Mathematical Modeling and Analysis of Torsional Surface Waves in a Transverse Isotropic Elastic Solid Semi-Infinite Medium with Varying Rigidity and Density under a Rigid Layer

Abstract

In this paper,
mathematical modeling of the propagation of torsional surface waves in a
transverse isotropic elastic medium with varying rigidity and density under a
rigid layer has been considered. The equation of motion has been formulated in
the elastic medium using suitable boundary conditions. The frequency equation
containing Whittaker’s function for phase velocity due to torsional surface
waves has been derived. The effect of rigid layer in the propagation of
torsional surface waves in a transverse isotropic elastic medium with varying
rigidity and density has been discussed. The numerical results have been
shown graphically. It is observed that the influence of transverse and
longitudinal rigidity and density of the medium have a remarkable effect on the
propagation of the torsional surface waves. Frequency equations have also been
derived for some particular cases, which are in perfect agreement with some
standard results.

Full text

Applied Mathematics, 2014, 5, 877-885
Published Online April 2014 in SciRes. http://www.scirp.org/journal/am
http://dx.doi.org/10.4236/am.2014.56083
How to cite this paper: Ghorai, A.P. and Tiwary, R. (2014) Mathematical Modeling and Analysis of Torsional Surface Waves
in a Transverse Isotropic Elastic Solid Semi-Infinite Medium with Varying Rigidity and Density under a Rigid Layer. Applied
Mathematics, 5, 877-885. http://dx.doi.org/10.4236/am.2014.56083
Mathematical Modeling and Analysis of
Torsional Surface Waves in a Transverse
Isotropic Elastic Solid Semi-Infinite Medium
with Varying Rigidity and Density under a
Rigid Layer
Anjana P. Ghorai1, Rekha Tiwary2
1Department of Applied Mathematics, BIT, Mesra, Ranchi, India
2Department of Mathematics, RVSCET, Jamshedpur, India
Email: rekha.tiwari68@gmail.com
Received 25 December 2013; revised 25 January 2014; accepted 5 February 2014
Copyright © 2014 by authors and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY).
http://creativecommons.org/licenses/by/4.0/
Abstract
In this paper, mathematical modeling of the propagation of torsional surface waves in a transverse
isotropic elastic medium with varying rigidity and density under a rigid layer has been considered.
The equation of motion has been formulated in the elastic medium using suitable boundary condi-
tions. The frequency equation containing Whittaker’s function for phase velocity due to torsional
surface waves has been derived. The effect of rigid layer in the propagation of torsional surface
waves in a transverse isotropic elastic medium with varying rigidity and density has been dis-
cussed. The numerical results have been shown graphically. It is observed that the influence of
transverse and longitudinal rigidity and density of the medium have a remarkable effect on the
propagation of the torsional surface waves. Frequency equations have also been derived for some
particular cases, which are in perfect agreement with some standard results.
Keywords
Torsional Surface Waves, Transverse Isotropic Elastic Medium, Phase Velocity
1. Introduction
Surface waves in elastic medium have been well recognized in the study of earthquake waves, seismology, geo-

A. P. Ghorai, R. Tiwary
878
physics and geodynamics. A good amount of work in respect of surface waves in classical elasticity is available
in the standard books of Bullen [1], Ewing et al. [2], Love [3] [4] and Stonely [5]. Much information is available
on the propagation of surface waves, such as Rayleigh waves, Love waves and Stonely waves but torsional sur-
face waves have not drawn much attention.
In seismogram, some disturbances are observed between the arrival of Rayleigh and Love wave disturbances.
As sufficient information was not available earlier, these disturbances were termed as “noise” and are ignored in
the study of seismic waves and these “noise” may be due to the torsional wave. A wave motion in which the vi-
brations of the medium are periodic rotational motions around the direction of propagation is known as torsional
wave. It gives twist to the medium during the propagation of earthquake thus producing torque in the medium
involve circumferential displacement which is independent of
θ
coordinate of cylindrical coordinate axes. But
only scanty of literature is available about torsional surface waves.
Lord Rayleigh [6], in his paper, showed that isotropic homogeneous elastic half space does not allow torsional
surface waves to propagate. Meissner [7] showed that torsional surface waves may propagate in an inhomoge-
neous elastic half space with quadratic variation of shear modulus and density varying with depth. Vardoulakis
[8] has observed that torsional surface waves also propagate in a Gibson half-space in which shear modulus va-
ries linearly with depth but the density remains unchanged. Georgiadis et al. [9] have studied tortional surface
waves in a gradient elastic halfspace. In a series of papers, Dey et al. [10]-[12] have studied torsional waves in
different media. They have discussed the existence and the propagation of torsional surface waves in an elastic
half-space with pores, in a homogeneous substrum over a heterogeneous half-space and in initially stressed ani-
sotropic porous media. The effect of irregularity on the propagation of torsional surface waves in a heterogene-
ous elastic half space has been studied by Selim [13] and concluded that the surface irregularity has a notable
effect on the propagation of torsional surface waves in heterogeneous medium with irregular free surface.
Propagation of torsional surface wave in anisotropic poroelastic medium under initial stress has been discussed
by Chattaraj et al. [14]. They observed that there is a significant effect of porosity, initial stress and inhomo-
geneity in the propagation of torsional surface wave in a layered anisotropic porous media under initial stress.
Gupta et al. [15] studied the propagation of torsional surface waves in a homogeneous layer of finite thickness
over a heterogeneous half space and observed that such a medium allows torsional surface wave to propagate.
Again Gupta et al. [16] studied the propagation of torsional surface waves in an initially stressed non-homoge-
neous layer over a non-homogeneous half space and observed that as the non-homogeneity parameter in the
layer as well as half space increases, the velocity of torsional surface wave also increases. It has also been ob-
served that an increase in compressive initial stresses decreases the velocity of torsional surface wave. Chatto-
padhyay et al. [17] studied the propagation of the same waves in a heterogeneous anisotropic half space under
the initial compressive stress. They found that the phase velocity of torsional waves decreases with increase of
initial stress and inhomogenity.
In this paper, we have discussed the effect of rigid layer in the propagation of torsional surface waves in a
transverse isotropic elastic medium with varying rigidity and density. It is found that in-homogeneity of rigidity
and density of the medium influence the velocity of the torsional surface wave. When the longitudinal and
transverse rigidity are the same, our result is similar to that of Dey et al. [18]. All results have been computed
and presented using MAT lab.
2. Formulation and Solution of the Problem
We consider a transversely isotropic elastic solid semi-infinite medium with varying rigidity and density under a
rigid layer. The constitute equation for transversely isotropic linear elastic material with preferred direction
a

is (A. J. M. Spencer [19])
( )
( )
( ) ( )
22
ij kk ij T ij k m km ij i j kk L T k i kj k j ki k m km i j
e e aae aae aae aae aae aa
τ λδ µ α δ µ µ β
= + + + +− + +
(1)
where
ij
τ
are the components of stress,
1
2
j
i
ij ji
u
u
exx


∂
∂

= +



∂∂


are the components of infinitesimal strain and
for homogeneous material,
T
µ
and
L
µ
are elastic shear modules in transverse and longitudinal shear respec-
tively,
λ
is elastic constant,
α
and
β
are reinforcement elastic coefficients,
( )
( )
123
,,
i
a aaa=
are the
components of
a

referred to the cylindrical coordinate system and
i
u
are the displacement vector compo-

A. P. Ghorai, R. Tiwary
879
nents. Taking the origin of cylindrical coordinate system at the interface of rigid layer and z-axis positive
downwards (Figure 1), we consider the following variations in rigidity and density:
( )
( )
( )
0
0
0
1
1
1
L
T
az
bz
cz
µµ
µµ
ρρ
= + 

= + 

= + 
(2)
where a, b, c are constants having dimensions of inverse of length.
It is assumed that torsional surface wave travels in the radial direction and all the mechanical properties asso-
ciated with it are independent of
θ
. So it is characterized by the displacements
( )
0, 0, , ,
rz
u u u vrzt
θ
= = =
(3)
and hence
0, 0, 0, 0,
2 ,2
rr zz zr
zr
ee e e
v vv
ee
z rr
θθ
θθ
= = = = 


∂∂
= = − 
∂∂

(4)
Using (3) and (4) in Equation (1) and setting
12 3
0, 1
aa a= = =
, we have the following non-zero stress com-
ponents
,
zL
rT
v
zvv
rr
θ
θ
τµ
τµ
∂
=
∂

∂


= −


∂


(5)
For the torsional surface wave motion in the radial direction, the equation of motion may be written as
( ) ( )
2
2
2
r zr v
r zr t
θ θθ
τ τ τρ
∂∂ ∂
+ +=
∂∂ ∂
(6)
Equation (6) with the help of (5) becomes
2 22
2 22 2
1
L
TL
v vv v v v
rr z z
r rz t
µ
µ µρ

∂
∂ ∂ ∂∂ ∂
+ −+ + =

∂ ∂∂
∂ ∂∂

(7)
For the wave propagating along the r-direction, we may assume the solution of (7) as
( ) ( )
1
e
it
v V z J kr
ω
=
(8)
where J1(kr) is the Bessel function of first kind.
Now substituting (8) into (7) and using (2) we have
2
21
2
2
11
10
1 11
c
a cz bz
V Vk V
az bz az
c

++
′′ ′
+ −− =

+ ++

(9)
Rigid layer
X
z = 0
Transversely isotropic elastic medium with
varying rigidity and density
Z
Figure 1. Geometry of the problem.

A. P. Ghorai, R. Tiwary
880
in which (') represents differentiation with respect to z,
1
ck
ω
=
and
12
0
20
c
µ
ρ

=

.
On substitution of
( ) ( )
( )
12
1
z
Vz az
φ
=+
, Equation (9) takes the form
2
21
2
2
11
10
1 11
c
a cz bz
k
az bz az
c
φφ


++
′′ + −− =


+ ++



(10)
Using the following dimensionless quantities
12
2
1
2
2
1c
bc
ab
c
γ


= −





and
( )
21k az a
ηγ
= +
,
in the Equation (10), we get
( )
2
11 0
24
4
P
φη φ
ηη

′′ + −+ =


(11)
where
( ) ( )
2
1
22
2
c
k
P ac ab
ac
γ

= −−−


(12)
Equation (11) is a standard Whittaker’s equation and whose solution is
( ) ( ) ( )
2,0 2,0PP
AW BW
φη η η
−
= +
where
( )
2,0P
W
η
and
( )
2,0P
W
η
−
are Whittaker functions.
As the lower medium is a half space, the solution should vanish at z→∝ i.e. for
η
→∝.
So in view of the above condition the solution may be taken as
( ) ( )
2,0P
AW
φη η
=
(13)
And hence we have displacement component as
( )( )
( ) ( )
2,0 1
12
21 e
1
Pit
AW k a az
v J kr
az
ω
γ
+


=+
(14)
3. Boundary Condition
At the interface, the displacement component vanishes i.e.
0 at 0.vz= =
(15)
Expanding the Whittaker function up to linear terms and substituting the boundary condition (15), we find the
velocity equation from (14) as
12
2 12
e1 0
2
ka
k Pk
aa
γ
γγ
−
−
  
+=
 
  
(16)
From which we have either
12
1
2
cb
cc

=

(17)
or,
22
42 242
11
42 2 4 2
22
1 21 1 23 1 23 0
cc
c c ba c a b a b
a aa a a a
cc k k k

     
−− − −+ − ++−+ − =

     
     

(18)
4. Observations
Case-1:
When c → 0 i.e. the medium is of constant density and rigidity components vary linearly with depth, the Equ-

A. P. Ghorai, R. Tiwary
881
ation (18) is reduced to
2
42 24 2
11
42 2 4 2
22
21 2 1 2 3 0
cc b aa ba b
a aa
cc k k k

   
− − + + +− + − =

   
   

(19)
Case-2:
When a = b the equation (18) reduced to
2
42
2 42
11
4 22 4 2
22
1 23 0
cc
c a c aa
aa
c ck k k

  
− − − +−=

  
  

(20)
which is similar to the result obtained by Dey et al. [18].
Case-3:
When a = b → 0 and c→ 0 i.e. the medium is of constant rigidity and density,
1
2
0
c
c→
i.e. torsional surface
wave does not propagate in such a medium.
5. Numerical Calculation and Discussions
The values of
1
2
c
c
have been computed from Equation (17) for
6,8,10
kc
=
for different values of
kb
and
shown in Figure 2.
The values of
1
2
c
c
have also been computed from Equation (18) for
1,1.6,2kb=
and
6,8,10kc=
for
different values of
ka
and shown in Figures 3-5.
When a = b i.e. longitudinal and transverse rigidity are same, the change of phase velocity
1
2
c
c
computed
from the Equation (20) with respect to
ka
for
6,8,10kc=
has been shown in Figure 6.
Figure 7 shows the change of phase velocity
1
2
c
c
with the changes in the longitudinal and transverse rigidity
for
8kc=
.
Figure 2. Torsional wave dispersion curve for different values of
kc
.

A. P. Ghorai, R. Tiwary
882
Figure 3. Torsional wave dispersion curve for different values of
kc
when
1kb=
.
Figure 4. Torsional wave dispersion curve for different values of
kc
when
1.6kb=
.
6. Conclusion
It has been found that there are two torsional wave fronts propagating in a transversely isotropic elastic solid
semi-infinite medium with varying rigidity and density under a rigid layer, one of which is shown in Figure 2
and the second one is given by Equation (18) [Figure 3-5]. It is also observed that as the density and the rigidity

A. P. Ghorai, R. Tiwary
883
Figure 5. Torsional wave dispersion curve for different values of
kc
when
2kb
=
.
Figure 6. Torsional wave dispersion curve for different values of
kc
when a = b.

A. P. Ghorai, R. Tiwary
884
Figure 7. Torsional wave dispersion curve for different values of
kb
when
8kc=
.
of the medium increases, the velocity of the torsional wave decreases. In the lower ranges of rigidity, the disper-
sions of torsional surface waves are less significant as compared to the higher ranges. Also from Figure 7, it has
been shown that the longitudinal and transverse rigidity has inverse effect in the propagation of torsional surface
waves in the medium of fixed density. Lastly when the medium is homogeneous and isotropic in nature, tor-
sional surface wave does not propagate there.
References
[1] Bullen, K.E. (1963) An Introduction to the Theory of Seismology. Cambridge University Press, London.
[2] Ewing, M., Jardetzky, W. and Press, F. (1957) Elastic Waves in Layered Media. McGraw-Hill, New York.
[3] Love, A.E.H. (1911) Some Problems of Geodynamics. Cambridge University Press, London.
[4] Love, A.E.H. (1944) A Treatise on the Mathematical Theory of Elasticity. 4th Edition, Dover Publication, New York.
[5] Stonely, R. (1934) Transmission of Love Waves in a Half-Space with a Surface Layer Whose Thickness Varies Hyper-
bolically. Bulletin of the Seismological Society of America, 54, 611.
[6] Rayleigh, L. (1885) On Wave Propagated along the Plane Surface of an Elastic Solid. Proceedings London Mathe-
matical Society, 17, 4-11. http://dx.doi.org/10.1112/plms/s1-17.1.4
[7] Meissner, E. (1921) Elastic Oberflanchenwellen MIT Dispersion in Einem Inhomogeneous Medium. Viertelgahrsschr
Naturforsch. Gses. Zurich, 66, 181-195.
[8] Vardoulakis, I. (1984) Torsional Surface Wave in Inhomogeneous Elastic Medium. International Journal for Numeri-
cal and Analytical Methods in Geomechanics, 8, 287-296. http://dx.doi.org/10.1002/nag.1610080306
[9] Georgiadis, H.G., Vardoulakis, I. and Lykotrafitis, G. (2000) Tortional Surface Waves in a Gradient-Elastic Half-Space.
Wave Motion, 31, 333-348. http://dx.doi.org/10.1016/S0165-2125(99)00035-9
[10] Dey, S., Gupta, S. and Gupta, A.K. (1993) Torsional Surface Waves in an Elastic Half-Space with Void Pores. Interna-
tional Journal for Numerical and Analytical Methods in Geomechanics, 17, 197-204.
http://dx.doi.org/10.1002/nag.1610170305
[11] Dey, S., Gupta, A.K. and Gupta, S. (1996) The Propagation of Torsional Surface Waves in a Homogeneous Substratum
over a Hetero-Geneous Half-Space. International Journal for Numerical and Analytical Methods in Geomechanics, 20,
287-294. http://dx.doi.org/10.1002/(SICI)1096-9853(199604)20:4<287::AID-NAG822>3.0.CO;2-2
[12] Dey, S. and Sarkar, M.G. (2002) Torsional Surface Waves in an Initially Stressed Anisotropic Porous Medium. Journal

A. P. Ghorai, R. Tiwary
885
of Engineering Mechanics, 128, 184-189. http://dx.doi.org/10.1061/(ASCE)0733-9399(2002)128:2(184)
[13] Selim, M.M. (2007) Propagation of Torsional Surface Waves in Heterogeneous Half-Space with Irregular Free Surface.
Applied Mathematical Sciences, 1, 1429-1437.
[14] Chattaraj, R., Samal, S. and Mahanti, N.C. (2011) Propagation of Torsional Surface Wave in Anisotropic Poroelastic
Medium under Initial Stress. Wave Motion, 48, 184-195. http://dx.doi.org/10.1016/j.wavemoti.2010.10.003
[15] Gupta, S., Chattopadhyay, A., Kundu, S.K. and Gupta, A.K. (2009) Propagation of Torsional Surface Waves in Gravi-
tating Anisotropic Porous Half-Space with Rigid Boundary. International Journal of Applied Mathematics and Me-
chanics, 6, 76-89.
[16] Gupta, S., Majhi, D.K. and Vishwakarma, S.K. (2012) Torsional Surface Wave Propagation in an Initially Stressed
Non-Homogeneous Layer over a Non-Homogeneous Half Space. Applied Mathematics and Computation, 219, 3209-
3218. http://dx.doi.org/10.1016/j.amc.2012.09.058
[17] Chattopadhyay, A., Gupta, S., Sahu, S.A. and Dhua, S. (2013) Torsional Surface Waves in Heterogeneous Anisotropic
Half-Space under Initial Stress. Archive of Applied Mechanics, 83, 357-366.
http://dx.doi.org/10.1007/s00419-012-0683-8
[18] Dey, S., Gupta, S., Gupta, A.K. and Prasad, A.M. (2001) Propagation of Torsional Surface Waves in a Heterogeneous
Half-Space under Rigid Layer. Acta Geophysica Polonica, XLLX, 113-118.
[19] Spencer, A.J.M. (1972) Deformation of Fiber-Reinforced Material. Oxford University Press, London.