Applied Mathematics and Computation 219 (2012) 3209–3218

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate/amc

Torsional surface wave propagation in an initially stressednon-homogeneous layer over a non-homogeneous half-space

Shishir Gupta, Dinesh Kumar Majhi, Sumit Kumar Vishwakarma ⇑Department of Applied Mathematics, Indian School of Mines, Dhanbad 826 004, India

a r t i c l e i n f o

Keywords:Torsional waveInitial stressHalf-spaceHomogeneous layer

0096-3003/$ - see front matter � 2012 Elsevier Inchttp://dx.doi.org/10.1016/j.amc.2012.09.058

⇑ Corresponding author.E-mail address: sumo.ism@gmail.com (S.K. Vishw

a b s t r a c t

It is of great interest to study torsional surface wave propagation in an initially stressednon-homogeneous layer over a non-homogeneous half-space. The method of separationof variables is applied to find the displacement field. It is well known in the literature thatthe earth medium is not at all initial stress free and homogeneous throughout, but it is ini-tially stressed and non-homogeneous. Keeping these things in mind, we have discussedpropagation of torsional surface wave in an initially stressed non-homogeneous layer overa non-homogeneous half-space. It has been observed that the inhomogeneity parameterand the initial stress play an important role for the propagation of torsional surface wave.It has been seen that as the non-homogeneity parameter in the layer increases, the velocityof torsional surface wave also increases. Similarly as the non-homogeneity parameter inthe half-space increases, the velocity of torsional surface wave increases. The initial stres-ses P present in the inhomogeneous layer also have effect in the velocity of propagation. Ithas been observed that an increase in compressive initial stresses decreases the velocity oftorsional surface wave.

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1. Introduction

The study of seismic waves gives important information about the layered earth structure and has been used to accu-rately determine the earthquake epicenter. Earthquakes generate waves on the grandest scale, with surface waves observa-ble after several trips around the world, and their systematic study has obvious implications for man’s safety, as well as forhis curiosity concerning the structure and evolution of the earth. Artificially generated seismic waves provide informationabout the configuration of rock layers for oil exploration and, on a smaller scale, information as to the rigidity of shallowlayers for engineering purposes. Properties of rocks penetrated by oil wells have been determined by observing seismicwaves at various depths, due either to a distant explosion or to a sound source nearby in the same well. Hence, the studyof the surface waves and their propagation in various media is of great geophysical significance. Thus, modeling of seismicwave propagation plays a significantly important role and is of great utility in the exploration of petroleum, earthquakedisaster prevention, civil engineering and signal processing. Ewing et al. [1] has given the basic literature on the propagationof elastic waves. A large number of papers have been published in different journals after the publication of this book. A de-tail study on elastic wave propagation and its generation in seismology had been made by Pujol [2] and Chapman [3]. Re-cently a lot of work has been done by many researchers in the field of wave propagation. Some of them are Singh [4],Ponnusamy and Rajagopal [5], Nayfesh and Abdelrahman [6], Tomar and Singh [7], Asfar and Hawwa [8] and Tomar [9].

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akarma).

3210 S. Gupta et al. / Applied Mathematics and Computation 219 (2012) 3209–3218

The problems related to initial stressed elastic medium has been a subject of continued interest due to its importance invarious fields, such as earthquake engineering, seismology and Geophysics. The development of initial stresses in the med-ium is due to many reasons, for example, resulting from the difference of temperature, process of quenching, shot peeningand cold working, slow process of creep, differential external forces, gravity variation etc. It is therefore of much interest tostudy the influence of these stresses on the propagation of surface waves. These stresses have a pronounced influence on thepropagation of waves as shown by Biot [10]. Biot showed that the acoustic wave propagation under initial stress was fun-damentally different from the stress free case and could not be represented by simply introducing into the classical theoryand the stress dependent elastic coefficients. In his treatment he considered the fluid as a particular case of an elastic med-ium under initial stress with rigidity zero. A detailed discussion based on this viewpoint is found in some of his remarkablepapers Biot [11–13]. Several authors have employed the theory of incremental deformation formulated by Biot [14] to studythe propagation of surface waves in pre-stressed elastic solids. Cauchy [15] used assumption that stress was due to centralforces between particles of the solid. Bromwich [16] examined the efforts of gravity on surface waves. The case of uniforminitial stress has been discussed by Southwell [17]. The equations for an incompressible solid under hydrostatic pressurehave been derived by Love [18].

A study of the effect of inhomogeneity on the propagation of surface waves provides an interesting field for the applica-tion of mathematical technique and in addition, is of practical importance to seismologists because in any realistic model ofthe earth there is a continuous change in the elastic properties of the material in the vertical direction giving rise toinhomogeneity.

Torsional surface wave is one kind of surface wave which is horizontally polarized but gives a twist to the medium whenit propagates. It has been confirmed that although homogeneous elastic half-space does not allow torsional surface waves topropagate but certain types of non-homogeneity in the layers allows it to propagate. Meissner [19] has shown that an inho-mogeneous elastic half-space with a quadratic variation of shear modulus and density varying linearly with depth, the tor-sional surface waves do exist. It was pointed out by Rayleigh [20] that an isotropic homogeneous elastic half-space doesn’tallow torsional surface waves to propagate. Vardoulakis [21] has studied that torsional surface waves also propagate inGibson’s half-space where the shear modulus varies linearly with depth but the density remains constant. Georgiadiset al. [22] have demonstrated that torsional surface waves do exist in a gradient elastic half-space. Selim [23] has discussedthe propagation of torsional surface waves in heterogeneous half-space with irregular free surface. The propagation of tor-sional surface waves in an elastic half-space with void pores has been studied by Dey et al. [24]. The propagation of torsionalsurface wave in an initially stressed cylinder has been discussed by Dey and Dutta [25]. The torsional wave propagation in atwo-layered circular cylinder with imperfect bond has been investigated by Bhattacharya [26]. Paul and Sarma [27] havediscussed the propagation of torsional wave in a finite piezoelectric cylindrical shell. Propagation of torsional surface wavesin a homogeneous layer of finite thickness over an initially stressed heterogeneous half-space has been studied by Guptaet al. [28]. Barton et al. [29] have studied nonlinear dynamics of torsional waves in a drill-string model with spatial extent.The earth is considered to be a layered elastic medium with variation in density and rigidity in constituent layers. Thereforethe torsional surface wave must propagate during earthquakes. The near surface of the earth consists of layers of differenttypes of material properties overlying a half-space of various types of rock, underground water, oil and gases. So, the studiesof the propagation of torsional surface waves will be of great interest to seismologists.

In the present paper torsional wave propagation in anisotropic initially stressed layer of sandstone over a non-homogeneous half-space has been studied. The inhomogeneity of the layer is taken into consideration by assumingN = N0 cosh2az, L = L0 cosh2az, q = q0 cosh2az and P = P0 cosh2az where N, L are directional rigidities, q is the density, P isthe compressive initial stress at any point in the layer which is assumed to be transversely isotropic with z-axis as the axisof symmetry and a is a constant having dimension that is inverse of length. The inhomogeneity of the half-space has beentaken along the z-direction. In the half-space polynomial variation in rigidity and density with depth has been considered.The velocities of torsional waves are obtained as complex ones, in which real part gives the phase velocity of propagation andcorresponding imaginary part gives the damping.

2. Formulation

For the study of torsional surface waves, a cylindrical co-ordinate system has been considered. The model consists of anon-homogeneous anisotropic layer of finite thickness H under compressive initial stress P along the radial direction andover an inhomogeneous elastic half-space. The interface is located at z = 0 and the z-axis is directed vertically downwardas shown in Fig. 1. N, L are the directional rigidities, q is the density and P is the compressive initial stress at any point inthe layer which is assumed to be transversely isotropic with z-axis as the axis of symmetry.

Consider the hyperbolic variation in elastic moduli, density and initial stress with depth z as

N ¼ N0 cosh2az

L ¼ L0 cosh2az

q ¼ q0 cosh2az

P ¼ P0 cosh2az

9>>>>=>>>>;

ð1Þ

Z

rO

( )( )

1

1

1

1

n

n

az

az

μ μ

ρ ρ

= +

= +

20

20

20

20

cos h

cosh

cosh

cosh

N N z

L L z

z

P P z

αα

ρ ρ αα

=

=

=

=

Fig. 1. Geometry of the problem.

S. Gupta et al. / Applied Mathematics and Computation 219 (2012) 3209–3218 3211

where a is the constant and having dimension that is inverse of length. The inhomogeneity of the half-space has been takenalong the z direction.

In the half-space following variation in rigidity and density has been considered

l ¼ l1ð1þ azÞn and q ¼ q1ð1þ azÞn ð2Þ

where a is constant and having dimension that is inverse of length and n is a positive integer. The wave is assumed to prop-agate along the radial direction.

3. Solution

3.1. Solution of pre-stressed inhomogeneous anisotropic layer

If r and h are the radial and circumferential co-ordinates respectively, the equation of motion for the initially stressedanisotropic layer is given by Biot [14]

@srh

@rþ @szh

@zþ 2

rsrh �

@

@zP2@m@z

� �¼ q

@2m@t2 ð3Þ

where v(r, z, t) is the displacement along the h direction; P is the initial compressive stress in the medium and q is the densityof the medium. srh and szh are the incremental stress components in the anisotropic elastic layer and given by srh ¼ Nð@m

@r � vrÞ

and szh ¼ L @m@z, where N and L are the rigidity of the medium along r and z directions respectively.

Using the above relations, Eq. (3) takes the form as

@2m@r2 þ

1r@m@r� v

r2

!þ 1

N@

@zG@m@z

� �¼ q

N@2m@t2 ð4Þ

where G ¼ L� P2 .

Introducing non-dimensional co-ordinates R = kr and f = zk + f0, where f0 is the constant and k the wave number, Eq. (4)takes the form

@2m@R2 þ

1R@m@R� m

R2

!þ 1

NdGdf

@m@fþ G

@2m@f2

!¼ q

Nk2

@2m@t2 ð5Þ

Assuming the solution of Eq. (5) as m ¼ m1ðfÞv2ðRÞeixt ; where x is the frequency, Eq. (5) reduces to

d2m2

dR2 þ1R

dm2

dRþ 1� 1

R2

� �m2

( )m1 þ

GN

d2m1

df2 þ1G

dGdf

dm1

dfþ qx2 � Nk2

Gk2 m1

!m2 ¼ 0 ð6Þ

The equation

d2m2

dR2 þ1R

dm2

dRþ 1� 1

R2

� �m2 ¼ 0

is the Bessel’s equation of the first kind with solution as v2 = J1(R), Eq. (6) takes the form

d2m1

df2 þ1G

dGdf

dm1

dfþ qx2 � Nk2

Gk2 m1 ¼ 0 ð7Þ

3212 S. Gupta et al. / Applied Mathematics and Computation 219 (2012) 3209–3218

Assuming m1 ¼VffiffiffiffiGp

Eq. (7) can be written as

d2V

df2 �1

2Gd2G

df2 �1

2GdGdf

� �2

þ NG

1� qc2

N

� �( )V ¼ 0 ð8Þ

where c ¼ xk is the phase velocity of the torsional surface waves in initial stress layer.

Using Eq. (1) in Eq. (6), it takes the form

d2V

df2 � k21V ¼ 0 ð9Þ

where

k21 ¼

a2

k2 þN0

L0ð1� n0Þð1� c2

c20

Þ

c0 ¼ffiffiffiffiN0q0

qis the shear wave velocity in the layer along the radial direction;

n0 ¼ P02L0

is the non-dimensional initial stress parameter.

The solution of Eq. (9) is given by

V ¼ A1e�k1f þ B1ek1f

Now, v1 ¼ VffiffiffiGp ¼ A1e�k1fþB1ek1fffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

G0cosh2ðakÞðf�f0Þp ; where G0 ¼ L0 � P0

2

Therefore the solution of Eq. (3) reduces to

m ¼ m0ðsayÞ ¼ A1e�k1f þ B1ek1fffiffiffiffiffiffiG0p

coshðakÞðf� f0ÞJ1ðRÞeixt ð10Þ

3.2. Solution of half-space

The dynamical equations of motion are

@rrr@r þ 1

r@rrh@h þ

@rrz@z þ

rrr�rhhr ¼ q @2u

@t2

@rrh@r þ 1

r@rhh@h þ

@rhz@z þ

2rrhr ¼ q @2m

@t2

@rrz@r þ 1

r@rhz@h þ

@rzz@z þ

rrzr ¼ q @2w

@t2

ð11Þ

where rrr, rhh, rzz, rrz, rrh and rhz are the respective stress components and u, v and w are the respective displacementcomponents.

The stress–strain relations are

rrr ¼ kXþ 2lerr; rhh ¼ kXþ 2lehh; rzz ¼ kXþ 2lezzrrh ¼ 2lerh; rrz ¼ 2lerz; rhz ¼ 2lehz ð12Þ

where k and l are Lame’s constants and X ¼ ð@u@r þ 1

r@m@hþ u

r þ @w@z Þ denotes the dilatation.

The strain–displacement relations are

err ¼@u@r; ehh ¼

1r@m@hþ u

r; ezz ¼

@w@z

; erh ¼1r@u@hþ @m@r� v

r; ehz ¼

@m@zþ 1

r@w@h

; ezr ¼@w@rþ @u@z

ð13Þ

Following the usual method for the problems having h symmetry it can easily be seen that 1st and 3rd equations of Eq.(11) are automatically satisfied as u = 0 w = 0. Therefore the torsional wave is characterized by the displacements

u ¼ 0; w ¼ 0; v ¼ vðr; z; tÞ ð14Þ

For torsional wave propagation in the radial direction, the equation of motion may be written as

@

@rrrh þ

@

@zrzh þ

2rrrh ¼ qðzÞ @

2m@t2 ð15Þ

with v(r, z, t) being the displacement along the h (azimuthal) direction.For an elastic medium the stresses are related to the displacement component v by

rrh ¼ lðzÞ @m@r� v

r

� �; rzh ¼ lðzÞ @m

@zð16Þ

S. Gupta et al. / Applied Mathematics and Computation 219 (2012) 3209–3218 3213

Using relation (16), Eq. (15) takes the form

@2m@r2 þ

@2m@z2 þ

1r@m@r� m

r2 þl0

l@m@z¼ q

l@2m@t2 ð17Þ

where l0 ¼ dldz

We assume a solution of (17) of the form

m ¼ VðzÞJ1ðkrÞeixt ð18Þ

Then Eq. (17) reduces to the form

V 00ðzÞ þ l0

lV 0ðzÞ � k2 1� c2

c2s

� �VðzÞ ¼ 0 ð19Þ

where x ¼ kc and cs ¼ffiffiffilq

qUsing Eqs. (2) and (19) takes the form

V 00ðzÞ þ na1þ az

V 0ðzÞ � k2 1� c2

c21

� �VðzÞ ¼ 0 ð20Þ

where c1 ¼ffiffiffiffil1q1

qputting VðzÞ ¼ /ðzÞ

ð1þazÞn=2 and m22 ¼ ð1� c2

c21Þ in (20) we get

/00ðzÞ þ k2m22

q

ð1þ azÞ2k2m22

� 1

" #/ðzÞ ¼ 0 ð21Þ

where q ¼ � n2 ðn2� 1Þa2

Using dimensionless quantitiesb ¼ km2

a ; p ¼ b2qk2m2

2

and g ¼ 2ðkm2zþ bÞ in Eq. (21) we get� �

/00ðgÞ þ �1

4þ p

g2 /ðgÞ ¼ 0 ð22Þ

which is a Whittaker equation, the solution of which is given by

/ðgÞ ¼ DW0;H1 ðgÞ þ D1W0;H1 ð�gÞ;

where D and D1 are arbitrary constants.The solution of Eq. (22) satisfying the condition limz!1VðzÞ ! 0 i.e. limg!1/ðgÞ ! 0 may be taken as

/ðgÞ ¼ DW0;H1 ðgÞ

where W0;H1 ðgÞ is the Whittaker function and H21 ¼ 1

4� pHence the displacement component v in the heterogeneous half-space is

m ¼ m1ðsayÞ ¼ DW0;H1 ½2ðkm2zþ bÞ�ð1þ azÞn=2 J1ðkrÞeixt ð23Þ

4. Boundary conditions

(i) at the upper boundary z = � H stress component L @m0@z ¼ 0

(ii) at the interface z ¼ 0; L @m0@z ¼ l1

@m1@z

(iii) at the interface z = 0, v0 = v1

From boundary condition (i) we have

A1A2 þ B1B2 þ 0:D ¼ 0 ð24Þ

where A2 ¼ e�k1ðf0�kHÞf�k1k cosh aH þ a sinh aHgB2 ¼ ek1ðf0�kHÞfk1k cosh aH þ a sinh aHg

From boundary condition (ii) we have

�A1L0k1ke�k1f0 þ B1L0k1kek1f0 � l1e�km2

a

ffiffiffiffiffiffiG0

pMD ¼ 0 ð25Þ

where M ¼ q2km2� f1� q

2akm2g � ðkm2 þ na

2 Þ and b ¼ km2a

From boundary condition (iii) we have

3214 S. Gupta et al. / Applied Mathematics and Computation 219 (2012) 3209–3218

A1e�k1f0 þ B1ek1f0 � ND ¼ 0 ð26Þ

where N ¼ffiffiffiffiffiffiG0p

e�km2

a f1� q2akm2g

Eliminating A1; B1 and D from (24)–(26) we get

A2 B2 0

�L0k1ke�k1f0 L0k1kek1f0 �l1e�km2

affiffiffiffiffiffiG0p

Me�k1f0 ek1f0 �N

�������������� ¼ 0

Expanding the above determinant, we get

tan

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiN0

L0ð1� n0Þðc

2

c20

� 1Þ � a2

k2

s� kH

" #¼ �iK1

K2ð27Þ

where K1 ¼ 2 ak B0C0 sinh aH þ 2C0 l1

L0� n

4 ðn2� 1Þ a2

k2ffiffiffiffiffiffiffiffi1�c2

c21

q � B0

8<:

ffiffiffiffiffiffiffiffiffiffiffiffi1� c2

c21

qþ n

2ak

� �g coshaH; K2 ¼ 2B0C0

2cosh aH � a

kl1L0

n2

n2� 1�

a2

k2ffiffiffiffiffiffiffiffi1�c2

c21

q sinhaH � 2B0 l1L0

ak

ffiffiffiffiffiffiffiffiffiffiffiffi1� c2

c21

qþ n

2ak

� �sinh aH,

B0 ¼ 1þ n2

n2� 1��

a

2kffiffiffiffiffiffiffiffi1�c2

c21

qand C 0 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2

k2 þ N0L0ð1�n0Þ

1� c2

c20

�r

Ignoring the damping term and equating real part from Eq. (27) we get

tan

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiN0

L0ð1� n0Þðc

2

c20

� 1Þ � a2

k2

s� kH

" #¼

l1R1R2L0

ak cosh aHðnþ 2Þ � 2R1R2a

k sinh aH � 2R1l1L0

ffiffiffiffiffiffiffiffiffiffiffiffic2

c21� 1

qcosh aH

2R21ð1þ R4Þ cosh aH � 2R4

l1L0

ak

ak sinh aH � 2l1

L0

ak ð1þ R4ÞR3 sinh aH

ð28Þ

where R1 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2

k2 þ N0L0ð1�n0Þ

1� c2

c20

�r

R2 ¼ n2

n2� 1�

a

2kffiffiffiffiffiffiffiffic2

c21

�1q

R3 ¼ffiffiffiffiffiffiffiffiffiffiffiffi1� c2

c21

qþ n

2 :ak

and R4 ¼ n2 ðn2� 1Þ a

2kffiffiffiffiffiffiffiffi1�c2

c21

q

Eq. (28) is the dispersion equation of the torsional surface wave in an initially stressed non-homogeneous layer over a

non-homogeneous half-space.

5. Particular cases

5.1. Case1

If a ? 0, a ? 0, P ? 0 i.e. n0 ? 0, N0 = L0 then Eq. (27) reduces to

tan

ffiffiffiffiffiffiffiffiffiffiffiffiffic2

c20

� 1

skH

" #¼ l1

L0

ffiffiffiffiffiffiffiffiffiffiffiffi1� c2

c21

qffiffiffiffiffiffiffiffiffiffiffiffic2

c20� 1

q

which is the well-known classical equation of Love wave, it is interesting to note that in the homogeneous layer over isotro-pic homogeneous half-space torsional wave changes to Love wave mode.

5.2. Case2:

When the layer is free from initial stress i.e. P ? 0 i.e. n0 ? 0 then Eq. (28) reduces to

tan

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiN0

L0

c2

c20

� 1� �

� a2

k2

s� kH

" #¼

l1R1R2L0

ak cosh aHðnþ 2Þ � 2R1R2a

k sinh aH � 2R1l1L0

ffiffiffiffiffiffiffiffiffiffiffiffic2

c21� 1

qcosh aH

2R21ð1þ R4Þ cosh aH � 2R4

l1L0

ak

ak sinh aH � 2l1

L0

ak ð1þ R4ÞR3 sinh aH

1.5 2 2.5 3 3.5 4 4.5 52

2.5

3

3.5

4

4.5

5

5.5

6

kH

c2/c

02

1234

Fig. 3. Torsional surface wave dispersion curve in the absence of initial stress for a=k ¼ 0:8; N0=L0 ¼ 0:9; a=k ¼ 0:4; n0 ¼ 0:0; l1=L0 ¼ 0:5;c2

0=c21 ¼ 0:1 and n ¼ 1 for curve1; n ¼ 2 for curve2; n ¼ 3 for curve3; n ¼ 4 for curve4.

3 3.5 4 4.5 5 5.51.6

1.65

1.7

1.75

1.8

1.85

1.9

1.95

kH

c2/c02

1234

Fig. 2. Dimensionless phase speed c2=c20 as function of dimensionless kH for a=k ¼ 0:8; N0=L0 ¼ 0:9; a=k ¼ 0:4; n0 ¼ 0:4; l1=L0 ¼ 0:5; c2

0=c21 ¼ 0:1

and n ¼ 1 for curve1; n ¼ 2 for curve2; n ¼ 3 for curve3; n ¼ 4 for curve4 .

S. Gupta et al. / Applied Mathematics and Computation 219 (2012) 3209–3218 3215

which is the dispersion equation of torsional surface wave in the absence of initial stress,

where R1 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2

k2 þN0

L01� c2

c20

� �s; R2 ¼

n2

n2� 1

� a

2kffiffiffiffiffiffiffiffiffiffiffiffic2

c21� 1

q ; R3 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffi1� c2

c21

sþ n

2� ak; R4 ¼

n2

n2� 1

� a

2kffiffiffiffiffiffiffiffiffiffiffiffi1� c2

c21

q

5.3. Case3:

When the layer is homogeneous and free from initial stress i.e. a ? 0, N0 = L0 and P ? 0 i.e. n0 ? 0 then Eq. (28) reduces to

tan

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic2

c20

� 1� �s

� kH

" #¼

l1R1R2L0

ak ðnþ 2Þ � 2R1l1

L0

ffiffiffiffiffiffiffiffiffiffiffiffic2

c21� 1

q2R2

1ð1þ R4Þ

which is the dispersion equation of torsional surface wave in homogeneous layer over a heterogeneous half-space in theabsence of initial stress,

1 1.5 2 2.5 3 3.5 4 4.5 51.5

2

2.5

3

3.5

4

4.5

5

5.5

6

kH

c2/c02

1

234

Fig. 4. Torsional wave dispersion curve in the presence of compressive initial stress (n0 > 0) for a=k ¼ 0:8; N0=L0 ¼ 0:9; a=k ¼ 0:4; n ¼ 4; l1=L0 ¼ 0:5;c2

0=c21 ¼ 0:1 and n0 ¼ 0:2 for curve1; n0 ¼ 0:4 for curve2; n0 ¼ 0:5 for curve3; n0 ¼ 0:6 for curve4.

1.5 2 2.5 3 3.5 4 4.5 52

2.5

3

3.5

4

4.5

5

5.5

6

kH

c2/c

02

1234

Fig. 5. Torsional surface wave dispersion curve in the presence of tensile initial stress (n0 < 0) for a=k ¼ 0:8; N0=L0 ¼ 0:9; a=k ¼ 0:4; n ¼ 3;l1=L0 ¼ 0:5; c2

0=c21 ¼ 0:1 and n0 ¼ 0:0 for curve1; n0 ¼ �0:1 for curve2; n0 ¼ �0:2 for curve3; n0 ¼ �0:3 for curve4.

3216 S. Gupta et al. / Applied Mathematics and Computation 219 (2012) 3209–3218

where R1 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffi1� c2

c20

s; R2 ¼

n2

n2� 1

� a

2kffiffiffiffiffiffiffiffiffiffiffiffic2

c21� 1

q ; R3 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffi1� c2

c21

sþ n

2� a

k; R4 ¼

n2

n2� 1

� a

2kffiffiffiffiffiffiffiffiffiffiffiffi1� c2

c21

q

6. Numerical computation and discussion

In order to show the effect of different values of n, initial stresses and non-homogeneity on the propagation of torsionalsurface waves, numerical computation of Eq. (28) were performed with different values of parameter representing the abovecharacteristic. The value of N0=L0; l1=L0 and c2

0=c21 have been taken as 0.9, 0.5 and 0.1 respectively in all the figures from

Fig. 2 to Fig. 7.In Fig. 2 we have plotted different curves for different values of n and it has been found that as n increases, the velocity of

torsional surface wave increases. In Fig. 2 the value of a/k, a/k and n0 have been taken as 0.8, 0.4 and 0.4 respectively whereasthe value of n for curve1, curve2, curve3 and curve4 have been taken as 1, 2, 3 and 4 respectively.

3 3.5 4 4.5 5 5.53

3.5

4

4.5

5

5.5

6

kH

c2/c02

12 3

4

Fig. 6. Dimensionless phase speed c2=c20 as function of dimensionless kH for a=k ¼ 0:4; N0=L0 ¼ 0:9; n ¼ 3; l1=L0 ¼ 0:5; n0 ¼ 0:5; c2

0=c21 ¼ 0:1 and

a=k ¼ 0:6 for curve1; a=k ¼ 0:7 for curve2; a=k ¼ 0:8 for curve3; a=k ¼ 0:9 for curve4.

3 3.5 4 4.51.6

1.65

1.7

1.75

1.8

1.85

kH

c2/c02

1234

Fig. 7. Dimensionless phase speed c2=c20 as function of dimensionless kH for a=k ¼ 0:8; N0=L0 ¼ 0:9; n ¼ 6; l1=L0 ¼ 0:5; n0 ¼ 0:5; c2

0=c21 ¼ 0:1

and a=k ¼ 0:01 for curve1; a=k ¼ 0:02 for curve2; a=k ¼ 0:05 for curve3; a=k ¼ 0:07 for curve4.

S. Gupta et al. / Applied Mathematics and Computation 219 (2012) 3209–3218 3217

Fig. 3 represents the dispersion curve in the absence of initial stress in the layer. In Fig. 3 the value of a/k, a/k and n0 havebeen taken as 0.8, 0.4 and 0.0 respectively whereas the value of n for curve1, curve2, curve3 and curve 4 have been taken as 1,2, 3 and 4 respectively. The figure shows that when the layer is free from initial stress and when the value of n increases, thevelocity of torsional surface wave increases.

Fig. 4 shows the effect of compressive initial stress (n0 > 0) in the non-homogeneous layer. In Fig. 4 the value of a/k, a/kand n have been taken as 0.8, 0.4 and 4 respectively whereas the compressive initial stress n0 for curve1, curve2, curve3 andcurve 4 have been taken as 0.2, 0.4, 0.5 and 0.6 respectively. It has been observed that an increase in compressive initialstress (n0 > 0) decreases the velocity of torsional surface wave for the same frequency.

Fig. 5 shows the effect of tensile initial stress (n0 < 0) in the non-homogeneous layer. In Fig. 5 the value of a/k, a/k and nhave been taken as 0.8, 0.4 and 3 respectively whereas the tensile initial stress n0 for curve1, curve2, curve3 and curve 4 havebeen taken as 0.0, �0.1, �0.2 and �0.3 respectively. In Fig. 3.3.5, it has been observed that decrease in tensile stress (n0 < 0)decreases the velocity of torsional surface wave for the same frequency.

Fig. 6 gives the dispersion curves for different values of non-homogeneity parameter in the initially stressed layer. InFig. 6 the value of a/k, n and n0 have been taken as 0.4, 3 and 0.5 respectively whereas the value of a/k for curve1, curve2,curve3 and curve 4 have been taken as 0.6, 0. 7, 0.8 and 0.9 respectively. The figure shows that as the non-homogeneityparameter in the layer increases, the velocity of torsional surface wave increases.

3218 S. Gupta et al. / Applied Mathematics and Computation 219 (2012) 3209–3218

Fig. 7 gives the dispersion curves for different values of non-homogeneity parameter in the half-space. In Fig. 7 the valueof a/k, n and n0 have been taken as 0.8, 6 and 0.5 respectively whereas the value of a/k for curve1, curve2, curve3 and curve 4have been taken as 0.01, 0.02, 0.05 and 0.07 respectively. In this case it is seen that as the non-homogeneity parameter in thehalf-space increases, the velocity of torsional surface wave also increases.

Acknowledgments

The authors convey their sincere thanks to Indian School of Mines, Dhanbad, for providing Senior research fellowMr. Dinesh Kumar Majhi and Mr. Sumit Kumar Vishwakarma and also facilitating us with best facility.

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