Dispersion of Torsional Surface Wave in a Pre-Stressed Heterogeneous Layer Sandwiched Between Anisotropic Porous Half-Spaces Under Gravity

Document Type : Research Paper


1 Department of Mathematics , S. N. Sinha College, Tekari, Magadh University, Bodh-Gaya, India

2 Department of Mathematics & Computing, Indian Institute of Technology (Indian School of Mines), Dhanbad, India



The study of surface waves in a layered media has their viable application in geophysical prospecting. This paper presents an analytical study on the dispersion of torsional surface wave in a pre-stressed heterogeneous layer sandwiched between a pre-stressed anisotropic porous semi-infinite medium and gravitating anisotropic porous half-space. The non-homogeneity within the intermediate layer and upper semi-infinite medium is assumed to rise up, because of quadratic variation and exponential variation in directional rigidity, pre-stress, and density respectively. The displacement dispersion equation for the torsional wave velocity has been expressed in the term of Whitaker function and their derivatives. Dispersion relation and the closed-form solutions have been obtained analytically for the displacement in the layer and the half-spaces. It is determined that the existing geometry allows torsional surface waves to propagate and the observe exhibits that the layer width, layer inhomogeneity, frequency of heterogeneity in the heterogeneous medium has a great impact on the propagation of the torsional surface wave. The influence of inhomogeneities on torsional wave velocity is also mentioned graphically by means plotting the dimensionless phase velocity against non-dimensional wave number for distinct values of inhomogeneity parameters.


[1] Meissner R., 2002, The Little Book of Planet Earth, Springer-Verlag, New York.
[2] Chattopadhyay A., Gupta S., Kumari P., 2011, Propagation of torsional waves in an inhomogeneous layer over an inhomogeneous half-space, Meccanica 46: 671-680.
[3] Gupta S., Majhi D.K., Kundu S., Vishwakarma S.K., 2013, Propagation of Love waves in non-homogeneous substratum over initially stressed heterogeneous half-space, Applied Mathematics and Mechanics 34(2): 249-258.
[4] Gupta S., Kundu S., Verma A.K., Verma R., 2010, Propagation of S-waves in a non-homogeneous anisotropic incompressible and initially stressed medium, International Journal of Engineering Science and Technology 2(2): 31-42.
[5] Dey S., Gupta S., Gupta A.K., Prasad A.M., 2001, Propagation of torsional surface waves in a heterogeneous half-space under a rigid layer, Acta Geophysica Polonica 49(1): 113-118.
[6] Kumari P., Sharma V.K., 2014, Propagation of torsional waves in a viscoelastic layer over an inhomogeneous half space, Acta Mechanica 225: 1673-1684.
[7] Biot M.A., 1941, General theory of three-dimensional consolidation, Journal of Applied Physics 12(2): 155-164.
[8] Biot M.A., 1956, Theory of propagation of elastic wave in a fluid-saturated porous solid, Journal of Acoustical Society of America 28(2): 168-178.
[9] Wang H., Tian J., 2014, Acoustoeelastic theory for fluid-saturates porous media, Acta Mechanica Solida Sinica 27(2): 41-53.
[10] Arani A.G., Zamani M.H., 2018, Nonlocal free vibration analysis of FG-porous shear and normal deformable sandwich nanoplate with piezoelectric face sheets resting on silica aerogel foundation, Arabian Journal for Science and Engineering 43: 4675-4688.
[11] Arani A.G., Khani M, Maraghi Z.K.., 2017, Dynamic analysis of a rectangular porous plate resting on an elastic foundation using high-order shear deformation theory, Journal of Vibration and Control 24: 3698-3713.
[12] Chattaraj R., Samal S.K., 2016, On dispersion of Love type surface wave in anisotropic porous layer with periodic non uniform boundary surface, Meccanica 51(9): 2215-2224.
[13] Konczak Z., 1989, The propagation of Love waves in a fluid-saturated porous anisotropic layer, Acta Mechanica 79: 155-168.
[14] Saroj P.K., Sahu S.A., 2017, Reflection of plane wave at traction-ffree surface of a pre-stressed functionally graded piezoelectric material (FGPM) half-space, Journal of Solid Mechanics 9(2): 411-422.
[15] Ke L.L., Wang J.S., Zhang Z.M., 2006, Love wave in an inhomogeneous fluid porous layered half-space with linearly varying properties, Soil Dynamics and Earthquake Engineering 26: 574-581.
[16] Prasad R.M., Kundu S. 2017, Torsional surface wave dispersion in pre-stressed dry sandy layer over a gravitating anisotropic porous half-space, Zeitschrift für Angewandte Mathematik und Mechanik 97(5): 550-560.
[17] Ghorai A.P., Samal S.K., Mahanti N.C., 2010, Love waves in a fluid-saturated porous layer under a rigid boundary and lying over an elastic half-space under gravity, Applied Mathematical Modelling 34: 1873-1883.
[18] Gupta S., Vishwakarma S.K., Majhi D.K., Kundu S., 2013, Possibility of Love wave propagation in a porous layer under the effect of linearly varying directional rigidities, Applied Mathematical Modelling 37: 6652-6660.
[19] Arani A.G., Maraghi Z.K.., Khani M., Alinaghian I., 2017, Free vibration of embedded porous plate using third-order shear deformation and poroelasticity theories, Journal of Engineering 2017:1474916.
[20] Biot M.A., 1965, Mechanics of Incremental Deformation, John Wiley and Sons, New York.
[21] Whittaker E.T., Watson G.N., 1991, Acourse of Modern Analysis, Cambridge University Press, Cambridge.
[22] Samal S.K., Chattaraj R., 2011, Surface wave propagation in fiber-reinforced anisotropic elastic layer between liquid saturated porous half-space and uniform liquid layer, Acta Geophysica 59(3): 470-482.
[23] Gubbins D., 1990, Seismology and Plate Tectonics, Cambridge University Press, Cambridge, New york.