dif-
ferent layers of different material properties. This study
may be helpful to understand the cause of damages dur-
ing large earthquake s; also it may be useful to predict the
nature of long period Love waves. These results can also
be utilized in the interpretation and analysis of data of
geophysical studies. The findings will be useful in fore-
casting formation details at greater depth through signal
processing and seismic data analysis. The present study
may be effectively utilized to generate initial data prior
to exploitation of the formation. This stud y may be useful
to geophysicist and metallurgist for analysis of rock and
material structures through Non-Destructive Testing
(NDT).
6. Acknowledgements
The authors convey their sincere thanks to Indian School
of Mines, Dhanbad for providing JRF to Mr. Abhishek
Kr. Singh and also facilitating us with its best facility.
Acknowledgement is also due to DST, New Delhi for the
providing financial support through Project No. SR/S4/
MS: 436/07, Project title: “Wave propagation in anisot-
ropic media”.
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