Propagation of torsional waves in a hetrogeneous medium with irregular boundary.

Gupta, S. ; Chattopadhyay, A. ; Kundu, S. 等

Introduction

The study of wave propagation in elastic media with non-parallel boundaries is important to geophysicists since it helps them to understand and predict seismic behavior at continental margins and mountain roots. Review of literature indicates that numerous studies were conducted in the past to study the surface waves with different forms of irregularities at the interface. References may be made to Sato[1], De Noyer [2], Mal [3],

Bhattacharya [4] and Chattopadhyay [5]. Sato studied the propagation of Love waves in a layer with sharp change in thickness while De Noyer considered the same in a layer over a half space with sinusoidal interface. Mal studied the problem when the thickness of the layer abruptly increases throughout a certain length of the path. Bhattacharya [4] has shown the effect of irregularity on the dispersion of Love waves in the thickness of the transversely isotropic crustal layer. Chattopadhyay discussed the effects of irregularities and non-homogeneities in the crystal layer on the propagation of Love waves. Wolf [6] discussed the propagation of Love waves in an isotropic layer with irregular boundary. Sezawa [7] discussed Love waves generated from a source of a certain depth. Chattopadhyay and Pal [8] discussed the propagation of SH waves in an anisotropic layer with irregular boundary, and the displacement fields are obtained. They determined the reflected field in the anisotropic layer when an SH wave is incident on an irregular boundary in the shape of triangular notch.

As the earth's crust and mantle are not homogeneous, it is desirable to have information about torsional wave propagation in an inhomogeneous medium. Rayleigh [9] has shown that an isotropic homogeneous elastic half-space does not allow torsional surface waves to propagate. Meissner [10] has shown that, in an inhomogeneous elastic half-space with a quadratic variation in depth of the shear modulus and the mass density varying linearly with depth, torsional surface waves do exist. Vardoulakis [11] has demonstrated that torsional surface waves can also propagate in the Gibson half-space, that is, an elastic half-space in which shear modulus varies linearly with depth but the mass density remains constant. The two studies by Verttos [12,13] have provided an insight into the role of material inhomogeneity in the propagation of torsional surface waves generated by line loads. Torsional waves in an initially stressed cylinder have been treated by Dey and Dutta [14]. The propagation of torsional surface waves in an elastic half-space with void pores has been studied by Dey et al.[15], and Georgiadis et al. [16] have demonstrated that torsional surface waves do exist in a gradient elastic half-space.

Selim [17] studied the propagation of torsional surface waves in heterogeneous half- space with irregular free surface.

Since the composition of the Earth is heterogeneous including irregularity on the boundary, medium heterogeneity and irregular boundary play a significant role in the propagation of seismic waves. Therefore the present paper treats the propagation of torsional waves in heterogeneous half-space with irregular boundary. Both rigidity and density of the half-space are assumed to vary exponentially with depth. It is observed that such a medium allows torsional waves to propagate. The velocities of torsional surface waves have been calculated numerically and are presented in number of graphs. Prominent effect of variation of density on the propagation of torsional wave is observed in the low frequency range.

Formulation

Consider a heterogeneous half-space with irregularity as shown in fig.1. The heterogeneity has been considered both in density and rigidity. The free surface is considered to be of the form z = [epsilon] f (r), where e is small quantity.

The origin of the cylindrical co-ordinate system is located at the free surface the layer and the z-axis is directed downwards. The following variation in rigidity and density is assumed:

[mu] = [[mu].sub.1] (mz) and [rho] = [[rho].sub.1] exp (nz) (1)

In the above, [mu] and [rho] are rigidity and density of the media, respectively, and m and n are constants having dimensions that are inverse of length.

[FIGURE 1 OMITTED]

The equation of irregularity has been taken as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and [epsilon] = h/[d.sup.2] <<1, h is the maximum height of irregularity and is the period of irregularity.

The dynamical equations of motion are [18]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

where [[sigma].sub.rr], [[sigma].sub.[theta][theta]], [[sigma].sub.zz], [[sigma].sub.rz], [[sigma].sub.r[theta], and [[sigma].sub.[theta]z] are the respective stress components and u, v and are the respective displacement components.

The stress-strain relations are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

where [lambda] and [mu] are Lame's constants and [OMEGA] = ([partial derivative]u/[partial derivative]r + [partial derivative]v/[partial derivative][theta] + u/r + [partial derivative]w/[partial derivative]z denotes the dilatation.

The strain-displacement relations are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

The torsional wave is characterized by the displacements u = 0, w = 0, v = v (r, z, t) (5)

In view of eqs. (3), (4) and (5), the dynamical equations of motion governing torsional waves reduce to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

with v(r,z,t) being the displacement along the [theta] (azimuthal) direction. For an elastic medium the stresses are related to the displacement component by v

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

The boundary condition is:

[mu] ([partial derivative]v/[partial derivative]z) = 0 at z = [epsilon] f(r) (8)

Solution of the Problem

Using eq. (7), eq. (6) takes the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

We assume a solution of eq. (9) of the form

v = V (z) [J.sub.1](Kr)exp(i[omega]t) (10)

where V(z) is the solution of the following equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

in the above [c.sub.1] = [omega]/K and [c.sub.5] = [square root of [mu]/[rho] and [J.sub.1] (Kr) is the Bessel function of first kind and dash denotes the differentiation with respect to z.

Using eq.(1) the eq. (11) takes the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

where [c.sub.0] = [absolute value of [[mu].sub.1]/[[rho].sub.1].

Substituting V(z)=[sigma](z)exp(mz/2) in eq (12) to eliminate the term dv/dz, we

obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

Introducing the dimensionless quantities [gamma] = [(1+[m.sup.2]/[4K.sup.2] - [nc.sup.2.sub.1]/[mc.sup.2.sub.0]).sup.1/2] and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

Equation (14) is known as Whittaker equation [19].

We are interested in the solution of Eq. (14) which is bounded and vanishes as z [right arrow] [infinity], therefore we search for the solution which gives V (z) [right arrow] 0 as z [right arrow] [infinity]. This condition is equivalent to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Therefore the solution of Eq. (14) satisfying the above may be written as [phi]([eta]) = [DW.sub.R,1/2]([eta])

where [W.sub.R,1/2]([eta]) is the Whittaker function.

Hence, the displacement component v in the heterogeneous half space is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

Expanding Whittaker's function up to linear terms and using boundary condition (8), we get

[X.sub.0] + [X.sub.1]c + [X.sub.2][c.sup.2] + [X.sub.3][c.sup.3] = 0 (16)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

Particular Case

When n [right arrow] 0, i.e. the half-space has constant density, the velocity equation (16) takes the form

[X.sub.0] + [X.sub.1]c + [X.sub.2][c.sup.2] = 0 (18)

Where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

Example of Applications and Discussion of the Results

In order to demonstrate the effect of irregularity on the propagation of torsional surface wave in the heterogeneous isotropic elastic half-space, numerical computations of equation (16) were performed. Here it has been seen that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], for

assumed values of KH, [(r/d).sup.2] and n/K which shows that equation (16) has one real value of . Thus equation (16) gives only one wave front of torsional surface wave. The effect of irregularity and heterogeneity on the phase velocity of torsional surface wave is shown in Fig.2-7. In these figures, the variation of [c.sub.1] / [c.sub.0] has been ploted against K/m for different sizes of irregularity (Kh = 0.0, [+ or -] 0.2, and [+ or -] 0.4; [(r/d).sup.2] = 0.5 , 1.0 and 1.5; n/K = 1.75) using Matlab. Fig. 2-4 gives the variation of phase velocity of torsional surface wave against K/m for the positive value Kh = 0.2 and 0.4. Comparing with the case of Kh = 0.0, it is observed that the phase velocity of torsional surface waves decreases with an increase in irregularity parameter Kh and [(r/d).sup.2] for the same value of K/m. Fig. 5-7 gives the same for negative values of Kh = -0.2 and - 0.4.

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

[FIGURE 7 OMITTED]

Acknowledgement

The authors convey their sincere thanks to Indian School of Mines University, Dhanbad for providing JRF to Mr. Santimoy Kundu and also facilitating us with its best facility. Acknowledgement is also due to DST, New Delhi for the providing financial support through Project No.SR/S4/ES-246/2006, Project title: "Investigation of torsional surface waves in non-homogeneous layered earth".

References

[1] Y. Sato, Study on surface waves. VI Generation of Love and other types of SH waves Bull Earthqu. Res Inst., 30 (1952), 101

[2] J. De Noyer, The effect of variations in layer thickness of Love waves . Bull Seis Soc. Amer., 51 (1961), 227

[3] A. K. Mal, on the frequency equation for love wave due to abrupt thickening of the crustal layer. Geof Pure. E. Appl. 52 (1962), 59

[4] J. Bhattacharya, On the dispersion curve for Love waves due to irregularity in the thickness of the transversely isotropic crustal layer. Gerl Beitr Z Geophys . 71 (1962), 324

[5] A. Chattopadhyay, On the dispersion equations for Love waves due to irregularity in the thickness of non homogeneous crustal layer, Acta Geophysica Polonica , 23 (1975), 307

[6] B. Wolf , Propagation of Love waves in Layers with irregular boundaries, Pure and Appl. Geophysics, 78 (1970). 48

[7] K. Sezawa, Love waves generated from a source of a certain depth, Bull Earthqu. Res Inst., 13 (1935), 1.

[8] A. Chattopadhyay, A.K.Pal , SH Waves in Anisotropic Layer with Irregular Boundary, Bulletin De L'academie Polonaise Des Sciences serie des sciences techniques . Volume-30, No. 5-6,1982

[9] Lord Rayleigh, Theory of sound, Dover, New York 1945.

[10] E. Meissner,' Elastic oberflachenwellen mit dispersion in einem inhomogenen mmedium', Viertlagahrsschriftder Naturforschenden Gesellschaft, 66, 181195(1921)(in Zurich).

[11] J. Vardoulakis, ' torsional surface waves in inhomogeneous elastic media', Int. j. numer. Anal. Methods Geomech.,8, 287-296(1984).

[12] Vrettos, Ch.: In-plane vibrations of soil deposits with variable share modulus: II. Line load, Int. j. numer. anal. methods Geomech.,14,649-662(1990).

[13] Vrettos, Ch.: In-plane vibrations of soil deposits with variable share modulus: I. Line load, Int. j. numer. anal. methods Geomech.,14,209-222(1990).

[14] S.Dey and A. Dutta , ' Torsional wave propagation in an initially stressed cylinder', Proc. Indian Natn. Sci. Acad., 58, 425-429(1992).

[15] S.Dey, S. Gupta and A.K.Gupta, ' Torsional surface wave in an elastic half-space with void pores', Int. j. numer. anal. methods Geomech., 17, (1993).

[16] Georgiadis, H.G., Vardoulakis, I., Lykotrafitis, G.: Torsional surface waves in a gradient-elastic half space. Wave Motion. Vol. 31, pp. 333-348(2000).

[17] M. M. Selim, 'Propagation of Torsional Surface waves in heterogeneous half-space with irregular free surface', Applied Mathematical Sciences, 1, 29, 14291437(2007).

[18] Love, A.E.H.: The mathematical theory of elasticity, Cambridge University Press, 1927.

[19] W. W. Bell.: Special function for scientists and engineers, Courier Dover Publications, 1968.

S. Gupta, A. Chattopadhyay and S. Kundu

Department of Applied Mathematics

Indian School of Mines University, Dhanbad- 826 004, India

Table 1

Curve
  No     [(r/d).sup.2]    Kh

1             0.5        0.0
2             0.5        0.2
3             0.5        0.4

Table 2

Curve
  No     [(r/d).sup.2]     Kh

1             1.0         0.0
2             1.0         0.2
3             1.0         0.4

Table 3

Curve
  No     [(r/d).sup.2]     Kh

1             1.5          0.0
2             1.5          0.2
3             1.5          0.4

Table 4

Curve
 No.     [(r/d).sup.2]     Kh

1             0.5         -0.2
2             0.5         -0.4

Table 5.

Curve
 No.     [(r/d).sup.2]     Kh

1             1.0         -0.2
2             1.0         -0.4

Table 6.

Curve
 No.     [(r/d).sup.2]     Kh

1             1.5         -0.2
2             1.5         -0.4