Transference of SH-Waves in Fluid Saturated Porous Medium Sandwiched Between Heterogeneous Half-Spaces

Subject Areas : Engineering

S.A Sahu ¹ , S Chaudhary ² , P.K Saroj ³ , A Chattopadhyay ⁴

1 - Department of Applied Mathematics, Indian Institute of Technology (ISM), Dhanbad, India
2 - Department of Applied Mathematics, Indian Institute of Technology (ISM), Dhanbad, India
3 - Department of Applied Mathematics, Indian Institute of Technology (ISM), Dhanbad, India
4 - Department of Applied Mathematics, Indian Institute of Technology (ISM), Dhanbad, India

Received: 2017-06-28 Accepted : 2017-08-25 Published : 2017-09-01

Keywords: Whittaker function, Frequency equation, Porosity, Heterogeneity, SH-waves, Fluid saturated medium,

Abstract :

A mathematical model is considered to investigate the behavior of horizontally polarized shear waves (SH-waves) in fluid saturated porous medium sandwiched between heterogeneous half-spaces. Heterogeneity in the upper half-space is due to linear variation of elastic parameters, whereas quadratic variation has been considered for lower half-space. The method of separation of variables and Whittaker’s function are used to get an analytical solution for the considered problem. Frequency equation of SH waves in considered model has been obtained. Also, frequency equations have been derived for several particular cases. It is observed that the heterogeneity and porosity have significant effect on the phase velocity of SH-waves. In particular, heterogeneity and porosity increases the phase velocity of SH-waves. Obtained result is matched with classical Love wave equation. Graphical representation is done efficiently to explain the findings. Also the surface plot is added to exhibit the velocity profile of SH-waves in different cases.

References:

[1] Biot M.A., 1956, Theory of propagation of elastic waves in a fluid saturated porous solid I: Low frequency range, The Journal of the Acoustical Society of America 28(2): 168-178.
[2] Biot M.A., 1956, Theory of propagation of elastic waves in a fluid saturated porous solid II: High frequency range, The Journal of the Acoustical Society of America 28(2): 179-191.
[3] Biot M.A., 1956, Propagation of elastic waves in a liquid filled porous solid, Journal of Applied Physics 27: 459-467.
[4] Bhattacharya S.N., 1970, Exact solution of SH-wave equation for inhomogeneous media, Bulletin of the Seismological Society of America 60(6): 1847-1859.
[5] Chattopadhyay A., De R.K., 1983, Love type waves in a porous layer with irregular interface, International Journal of Engineering Science 21: 1295-1303.
[6] Chattopadhyay A., Chakraborty M., Mahata N.P., 1986, SH waves in a porous layer of non uniform thickness, Trans Engineers 34: 3-13.
[7] Dey S., 1987, Longitudinal and shear waves in an elastic medium with void pores, Proceedings of the Indian National Science Academy, India.
[8] Gubbins D., 1990, Seismology and Plate Tectonics, Cambridge University, Press Cambridge, New York.
[9] Pradhan A., Samal S.K., Mahanti N.C., 2002, Shear waves in a fluid saturated elastic plate, Sadhana 27(6): 595-604.
[10] Kumar R., Hundal B.S., 2003, Wave propagation in a fluid saturated incompressible porous medium, Indian Journal of Pure and Applied Mathematics 4: 651-665.
[11] Edelman I., 2004, Surface wave in porous medium interface: low frequency range, Wave Motion 39: 111-127.
[12] Chattopadhyay A., Kumari P., 2007, Propagation of shear waves in an anisotropic medium, International Journal of Applied Mathematics and Science 55(1): 2699-2706.
[13] Roy A., 2009, SH wave propagation in laterally heterogeneous medium, Springer Proceedings in Physics 126: 335-338.
[14] Gaur V.K., Rani S., 2010, Surface wave propagation in non-dissipative porous medium, International Journal of Educational Administration 2: 443-454.
[15] Chattopadhyay A, Gupta S, Sharma V.K., Kumari P, 2010, Effect of point source and heterogeneity on the propagation of SH-waves, International Journal of Applied Mathematics and Mechanics 6: 76-89.
[16] Samal S.K., Chattaraj R., 2011, Surface wave propagation in fiber-reinforced anisotropic elastic layer between liquid saturated porous half space, Acta Geophysica 59(3): 470-482.
[17] Kakar R, Kakar S, 2012, Propagation of love waves in non-homogeneous elastic media, Journal of Academia and Industrial Research 1: 61-67.
[18] Sahu S.A., Saroj P.K., Dewangan N., 2014, SH-waves in viscoelastic heterogeneous layer over half-space with self-weight, Archive of Applied Mechanics 84: 235-245.
[19] Kakar R, 2015, SH-wave velocity in a fiber-reinforced anisotropic layer overlying a gravitational heterogeneous half-space, Multidiscipline Modeling in Materials and Structures 11(3): 386-400.
[20] Ewing W.M., Jardetzky W.S., Press F., 1957, Elastic Waves in Layered Media, McGraw-Hill, New York.
[21] Chattopadhyay A., Samal S.K., Banerjee D., 2002, Propagation of shear waves in fluid saturated porous layer with initial stress, Acta Ciencia Indicia 28(3): 423-430.
[22] Kundu S., Gupta S., Majhi D.K., 2013, Love wave propagation in porous rigid layer lying over an initially stressed half-space, International Journal of Applied Physics and Mathematics 3(2): 140-142.
[23] Kundu S., Manna S., Gupta S., 2014, Love wave dispersion in pre-stressed homogeneous medium over a porous half-space with irregular boundary surfaces, International Journal of Solid and Structures 51: 3689-3697.