Research ArticlePropagation of Love-Type Wave in Porous Medium over anOrthotropic Semi-Infinite Medium with Rectangular Irregularity

Pramod Kumar Vaishnav, Santimoy Kundu, Shishir Gupta, and Anup Saha

Department of Applied Mathematics, Indian School of Mines, Dhanbad 826004, India

Correspondence should be addressed to Pramod Kumar Vaishnav; pvaishnav.ism@gmail.com

Received 4 November 2015; Revised 12 January 2016; Accepted 14 January 2016

Academic Editor: Evangelos J. Sapountzakis

Copyright © 2016 Pramod Kumar Vaishnav et al. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

Propagation of Love-type wave in an initially stressed porous medium over a semi-infinite orthotropic medium with the irregularinterface has been studied.Themethod of separation of variables has been adopted to get the dispersion relation of Love-type wave.The irregularity is assumed to be rectangular at the interface of the layer and half-space. Finally, the dispersion relation of Love wavehas been obtained in classical form.The presence of porosity, irregularity, and initial stress in the dispersion equation approves thesignificant effect of these parameters in the propagation of Love-type waves in porous medium bounded below by an orthotropichalf-space. The scientific effect of porosity, irregularity, and initial stress in the phase velocity of the Love-type wave propagationhas been studied and shown graphically.

1. Introduction

The Earth contains fluid-saturated porous rocks on or belowits surface in the form of sandstone and other sedimentspermeated by groundwater or oil; the diffusion of fluidand readjustment of fluid pressure have been acting as atriggering mechanism for earthquakes. So, the study of wavepropagation in a porous medium has gained prime interest.The propagation of Love-type wave in porous media withirregular boundary surfaces is important leading to betterunderstanding and prediction of behaviour of seismic wave atmountain roots, continental margins, and so forth. Love-typewave propagation in layered media has long been a researchsubject because of its practical importance in exploration ofoil, geophysics, earthquake engineering, and undergroundwater. The current work is concerned with the propagationof Love-type waves in initially stressed porous layer overlyingsemi-infinite orthotropic medium with irregular interface. Ithas been noticed that the presence of porosity, irregularity,and initial stress in the dispersion equation approves thesignificant effect of these parameters in the propagation ofLove-type waves.

The intended applications of this theory may be found inthe field of geophysics and the manufactured porous solids.

Various problems of waves and vibrations based on thesetheories of elasticity have been attempted by the researchersand have appeared in the open literature. Following Biot ([1–4]), the frequency equation has been used from the dynamictheory of wave propagation in fluid-saturated porous media.The effect of porosity, initial stress, and gravity has beendescribed by many researchers in several Earth structuresas when the porosity of the porous half-space increases, thephase velocity decreases, whereas the sandy parameter hasincreasing effect in the propagation of Love waves concludedby Pal and Ghorai [5]. Abo-Dahab et al. [6] discussed therotation and magnetic field effect on surface waves propaga-tion in an elastic layer lying over a generalized thermoelasticdiffusive half-space with imperfect boundary. Ahmed andAbo-Dahab [7] pointed out the propagation of Love wavesin an orthotropic granular layer under initial stress overlyinga semi-infinite granular medium. Chattaraj and Samal [8]discussed the effect of gravity, porosity in the Love wavesin the fibre-reinforced layer over a gravitating porous half-space. Chen et al. [9] discussed a mixture theory analysisfor the surface wave propagation in an unsaturated porousmedium. Kalyani et al. [10] pointed out the finite differencemodeling of seismic wave propagation in monoclinic media.

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016, Article ID 2081505, 9 pageshttp://dx.doi.org/10.1155/2016/2081505

2 Mathematical Problems in Engineering

Ghorai et al. [11] considered the Love waves in a fluid-saturated porous layer under a rigid boundary and lyingover an elastic half-space under gravity. Chattopadhyay et al.[12] concluded the effect of point source and heterogeneityon the propagation of SH waves in a viscoelastic layerover a viscoelastic half-space. Gupta et al. [13] establishedthe possibility of Love wave propagation in a porous layerunder the effect of linearly varying directional rigidities.Gupta et al. [14] provided the effect of initial stress onpropagation of Love waves in an anisotropic porous layer.Ke et al. [15] discussed the propagation of Love wavesin an inhomogeneous fluid-saturated porous layered half-space with properties varying exponentially. Kończak [16]displayed the propagation of Love waves in a fluid-saturatedporous anisotropic layer. Analysis of wave motion at theboundary surface of orthotropic thermoelastic material withvoids and isotropic elastic half-spacewas studied byR.Kumarand R. Kumar [17]. Liu and Boer [18] explained the disper-sion and attenuation of surface waves in a fluid-saturatedporous medium. Chakraborty and Dey [19] discussed thepropagation of Love waves in water-saturated soil underlainby a heterogeneous elastic medium. Sharma [20] investigatedthe wave propagation in a general anisotropic poroelasticmedium with anisotropic permeability: phase velocity andattenuation. Wang and Zhang [21] discussed the propagationof Love waves in a transversely isotropic fluid-saturatedporous layered half-space. Abd-Alla et al. ([22–26]) pointedout the propagation of Love wave, Rayleigh wave in variousstructures of the Earth. The authors used distinct medium ofthe Earth to propagate the seismic waves, such as orthotropic,magnetoelastic, fibre-reinforced anisotropic, viscoelastic, andmagnetothermoelastic medium. In such a medium, theyconcluded the effect of various parameters such as rotationand transmission, influence of initial stress on orthotropicmedium, and magnetic and gravity field on the propagationof seismic waves. Abd-alla and Abo-dahab [27] investigatedthe Rayleigh waves in magnetothermoviscoelastic solid withthermal relaxation times. Abo-Dahab et al. [28] concludedthe effect of magnetism and rotation on surface waves infibre-reinforced anisotropic general viscoelastic media ofhigher order.

In this paper, we use the porous medium (layer) over anorthotropic half-space with the effect of irregular interfacein the propagation of Love-type waves. The main attentionis paid to the influence of irregularity of interface, porosity,and initial stress on the propagation of Love-type wavesin porous-orthotropic medium (Figure 1). The rectangularirregularity at the interface of layered half-space affected thephase velocity of Love-type waves. The classical dispersionrelation of Love wave has been obtained in particular casesas Love [29]. To study the effect of porosity, initial stress, andirregularity, we represent numerical data from Gubbins [30].The study shows that the irregularity and porosity and initialstress have significant effect on the phase velocity of Love-type wave. For graphical representation, MATLAB softwarehas been used to generate results. The study of surface wavepropagation with irregular interface helps civil engineers inbuilding construction, analysis of earthquake in mountain

H

h

z

x

o−a a

Initially stressed porous medium

Initially stressed orthotropicsemi-infinite medium

Figure 1: Geometry of the problem.

roots, continental margins, and so forth. It is also useful forthe study of seismic waves generated by artificial explosions.

2. Mathematical Formulation of the Problem

We have considered a model consisting of initially stressedporous layer of finite thickness 𝐻 overlying an orthotropichalf-space with irregular interface. The rectangular irregularsurface has been taken at the interface of the layered-half-space model with length 2𝑎 and depth ℎ. 𝑥-axis is parallelto the direction of wave propagation, and 𝑧-axis is verticallydownward to the direction of wave propagation. The uppersurface of the porous layer is stress-free. The shape of theirregularity at the interface of the porous layer is taken as𝑧 = 𝜖𝑓(𝑥), where

𝑓 (𝑥) =

{

{

{

0; |𝑥| > 𝑎,

2𝑎; |𝑥| ≤ 𝑎,

𝜖 =ℎ

2𝑎≪ 1.

(1)

3. Solution of the Initially StressedPorous Layer

Neglecting the viscosity, in the absence of body forces,the dynamical equations of motion for initially stressedanisotropic porous medium can be written as Biot [4]:

𝜕𝑠11

𝜕𝑥+𝜕𝑠12

𝜕𝑦+𝜕𝑠13

𝜕𝑧− 𝑃1(𝜕𝑤𝑧

𝜕𝑦−

𝜕𝑤𝑦

𝜕𝑧)

=𝜕2

𝜕𝑡2(𝜌11𝑢1+ 𝜌12𝑈𝑥) ,

𝜕𝑠21

𝜕𝑥+𝜕𝑠22

𝜕𝑦+𝜕𝑠23

𝜕𝑧− 𝑃1(𝜕𝑤𝑧

𝜕𝑥)

=𝜕2

𝜕𝑡2(𝜌11V1+ 𝜌12𝑉𝑦) ,

Mathematical Problems in Engineering 3

𝜕𝑠31

𝜕𝑥+𝜕𝑠32

𝜕𝑦+𝜕𝑠33

𝜕𝑧− 𝑃1(

𝜕𝑤𝑦

𝜕𝑥)

=𝜕2

𝜕𝑡2(𝜌11𝑤1+ 𝜌12𝑊𝑧) ,

(2)

𝜕𝑠

𝜕𝑥=𝜕2

𝜕𝑡2(𝜌11𝑢1+ 𝜌22𝑈𝑥) ,

𝜕𝑠

𝜕𝑦=𝜕2

𝜕𝑡2(𝜌11V1+ 𝜌22𝑉𝑦) ,

𝜕𝑠

𝜕𝑧=𝜕2

𝜕𝑡2(𝜌11𝑤1+ 𝜌22𝑊𝑧) ,

(3)

where (𝑢1, V1, 𝑤1) and (𝑈

𝑥, 𝑉𝑦,𝑊𝑧) are displacement compo-

nents of solid and liquid part of porous medium in 𝑥, 𝑦, and𝑧 direction, respectively, and 𝑃

1represents the initial stress in

porous medium. The incremental stress components of solidpart of porousmedium are 𝑠

𝑖𝑗(𝑖, 𝑗 = 1, 2, 3) and 𝑠 is the stress

vector of liquid part of porous medium, where 𝑠 = −𝑓𝑝, 𝑓 isthe porosity, and 𝑝 is the fluid pressure, and 𝑤

𝑥, 𝑤𝑦, and 𝑤

𝑧

are angular components, defined as

𝑤𝑥=1

2(𝜕𝑤1

𝜕𝑦−𝜕V1

𝜕𝑧) ,

𝑤𝑦=1

2(𝜕𝑢1

𝜕𝑧−𝜕𝑤1

𝜕𝑥) ,

𝑤𝑧=1

2(𝜕V1

𝜕𝑥−𝜕𝑢1

𝜕𝑦) .

(4)

The relations between mass coefficients 𝜌11, 𝜌12, and 𝜌

22and

the densities 𝜌, 𝜌𝑠, and 𝜌

𝑤of the layer in solid and liquid

porous media are given by

𝜌11+ 𝜌12= (1 − 𝑓) 𝜌

𝑠,

𝜌12+ 𝜌22= 𝑓𝜌𝑤,

(5)

and the mass density of the bulk material is

𝜌= 𝜌11+ 𝜌22+ 2𝜌12= 𝜌𝑠+ 𝑓 (𝜌

𝑤− 𝜌𝑠) . (6)

These mass coefficients also satisfy the following inequalities:

𝜌11> 0,

𝜌22> 0,

𝜌12< 0,

𝜌11𝜌12− 𝜌2

12> 0.

(7)

In thewater-saturated anisotropic porousmedium, the stress-strain relations are

𝑠11= (𝐴 + 𝑃

1) 𝑒𝑥𝑥+ (𝐴 − 2𝑁 + 𝑃

1) 𝑒𝑦𝑦+ (𝐹 + 𝑃

1) 𝑒𝑧𝑧

+ 𝑄𝜖,

𝑠22= (𝐴 − 2𝑁) 𝑒

𝑥𝑥+ 𝐴𝑒𝑦𝑦+ 𝐹𝑒𝑧𝑧+ 𝑄𝜖,

𝑠33= 𝐹𝑒𝑥𝑥+ 𝐹𝑒𝑦𝑦+ 𝐶𝑒𝑧𝑧+ 𝑄𝜖,

𝑠12= 2𝑁𝑒

𝑥𝑦,

𝑠23= 2𝐿𝑒

𝑦𝑧,

𝑠13= 2𝐿𝑒

𝑧𝑥,

𝑒𝑖𝑗=1

2(𝜕𝑢𝑖

𝜕𝑥𝑗

+

𝜕𝑢𝑗

𝜕𝑥𝑖

) ,

𝜖 =𝜕𝑈𝑥

𝜕𝑥+

𝜕𝑈𝑦

𝜕𝑦+𝜕𝑈𝑥

𝜕𝑧,

(8)

where𝑁 and 𝐿 represent the shear moduli of the anisotropiclayer in the 𝑥 and 𝑧 direction, respectively, whereas 𝐴, 𝐹, and𝐶 are elastic constants for the medium.The positive quantity𝑄 is the measure of coupling between the changes of volumeof solid and liquid.

For the propagation of Love waves along the 𝑥 direction,

𝑢1= 0,

𝑤1= 0,

V1= V1(𝑥, 𝑥, 𝑡) ,

𝑈𝑥= 0,

𝑊𝑧= 0,

𝑉𝑦= 𝑉 (𝑥, 𝑧, 𝑡) .

(9)

Thus, the stress-strain relations are

𝑠23= 2𝐿𝑒

𝑦𝑧,

𝑠12= 2𝑁𝑒

𝑥𝑦.

(10)

Using (10) in (2), the equations of motion which are notautomatically satisfied are

𝜕𝑠21

𝜕𝑥+𝜕𝑠22

𝜕𝑦+𝜕𝑠23

𝜕𝑧− 𝑃1(𝜕𝑤𝑧

𝜕𝑥)

=𝜕2

𝜕𝑡2(𝜌11V1+ 𝜌12𝑉𝑦) ,

𝜕𝑠

𝜕𝑦= 0 =

𝜕2

𝜕𝑡2(𝜌12V1+ 𝜌22𝑉𝑦) .

(11)

4 Mathematical Problems in Engineering

By using (9) and (10) with Love wave condition, the aboveequations reduced into

(𝑁 −𝑃1

2)𝜕2V1

𝜕𝑥2+ 𝐿𝜕2V1

𝜕𝑧2=𝜕2

𝜕𝑡2(𝜌11V1+ 𝜌12𝑉) , (12)

𝜕2

𝜕𝑡2(𝜌12V1+ 𝜌22𝑉) = 0. (13)

From (𝜕2/𝜕𝑡2)(𝜌12V1+ 𝜌22𝑉) = 0 and 𝜌

12V1+ 𝜌22𝑉 = 𝑑

(say) 𝑉 = (𝑑 − 𝜌12V1)/𝜌22. Now, (𝜕2/𝜕𝑡2)(𝜌

11V1+ 𝜌12𝑉) =

𝑑(𝜕2V1/𝜕𝑡2), where 𝑑 = 𝜌

11− 𝜌2

12/𝜌22.

Therefore, (12) can be written as

(𝑁 −𝑃1

2)𝜕2V1

𝜕𝑥2+ 𝐿𝜕2V1

𝜕𝑧2= 𝑑 𝜕2V1

𝜕𝑡2. (14)

From the above equation, the shear wave velocity along the𝑥 direction is √(𝑁 − 𝑃

1/2)/𝑑 and along the 𝑧 direction is

√𝐿/𝑑.In the anisotropic porous medium, the shear wave veloc-

ity along the 𝑥 direction can be expressed as

𝛽 = √𝑁 − 𝑃

1/2

𝑑= 𝛽1√1 − 𝜁

𝑑1

, (15)

where 𝑑1= 𝛾11− 𝛾2

12/𝛾22, 𝛽1= √𝑁/𝜌, 𝛽

1is the shear wave

velocity in the corresponding initial stress-free, nonporous,anisotropic, elasticmediumalong the𝑥direction, 𝜁 = 𝑃

1/2𝑁

is the nondimensional parameter due to the initial stress 𝑃1,

and

𝛾11=𝜌11

𝜌,

𝛾13=𝜌13

𝜌,

𝛾23=𝜌23

𝜌

(16)

are the dimensionless parameters for the materials of theporous layer as obtained by Biot [3].

Thus, one gets the following:

(i) 𝑑1→ 1, when the layer is nonporous solid.

(ii) 𝑑1→ 0, when the layer is fluid.

(iii) 0 < 𝑑1< 1, when the layer is poroelastic.

Assume the solution of (14) as

V1(𝑥, 𝑧, 𝑡) = 𝑉

1(𝑧) 𝑒𝑖𝑘(𝑥−𝑐𝑡)

. (17)

Substituting (17) into (14), we obtain

𝑑2𝑉1

𝑑𝑧2+ 𝑚2

1𝑉1= 0, (18)

where𝑚1= 𝑘√(1/𝐿)(𝑐2𝑑 − 𝑁 + 𝑃

1/2).

Therefore, the solution of (18) takes the form 𝑉1(𝑧) =

𝐴1cos{𝑚

1𝑧} + 𝐴

2sin{𝑚

1𝑧}, where 𝐴

1and 𝐴

2are arbitrary

constants. Hence, the displacement in the porous layer isgiven by

V1= [𝐴1cos {𝑚

1𝑧} + 𝐴

2sin {𝑚

1𝑧}] 𝑒𝑖𝑘(𝑥−𝑐𝑡)

. (19)

This is the displacement on an initially stressed anisotropicporous layer, where

𝑚1= 𝑘√[𝑐2𝑑− (𝑁 − 𝑃

1/2)]

𝐿

= 𝑘√𝛾𝑑1[𝑐2

𝛽2

1

−1 − 𝜁

𝑑1

],

(20)

𝛾 = 𝑁/𝐿, 𝜁 = 𝑃1/2𝑁, 𝛽2

1= 𝑁/𝜌

, and 𝑘 is the wave number.

4. Solution of Orthotropic Half-Space

The equations of motion for the orthotropic medium underinitial stress in the absence of body forces are

𝜕𝜏11

𝜕𝑥+𝜕𝜏12

𝜕𝑦+𝜕𝜏13

𝜕𝑧− 𝑃2(𝜕𝑤

𝑧

𝜕𝑦−

𝜕𝑤

𝑦

𝜕𝑧) = 𝜌

2

𝜕2𝑢2

𝜕𝑡2,

𝜕𝜏21

𝜕𝑥+𝜕𝜏22

𝜕𝑦+𝜕𝜏23

𝜕𝑧− 𝑃2

𝜕𝑤

𝑧

𝜕𝑥= 𝜌2

𝜕2V2

𝜕𝑡2,

𝜕𝜏31

𝜕𝑥+𝜕𝜏32

𝜕𝑦+𝜕𝜏33

𝜕𝑧− 𝑃2

𝜕𝑤

𝑦

𝜕𝑥= 𝜌2

𝜕2𝑤2

𝜕𝑡2,

(21)

where 𝑢2, V2, and𝑤

2are the displacement components in the

orthotropic medium and 𝑤𝑥, 𝑤𝑦, and 𝑤

𝑧are the rotational

components along 𝑥, 𝑦, and 𝑧 direction. Here, 𝜏𝑖𝑗are the

incremental stress components and 𝜌2is the density of the

material in the semi-infinite medium.The stress-strain relations in the orthotropic medium are

𝜏11= 𝐵11𝑒11+ 𝐵12𝑒22+ 𝐵13𝑒33,

𝜏12= 2𝑄3𝑒12,

𝜏22= 𝐵21𝑒11+ 𝐵22𝑒22+ 𝐵23𝑒33,

𝜏23= 2𝑄1𝑒23,

𝜏33= 𝐵31𝑒11+ 𝐵32𝑒22+ 𝐵33𝑒33,

𝜏31= 2𝑄2𝑒31,

(22)

where 𝐵𝑖𝑗are the incremental normal elastic coefficient and

𝑄𝑖are shear moduli, whereas 𝑒

𝑖𝑗are the strain components.

Mathematical Problems in Engineering 5

Again, using the Love waves conditions 𝑢2= 0, 𝑤

2= 0,

and V2= V2(𝑥, 𝑧, 𝑡), the only equation of motion from (21)

and (22) for the orthotropic half-space can be written as

𝜕

𝜕𝑥(𝑄3

𝜕V2

𝜕𝑥) +

𝜕

𝜕𝑧(𝑄1

𝜕V2

𝜕𝑧) − 𝑃2

𝜕

𝜕𝑥(1

2

𝜕V2

𝜕𝑥)

= 𝜌2

𝜕2V2

𝜕𝑡2,

(23)

(𝑄3−𝑃2

2)𝜕2V2

𝜕𝑥2+ 𝑄1

𝜕2V2

𝜕𝑧2= 𝜌2

𝜕2V2

𝜕𝑡2, (24)

and the stress components 𝜏12= 2𝑄3𝑒12

and 𝜏23= 2𝑄1𝑒23;

other components will be zero.For wave propagation along 𝑥 direction, it may be

assumed that V2= 𝑉2(𝑧)𝑒𝑖𝑘(𝑥−𝑐𝑡), where 𝑘 is the wave number

and 𝑐 is the phase velocity; then (23) can be reduced:

𝑑2𝑉2

𝑑𝑧2− 𝑚2

2𝑉2= 0, (25)

where𝑚22= (𝑘2/𝑄1)[(𝑄3− 𝑃2/2) − 𝑐

2𝜌2]. So, the solution for

the initially stressed semi-infinite orthotropicmediumwill beof the form

V2= 𝐴3𝑒−𝑚2𝑧𝑒𝑖𝑘(𝑥−𝑐𝑡)

, (26)

where 𝐴3is the arbitrary constant.

5. Boundary Conditions

The upper surface of the porous layer is stress-free; that is,

𝐿𝜕V1

𝜕𝑧= 0, at 𝑧 = −𝐻, (27)

at the irregular interface; that is, 𝑧 = 𝜖𝑓(𝑥),

V1(𝑧) = V

2(𝑧) ,

𝐿𝜕V1

𝜕𝑧= 𝑄1

𝜕V2

𝜕𝑧.

(28)

Now, applying the boundary conditions, we have

sin {𝑚1𝐻}𝐴1+ cos {𝑚

1𝐻}𝐴2= 0,

cos𝑚1(𝜖𝑓 (𝑥)) + 𝐴

2sin𝑚1(𝜖𝑓 (𝑥)) = 𝐴

3𝑒−𝑚2𝜖𝑓(𝑥),

𝐿 {−𝑚1𝐴1sin𝑚1(𝜖𝑓 (𝑥)) + 𝑚

1𝐴2cos𝑚

1(𝜖𝑓 (𝑥))}

= −𝑚2𝑄1𝑒−𝑚2𝜖𝑓(𝑥)𝐴3.

(29)

The generalized dispersion equation of Love-type wave willbe obtained by eliminating arbitrary constants from the aboveequations as

sin𝑚1𝐻 cos𝑚

1𝐻 0

cos𝑚1(𝜖𝑓 (𝑥)) sin𝑚

1(𝜖𝑓 (𝑥)) −𝑒

−𝑚2𝜖𝑓(𝑥)

−𝐿𝑚1sin𝑚1(𝜖𝑓 (𝑥)) 𝐿𝑚

1cos𝑚

1(𝜖𝑓 (𝑥)) 𝑄

1𝑚2𝑒−𝑚2𝜖𝑓(𝑥)

= 0,

(30)

or

sin𝑚1𝐻

⋅ {𝑄1𝑚2sin𝑚1(𝜖𝑓 (𝑥)) + 𝐿𝑚

1cos𝑚

1(𝜖𝑓 (𝑥))}

− cos𝑚1𝐻

⋅ {𝑄1𝑚2cos𝑚

1(𝜖𝑓 (𝑥)) − 𝐿𝑚

1sin𝑚1(𝜖𝑓 (𝑥))}

= 0.

(31)

The generalized dispersion relation of Love wave has beenobtained as

tan𝑚1𝐻 =

{𝑄1𝑚2cos (𝑚

1ℎ) − 𝐿𝑚

1sin (𝑚

1ℎ)}

{𝑄1𝑚2sin (𝑚

1ℎ) + 𝐿𝑚

1cos (𝑚

1ℎ)}. (32)

6. Particular Cases

Case 1. In case the porous layer has no irregularity, that is,ℎ = 0, (32) reduces to

tan 𝑘𝐻√𝛾𝑑1[𝑐2

𝛽2

1

−1 − 𝜁

𝑑1

]

=𝑄1

𝐿

√[(𝑄3/𝑄1− 𝑃/2𝑄

1) − 𝑐2/𝛽

2

2]

√𝛾𝑑1[𝑐2/𝛽2

1− (1 − 𝜁) /𝑑

1]

,

(33)

where 𝛽2= √𝑄

1/𝜌2, which is the dispersion relation of

Love-type wave when the interface of the layered half-spaceis regular.

Case 2. For the nonporous homogeneous layer 𝑑1= 1, 𝑁 =

𝐿 = 𝜇1, 𝛾 = 1, 𝑃

1/2𝜇1= 0, (33) becomes

tan{

{

{

𝑘𝐻√𝑐2

𝛽2

1

− 1

}

}

}

=𝑄1

𝜇1

√[(𝑄3/𝑄1− 𝑃/2𝑄

1) − 𝑐2/𝛽

2

2]

√𝑐2/𝛽2

1− 1

.

(34)

The above equation represents the dispersion equation ofLove-type wave for initial stress-free nonporous homoge-neous layer.

Case 3. When the semi-infinitemedium is initially stress-freeand homogeneous with rigidity 𝜇

2(i.e.,𝑄

1→ 𝑄2→ 𝜇2and

𝑃2/2𝜇2= 0), (34) becomes

tan{𝑘𝐻√( 𝑐2

𝛽2

1

) − 1} =𝜇2

𝜇1

√1 − 𝑐2/𝛽2

2

√𝑐2/𝛽2

1− 1

, (35)

which is the classical dispersion relation of Lovewave (as Love[29]) in a homogeneous layer over a homogeneous half-space.

6 Mathematical Problems in Engineering

7. Numerical Calculations and Discussions

Based on dispersion (32), numerical results are provided toshow the propagation characteristics of Love waves in aninitially stressed anisotropic porous layer over an orthotropichalf-space. The effect of porosity, initial stress, and irregu-larity of the porous layer in phase velocity 𝑐/𝛽

1has been

analyzed graphically.To study the effect of porosity, initial stress, and irregu-

larity, we represent the numerical data from Gubbins [30] asfollows:

(a) For the orthotropic half-space,

𝑄1= 5.82 × 10

10N/m2,

𝑄3= 3.99 × 10

10N/m2,

𝜌2= 4.5 × 10

3 kg/m3.

(36)

(b) For the anisotropic porous layer,

𝐿 = 0.1387 × 1010N/m2,

𝑁 = 0.2774 × 1010N/m2,

𝜌11= 1.926137 × 10

3 kg/m3,

𝜌12= −0.002137 × 10

3 kg/m3,

𝜌22= 0.215337 × 10

3 kg/m3,

𝑓 = 0.26.

(37)

In all the figures, curves have been plotted as phasevelocity 𝑐/𝛽

1along vertical axis against dimensionless wave

number 𝑘𝐻 along horizontal axis. It has been observed thatthe maximum changes happen in phase velocity between𝑘𝐻 = 0.1 and 𝑘𝐻 = 1.0. The phase velocity of Love-typewave affected by the porosity of the medium, initial stress,and irregular interface of the layer and half-space and thesignificant impact of the abovementioned parameters hasbeen shown in the figures.

Figure 2 shows the effect of height of irregularity (ℎ/𝐻)on the phase velocity of Love-type wave in anisotropic porousmedium.The presence of irregularity at the interface of layerand half-space has the significant impact on the propagationof Love-type wave; the height (ℎ/𝐻) of irregular surfacehas been taken as 0.1, 0.2, and 0.3 for curves 1, 2, and 3,respectively, whereas 𝑃

1/2𝜇1= 0.2, 𝑃

2/2𝜇2= 0.3, and 𝑑

1=

0.01 are constants. It is observed that the phase velocity 𝑐/𝛽1

of Love-type waves decreases as the height of irregularityincreases; that is, the speed of a Love-type wave depends onthe height of irregularity in porous medium and the obtainedresult may be helpful for civil construction and evaluation ofearthquake damage in mountain region.

Figure 3 depicts the influence of initial stress 𝑃1/2𝜇1

associated with porous layer on the phase velocity 𝑐/𝛽1of

Love-type wave. Curve 1, curve 2, and curve 3 demonstratethe impact of initial stress on the phase velocity of a Love-type wave for 𝑃

1/2𝜇1= 0.1, 0.2, and 0.3, respectively. It is

(1) h/H = 0.1(2) h/H = 0.2(3) h/H = 0.3

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.1Dimensionless wave number kH

1.6

1.65

1.7

1.75

1.8

1.85

1.9

Effect of height of irregularity h/H

Non

dim

ensio

nal p

hase

velo

city

c/𝛽1

Figure 2: Variation of phase velocity (𝑐/𝛽1) with the wave number

(𝑘𝐻) for different values of ℎ/𝐻 (ℎ/𝐻 = 0.1, 0.2, 0.3) when𝑃1/2𝜇1=

0.2, 𝑃2/2𝜇2= 0.3, 𝑑

1= 0.01, and 𝛾 = 1.

Effect of initial stress P1/2𝜇1

(1) P1/2𝜇1 = 0.1(2) P1/2𝜇1 = 0.2(3) P1/2𝜇1 = 0.3

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.1Dimensionless wave number kH

1.65

1.7

1.75

1.8

1.85

1.9

1.95

Non

dim

ensio

nal p

hase

velo

city

c/𝛽1

Figure 3: Variation of phase velocity (𝑐/𝛽1) with the wave number

(𝑘𝐻) for different values of 𝑃1/2𝜇1(𝑃1/2𝜇1= 0.1, 0.2, 0.3) when

ℎ/𝐻 = 0.1, 𝑃2/2𝜇2= 0.3, 𝑑

1= 0.01, and 𝛾 = 1.

observed that the presence of initial stress in porous mediumincreases the phase velocity of Love-type wave as the value ofinitial stress increases. It is also observed that the behaviourof Lovewave speed in initially stressed porousmedium and atirregular interface is different, so the term irregularity in theEarth plays an important role in the propagation of surfacewaves.

Figure 4 shows the effect of initial stress𝑃2/2𝜇2associated

with half-space on the phase velocity of Love-type waves

Mathematical Problems in Engineering 7

1.6

1.65

1.7

1.75

1.8

1.85

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.1Dimensionless wave number kH

Effect of initial stress P2/2𝜇2

(1) P2/2𝜇2 = 0.30(2) P2/2𝜇2 = 0.31(3) P2/2𝜇2 = 0.32

Non

dim

ensio

nal p

hase

velo

city

c/𝛽1

Figure 4: Variation of phase velocity (𝑐/𝛽1) with the wave number

(𝑘𝐻) for different values of 𝑃2/2𝜇2(𝑃2/2𝜇2= 0.30, 0.31, 0.32) when

ℎ/𝐻 = 0.1, 𝑃1/2𝜇1= 0.2, 𝑑

1= 0.01, and 𝛾 = 1.

in porous medium. The presence of initial stress in theorthotropic medium affected the phase velocity of Love-type wave significantly. Curve 1, curve 2, and curve 3 havebeen plotted for 𝑃

2/2𝜇2= 0.30 and 𝑃

2/2𝜇2= 0.31 and

0.32 in the presence of irregular interface. It is found thatthe phase velocity decreases as the value of initial stressincreases and has much dominance at large values of wavenumber. The presence of initial stress 𝑃

1/2𝜇1in the porous

medium increases the phase velocity of Love-type wave,whereas the phase velocity decreases in the presence of initialstress (𝑃

2/2𝜇2) in orthotropic medium. It is observed that

the presence of irregularity of the interface affected the phasevelocity of Love wave in different ways in both mediums.

Figure 5 pointed out the influence of porosity (𝑑1) of

the medium on the phase velocity of Love-type wave. Theporosity is taken as 𝑑

1= 0.01, 0.02, and 0.03 for curve 1, curve

2, and curve 3, respectively.The curves apart from each otherbetween 𝑘𝐻 = 0.1 and 1.0 show that𝑑

1has a perfect influence

over the phase velocity of Love-type wave. It is observed thatthe phase velocity increases rapidly as the value of porosityincreases. It has been found that, with the increase in wavenumber, the phase velocity decreases rapidly in each of thesefigures under the considered values of various parameters.

Figure 6 described the impact of height of irregularity inthe absence of initial stress (𝑃

1/2𝜇1) on the phase velocity of

Love-type wave. It has been observed that the phase velocitydecreases with the depth of irregularity in an orthotropicmedium.

The study of seismic waves gives important informationabout the layered Earth structure and has been used todetermine the epicenter of the earthquake. Seismologistsare able to learn about the Earth’s internal structure bymeasuring the arrival of seismic waves at stations around theworld because these waves travel at different speeds through

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.1Dimensionless wave number kH

1.65

1.7

1.75

1.8

1.85

1.9

Effect of porosity d1

(1) Porosity d1 = 0.01(2) Porosity d1 = 0.02(3) Porosity d1 = 0.03

Non

dim

ensio

nal p

hase

velo

city

c/𝛽1

Figure 5: Variation of phase velocity (𝑐/𝛽1) with the wave number

(𝑘𝐻) for different values of porosity (𝑑1= 0.01, 0.02, 0.03) when

ℎ/𝐻 = 0.1, 𝑃1/2𝜇1= 0.1, 𝑃

2/2𝜇2= 0.31, and 𝛾 = 1.

1.5

1.55

1.6

1.65

1.7

1.75

1.8

1.85

1.9

1.95

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.1Dimensionless wave number kH

Effect of irregularity in the absence of initial stress P1/2𝜇1

(1) h/H = 0.1(2) h/H = 0.2

Non

dim

ensio

nal p

hase

velo

city

c/𝛽1

Figure 6: Variation of phase velocity (𝑐/𝛽1) with the wave number

(𝑘𝐻) for different values of ℎ/𝐻 (ℎ/𝐻 = 0.1, 0.2)when𝑃1/2𝜇1= 0.0,

𝑃2/2𝜇2= 0.3, 𝑑

1= 0.01, and 𝛾 = 1.

different materials. Knowing how fast these waves travelthrough the Earth, seismologists can calculate the time whenthe earthquake occurred and its location by comparing thetimes when shaking was recorded at several stations. If a wavearrives late, it passed through a hot, soft part of the Earth.

8. Conclusions

Propagation of Love-type waves in an initially stressedanisotropic porous layer over an initially stressed orthotropic

8 Mathematical Problems in Engineering

medium with rectangular irregularity has been discussed.The method of separation of variables has been adoptedto solve the equation of motion, separately, for differentmedia using suitable boundary condition at the interfaceof anisotropic porous layer and orthotropic half-space withirregular interface. The dispersion relation of Love-typewave has been obtained and coincides with the classicaldispersion relation of Love wave in particular cases. Thepresence of porosity, irregularity, and initial stress in thedispersion equation approves the significant effect of theseparameters on the propagation of Love-type wave in porousmedium bounded below by an orthotropic half-space. It hasbeen observed that the maximum changes happen in phasevelocity between 𝑘𝐻 = 0.1 and 𝑘𝐻 = 1.0. The conclusionsare as follows:

(i) The height ℎ/𝐻 of the irregularity affected the phasevelocity of Love-type wave, and the phase velocity𝑐/𝛽1decreases with increases in the height of the

irregularity. It has been noticed that the rectangularirregularity of interface is more effective for highrange of wave number 𝑘𝐻.

(ii) It is observed that the porosity also has a dominantrole in the propagation of Love-type wave. Whenthe porosity of the porous layer increases, the phasevelocity of the Love wave also increases in such astructure.

(iii) The phase velocity increases with increases in initialstress (𝑃

1/2𝜇1) of the porous layer, whereas the phase

velocity gradually decreases with increases in initialstress (𝑃

2/2𝜇2) of orthotropic half-space.

(iv) The height of irregularity has the impact on the phasevelocity of Love-type wave in the absence of initialstress (𝑃

1/2𝜇1). It has been observed that the phase

velocity decreases with the depth of irregularity in theorthotropic medium.

It is observed that the presence of porosity, initial stress,and irregularity affected the phase velocity of Love-typewave and has much dominance at large values of wavenumber.The initial stress in the porousmedium increases thephase velocity of Love-type wave, whereas the phase velocitydecreases in orthotropic medium due to initial stress. Thephase velocity of Love-type wave also decreases with thedepth of irregularity in an orthotropic medium.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgment

The authors convey their sincere thanks to the Indian Schoolof Mines, Dhanbad, India, for providing them with the bestfacilities.

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