influence of imperfect interface on ...influence of imperfect interface on reflection and...

INFLUENCE OF IMPERFECT INTERFACE ON REFLECTION AND

TRANSMISSION COEFFICIENTS OF PLANE WAVES BETWEEN

TWO DIFFERENT FLUID SATURATED POROUS HALF SPACES

Rajneesh Kumar1, Aseem Meglani

2, Sanjay Garg

3*

1Department of Mathematics, Kurukshetra University, Kurukshetra, Haryana, India 2Department of Mathematics, Chaudhary Devilal University, Sirsa, Haryana, India

3Department of Mathematics, R.K.S.D. (P.G.) College Kaithal, Haryana, India

*e-mail: [email protected]

Abstract. The present study is concerned with the reflection and transmission of plane waves at an imperfect interface between two different fluid saturated porous half spaces. The expression for the reflection and transmission coefficients which are the ratios of the amplitude of reflected and transmitted waves to the amplitude of incident waves are obtained for an imperfect boundary and deduced for normal stiffness, transverse stiffness. The variations of amplitude ratios with angle of incidence are depicted graphically. A particular case of reflection at the free surface in fluid saturated porous half spaces has been deduced and discussed. A special case of interest has also been deduced from the present investigation. It is found that amplitude ratios of the reflected and transmitted waves are affected by the stiffness of the media. 1. Introduction

The dynamic response of porous media is of great interest in various areas such as geophysics, soil-mechanics, civil engineering, petroleum engineering and environmental engineering. As most of the modem engineering structures are generally made up of multiphase porous continuum, the classical theory, which represents a fluid saturated porous medium as a single phase material, is inadequate to represent the mechanical behavior of such materials especially when the pores are filled with liquid. In this context the solid and liquid phases have different motions. Due to these different motions and different material properties and the complicated geometry of pore structures; the mechanical behavior of a fluid saturated porous medium is very complex and difficult. So from time to time, researchers have tried to overcome this difficulty and considerable work has been done in this regard.

Based on the work of von Terzaghi [1, 2], Biot [3] proposed a general theory of three-dimensional consolidation. Taking the compressibility of the soil into consideration, the water contained in the pores was taken to be incompressible. Biot [4, 5] developed the theory for the propagation of stress waves in porous elastic solids containing a compressible viscous fluid and demonstrated the existence of two types of compressional waves (a fast and a slow wave) along with one shear wave. Biot's model was broadly accepted and some of his results have been taken as standard references and the basis for subsequent analysis in acoustic, geophysics and other such fields.

Based on the work of Fillunger model [6] (which is further based on the concept of volume fractions combined with surface porosity coefficients), Bowen [7] and De Boer and

Materials Physics and Mechanics 17 (2013) 142-157 Received: August 28, 2013

© 2013, Institute of Problems of Mechanical Engineering

Ehlers [8, 9] developed and used another interesting theory in which all the constituents of a porous medium are assumed to be incompressible. There are reasonable grounds for the assumption that the constituents of many fluid saturated porous media are incompressible. For example, taking the composition of soil, the solid constituents are incompressible and liquid constituents, which are generally water or oils are also incompressible. Moreover in an empty porous solid as a case of classical theory, the change in the volume is due to the changes in porosity during the propagation of longitudinal waves. The assumption of incompressible constituents does not only meet the properties appearing in many branches of engineering practice, but it also avoids the introduction of many complicated material parameters as considered in the Biot theory. So this model meets the requirements of further scientific developments. Based on this theory de Boer and Ehlers [10] and recently, Kumar and Hundal [11-15] studied some problems of wave propagation in fluid saturated incompressible porous media.

In the study of propagation of seismic waves in liquid saturated porous solids, problem of reflection of seismic waves is useful both for its seismological interest and geophysical exploration such as hydrocarbons prospecting, ground water prospecting or mineral prospecting. Employing Biot’s theory, reflection of waves at free permeable boundary was studied by Deresiewicz [16] and Malla Reddy and Tajuddin [17]. Expression for reflection coefficients were obtained but neither numerical work was carried out nor discussed. Kumar and Deswal [18] studied the reflection of waves in micro-polar liquid-saturated porous solid for a free boundary and calculated only reflection coefficients. Tomar and Singh [19] studied the problem of transmission of longitudinal waves through a plane interface between two dissimilar porous elastic solid half spaces. Tomar and Arora [20] investigated the reflection and transmission of elastic waves at an elastic / porous solid saturated by two immiscible fluids. Tajuddin and Hussaini [21] studied the reflection of plane waves at boundaries of a liquid filled poroelastic half- space. However, no attempt has been made to study the problem of reflection and transmission of plane waves at the imperfect interface between two fluid saturated incompressible porous media.

Imperfect bounding considered in the present investigation is to mean that the stress components are continuous and small displacement field is not. The small vector difference in the displacement is assumed to depend linearly on the traction vector. Significant work has been done to describe the physical conditions on the interface by different mechanical boundary conditions by different investigators. Notable among them are Jones and Whitter [22], Murthy [23], Nayfeh and Nasar [24], Rokhlin et al. [25], Rokhlin [26].

In the present investigation, we studied the reflection and transmission of plane waves at an imperfect interface between two fluid saturated incompressible porous half spaces and deduced different cases. 2. Basic equations

Following De Boer and Ehlers [9], the equations governing the deformation of an incompressible porous medium saturated with non-viscous fluid in the absence of body forces are

( S Su + F

Fu ) =0, (1)

( S + S ) ( Su ) + S 2Su - S p - S

Su + S V ( Fu - Su ) =0, (2)

F p + F Fu +S V ( Fu - Su ) =0, (3)

S

ET = 2 S SE + S (E S

.I)I, (4)

143Influence of imperfect interface on reflection and transmission coefficients of plane waves...

SE = 21 (grad Su + grad T

Su ), (5)

where i iu , u , ui i = F,S denote the displacement, velocities and acceleration of fluid and solid

phases respectively and p is the effective pore pressure of the incompressible pore fluid; s and F are the densities of the solid and fluid respectively; S

ET is the stress in the solid phase and SE is the linearized langrangian strain tensor; s and s are the macroscopic Lame’s parameters of the porous solid and s and F are the volume fractions satisfying

s + F = 1. The case of isotropic permeability, the tensor S V describing the coupled interaction

between the solid and fluid is given by De Boer and Ehlers [9], as

S V = F

FRF

K

2)(I,

where FR is the effective specific weight of the fluid and FK is the Darcy’s permeabilitbility coefficient of the porous medium. 3. Formulation of the problem We consider two fluid saturated incompressible porous half spaces being in contact with each other at the plane surface which we designate as plane z = 0 of rectangular Cartesian coordinate system OXYZ. The Z- axis is taken downward pointing into the medium.

For two dimensional problem, we assume the displacement vector iu (i=F, S) as

iu =( ,0,iu iw ), where i=F,S. (6) Using equation (6) in equations (1) - (3) we obtain the following equations for fluid saturated incompressible porous medium as:

( S + S ) x

S

+ S 2 u S - Sx

p

- S 2

2

t

u S

+ VS

t

u

t

u SF

= 0, (7)

( S + S ) z

s

+ S 2 Sw - Sz

p

- S 2

2

t

wS

+ VS

t

w

t

w sF

= 0, (8)

x

pF

+ 2

2

t

uFF

+

t

u

t

uS

SF

V =0, (9)

z

pF

+ 2

2

t

wFF

+

t

w

t

wS

sF

V =0, (10)

tz

w

tx

u SSS

22

+

tz

w

tx

u FFF

22

=0, (11)

where S = z

w

x

u SS

)()( . We define the dimensionless quantities defined as:

144 Rajneesh Kumar, Aseem Meglani, Sanjay Garg

xC

x1

, z

Cz

1

, tt , Su =

1

2CE

SS

Su , sw = 1

2CE

SS

Sw ,

Fu =1

2CE

SS

Fu , Fw =1

2CE

SS

Fw , E

pp ,

31T =E

T31 , 33T =

E

T33 ,

nK = nSK

C

1, tK = tS

KC

1 . (12)

In these relations E is the Young’s modulus of the solid phase, is a constant having the dimensions of frequency, 1C is the velocity of a longitudinal wave propagating in a fluid saturated incompressible porous medium and is given by

1C =FSSF

SSF

22

2

)()()2()(

, (13)

If pore is absent or gas is filled in the pores then F is very small as compare to S and can be neglected so the relation reduce to

0C =S

SS

2 . (14)

This gives the velocity of the longitudinal wave propagating in an incompressible empty porous solid where the change in volume is due to the change in porosity and well known result of the classical theory of elasticity .In an incompressible non porous solid F 0, then (13) becomes 1C = 0 and physically acceptable as longitudinal wave cannot propagate in an incompressible medium. The displacement components iu and iw are related to the non dimensional potential i and

i as

iu =x

i

+z

i

, iw =xz

ii

. i = F, S. (15)

Making use of non dimensional quantities given by (12) in equations (7) – (11) and with the help of (15) we obtain the following equations determining pS F S F as:

S2 - 2

2

t

S

-t

S

F

2

2

)(=0, (16)

S

F

SF

, (17)

2)( F p-S

FS

21 2

2

t

S

- 2t

S

=0, (18)

S 22 - 2

22

1t

S

+

tt

SF 2 =0, (19)


2

22

1t

F

S

F

+

tt

SF 2 =0, (20)

where 1 =OC

C1 , = OC

0 , S

S

O

, 2 = 2

21

O

S

V

Cw

CS

.

4. Reflection and transmission of the waves

We consider a plane waves propagating through a medium 1M which we designate as the region Z > 0 and incident at the plane z = 0 and making an angle 0 with normal to the surface. Corresponding to each incident wave (longitudinal / transverse wave) we get two reflected waves in the medium 1M and two transmitted waves in medium 2M . We write all the variables without bar in the region Z > 0 (medium 1M ) and attach a bar to denote the variables in the region Z0 (medium 2M ) as shown in the Fig. 1 (Geometry of the problem). 1B M2 1A

2

1

X

M1 0 1

2 A1

A B1

Z Fig. 1. Geometry of the problem.

We assume the solution of the system of equations (16) - (20) in the form

( FSFS ,,, , p) = ( 11111 ,,,, pFSFS

) exp[i k (x sin - z cos ) – i t ], (21) where k is the wave number and is the complex circular frequency. Making the use of (21) in equations (16), (19) - (20) we obtain two quadratic equations in V given by V2 + AV + B = 0, (22)


V2 + CV + D = 0, (23)

where A = 22

)( Fk

i

, B = -1, C =

SF

S

ii

k

22

1

22

1

2 , D = 21

2

.

The velocities of the longitudinal waves are the roots of equation (22):

1V = 2

42 BAA ,

and the velocities of the transverse waves are the roots of equation (23):

2V = 2

42 DCC ,

where the upper sign correspond to the incident wave and lower sign correspond to the reflected wave. 5. Boundary conditions

We have considered two fluid saturated incompressible porous half spaces .Imperfect bounding considered here mean that the traction is continuous across the interface but the small displacements are assumed to depend linearly on the traction vector. Boundary conditions at the interface z = 0 are:

ST33 = Kn

Sw( - Sw ), (24)

)( 31S

T = Kt Su( )Su , (25)

)( 33 pTS = )( 33 pT

S , (26)

)( 31S

T = )( 31S

T , (27) where Kn, Kt denote normal and transverse stiffness of layer having dimension N/m3.

In view of (21) we assume the values of FSFS ,,, and p satisfying the boundary conditions for medium M1 and M2 as:

Medium M1:

{ ,, FS p} = {1, m1, m2}[A01 exp{i k1 (x sin 0 - z cos 0 ) – i 1 t }+ A1 exp{i k1 (x sin 1 +

z cos 1 ) – i 1 t }], (28)

{ FS , } = {1, m3}[B01 exp{i k2 (x sin 0 - z cos 0 ) – i 2 t }+ B1 exp{i k2 (x sin 2 +

z cos 2 ) – i 2 t }]. (29) Medium M2:

{ pFS ,, } = {1, 21,mm } [ 1A exp {i 1k (x sin 1 - z cos 1 ) – i 1 t }], (30)

{ FS , } = {1, 3m } [ 1B exp{i 2k (x sin 2 - z cos 2 ) – i 2 t }], (31) where


m1 = - F

S

, m2 = -

SF

SFS i

2

222

1 , m3 = SF

S

i

i

22

1

2

,

1m = - F

S

, 2m = -

SF

SFS i

2

222

1 , 3m = SF

S

i

i

22

1

2

.

and A01, B01 are amplitudes of the incident longitudinal and transverse waves respectively. A1, B1 and 1A , 1B are amplitudes of the corresponding reflected and transmitted waves respectively.

In order to satisfy the boundary conditions, the extension of the Snell’s law will be

0

0sinV

=

1

1sinV

=

2

2sinV

=

1

1sinV

=

2

2sinV

, (32)

where

1k 1V = 2k 2V = 1k 1V = 2k 2V = at z = 0. (33) For longitudinal wave,

0V = 1V , 10 . (34) For transverse wave,

0V = 2V , 20 . (35) Making the use of potentials given by equations (28) – (31) in boundary conditions (24) –(27) and using the equations (33) - (35), we get a system of four non- homogenous equations which can be written as

j

i

ij Za

4

1= Yi , (j = 1,2,3,4), (36)

where

a11 = (ik1 1cos ) l1, a12 = (-ik2 2sin ) l2, a13 = (i 1k 1cos ) l1+ ( r32

112

412 )cossin kr ,

a14 = (i 2k 2sin ) l1+( r4 – r3) ( 2222 )cossin k ,a21 = (ik1 1sin ) l2, a22 = (ik2 2cos ) l2,

a23 = -(i 1k 1sin ) l2 + (-2r52

111 )cossin k ,

a24 = (i 2k 2cos ) l2 + r5 ( 222

22

2 )sincos k ,

a31 = - ( r12

112

12 )cossin k - m2 , a32 = - ( r1 - 1) ( 2

222 )cossin k ,

a33 = ( r32

112

412 )cossin kr + 2m , a34 = ( r4 – r3) ( 2

222 )cossin k ,

a41 = (-2r22

111 )cossin k , a42 = r2 ( 222

22

2 )cossin k , a43 = (-2r52

111 )cossin k ,

a44 = r5 ( 222

22

2 )sincos k , and


r1 = SS

S

2, r2 = SS

S

2, r3 = SS

S

2, r4 = SS

SS

2

, r5 = SS

S

2, l1= r1Kn,

l2 = r1Kt, Z1= A

A1 , Z2= A

B1 , Z3= A

A1 , Z4= A

B1 . (37)

(i) For incident longitudinal wave : A = A01, B01 = 0, Y1 = a11, Y2 = - a21, Y3 = - a31, Y4 = a41.

(ii) For incident transverse wave: A = B0, A01 = 0, Y1 = - a12, Y2 = a 22, Y3 = a32, Y4 = - a42.

where Z1, Z2, are amplitude ratio’s of reflected longitudinal wave making an angle 1 and transverse wave making an angle 2 and Z3, Z4 are amplitudes ratio’s of the transmitted longitudinal wave making an angle 1 and transmitted transverse wave making an angle

2 with the normal to the surface. Case I. Normal stiffness:

Kn 0, Kt correspond to the case of normal stiffness and we obtain a system of four non- homogenous equations which can be written as given by (36) with the changed values of amn as:

a31 = (ik1 1sin ), a32 = (ik2 2cos ), a33 = -(i 1k 1sin ), a34 = (i 2k 2cos ).

Case II. Transverse stiffness:

Kt 0, Kn correspond to the case of transverse stiffness and we obtain a system of four non- homogenous equations which can be written as given by (36) with the changed values of amn as

a11 = (ik1 1cos ), a12 = (-ik2 2sin ), a13 = (i 1k 1cos ), a14 = (i 2k 2sin ).

Case III. Free surface:

Kt = 0, Kn = 0 correspond to the case of free surface and boundary conditions in this case reduces to

pTS33 = 0, (38)

S

T31 = 0, (39)

and we obtain a system of two non- homogenous equations which can be written as

j

i

ij Za

2

1= Yi, (j = 1,2), (40)

where a11, a12, a21, a22 are given by

a11 = - ( r12

102

02 )cossin k - m2 , a12 = - ( r1 - 1) ( 2

222 )cossin k ,

a21 = (-2r22

111 )cossin k , a22 = r2 ( 222

22

2 )cossin k . 6. Particular case. Reflection and transmission at the imperfect interface between fluid

saturated porous half space and elastic half space

If F is very small as compare to S and can be neglected then medium 2M reduces to elastic half space and we obtain a system of four non- homogenous equations which can be


written as given by (39) with the changed values of amn as:

a13 = (i 1k 1cos ) l1+ ( r62

112

712 )cossin kr ,

a14 = (i 2k 2sin ) l1+( r7 – r6) ( 2222 )cossin k ,

a23 = -(i 1k 1sin ) l2 + (-2r82

111 )cossin k ,

a24 = (i 2k 2cos ) l2 + r8 ( 222

22

2 )sincos k ,

a33 = (r62

112

712 )cossin kr , a34 = (r7 – r6) ( 2

222 )cossin k ,

a43 = (-2r82

111 )cossin k , a44 = r8 ( 222

22

2 )sincos k ,

r6 = SS

e

2, r7 =

SS

ee

2

, r8 = SS

e

2.

7. Numerical results and discussion

With the view of illustrating the theoretical results and for numerical discussion we take a model for which the values of the various physical parameters are taken from De Boer and Ehlers [10] as follows:

S = 0.67, F = 0.33, S =1.34 Mg / 3m , F 0.33 Mg / 3m ,

S = 5.5833 MN / 2m , FK = 0.01m /s, FR = 10.00KN / 3m , S = 8.3750 MN/ 2m ,

S = 0.6, F = 0.4, S =2.0 Mg / 3m , F 0.01 Mg / 3m , S = 4.2368 MN / 2m ,

FK = 0.02 m /s, FR = 9.00KN / 3m , S = 3.3272 MN/ 2m , e =2.65Mg / 3m ,

e = 2.238 MN / 2m , e = 2.238 MN / 2m .

A computer program has been developed and amplitude ratios of various reflected and transmitted waves has been computed. The variations of amplitude ratios for fluid saturated porous half spaces with stiffness (ST), with normal stiffness (NS) and with tangential stiffness (TS) have been shown by solid line, long dashed line and small dashed lines respectively. Solid lines without central symbols correspond to the case of fluid saturated porous half space (FS) whereas Solid lines with central symbols correspond to the case of elastic half space (ES).

The variations of amplitude ratios iZ (i = 1, 2, 3, 4) for ST, NS, and TS with angle of incidence for longitudinal wave (LW) and transverse wave (TW) are shown graphically in Figs. 1-8. Figures 9 -12 show the variations of amplitude ratios iZ (i = 1, 2) for longitudinal waves (LW) and transverse waves (TW) at the free surface .

7.1. Incident longitudinal wave. The values of amplitude ratios 1Z in case of ES are more

than the values for FS in all the three cases of ST, NS and TS in the range 000 900 . Its values in case of FS increase in the range 000 130 , 000 3528 , 000 9038 and decrease in the remaining range for ST and NS whereas its values decrease in the range

000 130 , 000 4533 and increase in the remaining range for TS. These variations are shown in Fig. 2.

The variations of amplitude ratios 2Z are shown in Fig. 3. The values of amplitude ratios

2Z in case of ES are less than the values of FS in all the three cases of ST, NS and TS in the


whole range 000 900 .

0

Fig. 2. Variation of amplitude ratios 1Z with angle of incidence for longitudinal wave.

0


Figure 4 depicts the behavior of variations of amplitude ratios 3Z for FS and ES.

Values of amplitude ratios 3Z for TS boundary are less than for ST and NS in both the cases

FS and ES. The values of amplitude ratios 3Z in case of ES are more than the values of FS

for ST and NS in the whole range 000 900 whereas in case TS these values are more in the range 000 430 and less in the range 000 9043 .


0


The values of amplitude ratios 4Z for FS increase in the range 000 230 , 000 5037 and decrease in the remaining range. The values of amplitude ratios 4Z in

case of ES increase in the range 000 200 then decrease in the remaining range. These variations are shown in Fig. 5.

0


7.2. Incident transverse wave. The values of amplitude ratios 1Z in case of FS have an

oscillatory behavior for TS in the whole range 000 900 and increase in the whole range for


ST and NS. The values of amplitude ratios 1Z in case of ES increase in the whole range for TS

and these values increase for ST and NS except in the range 000 3530 . These variations are shown in Fig. 6.

0

Fig. 6. Variation of amplitude ratios 1Z with angle of incidence for transverse wave.

It is evident from Fig. 7 that values of amplitude ratios 2Z in case of FS increase

gradually in the whole range 000 900 . The values of amplitude ratios 2Z in case of ES

decrease in the range 000 200 , 000 4032 and increase in the remaining range for ST and NS whereas these values increase in the whole range for TS.

0



The values of amplitude ratios 3Z in case of ES decrease in the range 000 3530 and increase in the remaining range for ST, NS and these values increase in the range

000 450 and decrease in the range 000 9045 for TS. The values of amplitude ratios

3Z in case of FS have an oscillatory behavior for TS in the range 000 900 and increase in the whole range for ST and NS. These variations are shown in Fig. 8.

0


Figure 9 shows that the values of amplitude ratios 4Z for FS are close to each other in all three cases of ST, NS and TS whereas for ES these values in case of TS are more than the values of ST and NS.

0



7.3. Incident longitudinal wave at the free surface. Figure 10 depicts the behavior of variations of amplitude ratios 1Z for FS and ES. Values of amplitude ratios in case of FS increase in

the range 000 370 , 000 9060 and decrease in the remaining range. The values of amplitude ratios 1Z in case of ES are constant in the whole range 000 900 .

0

Fig. 10. Variation of amplitude ratios 1Z with angle of incidence for longitudinal wave at the free surface 0z .

It is noticed from Fig. 11 that values of amplitude ratios 2Z in case of ES start with small

initial increase and oscillates in the range 000 3530 and decrease slowly in the remaining range. The values of amplitude ratios 2Z in case of FS decrease sharply in the range

000 150 and then decrease slowly in the remaining range.

0

Fig. 11. Variation of amplitude ratios 2Z with angle of incidence for longitudinal wave at the free surface 0z .


7.4 Incident transverse wave at the free surface. Figure 12 shows that the values of amplitude ratios 1Z for ES start with small initial increase then oscillate in the range

000 2015 and 000 4335 and decrease slowly in the remaining range whereas for FS its values increase monotonically in the whole range.

0

Fig. 12. Variations of amplitude ratios 1Z with angle of incidence for transverse wave at the free surface 0z .

It is observed from Fig. 13 that values of amplitude ratios 2Z in case of FS oscillates

about the value zero in the whole range 000 900 whereas for ES its values start with small initial increase then oscillate in the range 000 1915 and 000 4240 and decrease slowly in the remaining range .

0

Fig. 13. Variations of amplitude ratios 2Z with angle of incidence for transverse wave at the free surface 0z .


8. Conclusions

The analytic expressions of the amplitude ratios for various reflected and transmitted waves are obtained for an imperfect boundary between two different fluid saturated porous half spaces. Different types of special cases have been deduced and discussed. It is concluded that imperfect boundary and porosity have significant effect on amplitude ratios. The model presented in this paper is one of the more realistic forms of the earth models. It may be of some use in seismology, engineering and geophysics etc. References

[1] K. Von Terzaghi // Sitzungsber Akad. Wiss. (Wien), Math. – Naturwiss. Kl, Abt. IIa 132

(1923) 125. [2] K. Von Terzaghi, Erdbaumechanik auf Bodenphysikalischer Grundlage (Franz Deuticke,

Leipzig – Wien, 1925). [3] M.A. Biot // J. Appl. Phys. 12, Iss. 2 (1941) 144. [4] M.A. Biot // J. Acoust. Soc. Am. 28 (1956) 168. [5] M.A. Biot // J. Acoust. Soc. Am. 28 (1956) 179. [6] P. Fillunger // Osterr. Wochenschrift fur den Offent. Baudienst. Teile I-III (1913) 532. [7] R.M. Bowen // J. Int. J. Eng. Sci. 18 (1980) 1129. [8] R. De Boer, W. Ehlers // Acta Mechanica 83 (1990) 77. [9] R. De Boer, W. Ehlers // Int. J. Solids Struct. 26 (1990) 43. [10] R. De Boer, W. Ehlers, Zh. Liu // Arch. App. Mech. 63 (1993) 59. [11] R. Kumar, B.S. Hundal, In: Currents Trends in Industrial and Applied Mathematics,

ed. by P. Manchanda, K. Ahmad, A.H. Siddiqi (Anamya Publishers, New Delhi, 2002), p. 181.

[12] R. Kumar, B.S. Hundal // Indian J. Pure and Applied Math. 4 (2003) 51. [13] R. Kumar, B.S. Hundal // Bull. Allahabad Math. Soc. 18 (2003) 1. [14] R. Kumar, B.S. Hundal // Bull .Cal. Math. Soc. 96 (3) (2004) 179. [15] R. Kumar, B.S. Hundal // J. of Sound and Vibration 288 (2005) 361. [16] H. Deresiewicz // Bull. Seismol. Soc. Am. 50 (1960) 599. [17] P. Malla Reddy, M. Tajuddin // Int. J. Solids Struct. 37 (2000) 3439. [18] R. Kumar, S. Deswal // Indian J. Pure Appl. Math. 31(10) (2000) 1317. [19] S.K. Tomar, Jaswant Singh // Applied Mathematics and Computation 169 (2005) 671. [20] S.K. Tomar, Ashish Arora // Int. J. Solids Struct. 43 (2006) 1991. [21] M. Tajuddin, S.J. Hussaini // Journal of Applied Geophysics 58 (2006) 59. [22] J.P. Jones, J.S. Whittier // J. Appl. Mech. 34 (1967) 905. [23] G.S. Murthy // Phys. Earth and Planetary Interiors 11 (1975) 65. [24] A.H. Nayfeh, E.M. Nassar // J. Appl. Mech. 45 (1978) 822. [25] S.I. Rokhlin, M. Hefets, M. Rosen // J. Appl. Phys. 51 (1980) 3579. [26] S.I. Rokhlin, In: Adhesive Joints: Formulation, Characteristics and Testing, ed. by

K.L. Mittal (Plenum, New York, 1984), p. 307.


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