In this paper, mathematical modeling of the propagation of torsional surface waves in a transverse isotropic elastic medium with varying rigidity and density under a rigid layer has been considered. The equation of motion has been formulated in the elastic medium using suitable boundary conditions. The frequency equation containing Whittaker’s function for phase velocity due to torsional surface waves has been derived. The effect of rigid layer in the propagation of torsional surface waves in a transverse isotropic elastic medium with varying rigidity and density has been discussed. The numerical results have been shown graphically. It is observed that the influence of transverse and longitudinal rigidity and density of the medium have a remarkable effect on the propagation of the torsional surface waves. Frequency equations have also been derived for some particular cases, which are in perfect agreement with some standard results.
 Rayleigh, L. (1885) On Wave Propagated along the Plane Surface of an Elastic Solid. Proceedings London Mathematical Society, 17, 4-11.
 Vardoulakis, I. (1984) Torsional Surface Wave in Inhomogeneous Elastic Medium. International Journal for Numerical and Analytical Methods in Geomechanics, 8, 287-296.
 Georgiadis, H.G., Vardoulakis, I. and Lykotrafitis, G. (2000) Tortional Surface Waves in a Gradient-Elastic Half-Space. Wave Motion, 31, 333-348.
 Dey, S., Gupta, S. and Gupta, A.K. (1993) Torsional Surface Waves in an Elastic Half-Space with Void Pores. International Journal for Numerical and Analytical Methods in Geomechanics, 17, 197-204.
 Dey, S., Gupta, A.K. and Gupta, S. (1996) The Propagation of Torsional Surface Waves in a Homogeneous Substratum over a Hetero-Geneous Half-Space. International Journal for Numerical and Analytical Methods in Geomechanics, 20, 287-294.
 Dey, S. and Sarkar, M.G. (2002) Torsional Surface Waves in an Initially Stressed Anisotropic Porous Medium. Journal of Engineering Mechanics, 128, 184-189.
 Chattaraj, R., Samal, S. and Mahanti, N.C. (2011) Propagation of Torsional Surface Wave in Anisotropic Poroelastic Medium under Initial Stress. Wave Motion, 48, 184-195.
 Gupta, S., Chattopadhyay, A., Kundu, S.K. and Gupta, A.K. (2009) Propagation of Torsional Surface Waves in Gravitating Anisotropic Porous Half-Space with Rigid Boundary. International Journal of Applied Mathematics and Mechanics, 6, 76-89.
 Gupta, S., Majhi, D.K. and Vishwakarma, S.K. (2012) Torsional Surface Wave Propagation in an Initially Stressed Non-Homogeneous Layer over a Non-Homogeneous Half Space. Applied Mathematics and Computation, 219, 32093218.
 Chattopadhyay, A., Gupta, S., Sahu, S.A. and Dhua, S. (2013) Torsional Surface Waves in Heterogeneous Anisotropic Half-Space under Initial Stress. Archive of Applied Mechanics, 83, 357-366.