Reflection of Plane Waves from a Free Surface of an Initially Stressed Transversely Isotropic Dissipative Medium

The governing equations of a transversely isotropic dissipative medium are solved analytically to obtain the speeds of plane waves. The appropriate solutions satisfy the required boundary conditions at the stress-free surface to obtain the expressions of the reflection coefficients of reflected quasi-P (qP) and quasi-SV (qSV) waves in closed form for the incidence of qP and qSV waves. A particular model is chosen for numerical computation of these reflection coefficients for a certain range of the angle of incidence. The numerical values of these reflection coefficients are shown graphically against the angle of incidence for different values of initial stress parameter. The impact of initial stress parameter on the reflection coefficients is observed significantly.


Introduction
We can not know the Earth completely by assuming mere an elastic body.If we consider various additional parameters, e.g.porosity, initial stress, viscosity, dissipation, temperature, voids, diffusion, etc., then we can understand better the interior of the Earth.Initial stresses in a medium are caused by various reasons such as creep, gravity, external forces, difference in temperatures, etc.The reflection of plane waves at free surface, interface and layers is important in estimating the correct arrival times of plane waves from the source.Various researchers studied the reflection and transmission problems at free surface, interfaces and in layered media [1][2][3][4][5][6][7][8][9][10][11][12].The study of reflection of plane waves in the presence of initial stresses as well as dissipation is interesting.With the help of Biot [13] theory of incremental deformation, Selim [14] studied the reflection of plane waves at a free surface of an initially stressed dissipative medium.In the present paper, we studied the problem on reflection of plane waves at a stress-free surface of an initially stressed transversely isotropic solid half-space with dissipation.The reflection coefficients of reflected waves are com-puted numerically to observe the effect of initial stress.

. Formulation of the Problem and Solution
Following Biot [13], the basic dynamical equations of motion in x-z plane for an infinite, initially stressed medium, in the absence of external body forces are, where  is the density, ) are incremental stress components, u and w arethe displacement components.Following Biot [13], the stress-strainrelations are      where C ij are the incremental elastic coefficients.
where, i 1   , are real.Following Fung [15], the stress and strain components in dissipative medium are, where (i, j = 1, 3) and  being the angular frequency.
With the help of Equations ( 3) and ( 4), the Equation (2) becomes,  z x With the help of Equation ( 5), the Equation (1) becomes,   The displacement vector is given by, where (n) assigns an arbitrary direction of propagation of waves, is the unit displacement vector is the phase factor, in is the unit propagation vector, c n is the velocity of propagation = (x, z), and k n is corresponding wave number, which is related to the angular frequency by The displacement components u n and w n are written Making use of Equation ( 9) into the Equations ( 6) and (7 ), we obtain a system of two homogeneous equations, which as non-trivial solution if where, The roots co si-P (qP) waves and quasi-SV (qSV) rrespond to qua waves respectively.
The above two roots give the square of velocities of propagation as well as damping.Real parts of the right hand sides correspond to phase velocities and the respective imginary parts correspond to damping velocities of qP and qSV waves, respectively.It is observed that both Surface consider W occupying the region z > 0 (Figure 1).In this section, we shall drive the closed form expressions for the reflection coefficients for incident qP or qSV waves.
The displacement components of incident and reflected waves are as, where, Here, subscripts 1, 2, 3 and 4 correspond to incident qP nt and stress compone wave, incident qSV wave, reflected qP wave and reflected qSV wave, respectively.
In the x-z plane, the displaceme nts due to the incident qP wave where, .In the x-z plane, the displacement and stress compone itten as , nts due to the incident qSV wave In the x-z plane, the displacement and stress com ne ritten as , , In the x-z plane, the displacement and stress components due to the reflected qSV wave The boundary conditions required to be satisfied at the free surface z = 0, , The above boundary conditions are written as The Equations ( 11) to (16) will satisfy the boundary conditions (18), if the following Snell, s law holds sin e sin e sin e sin e where and the following relations hold 4 = 0, (20) For incident qP wave (A 2 = 0), 2) For incident qSV waves (A 1 = 0),

Numerical Example
For numerical purpose, a particular example of the material is chosen with the following physical constants, From Equations ( 23) and ( 24), the reflection cients of reflected qP and qSV waves are computed for th own graphically in Figure 2 for incident qP wave and in Figure 3 for incident qSV wave.coefficient of reflected qP ge of the angle of incir P = 1 and P = 2 also change at each angle of incidence as shown by solid line with asters and solid lines with triangles, respectively.The comparison of the variations of reflection coefficients for P = 0, P = 1 and P = 2, show the significant effect of initial stress on reflected qP wa for incident qP wave.Similarly, reflected qSV is also affected significantly due to the presence of initial stress as show  For incident qSV wave, the reflection coefficient of reflected qP wave first increases to its maximum value and then decreases to its minimum value at angle e 2 = 45˚ when P = 0. Thereafter, it oscillates as shown by solid line in Figure 3.The variations of reflection coefficients of qP wave for P = 1 and P = 2 are similar to that for P = 0.The comparison of solid line, solid line with asters and solid line with triangles shows the significant effect of initial stress on reflected qP wave for incident qSV wave.Similarly, reflected qSV is also affected significantly due to the presence of initial stress as shown in Figure 3.

Conclusions
The reflection from the stress-free surface of a transversely isotropic dissipative medium is considered.The expressions for the reflection coefficients of reflected qP and qSV waves are obtained in closed form for the incidence of qP and qSV waves.For a particular material, these coefficients are computed and depicted graphically against the angle of incidence for different values of initial stress parameter.From the figures, it observed that 1) the initial stresses affect significantly the reflection coefficients of all reflected waves.2) For incident qP wave, the critical angle for reflected qSV wave is observed at e 1 = 45˚ and for incident qSV wave, the critical angle f d or reflected qP wave is observed also at e 2 = 45˚.3) The effect of initial stresses on the reflection coefficients is minimum at e 1 = 45˚ for incidence qP wave and at e 2 = 45˚ for incidence qSV wave.
on initial stresses, damping and ction propagation n dire of  .In the absence of initial stresses and damping, the ve analysis corresponds to the case of transversely isotropic elastic solid.

Figure 1 .
Figure 1.Geometry of the problem.
sine x cos e z , k c t sine x cos e z , k c t sine x cos e z , k c t sine x cos e z , isotropic case, C 11 =  + 2 + P, C 13 = P = -S 11 , then, the above theoretical derivations reduce to Selim[14].

3 7
and qSV waves.The numerical values of the reflection coefficients of reflected qP and qSV waves are sh For P = 0, the reflection ave oscillates for the whole ran w dence of qP wave as shown by solid line in Figure 2. The variations of reflection coefficients of qP wave fo s ve n in Figure 2.

Figure 2 .
Figure 2. Variation of the reflection coefficients of qP an qSV waves against the angle of incidence for incidence of qP wave.

Figure 3 .
Figure 3. Variation of the reflection coefficients of qP and qSV waves against the angle of incidence for incidence of qSV wave.