G-Type Seismic Wave in Magnetoelastic Monoclinic Layer

This paper deals with the study of propagation of G type waves along the plane surface at the interface of two different types of media. The upper medium is taken as monoclinic magnetoelastic layer whereas the lower half-space is inhomogeneous isotropic. Dispersion equation and condition for maximum energy flow near the surface are obtained in compact form. The dispersion equation is in assertion with the classical Love-type wave equation for the isotropic case. Effect of magnetic field and inhomogeneity on phase velocity and variation of group velocity with scaled wave number has been depicted by means of graphs. It is observed that inhomogenetity decreases phase velocity and the magnetic field has the favouring effect. A comparative study for the case of isotropic layer and monoclinic layer over the same isotropic inhomogeneous half space has been made through graphs.


Introduction
Horizontally polarized surface wave of shear type is known as G-type wave after B. Gutenberg [1][2], who established the existence of a low velocity layer in the earth mantle.It has been studied by researchers that such waves propagate with a group velocity of 4.4 km/s [3][4][5] and since the group velocity of Love waves over the period range from about 40 to 300 s is same, so on other hand Love waves with long periods (60 to 300 s) may also be recognized as G-waves.These waves are followed by dispersed Love waves, especially for continental paths and they exhibit a transient pulse-like character in record.A sequence of G-waves may be observed after a large earthquake.An outstanding case of well developed G-type waves was provided by the earthquake in Peru (January, 1960).Bath and Arroyo [6] presented the result obtained from this earthquake, especially with regard to absorption and velocity dispersion of G-waves.Aki [7] discussed the generation and propagation of Gwaves from the Niigata earthquake of June 16, 1964.Some other notable works in this field are done by Jeffreys [8], Bhattacharya [9], Chattopadhyay [10], Haskell [11] and others.Possibility of generation of G-waves in different medium has been investigated by different authors.Mal [12] studied the generation of G-waves taking the medium to be isotropic.The wave propagation in crystalline media plays a very interesting role in geophy-sics and also in ultrasonic and signal processing.Chattopadhyay and Keshri [13] presented the low velocity layer by assuming the law of variation in the lower semiinfinite anisotropic medium of two different media of monoclinic symmetry.Recently, Chattopadhyay et al. [14] discussed the dispersion of G-type waves in low velocity layer.The variation for the half-space in elastic constants and density reduces the equation of motion into Hill's equation with periodic coefficients which is solved by the method given by Valeev [15].Valeev considered a certain class of system of linear differential equations with periodic coefficients which have the property that, by means of Laplace transformation, they may be converted to a system of linear difference equations, which in turn may be solved by the method of infinite determinants.This method of solving Hill's differential equation has been successfully employed by Bhattacharya [9], Mal [12] and Chattopadhyay [10].
Theoretical and experimental studies regarding to better understanding of the real earth has led a need for more realistic representation of the medium through which seismic waves propagate.The propagation of seismic waves is affected by the elastic properties of the layered materials.Moreover, the materials of the layer might be magnetoelastic in nature and the interplay of electromagnetic field with the motion of a deformable solid has its importance in various fields of science and technology.
In the present paper we study of the propagation of G type waves along the plane surface at the interface of two different types of media.The upper medium is taken as monoclinic magnetoelastic layer whereas the lower halfspace is inhomogeneous isotropic.Keeping terms up to first order, the Laplace transform of the displacement is obtained.Dispersion equation and condition for maximum energy flow near the surface are obtained in compact form.The dispersion equation is in assertion with the classical Love-type wave equation for the isotropic case.Effect of magnetic field and inhomogeneity on phase velocity and variation of group velocity with scaled wave number has been depicted by means of graphs.It is observed that inhomogenetity decreases phase velocity and the magnetic field has the favouring effect.A comparative study for the case of isotropic layer and monoclinic layer over the same isotropic inhomogeneous half space has been made through graphs.
The formulation part and solution part of the problem has been dealt in Section 2. In the same section dispersion equation, the condition for maximum energy flow near the surface and the expression for group velocity have been obtained.Special cases for the dispersion equation obtained in Section 2 are considered in Section 3. Section 4 deals with the numerical calculation and graphical illustration for the problem.Finally, Section 5 concludes the study.

Formulation and Solution of the Problem
We consider a monoclinic magnetoelastic layer of thickness H lying over an inhomogeneous isotropic half-space.The variation for half-space is taken in following manner where  is small positive constant and s is real depth parameter.The axes Z and Y are taken horizontally and vertically downwards respectively (Figure 1).At first, we deduce the equation governing the propagation of shear wave in monoclinic magnetoelastic crustal layer.
The strain-displacement relations for monoclinic medium are , where , , u v w are displacement components in the direction x, y, z respectively, and Also, the stress-strain relation for a rotated y-cut quartz plate, which exhibits monoclinic symmetry with x being the diagonal axis are

T C S C S C S C S T C S C S C S C S T C S C S C S C S T C S C S C S C S T C S C S T C S C S
where   are the elastic constants.Equations governing the propagation of small elastic disturbances in a perfectly conducting monoclinic medium having electromagnetic force  J B (the Lorentz force, J being the electric current density and B being the magnetic induction vector) as the only body forces are where  is the density of the layer.
For SH wave propagating in the z-direction and causing displacement in the x-direction only, we shall assume that   , , , 0 and 0. u u y z t v w x Using Equations ( 2) and ( 5), the stress-strain relation (3) becomes Using Equation (6) in Equation ( 4), the only non-vanishing equation we have The well known Maxwell's equations governing the electromagnetic field are 0, , with , where E is the induced electric field, J is the current density vector and magnetic field H includes both primary and induced magnetic fields.e  and  are the induced permeability and conduction coefficient respectively.
The linearized Maxwell's stress tensor   where i h is the change in the magnetic field.In writing the above equations, we have neglected the displacement current.From Equation (8), we get In component form, Equation ( 9) can be written as For perfectly conducting medium i.e.    , the Equation (10) becomes .
It is clear from Equation ( 11) that there is no perturbation in 2 H and 3 H , however from Equation ( 12) there may be perturbation in 1 H . Therefore, taking small perturbation, say 1 h in 1 H , we have , where 0 0 H  H and  is the angle at which the wave crosses the magnetic field.Thus we have   We shall take initial value of 1 h as 1 0 h  .Using Equation (13) in Equation ( 12), we get Integrating with respect to t , we get Considering Equation ( 8), we get In the component form Equation ( 16) can be written as 0 and sin sin 2 cos .
Using Equations ( 7) and ( 17), we find the equation of motion for the magnetoelastic monoclinic medium in the form where    Hence, the equation of motion for the propagation of shear wave in monoclinic magnetoelastic crustal layer is where k is wave number and c is wave velocity.Substituting Equation (21) in Equation (20), we get Solution of Equation ( 22) is and , A B are constants.
Since the upper surface is stress free therefore, In lower inhomogeneous half-space the displacement   2 , , u y z t satisfy the differential equation we considered , , Using ( 25) and ( 26), the equation of motion for lower inhomogeneous medium may be written as This is Hill's differential equation, which we will solve by the method given by Valeev [15].We apply Laplace transform with respect to y, i.e. we multiply (27) by py e  and integrate with respect to y from 0 to  , we get Now the boundary conditions are (i) at 0 (ii) at 0 (iii) 0 at .
From (i) of Equation (29) we get Now we consider Now defining the Laplace transform of Using the boundary conditions we obtain the following system of equations where To find

 
F p from Equation (33), we replace p by We then obtain the following infinite system of linear algebraic equations in the quantities   where p may be considered as a parameter in the coefficients.It should be noted that in order not to have to consider the special case 0 m  separately, we include Equation ( 33

 
F p as the ratio of two infinite determinants, viz. where The first approximation of Equation ( 36) is where The second approximation of Equation ( 36) is where Neglecting the terms containing 2  and higher powers, we get Then   2 U y will be obtained from the following inversion formula: The residues and where Equations (41), ( 42) and (43) show the conditions for a large amount of energy to be confined near the surface are and Equations ( 44) and ( 46) give which finally gives the dispersion relation as We will consider only the positive sign for further discussion.Now Equation (45) gives and hence we get the expression for group velocity as It follows from Equation (49) that 2 G   , i.e. the group velocity is less than the shear wave velocity in the upper mantle.(1 ) 1

Particular cases
which is the result obtained by Chattopadhyay et al. [14] for isotropic layer lying over an inhomogeneous isotropic half space.

Numerical Examples and Discussion
For the case of monoclinic magnetoelastic layer lying over a non homogeneous isotropic half-space, we select the following data: 1) For monoclinic magnetoelastic layer (Tiersten [16]) wave number kH in monoclinic magnetoelastic layer, for the case when magnetic field is absent and present respectively.By the comparative study of these two graphs we can conclude that presence of magnetic field increases the phase velocity, whereas increment in non-homogeneity parameter  decreases the phase velocity.layer, for the case when magnetic field is absent and present respectively.The presence of magnetic field increases the phase velocity, whereas increase in the nonhomogeneity parameter  decreases the phase velocity.
It is observed from Figures 2-4 that presence of monoclinic medium favours more to the phase velocity as compared to simply isotropic one.The magnetic field and inhomogeneity parameter show similar type of tendency for both the isotropic and monoclinic medium.
Figure 6 shows the variation in dimensionless group velocity 2 G  with respect to scaled wave number k s .This graph explains that group velocity increases with scaled wave number and approaches to shear wave velocity asymptotically.Keeping in the mind the dependence of group velocity G on wave number k and depth parameter s, surface plot of group velocity against varying k and s has been drawn in Figure 7.
For the case of magnetoelastic isotropic layer lying over a non homogeneous isotropic half-space, we select the following data: 3) For isotropic magnetoelastic layer (Gubbins [17])

Conclusions
Dispersion equation for the propagation of G-type seismic wave in monoclinic magnetoelastic layer lying over an inhomogeneous isotropic half space is obtained, using the transform technique and Valeev's method [15].Condition for maximum energy to be confined near the surface and expression for group velocity are found.Phase and group velocity curves against wave number are plotted taking variation in inhomogeneity parameter.It is observed that the presence of non-homogeneity decreases the phase velocity whereas the presence of magnetic field increases the phase velocity, in both the cases when there is isotropic magnetoelastic layer or monoclinic magnetoelastic layer.It is also observed from the comparative study that presence of monoclinic medium favours more to the phase velocity as compared to simply isotropic one.Group velocity is found lower than the shear wave velocity in the upper mantle.The present study has its application especially to the problem of waves and vibrations where the wave signals have to travel through different layers of different material properties.This study may be helpful to understand the cause of damages during large earthquakes; also it may be useful to predict the nature of long period Love waves.These results can also be utilized in the interpretation and analysis of data of geophysical studies.The findings will be useful in forecasting formation details at greater depth through signal processing and seismic data analysis.The present study may be effectively utilized to generate initial data prior to exploitation of the formation.This study may be useful to geophysicist and metallurgist for analysis of rock and material structures through Non-Destructive Testing (NDT).

Figure 1 .
Figure 1.Geometry of the problem.
) in (35) by agreeing to regard   1 Solving the system of difference Equation (35), we obtain is the dispersion relation for the case when monoclinic magnetoelastic layer lying over an isotropic halfspace.

Figure 2 and 1 c
Figure 2 and Figure 3 represent the variation in dimensionless phase velocity 1 c  against dimensionless

Figure 3 and
Figure 4 represent the variation in dimensionless phase velocity 1 c  against dimensionless wave number kH in isotropic magnetoelastic

Figure 2 .Figure 3 .
Figure 2. Variation of dimensionless phase velocity against dimensionless wave number in an monoclinic magnetoelastic layer over an inhomogeneous isotropic semi-infinite medium when 0.0  H m .

Figure 4 .
Figure 4. Variation of dimensionless phase velocity against dimensionless wave number in an isotropic magnetoelastic layer over an inhomogeneous isotropic semi-infinite medium when 0.0  H 

Figure 5 .Figure 6 .
Figure 5. Variation of dimensionless phase velocity against dimensionless wave number in an isotropic magnetoelastic layer over an inhomogeneous isotropic semi-infinite medium when 0.8  H 

Figure 7 .
Figure 7. Variation of group velocity   G with respect to parameter k and s.