INTRODUCTION
The problems concerning the propagation of seismic waves through crustal layer
of earth have been of considerable importance for seismologists since a long
time. Such type of study helps the scientists in understanding the internal
structure of earth, which in turn can be used for exploration of valuable materials
like oil, hydrocarbons, minerals etc. The problems of propagation of Love waves
in different media have been discussed by Deshwal and Mudgal
(1998), Kar et al. (1986), Niazy
and Kazi (1980), Wong et al. (1995) and Singh
(1998). Jardaneh (2004) has considered the expected
source of earthquake evaluating the ground source response spectra taking into
account local soil properties to evaluate seismic forces. Kaur
et al. (2005) have studied the reflection and refraction of SHwaves
at a corrugated interface between two laterally and vertically heterogeneous
viscoelastic solid halfspace. Dhaimat and Dhaisat (2006)
have studied the sharp cut decrease of Dead Sea. The propagation of wave in
inhomogeneous thin film has been discussed by Ugwu et
al. (2007) using the series expansion solution method of Green’s
function. Tomar and Kaur (2007) have studied the problem
of reflection and transmission of a plane SHwave at a corrugated interface
between a dry sandy half space and an anisotropic elastic half space. They used
the Rayleigh method of approximation for studying the effect of sandiness, the
anisotropy, the frequency and the angle of incidence on the reflection and transmission
coefficients. Ademeso (2009) has discussed the deformation
traits in Charnockitic rocks by analyzing the direction of maximum compressional
and tensional stresses inferred from the rose diagram. Chattopadhyay
et al. (2009) has studied the reflection of shear waves in viscoelastic
medium at parabolic irregularity. The authors found that the amplitude of reflected
wave decreases with increasing length of notch and increases with increasing
depth of irregularity. The finite element method analysis has been used by Adedeji
and Ige (2011) to investigate and compare the performance of a reinforce
concrete bare frame infilled with or without straw bale wall shape memory alloy
diagonal wires subjected to seismic loads and earthquake ground excitation.
Ramli and Dawood (2011) have studied the effect of steel
fibers on the engineering performance of concrete. A computational technique
has been applied to study the field propagation through an inhomogeneous thin
film using LippmannSchwinger equation by Ugwu (2011).
The propagation of seismic waves has also been studied by Zaman
(2001), HaiMing and XiaoFei (2003), Balideh
et al. (2009) and Aziz et al. (2011).
The problem of Love wave excitation due to interaction between ocean wave and
sea bottom topography gas been studied by Saito (2010).
Here, we discuss the propagation of Love waves through irregularity in form
of an infinite rigid strip present in the surface layer.
This study is based on a paper by Sato (1961) who studied
the problem of reflection and transmission of Love waves in case the surface
layer is variable in thickness. Here we discuss the propagation of Love waves
through irregularity in form of a finite rigid horizontal barrier present in
the surface layer at finite depth.
THE PROBLEM AND ITS SOLUTION
The propagation of Love waves due to a finite rigid horizontal barrier in the surface layer has been discussed in the present paper. The problem is being analyzed in zxplane. The zaxis has been taken vertically downwards and xaxis along the interface. The Love wave is normally incident from right to left on a perfectly rigid screen l<x<0; z = h. The geometry of the problem is shown in Fig. 1. The incident Love wave is given by:
Where:
and C_{1N} is the phase velocity of nth mode and k_{1N} is the root of equation:
μ_{1} and μ_{2} being the rigidities of shear waves in the half space and in the crustal layer, respectively.
The wave equation in two dimensions is given as:
The wave equation in the present study for the surface layer can be written
as:

Fig. 1: 
Geometry of the problem 
Where:
V_{1} and V_{2 }are the velocities of shear waves in the half space z≥0 and in the layer H≤z≤0, respectively.
Let the total displacement be given by:
The boundary conditions are:
The boundary conditions (11) and (12) specify that the barrier is rigid and no displacement takes place across the barrier. Using boundary conditions (11) and (12), we have:
Taking Fourier transform of Eq. 6, we obtain:
where,
represents Fourier transform of v_{j} (x, z)which can be defined as:
If for a given z, asx→ ∞ and M,τ>0, v_{j} (x,
z) ~ Me^{τx}, then +
(p, z) is analytic in β>τ and 
(p, z) is analytic in β<τ (= Im (k_{j})). So by analytic
continuation
(p, z) and its derivatives are analytic in the strip τ<β<τ
in the complex pplane. Solving Eq. 18 and choosing the sign
of θ_{j} such that its real part is always positive, we obtain:
Solving Eq. 21 by using boundary condition (15), we get:
Differentiating above equation with respect to z and putting z = h and denoting
^{,}
(p, h) by ^{,}
(p) etc. and then eliminating A (p), we obtain:
Solving Eq. 18 for j = 3, using boundary condition (13), we have
Differentiating above equation with respect to z and putting z = h and denoting
^{,}
(p, h) by ^{,}
(p) etc. and then eliminating D (p) we obtain:
where, δ = Hh.
Taking Fourier transformation of Eq. 14:
From Eq. 16 and 17, we can write in usual
notation as:
Now using Eq. 26 and 27 in Eq.
25 for z = h we have:
Now from Eq. 23 and 28 for z = h, we
can write:
Where:
The Eq. 29 is the WienerHopf type differential equation
discussed by Noble (1958).
SOLUTION OF WIENERHOPF EQUATION
Now using Eq. 27 and 29 may be written
as:
Where:
Now we factorize
as discussed by Sato (1961) as:
Let p = ± k_{1m} and p = ±k_{2m} are the zeros of f_{1} (p) and f_{2} (p), respectively. Then we write:
Where:
and G_{1} (p)and G_{2} (p) have no zeros. Also we can write:
Where:
Where:
tan φ_{2} and tan n_{2} are obtained from Eq.
39 and 40 by replacing H by h.
Now, we write:
Where:
and K_{+} (p)→p^{1/2}, as p→ ∞.
Now
can be factorized as:
Using Eq. 34 and 43, Eq.
32 can be written as:
Where:
Similarly, Eq. 34 may also be written as:
Where:
and:
Also, multiplying Eq. 32 by e^{ipl} and rearranging the terms, we have:
Where:
and:
Also, we make following substitutions for simplicity:
Where:
and:
Now Eq. 23 and 25 may respectively be
written as:
and:
Where:
The displacement v_{2} (x, z) is obtained by inversion of Fourier transform
of Eq. 55 as given below:
Also, the displacement v_{3} (x, z) is obtained by inversion of Fourier
transform of Eq. 56 as given below:
where, (p)
is given in Eq. 57.
RESULTS AND DISCUSSION
For finding different component of waves, we evaluate the integral in Eq.
58 and Eq. 59 in different regions. For evaluation of
Eq. 58, as anticipated, there is a pole at p = k_{1N}
having no contribution. Furthermore, let p = k_{2m} be the roots of
Eq. f_{2} (p) = θ_{2} cosh θ_{2}h+γθ_{1}
sinh θ_{2}h = 0. The residue due to this pole contribute to:
where,
Equation 60 represents the transmitted waves in the region
h≤z≤0; l≤x≤0. The first term in this equation represents the component
of diffracted wave at the edge x = 0, while other terms are the interaction
terms due to second edge. Similarly, the pole at p = k_{1m} contributes
to:
The Eq. 61 represents the transmitted waves in the region
h≤z≤0, ∞≤x≤l and the first term represents the diffracted
component of waves at the edge x = l.
Also:
and:
are the phase velocities of Love type waves of mth mode in the layered structure with a layer of uniform thickness h.
Now we find the transmitted waves in the region H≤z≤h; x≤0. There is a pole at p = k_{1N}, the contribution due to which is zero. The poles at:
n = 1, 2, 3...., contributes to:
Equation 62 represents the transmitted waves in the region
H≤z≤h; l≤x≤0. The poles at p = k_{2m} contributes to:
Equation 63 represents the transmitted waves in the region
H≤z≤h; ∞≤x≤l. Clearly, this result is an analytic continuation
of the result obtained in Eq. 61.
NUMERICAL COMPUTATION AND ANALYSIS OF RESULTS
The transmitted waves in different regions have been found out in Eq.
60, 61, 62 and 63.
For computation purpose we have considered k_{2}δ small as compared
to the wavelength of the incident Love waves. We have taken, V_{1} =
4.6 km sec^{1}, V_{2} = 3.9 km sec^{1}, μ_{1}
= 7.98x10^{10} pa, μ_{2} = 4.11x10^{10} pa, k_{2}δ<0.01
for calculation purpose and different waves have been obtained. In the present
discussion, we have taken the barrier of finite length. As a limiting case of
this problem, if we take l→ ∞ , then we have only one edge at x =
0 and Eq. 60 will have only the first term, which is same
as obtained by Kazi (1975) for a semiinfinite barrier.
Also, if we take l = 0, i.e., the whole of the surface layer is rigid, the Love
wave moves with the velocity of shear waves in the surface layer.
CONCLUSIONS
We studied the problem of propagation of Love waves in the layered media with a surface layer having a horizontal rigid barrier of finite length. The numerical computation shows that the diffracted Love wave decreases as the distance from the barrier increases. It is clear from the discussion that if the barrier of large size is considered, the diffracted wave of larger intensity is observed. So by measuring the component of the wave obtained, the internal structure of earth can be predicted for specific study. The case of semiinfinite barrier is obtained as a special case.