Table of Contents Author Guidelines Submit a Manuscript
Advances in Mathematical Physics
Volume 2014, Article ID 163505, 9 pages
http://dx.doi.org/10.1155/2014/163505
Research Article

Shear Wave Propagation in Multilayered Medium including an Irregular Fluid Saturated Porous Stratum with Rigid Boundary

1Department of Mathematics, Maharshi Dayanand University, Rohtak, Haryana 124 001, India
2Department of Mathematics, TIT&S, Bhiwani, Haryana 127 021, India

Received 25 June 2014; Revised 5 September 2014; Accepted 8 September 2014; Published 12 November 2014

Academic Editor: Klaus Kirsten

Copyright © 2014 Ravinder Kumar et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The present investigation is concerned with the study of propagation of shear waves in an anisotropic fluid saturated porous layer over a semi-infinite homogeneous elastic half-space lying under an elastic homogeneous layer with irregularity present at the interface with rigid boundary. The rectangular irregularity has been taken in the half-space. The dispersion equation for shear waves is derived by using the perturbation technique followed by Fourier transformation. Numerically, the effect of irregularity present is analysed. It is seen that the phase velocity is significantly influenced by the wave number and the depth of the irregularity. The variations of dimensionless phase velocity against dimensionless wave number are shown graphically for the different size of rectangular irregularities with the help of MATLAB.

1. Introduction

The earth has a layered structure, and this exerts a significant influence on the propagation of elastic waves. Inside the Earth, a very hard layer (also known as “rigid”) is present. Since the composition of the Earth is heterogeneous including a very hard layer, the medium porosity and the rigid interface play significant roles in the propagation of the seismic waves. The propagation of elastic waves in homogeneous layer is of considerable importance in earthquake engineering and seismology. The study of wave propagation in elastic medium with different boundaries is of great importance to seismologists as well as to geophysicists to understand and predict the seismic behavior at different margins of earth. The propagation of shear waves has been studied by many authors assuming different forms of irregularities at the interface. Bhattacharya [1] discussed the dispersion curves for Love wave propagation in a transversely isotropic crustal layer with an irregularity in the interface. Jones [2] discussed wave propagation in a two-layered medium. Chattopadhyay [3] studied the effect of irregularities and nonhomogeneities in the crustal layer on the propagation of Love waves. Biot [4] and Deresiewcz [5] investigated the propagation of surface waves in liquid filled porous solids. Chattopadhyay et al. [6] studied the propagation of SH guided wave in an internal stratum with parabolic irregularity in the lower interface. Chattopadhyay and De [7] studied the dispersion equation for Love waves in a nondissipative liquid filled with porous solid underlain by an isotropic and homogeneous half-space. They derived the dispersion equation by applying the perturbation method, and the phase velocity curve was obtained for different irregularities by using the parameters of the porous medium which were suggested by Biot [8]. Kończak [9] derived dispersion equation for shear waves in a multilayered medium including a fluid saturated porous stratum. The influence of irregularity and rigidity on the propagation of torsional waves has been discussed by Gupta et al. [10]. Love wave propagation in a porous rigid layer lying over an initially stressed half-space is discussed by Kundu et al. [11]. Madan et al. [12] also discussed the propagation of Love waves in an irregular fluid saturated porous anisotropic layer with rigid boundary. For the elastic and viscoelastic waves, a long list of references is available in the monographs of Biot [13], Lamb [14], Miklowitz [15], and Keolsky [16].

In this paper we have discussed the propagation of shear waves in a transversely isotropic fluid saturated porous layer resting on a homogeneous elastic half-space, lying under an elastic isotropic and homogeneous layer with rigid boundary with irregularity at the interface. The irregularity is in the form of rectangle. The dispersion curves are depicted by means of graphs for different size of irregularity and different values of common wave velocity. The influence of depth of irregularity on phase velocity and some special cases have been studied.

2. Formulation of the Problem

A transversely isotropic fluid saturated porous layer of thickness resting on a homogeneous elastic half-space and lying under an elastic isotropic and homogeneous layer of thickness has been considered. The Cartesian coordinate system (, , ) is chosen with -axes taken vertically downward in the half-space and -axes is chosen parallel to the layer in the direction of propagation of the disturbance. We assume the irregularity in the form of a rectangle with length and depth . The origin is placed at the middle point of the interface irregularity. The source of the disturbance is placed on positive -axes at a distance from the origin. Therefore, the upper layer describes the medium : , the intermediate layer describes the medium : ,  and the homogeneous elastic half-space describes the medium : . The geometry of the problem is shown in Figure 1.

163505.fig.001
Figure 1: Geometry of the problem.

The interface between the layer and half-space is defined as where where and .

3. Basic Equations

The basic equations for the medium considered are as follows.

3.1. Medium

The equations of motion, without body force [17], are given by where are the components of stress tensor,   are the components of displacement vector, and   is the density. The comma denotes differentiation with respect to position and dot denotes differentiation with respect to time.

The constitutive relations are given by where   and are Lame’s elastic coefficients and is the Kronecker delta and

3.2. Medium

The equations of motion for the intermediate fluid saturated porous layer in the absence of body forces are [18] where are the components of stress tensor in the solid skeleton, is the reduced pressure of the fluid ( is the pressure in the fluid, and is the porosity of the medium), and   are the components of the displacement vector of the solid skeleton and   are those of fluid.

The stress-strain relations for the transverse-isotropic fluid saturated porous layer are where and , , , , , , , and are the material constants.

3.3. Medium

For the lower homogeneous half-space the basic equations of motion, without body force, are [17] where   are the components of stress tensor,   are the components of displacement vector, and   is the density.

The constitutive relations are given by where   and   are Lame’s elastic coefficients and are functions of , , and .

In this paper, attention is confined to shear waves propagating in the -plane. The displacements are parallel to direction and are independent of the coordinate. Thus, and the equations of motion (3), (6), and (9) with the help of (4), (5), (7), (8), and (10), respectively, reduce to the form The appropriate boundary conditions for the considered problem are as follows.(i)At the rigid surface , the displacement component vanishes; that is, (ii)At the interface , the displacements are continuous; that is, (iii)At the interface , the shear stress components are continuous; that is, (iv)At the interface , the displacements are continuous; that is, (v)The stresses are continuous at the interface ; that is, Thus (12)–(14) with the previous boundary conditions are the governing equations of the problem considered.

4. Solution of the Problem

For waves changing harmonically with time and propagating in -direction, we obtain where is the angular frequency.

Thus equations of motion (12)–(14) take the form of where , and are given in Appendix section.

Define Fourier transformation of as And inverse Fourier transformation is given by and so forth.

The Fourier transformation of (21) then is where The solutions of (24) are where , , , , and are functions of .

Thus, by inverse Fourier transformation, we obtain where the second term in the integrand of is introduced due to the source in the lower half-space [19].

The relations between the constants , and , are provided by (13).

We set the following approximations due to small value of : Since the boundary is not uniform, the terms , , , , and in (30) are also functions of. Expanding these terms in ascending powers of and keeping in view that is small and so retaining the terms up to the first order of  , , , , , and can be approximated as in (30). In physical situations, when the depth of the irregular boundary is too small with respect to the length of the boundary , the above assumptions are justified. Further, for small , where is any quantity.

Define Fourier transformation of as and the inverse Fourier transformation is Therefore, Now, by using boundary conditions (15)–(19) along with (27) and (29)-(30) we obtain a system of ten equations (after equating the absolute term (terms not containing ) and the coefficients of ): where and are given in Appendix section.

Solve the system of equations for , , , , , , , , , and and the corresponding values are given in Appendix section.

The displacement in the anisotropic layer is where and are given in Appendix section.

Now from (1), (2), and (32), we have Using values of and as given in Appendix section, we obtain where is given in Appendix section.

Using asymptotic formula of Willis [19] and Tranter [20] and neglecting the terms containing and highest powers of for large , we obtain Now using (38) and (39), we obtain Therefore the displacement in the anisotropic layer is where .

The value of this integral depends entirely on the contribution of the poles of the integrand. The poles are located at the roots of the equation: This equation is the dispersion equation for SH waves.

If is the common wave velocity of wave propagating along the surface, then we can set in (42a)   ( is the circular frequency and is the wave number), , , and where Solving (42a), we obtain Since the quantity is complex, so we have where and are the real and imaginary parts of and are given in Appendix section.

Equation (43) is complex and its real part gives the dispersion equation for shear waves.

5. Numerical Results and Discussions

In order to investigate the effect of irregularity present in the transversely isotropic fluid saturated porous layer and to compare the results numerically between the phase velocity and the wave number, we will use the values of elastic constants given by Ding et al. [21] for medium and by Koczak [9] for media and. And by using MATLAB, we obtain the following graph for different values of common wave velocity for two special cases.

Case I. When which is the wave propagation in elastic homogeneous layer lying over a homogeneous half-space, the variations of the dimensionless phase velocity () against the dimensionless wave number () in an elastic isotropic homogeneous layer over a homogeneous elastic half-space for different values of and , , and   are shown in Figures 2, 3, and 4.

163505.fig.002
Figure 2
163505.fig.003
Figure 3
163505.fig.004
Figure 4

Case II. When which is the wave propagation in a transversely isotropic fluid saturated porous layer lying over a homogeneous half-space, the variations of the dimensionless phase velocity () against the dimensionless wave number () in a transversely isotropic fluid saturated porous layer over a homogeneous elastic half-space for different values of   (0, 0.15, 0.30, 0.45) and , , and are shown in Figures 5, 6, and 7.

163505.fig.005
Figure 5
163505.fig.006
Figure 6
163505.fig.007
Figure 7

The dimensionless phase velocity () is plotted against the dimensionless wave number () in Figures 2, 3, 4, 5, 6, and 7. We conclude that the multilayered medium with irregularity and rigid boundary have significant effect on the propagation of shear waves, and the phase velocity in a layer with irregularity is affected by not only the shape of irregularity but also the wave number, the ratio of the depth of the irregularity to layer width and layer structure.

Appendix

Consider the following: where is the dimensionless frequency and is the velocity of shear wave in the porous layer.

Consider the following: where When then therefore, from (42b) and (44),

Conflict of Interests

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

Acknowledgments

The authors are grateful to the reviewers and Editor Klaus Kirsten for constructive suggestions for the improvement of this paper.

References

  1. J. Bhattacharya, “On the dispersion curve for Love wave due to irregularity in the thickness of the transversely isotropic crustal layer,” Gerlands Beiträge zur Geophysik, vol. 6, pp. 324–334, 1962. View at Google Scholar
  2. J. P. Jones, “Wave propagation in a two-layered medium,” Journal of Applied Mechanics, vol. 31, pp. 213–222, 1964. View at Google Scholar · View at MathSciNet
  3. A. Chattopadhyay, “On the dispersion equation for Love wave due to irregularity in the thickness of the non-homogeneous crustal layer,” Acta Geophysica Polonica, vol. 23, pp. 307–317, 1975. View at Google Scholar
  4. M. A. Biot, “Propagation of elastic waves in liquid filled porous solid,” Journal of Applied Physics, vol. 27, pp. 459–467, 1956. View at Publisher · View at Google Scholar
  5. H. Deresiewicz, “The effect of boundaries on wave propagation in a liquid-filled porous solid, I: reflection of plane waves at a free plane boundary (non-dissipative case),” Bulletin of the Seismological Society of America, vol. 50, pp. 599–607, 1960. View at Google Scholar · View at MathSciNet
  6. A. Chattopadhyay, M. Chakraborty, and A. K. Pal, “Effects of irregularity on the propagation of guided SHwaves,” Journal de Mecanique Theorique et Appliquee, vol. 2, no. 2, pp. 215–225, 1983. View at Google Scholar · View at Scopus
  7. A. Chattopadhyay and R. K. De, “Love type waves in a porous layer with irregular interface,” International Journal of Engineering Science, vol. 21, no. 11, pp. 1295–1303, 1983. View at Publisher · View at Google Scholar · View at Scopus
  8. M. A. Biot, Mechanics of Incremental Deformation, John Wiley & Sons, New York, NY, USA, 1961.
  9. Z. Kończak, “On propagation of shear waves in a multilayer medium including a fluid-saturated porous stratum,” Acta Mechanica, vol. 79, no. 3-4, pp. 169–181, 1989. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. S. Gupta, A. Chattopadhyay, and S. Kundu, “Influence of irregularity and rigidity on the propagation of torsional wave,” Applied Mathematical Sciences, vol. 4, no. 17–20, pp. 805–816, 2010. View at Google Scholar · View at MathSciNet · View at Scopus
  11. S. Kundu, S. Gupta, and D. K. Majhi, “Love wave propagation in porous rigid layer lying over an initially stressed half space,” Applied Physics & Mathematics, vol. 3, no. 2, pp. 140–142, 2013. View at Google Scholar
  12. D. K. Madan, R. Kumar, and J. S. Sikka, “Love wave propagation in an irregular fluid saturated porous anisotropic layer with rigid boundary,” Journal of Applied Sciences Research, vol. 10, no. 4, pp. 281–287, 2014. View at Google Scholar
  13. M. A. Biot, “Generalized theory of acoustic propagation in porous dissipative media,” Acoustical Society of America, vol. 34, pp. 1254–1264, 1962. View at Publisher · View at Google Scholar · View at MathSciNet
  14. H. Lamb, “On waves in elastic plate,” Proceedings of the Royal Society of London A, vol. 93, pp. 114–128, 1926. View at Google Scholar
  15. J. Miklowitz, “Elastic wave propagation,” in Applied Mechanics Surveys, pp. 830–836, Spartan Books, Washington, DC, USA, 1966. View at Google Scholar
  16. H. Keolsky, “The propagation of stress waves in viscoelastic solids,” in Applied Mechanics Surveys, Spartan Books, 1966. View at Google Scholar
  17. W. M. Ewing, W. S. Jardetzky, and F. Press, Elastic Waves in Layered Media, McGraw-Hill, London, UK, 1957. View at MathSciNet
  18. M. A. Biot, “Theory of elasticity and consolidation for a porous anisotropic solid,” Journal of Applied Physics, vol. 26, pp. 182–185, 1955. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  19. H. F. Willis, “A formula for expanding an integral as a series,” Philosophical Magazine, vol. 39, pp. 455–459, 1948. View at Google Scholar
  20. C. J. Tranter, Integral Transform in Mathematical Physics, Mathuen, 1966.
  21. H. Ding, W. Chen, and L. Zhang, Elasticity of Transversely Isotropic Materials, Springer, 2006. View at MathSciNet