Influences of Heterogeneities and Initial Stresses on the Propagation of Love-Type Waves in a Transversely Isotropic Layer Over an Inhomogeneous Half-Space

Document Type: Research Paper


Department of Applied Mathematics, Indian Institute of Technology (Indian School of Mines), Dhanbad Jharkhand-826004, India


In the present paper, we are contemplating the influences of heterogeneities and pre-stresses on the propagation of Love-type waves in an initially stressed heterogeneous transversely isotropic layer of finite thickness lying over an inhomogeneous half space. The material constants and pre-stress have been taken as space dependent and arbitrary functions of depth in the respective media. To simplify the problem, we have used Whittaker’s function and separation of variables method. We present a general dispersion relation to describe the impacts on the propagation of Love-type waves in the structure. The present dispersion relation is analyzed case wise and also validated by comparison of the standard Love wave equation. Further, numerical computations are demonstrated graphically for the set of dimensionless parameters between dimensionless phase velocity and dimensionless wave number of the wave.


[1] Love A.E.H., 1920, Mathematical Theory of Elasticity, Cambridge University Press, UK.
[2] Ewing W.M., Jardetzky W.S., Press F., 1957, Elastic Waves in Layered Media, McGraw-Hill, New York.
[3] Biot M.A., 1965, Mechanics of Incremental Deformations, Wiley, New York.
[4] Gubbins D., 1990, Seismology and Plate Tectonics, Cambridge University Press, Cambridge.
[5] Ding H., Chen W., Zhang L., 2006, Elasticity of Transversely Isotropic Materials, Springer Science & Business Media.
[6] Singh A.K., Kumari N., Chattopadhyay A., Sahu S.A., 2015, Smooth moving punch in an initially stressed transversely isotropic magnetoelastic medium due to shear wave, Mechanics of Advanced Materials and Structures 23: 774-783.
[7] Acharya D.P., Roy I., Sengupta S., 2009, Effect of magnetic field and initial stress on the propagation of interface waves in transversely isotropic perfectly conducting media, Acta Mechanica 202: 35-45.
[8] Ahmad F., Khan A., 2001, Effect of rotation on wave propagation in a transversely isotropic medium, Mathematical Problems in Engineering 7(2): 147-154.
[9] Singh B., 2016, Wave propagation in a rotating transversely isotropic two-temperature generalized thermoelastic medium without dissipation, International Journal of Thermophysics 37(1): 1-13.
[10] Kundu S., Gupta S., Manna S., 2014, SH-type waves dispersion in an isotropic medium sandwiched between an initially stressed orthotropic and heterogeneous semi-infinite media, Meccanica 49(3): 749-758.
[11] Zhu H., Zhang L., Han J., Zhang Y., 2014, Love wave in an isotropic homogeneous elastic half-space with a functionally graded cap layer, Applied Mathematics and Computation 231: 93-99.
[12] Kakar R., 2015, Dispersion of love wave in an isotropic layer sandwiched between orthotropic and prestressed inhomogeneous half-spaces, Latin American Journal of Solids and Structures 12(10): 1934-1949.
[13] Dey S., De P.K., 2010, Propagation of channel wave in an incompressible anisotropic initially stressed plate of finite thickness, Tamkang Journal of Science and Engineering 13(2): 127-134.
[14] Dhua S., Chattopadhyay A., 2015, Torsional wave in an initially stressed layer lying between two inhomogeneous media. Meccanica 50(7): 1775-1789.
[15] 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 Solids and Structures 51: 3689-3697.
[16] Chattaraj R., Samal S.K., Mahanti N., 2013, Dispersion of love wave propagating in irregular anisotropic porous stratum under initial stress, International Journal of Geomechanics 13(4): 402-408.
[17] Dey S., Addy S.K., 1978, Love waves under initial stresses, Acta Geophysica Polonica 26(1):47-54.
[18] Mahmoud S.R., 2012, Influence of rotation and generalized magneto-thermoelastic on Rayleigh waves in a granular medium under effect of initial stress and gravity field, Meccanica 47(7): 1561-1579.
[19] Kepceler T., 2010, Torsional wave dispersion relations in a pre-stressed bi-material compounded cylinder with an imperfect interface, Applied Mathematical Modelling 34(12): 4058-4073.
[20] Biot M. A., 1940, The influence of initial stress on elastic waves, Journal of Applied Physics 11: 522-530.
[21] Bullen K.E., 1940, The problem of the earth’s density variation, Bulletin of the Seismological Society of America 30(3): 235-250.
[22] Birch F., 1952, Elasticity and constitution of the earth's interior, Journal of Geophysical Research 57(2): 227-286.
[23] Dey S., Gupta A.K., Gupta S., 1996, Torsional surface waves in nonhomogeneous and anisotropic medium, The Journal of the Acoustical Society of America 99(5): 2737-2741.
[24] Gupta S., Chattopadhyay A., Vishwakarma S.K., Majhi D.K., 2011, Influence of rigid boundary and initial stress on the propagation of love wave, Applied Mathematics 2: 586-594.
[25] 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.
[26] Whittaker E., Watson G.N., 1990, A Course of Modern Analysis, Universal Book Stall, New Delhi.