An Analytic Study on the Dispersion of Love Wave Propagation in Double Layers Lying Over Inhomogeneous Half-Space

Document Type : Research Paper


Indian Institute of Technology (Indian School of Mines), Dhanbad-826004, India



In this work, attempts are made to study the dispersion of Love waves in dry sandy layer sandwiched between fiber reinforced layer and inhomogeneous half space.Inhomogeneity in half space associated with density and rigidity and considered in exponential form. Displacement components for fiber reinforced layer, dry sandy layer and inhomogeneous half-space have been obtained by using method of separable variables. Boundary conditions are defined at the free surface of the fiber reinforced layer and at the interfaces between layers and half space. The dispersion equation has derived in closed form. Numerical calculations for dispersion equation are performed. The study results show the effect of parameters on the velocity of Love waves and presented graphically. Graphs are plotted between wave number and phase velocity to show the effect of reinforced parameter, sandiness parameter and inhomogeneity on the phase velocity of Love waves. From the graphs it can be concluded that phase velocity decreases with respect to wave number.


[1] Love A.E.H., 1920, Mathematical Theory of Elasticity, Cambridge University Press, Cambridge.
[2] Ewing M., Jardetzky W., Press F., 1957, Elastic Waves in Layered Media, McGraw-Hill Book Company, New York.
[3] Bullen K.E., 1965, An Introduction to the Theory of Seismology, Cambridge University Press, London.
[4] Pipkin A.C., Rogers T.G., 1971, Plane deformations of incompressible fiber-reinforced materials, Journal of Applied Mechanics 38(3): 634-640.
[5] Belfield A.J., Rogers T.G., Spencer A.J.M.,1983, Stress in elastic plates reinforced by fibres lying in concentric circles, Journal of the Mechanics and Physics of Solids 31(1): 25-54.
[6] Pradhan A. , Samal S.K. , Mahanti N.C., 2003, Influence of anisotropy on the love waves in a self-reinforced medium, Tamkang Journal of Science and Engineering 6(3):173-178.
[7] Kumar R., Gupta R.R.,2010, Study of wave motion in an anisotropic fiber-reinforced thermoelastic solid, Journal of Solid Mechanics 2(1): 91-100.
[8] Singh B., Singh S.J., 2004, Reflection of plane waves at the free surface of a fibre-reinforced elastic half-space, Sadhana 29(3): 249-257.
[9] Paswan B., Sahu S.A., Saroj P.K., 2017, Dynamic response of heterogeneity and reinforcement on the propagation of torsional surface waves, Technische Mechanic 37(1): 69-81.
[10] Kar B.K., Pal A.K., Kalyani V.K.,1986, Propagation of love waves in an irregular dry sandy layer, Acta Geophysica Polonica 34(2): 157-170.
[11] Pal J., Ghorai A.P., 2015, Propagation of love wave in sandy layer under initial stress above anisotropic porous half-space under gravity, Transport in Porous Media 109(2): 297-316.
[12] Dey, S., Gupta A.K., Gupta S., 2002, Effect of gravity and initial stress on torsional surface waves in dry sandy medium, Journal of Engineering Mechanics 128(10): 1115-1118.
[13] Tomar S. K., Kaur J., 2007, SH-waves at a corrugated interface between a dry sandy half-space and an anisotropic elastic half-space, Acta Mechanica 190(1-4): 1-28.
[14] Souriau A., 1998, New seismological constraints on differential rotation of the inner core from Novaya Zemlya events recorded at DRV, Antarctica, Geophysical Journal International 134(2): F1-F5.
[15] Deresiewicz H.,1962, A note on love waves in a homogeneous crust overlying an inhomogeneous substratum, Bulletin of the Seismological Society of America 52(3): 639-645.
[16] Dutta S.,1963, On the propagation of love waves in a non-homogeneous internal stratum of finite depth lying between two semi-infinite isotropic media, Geofisica Pura e Applicata 55(3): 31-36.
[17] Abd-Alla A.M., Ahmed S.M., 1999, Propagation of love waves in a non-homogeneous orthotropic elastic layer under initial stress overlying semi-infinite medium, Applied Mathematics and Computation 106(2-3): 265-275.
[18] Selim M. M., 2007, Propagation of torsional surface waves in heterogeneous half-space with irregular free surface, Applied Mathematical Sciences 29(1): 1429-1437.
[19] Saroj P. K., Sahu S. A., Chaudhary S., Chattopadhyay A., 2015, Love-type waves in functionally graded piezoelectric material (FGPM) sandwiched between initially stressed layer and elastic substrate, Waves in Random and Complex Media 25(4): 608-627.
[20] Abo-Dahab S. M., 2018, Surface waves in fiber-reinforced anisotropic general viscoelastic media of higher orders with voids, rotation, and electromagnetic field, Mechanics of Advanced Materials and Structures 25(4): 319-334.
[21] Sahu S. A., Saroj P. K., Paswan B., 2014, Shear waves in a heterogeneous fiber-reinforced layer over a half-space under gravity, International Journal of Geomechanics 15(2): 04014048.
[22] Pal P. C., Kumar S, Mandal D., 2016, Surface wave propagation in sandy layer overlying a liquid saturated porous half-space and lying under a uniform liquid layer, Mechanics of Advanced Materials and Structures 23(1): 59-65.
[23] Sushant S., Parvez I.A., 2016, Propagation of Torsional surface waves in an inhomogeneous anisotropic fluid saturated porous layered half space under initial stress with varying properties, Applied Mathematical Modelling 40(2): 1300-1314.
[24] Alam, P., Kundu S., 2017, Influences of heterogeneities and initial stresses on the propagation of love-type waves in a transversely isotropic layer over an inhomogeneous half-space, Journal of Solid Mechanics 9(4): 783-793.
[25] Biswas S., Mukhopadhyay B., 2018, Rayleigh surface wave propagation in transversely isotropic medium with three-phase-lag model, Journal of Solid Mechanics 10(1): 175-185.
[26] Spencer A.J.M., 1972, Deformations of Fibre-Reinforced Materials, Oxford University Press, New York.
[27] Biot M.A., 1965, Mechanics of Incremental Deformation, Wiley, New York.
[28] Weiskopf W. H., 1945, Stresses in soils under a foundation, Journal of the Franklin Institute 239(6): 445-465.
[29] Gubbins D., 1990, Seismology and Plate Tectonics, Cambridge University Press.
[30] Markham M.F., 1969, Measurements of elastic constants of fibre composites by ultrasonics, Composites 1(2): 145-149.