Abstract
Reflection and transmission coefficients due to incident plane SH-waves at an anisotropic magnetoelastic layer sandwiched between two inhomogeneous viscoelastic half-spaces are obtained. The inhomogeneity in viscoelastic coefficients of the lower half-space is considered as quadratic, while that of the upper half-space is considered as exponential. The solutions for the layer and half-spaces are obtained analytically. The formulae for reflection and transmission coefficients are obtained for viscoelastic half-spaces subjected to continuity conditions of interfaces of an anisotropic magnetoelastic layer with viscoelastic half-spaces and Snell’s law. These coefficients are found to be the function of phase velocity, wave number, inhomogeneity, thickness of layer, and angle of incidence. Two special cases are considered. In case I, the problem is reduced to reflection and transmission in two viscoelastic half-spaces, whereas case II considers two isotropic half-spaces. Numerical computations are carried out for a specific model. The graphs are plotted for reflection and transmission coefficients against the angle of incidence and non-dimensional wave number for inhomogeneity parameters, the angle at which the wave crosses the magnetic field, the thickness of the layer, and frequency. In general, as the inhomogeneity parameter increases, the reflection and transmission coefficients decrease, while the angle at which the wave crosses the magnetic field, the thickness of the layer and frequency have systematic effects. The effect of angle of incidence on the reflection and transmission coefficients is not monotonic, but converges for an angle of incidence for all the parameters. Also, graphs are plotted for phase shift versus angle of incidence. The effects of inhomogeneity, the angle at which the wave crosses the magnetic field, thickness of the layer and frequency are considered. The phase shift exhibits a similar nature for these parameters.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Achenbach, J.D. (1973), Wave propagation in elastic solids. North-Holland, Amsterdam.
Bullen, K.E. and Bolt, B.A. (1985), An introduction to the theory of seismology. Cambridge University Press, Cambridge.
Carcione, J.M. (1990), Wave propagation in anisotropic linear viscoelastic media: theory and simulated wavefields. Geophysical Journal International, 101(3), 739–750.
Chakrabarty, S.K. and De, T.K. (1971), On stresses and displacements due to a moving line load over the plane boundary of a heterogeneous elastic half-space. Pure and Applied Geophysics, 85, 214–218.
Chattopadhyay, A. and Choudhury, S. (1990), Propagation, reflection and transmission of magnetoelastic shear waves in a self-reinforced medium. International Journal of Engineering Science, 28(6), 485–495.
Chattopadhyay, A., Gupta, S., Singh, A.K. and Sahu, S.A. (2011), G-type Seismic wave in Magnetoelastic Monoclinic Layer. Applied Mathematics, 2(2), 145–154.
Chaudhary, S., Kaushik, V.P. and Tomar, S.K. (2004), Reflection/transmission of plane Sh-waves through a self-reinforced elastic layer between two half-spaces. Acta Geophysica Polonica, 52(2), 219–235.
Chaudhary, S., Kaushik, V.P. and Tomar, S.K. (2005), Transmission of shear waves through a self-reinforced layer sandwiched between two inhomogeneous viscoelastic half-spaces. Int. J. Mech. Sci., 47(9), 1455–1472.
Das, T.P. and Sengupta, P.R. (1990), Surface waves in general visco-elastic media of higher order. Indian J. pure appl. Math., 21(7), 661–675.
Deresiewicz, H. (1962), A note on Love waves in a homogeneous crust overlying an inhomogeneous substratum. Bulletin of the Seismological Society of America, 52, 639-645.
Dey, S., Gupta, A.K., Gupta, S. and Prasad, A.M. (2000), Torsional surface waves in nonhomogeneous anisotropic medium under initial stress. Journal of Engineering Mechanics (ASCE), 126, 1120–1123.
Dunkin, J.W. and Eringen, A.C. (1963), On the propagation of waves on electromagnetic elastic solids. Int. J. Eng. Sci., 1, 461–495.
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(1), 31–36.
Flugge, W. (1967), Visco-elasticity. Blaisdell Publishing Co., Michigan.
Gazetas, G. (1980), Static and dynamic displacements of foundations on heterogeneous multilayered soils. Géotechnique, 30, 159–177.
Knopoff, L. (1955), The interaction between elastic wave motion and a magnetic field in electrical conductors. Journal of Geophysical Research, 60, 441–456.
Kumar, S. and Pal, P.C. (2014), Wave Propagation in an inhomogeneous anisotropic generalized thermoelastic solid under the effect of gravity. Computational Thermal Sciences, 6(3), 241–250.
Kumar, S., Pal, P.C. and Bose, S. (2014), Propagation of SH-type waves in inhomogeneous anisotropic layer overlying an anisotropic viscoelastic half-space. International Journal of Engineering, Science and Technology, 6(4), 24–30.
Liu, X. and Greenhalgh, S. (2014), Reflection and transmission coefficients for an incident plane shear wave at an interface separating two dissimilar poroelastic solids. Pure Appl. Geophys. 171 (2014), 2111–2127.
Manolis, G.D. and Shaw, R.P. (1996), Green’s function for the vector wave equation in a mildly heterogeneous continuum. Wave Motion, 24, 59–83.
Manolis, G.D., Rangelov, T.V. and Shaw, R.P. (2002), Conformal mapping methods for variable parameter elastodynamics. Wave Motion, 36, 185–202.
Pal, P.C., Kumar, S. and Mandal, D. (2014), Wave Propagation in an inhomogeneous anisotropic generalized thermoelastic solid. Journal of Thermal Stresses, 37(7), 817–831.
Pal, P.K. and Acharya, D. (1998), Effects of inhomogeneity on the surface waves in anisotropic media. Sadhana, 23, 247–258.
Tomita, S. and Shindo, Y. (1979), Rayleigh waves in magneto-thermo-elastic solid with thermal relaxation. Int. J. Eng. Sci., 17, 227–232.
Udias, A. (1999), Principles of seismology. Cambridge University Press, United Kingdom.
Wilson, J. T. (1942), Surface waves in a heterogeneous medium. Bulletin of the Seismological Society of America, 32, 297–304.
Yu, C.P. and Tang, S. (1966), Magneto-elastic waves in initially stressed conductors. J. Appl Math Phys ZAMP, 17, 766–775.
Acknowledgments
Authors Mr. S. Kumar and Miss S. Majhi are grateful to the Indian School of Mines, Dhanbad, India authorities for financial support in the form of a Research Fellowship and for providing us with the best facility. Authors are also thankful to the Editor and reviewers for their valuable comments and suggestions to improve the quality of the paper.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kumar, S., Pal, P.C. & Majhi, S. Reflection and Transmission of Plane SH-Waves Through an Anisotropic Magnetoelastic Layer Sandwiched Between Two Semi-Infinite Inhomogeneous Viscoelastic Half-Spaces. Pure Appl. Geophys. 172, 2621–2634 (2015). https://doi.org/10.1007/s00024-015-1048-3
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00024-015-1048-3