International Journal of Applied Engineering Research ISSN 0973-4562 Volume 11, Number 10 (2016) pp 7119-7124
© Research India Publications. http://www.ripublication.com
Comparison and Analysis of Surface Waves Propagation in Initially
Stressed Dry Sandy Layer Using Conventional and Time Dependent Finite
Difference Scheme
Jayantika Pal and Anjana P. Ghorai
Department of Mathematics, Birla Institute of Technology,
Mesra-835215, Ranchi, Jharkhand, India.
any type of damage and destruction associates with
earthquakes. The surface waves propagating in different
media is of great importance to seismologists due to its
possible applications in geophysical prospecting, survey
techniques and reservoir engineering for understanding the
cause and estimation of damage due to natural and manmade
hazards. These waves provide seismologists most direct and
rich information regarding the structure of earth’s interior.
Surface wave which appear to travel along the earth’s surface
consist approximately of harmonic waves of varying
amplitudes and of progressively diminishing periods. There
are mainly two types of surface waves as Love waves and
Rayleigh waves. The existence of Rayleigh waves was
predicted by Lord Rayleigh, in 1885 and most of the shaking
felt from an earthquake is due to these waves. Love waves are
surface waves that cause circular shearing of the ground, the
particle motion during the passage of these waves is purely
horizontal and normal to the direction of wave propagation
and have the largest amplitude creating most of the damage
and destruction associate with earthquakes. In 1911, A. E. H.
Love, a British mathematician created a mathematical model
of these waves when there is a layered structure.
The layer of the soil in the earth is supposed to be more sandy
than elastic. A dry sandy mantle may be defined as a layer
consists of sandy particles retaining no moistures or water
vapours. Weiskopf has discussed the mechanics of dry sandy
soil [30] and it reveals that due to the slippage of granular
Abstract
In this reported manuscript, a mathematical modeling of
propagation of surface waves in dry sandy layer under initial
stress is presented. The equation of motion for the surface
wave has been formulated following Biot, using suitably
chosen boundary conditions. The dispersion equation of phase
velocity of this proposed layer has been derived using
conventional method of separation of variables and also by
using time dependent finite difference method of
approximation. The study reveals that initial stress and the
sandiness on the layer play important roles on the propagation
of surface waves. It is observed that with the increase of initial
stress parameter, the phase velocity of the surface wave
decreases, whereas it increases for the increase in the value of
the sandy parameter. The phase velocity curve has been
displays for different time-steps. Comparison between the
methods (conventional and time dependent finite difference
scheme) has been studied and presented graphically to show
the effects of these above mentioned parameters and it has
been shown that in every case the dispersion of phase velocity
curves are less dispersed in time dependent finite difference
method than in conventional method.
Keywords: Surface Waves, Dry Sandy Layer, Initial Stress,
Time-Space Domain, Finite Difference Method, Dispersion
Analysis and Phase Velocity.
particles in a dry sandy soil,
INTRODUCTION
Mathematical modeling plays an important role about how we
use the language of mathematics to express, communicate and
think about the real world. A model may help to explain a
system and to study the effects of different component and to
make predictions about behavior. Geologic modeling in the
applied science of creating computerized representations of
portions of the earth’s crust based on geophysical and
geological observations made on or below the earth surface.
There are two basic types of elastic waves or seismic waves
generated by an earthquake which is caused by sudden release
of energy in the earth’s crust; these are body waves and
surface waves. These wave causes shaking, damaging in
various ways. Traveling through the interior of the earth, body
waves arrive before the surface waves and are of higher
frequency than surface waves. In addition to the body waves,
mentioned above, there are surface waves which are of a
lower frequency than body waves. Although they arrive after
body waves then also they are almost entirely responsible for
E
 2(1   ) where E is the

Young’s modulus of elasticity, 
is the rigidity and  is the
Poisson’s ratio. Weiskopf suggested that the relationship
E

 2 (1   ) could be appropriate for the dry sandy soil,
where   1 is termed as sandy parameter and   1
corresponds to an elastic solid. Due to the factors like external
pressure, slow process creep, difference in temperature,
manufacturing processes etc. the media stay under high stress,
known as initial stress for which the layers of the earth exhibit
anisotropy, i.e.; deviates from the directionally regular elastic
behavior of an isotropic material. So it is more worthy to
study the propagation of surface waves in an initially stressed
dry sandy layer.
Much information about the propagation of seismic waves is
available in the well-known book by Ewing et al. [1].
Following Biot’s theory [2, 3] of Mechanics of Incremental
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International Journal of Applied Engineering Research ISSN 0973-4562 Volume 11, Number 10 (2016) pp 7119-7124
© Research India Publications. http://www.ripublication.com
truncated finite difference method for seismic modeling and
also considered a new time-space domain high-order
difference method for the acoustic wave equation. Dublain
[24] has demonstrated the advantages of the higher order
difference scheme. Liu and Sen [25] designed spatial finite
difference stencils on a time–space domain to simulate wave
propagation in acoustic vertically transversely isotropic
medium. Two-dimensional dispersion analysis and numerical
modeling demonstrate that this stencil has greater precision
than one used in a conventional FD. Liu and Xiucheng [26]
discussed finite difference numerical modeling in two phase
anisotropic media with even order accuracy. Zhu and
McMechan [27] developed finite difference modeling of the
seismic response of fluid saturated, porous, elastic solid using
Biot theory. Ghorai et al.[28] discussed higher order finite
difference method for modeling of Surface Waves in a fluid
saturated poro-elastic medium under initial stress. Liu and Sen
[29] developed a time–space domain dispersion-relation-based
staggered-grid finite-difference schemes for modeling the
scalar wave equation.
In the present paper, mathematical modeling, analysis and
studies of surface waves, propagating in initially stressed dry
sandy layer has been considered using conventional method
and time-space domain finite difference method as it has
remained less attempted. The dispersion equation of phase
velocity has been derived using suitably chosen boundary
conditions. The study reveals that initial stress and the
sandiness on the layer play important roles on the propagation
of surface waves. It is observed that with the increase of initial
stress parameter, the phase velocity of the surface wave
decreases, whereas it increases for the increase in the value of
the sandy parameter. The phase velocity curve has been
displays for different time-steps. Comparison between the
methods (conventional and time dependent finite difference
scheme) has been studied and presented graphically to show
the effects of these above mentioned parameters and it has
been shown that in every case the dispersion of phase velocity
curves are less dispersed in time dependent finite difference
method than in conventional method.
Deformations: Theory of Elasticity and Viscoelasticity of
Initially Stressed Solids and Fluids, including Thermodynamic
Foundations and Applications to Finite Strain, quite a good
amount of literature is available to analyze the effect of initial
stress on propagation of surface waves. In series of papers,
Dey et al.[4, 5], A.M.Abd-Alla et al. [6], Gupta et al. [7], Pal
et al.[8] have analyzed the effect of gravity and initial stresses
on the propagation of surface waves in a transversely isotropic
medium. Propagation of Love waves in an initially stressed
medium consisting of a slow elastic layer over a liquid–
saturated porous half-space with small porosity discussed by
Sharma et al.[9].
Many problems in the wave-field analysis are described by
parameters which depend on coordinates, time, etc. Such
systems are mathematically modeled by partial differential
equations. The theoretical study of wave propagation consists
of finding the solution of a partial differential equation or a
system of partial differential equations, under initial and
boundary conditions. The mathematical expression may
provide the bridge between modeling results and field
application.
Only a few of these equations can be solved by analytical
methods which are also complicated, require application of
advanced mathematical techniques. The finite difference
method is one of the most popular methods for numerically
solving seismic wave equations and has been used widely in
seismic modeling and migration since these methods are easy
to implement. This method depends upon how the temporal
and spatial derivatives in these equations are calculated. This
method requires relatively small memory and computation
time compared to other purely numerical methods. It specifies
the model at a series of grid points and approximates the
spatial and temporal derivatives by using the model values at
nearby grid points. A second order finite difference scheme is
usually used to approximate temporal derivatives which limit
the accuracy of modeling. A smaller time step or grid size
may increase the modeling accuracy but also increase the
computation time.
Virieux [10, 11] have used velocity-stress finite difference
method for the propagation of P-SV wave and SH wave in
heterogeneous media. To improve the accuracy and stability
of finite difference scheme, many authors have used and
developed different types of difference schemes. Levander
[12] applied 4th order approximation in space to the P-SV
scheme. Hayashi and burns [13] developed finite difference
scheme with variable grids. R.W. Graves[14] discussed the
simulation of seismic wave propagation in 3D elastic media
using staggered-grid finite difference method. Kristek and
Moczo [15] considered seismic wave propagation in viscoelastic media using 3D forth order staggered-grid finite
difference scheme. Saenger & Bohlen[16] have considered the
rotated staggered grid finite difference method. Bohlen and
Saenger [17] discussed the accuracy of heterogeneous
staggered-grid finite-difference modeling. Kristek and Moczo
[18] discussed on the accuracy of the finite-difference
schemes for the one dimensional elastic problem.
Tessmer[19]discussed Seismic finite difference modeling with
spatially variable time steps. Finkelstein and Kastner [20]
developed finite difference time domain dispersion reduction
schemes. Liu and Sen [21, 22, 23] discussed advanced and
FORMULATION OF THE PROBLEM
We have considered a model consisting of initially stressed
dry sandy layer of finite thickness h. To study the propagation
of surface waves, the cylindrical co-ordinates has been
considered where r and θ are the radial and circumferential
coordinates respectively and the z axis is taken positive
vertically downwards. The region h z 0is occupied by
the dry sandy layer and the wave is assumed to propagate
along the radial direction.
Dynamics of pre stressed dry sandy layer and its solution
As the surface waves propagate along the radial direction r
only, all material properties are independent of θ and
neglecting the body forces, the dynamic equation of motion
for the initially stressed dry sandy layer according to Biot
(1965) can be written as
 

2


2 

u
r

z





Pe
2
r


z

r 
zr 
z

t
7120
(1)
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 11, Number 10 (2016) pp 7119-7124
© Research India Publications. http://www.ripublication.com
Where P is the initial compressive stress along r and is the
c   k , the velocity of surface waves,
c1    , the shear wave velocity along the radial direction
in initially stressed sandy layer,
k = the wave number and
ξ  P 2, the non-dimensional initial stress parameter.
The solution of the equation (5) is given by
density of the medium.  r and  z are the incremental
stress components for dry sandy layer under initial stress and
u(ur,u,uz) displacement vector.
The displacement components are (independent of)
0
,
u
0
,
u
v
(
r
,
z
,
t
)
i.e. u
r
z

1
The non-zero stress components related to strain components
are given by
r 2Ner , z 2Nez
i
z

i
z
f2Ae

Be
where A and B are arbitrary constants.
And hence the solution of the equation (2) is

(7)
We have assumed the surface of the sandy layer is stress free,
which leads to the following boundary condition.
and the strain-displacement relation is given by
1
v
1 v1

1 v
e
1, e z 
and N   
(1b)
r
 
2
r r
2 z

where  , the sandy parameter and  , the shear modulus.
v1
0 at z  0 (stress free boundary)
z




v
1

v
v

v
v
P









(2)





r
r

r
r
2

z

t
 


11
2
2
1
2
A B0
2
1
2

cos

cos


v

J
kr

z

t
1
1




c
 1 
k 
1
2



c
 1
1 P ,

 2

2
where  is the angular frequency.
Then, the equation (2) is reduced to
 d 2 f1 (r ) 1 df1 (r ) f1 (r ) 

 2  f 2 ( z )
 
2
r
dr
d
r
r 

P
d 2 f 2 ( z)

    f1 (r )
   2 f1 (r ) f 2 ( z )  0 (3)
2
2
dz

Further simplifying, the equation (10) reduced to


2
2

  c 2

.
R
1
c
1h





cos
kh 

1
cos
kh


R


1
2 

c
P
r (11)
 1
 
1  kh
 c
1 
 

2




Introducing the non-dimensional parameter R  k r , the
equation (3) becomes
Where
 d 2 f1 1 df1 
1  

 1  2  f1  f 2
2
R dR  R  
dR
 k 2 


(10)
where
i
t
v
f1(r)f2(z)e
1
2
(9)
After simplifying the equation (7) reduced to
Solution using conventional method of separation of
variables
To solve the equation (2), let’s assume
Pd f

    22 f1    2   k 2 f1 f 2  0
2  dz

(8)
Using the boundary conditions (8) we have
Using (1a) and (1b) the equation (1) can be written as
2
1
2

i

z

i

z i

t
v

J
(
kr
)Ae

Be
e
1
1
(1a)
R1 
c1 t
.
h
Solution of the layer by Finite Difference approximation
method
Let’s revisited the dynamic equation of motion for the initially
stressed dry sandy layer as
(4)
The equation

d
f 1
df
 1

 1
1

f

0


1
2
d
RR
dR
R

We have considered here the 2nd order finite difference
scheme for spatial derivatives as
2
1
2

2v 1
 2 2v00,0 v10,0 v01,0 
2
r h


2v 1
0
0
0
 2 2v0,0 v0,1 v0,1 
2
z h

v 1 0

0
 v1,0 v1,0

r 2h

is Bessel’s equation of first kind with solution as
R
J1kr
.
f1J
1
Hence eq. (4) reduces to
d2f2
2

f2 0
2
dz
(5)

Where
2

1 c




1
2

k
c
 P
1


1
 


2




2
2
2



v
1

v
v

v
v
P








2
2
2
2 (2)



r

rr
r

z 
t
 2







(12)

As generally higher order finite difference on temporal
derivative scheme requires large space in the computer
(6)
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International Journal of Applied Engineering Research ISSN 0973-4562 Volume 11, Number 10 (2016) pp 7119-7124
© Research India Publications. http://www.ripublication.com
memory and usually unstable, 2nd order finite difference
scheme is used for temporal derivatives as

the increase of wave number, the phase velocity decreases
rapidly in each of these figures under the considered values of
various parameters. The phase velocity of Love wave verses
kh has been computed for different values of sandy

2

v1 0 1 
1


2
v

v

v
(13)
0
,
0
0
,
0
0
,
0
2
2

t 
n


v
r

mh
,z

jh
,t

n

, h is the grid size
Where v
m
,j
and  is the time step.
parameter Ita ( ) and
ξ
Using (12) and (13) into (2) we have
0
1
,
0
0
1
,0
0

1
,
0
0

1
,0
2
r
2
h
h


1 0 0 0
1 0   p





2
v
v
v



- 2 v0,0  +  
=
0
,
0
0
,
1
0
,

1
2
r
h
  2



1 0 1 

1


2
v

v

v


(14)
0
,
0
0
,
0
0
,
0
2



  



Using the plane wave theory, let us consider
i

k
(
r

mh
)

k
(
z

jh
)


(
t

n

)

n
r
z
v
e
m
,j
(15)
As the surface of the sandy layer is assumed to be stress free,
we have the boundary condition as follows:
velocity
v
 0 at z  0 (stress free boundary),
z

v 10 0
 v
0for finite
v
which reduce to
0
,
1
0
,
1

z2
h


parameter
due to the initial stress.
c
c1 increases at a particular wave number in both
conventional & finite difference method when other
parameters are fixed.
For  = 1, the medium turn out to be perfect elastic thereby
allowing surface waves to propagate with less velocity as
compared to other values of  (i.e., 1.5) for which the
medium becomes sandy.

difference scheme.
(16)
Substituting (15) into (14) and using (16), after simplifying,
we have
c
2
2

1


sin

R
2A

c
kh
R
1 
2
2
non-dimensional
Figure 1 and 1a has been plotted to understand the effect of
initial stress field on the propagation of surface wave velocity.
The value of non-dimensional initial stress parameter ξ has
been varied from 0.7 to.9 and other parameters are fixed. We
noticed that as the initial stress field increases in the abovementioned range, the phase velocity of surface waves in the
layer decreases significantly in both cases (conventional &
finite difference method).
Figure 2 and Figure 2a display the effect of sandy parameter
in the dispersion of phase velocity of surface waves under the
initial stress. The value of  has been taken as 1and 1.5. It is
observed that as the sandy parameter  increases, the phase
1
1

1



2
v
v
v
v 
v 

+  

0
0
,
0
P
the
(17)
2
 P
2
kh
kh

h




1

sin



,
Where A = sin


2
2
2
4
r



2
 
c
R2  1 .
h
2

NUMERICAL CALCULATIONS AND DISCUSSIONS
Based on the dispersion equations (11) & (17), numerical
results are provided to show the propagation characteristics of
surface waves in dry sandy layer under initial stress. In all the
Figure 1: (Conventional method): Variation of phase velocity
with kh for different values of ξ when  1 , t  0.0005and
c
figures, curves have been plotted as phase velocity
c1 along
vertical axis against dimensionless wave number kh along
horizontal axis. Range of kh is taken between the value of
k [1,0,1]
0.3 and 0.75. The numerical calculations have been done by
.0005
taking c1  3000
and time step t  0
with
k [1,0,1]. For all the plots of phase velocity (Figure 1 &
1a to Figure 2 &2a), it has been observed that the maximum
changes happened in phase velocity between kh = 0.3 to
kh =0.55. In order to get a better view of this range, more
number of data points are taken. It has been found that with
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International Journal of Applied Engineering Research ISSN 0973-4562 Volume 11, Number 10 (2016) pp 7119-7124
© Research India Publications. http://www.ripublication.com
motion have been formulated according to Biot. The presence
of initial stress and sandy parameter in the frequency equation
approve the effects of these parameters in the propagation of
surface waves in a dry sandy layer. We have observed that the
presence of initial stress allows surface waves to propagate
and with the increase of initial stress parameter, the phase
velocity decreases, whereas the sandy parameter has the
increasing effect in the propagation of surface waves. From
the graphs of the effects of these above mentioned parameters,
in every case, the dispersion of phase velocity curves are less
dispersed in time dependent finite difference method than in
conventional method.
So the time dependent finite difference method is more
trustworthy numerical method than conventional method.
Figure 1a. (Finite difference method): Variation of phase
velocity with kh for different values of ξ when
t  0.0005and k [1, 0,1]
 1 ,
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
Figure 2: (Conventional method): Variation of phase velocity
with kh for different values of
t  0.0005and

when ξ 
k [1,0,1]
0.8 ,
[7]
[8]
[9]
[10]
Figure 2a: (Finite difference method): Variation of phase
velocity with kh for different values of
t  0.0005and k [1, 0,1]

when ξ 
0.8 ,
[11]
CONCLUSION
A mathematical modeling of propagation of surface waves in
dry sandy layer with initial stress is presented. The equation of
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