Reflection and refraction phenomena of shear horizontal waves at the interfaces of sandwiched anisotropic magnetoelastic medium with corrugated boundaries


The present article analyses the reflection and refraction phenomena of an incident shear horizontal wave at the lower interface of corrugated anisotropic magnetoelastic medium sandwiched between an upper self-reinforced half-space and a lower heterogeneous viscoelastic half-space. Rayleigh’s approximation method and Snell’s law are employed to obtain an algebraic equation, which is solved by the application of Cramer’s method to determine the reflection and refraction coefficients for the reflected and refracted shear horizontal waves up to first order. The boundary conditions involve the continuity of displacements and stresses at the lower and upper common corrugated interfaces of the sandwiched anisotropic magnetoelastic medium. The deduced reflection and refraction coefficients depend on various parameters, viz. width of the sandwiched layer, anisotropic magnetoelastic constants, heterogeneity parameter, viscoelastic parameter and wave number. These reflected and transmitted coefficients are computed numerically, and their variation due to the affecting parameters is illustrated graphically against angle of incidence. The critical angle and slowness section are also determined for the shear horizontal wave propagating in the considered medium. Moreover, magnitude of energy distributed among each of the reflected and refracted waves is also computed. As a special case, the obtained results are found in well agreement with the pre-established results existing in the literature.

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  1. 1.

    Achenbach, J. (2012). Wave propagation in elastic solids (Vol. 16). Elsevier

  2. 2.

    H. Deresiewicz, R.D. Mindlin, Waves on the surface of a crystal. J. Appl. Phys. 28(6), 669–671 (1957)

    ADS  MathSciNet  Article  Google Scholar 

  3. 3.

    V. Thapliyal, Reflection of SH waves from anisotropic transition layer. Bull. Seismol. Soc. Am. 64(6), 1979–1991 (1974)

    Google Scholar 

  4. 4.

    Z. Wesolowski, Wave reflection on a continuous transition zone between two homogeneous materials. Acta Mech. 105(1–4), 119–131 (1994)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Y. Pang, J.S. Gao, J.X. Liu, SH wave propagation in magnetic-electric periodically layered plates. Ultrasonics 54(5), 1341–1349 (2014)

    Article  Google Scholar 

  6. 6.

    X. Guo, P. Wei, Effects of initial stress on the reflection and transmission waves at the interface between two piezoelectric half spaces. Int. J. Solids Struct. 51(21–22), 3735–3751 (2014)

    Article  Google Scholar 

  7. 7.

    C. Modi, P. Kumari, V.K. Sharma, Reflection/refraction of qP/qSV wave in layered self-reinforced media. Appl. Math. Modell. 40(19–20), 8737–8749 (2016)

    MathSciNet  Article  Google Scholar 

  8. 8.

    S.K. Tomar, J. Kaur, SH-waves at a corrugated interface between a dry sandy half-space and an anisotropic elastic half-space. Acta Mech. 190(1–4), 1–28 (2007)

    Article  Google Scholar 

  9. 9.

    J. Lalvohbika, S.S. Singh, Effect of corrugation on incident qP/qSV-waves between two dissimilar nematic elastomers. Acta Mech. 230(9), 3317–3338 (2019)

    MathSciNet  Article  Google Scholar 

  10. 10.

    S.A. Sahu, S. Mondal, N. Dewangan, Polarized shear waves in functionally graded piezoelectric material layer sandwiched between corrugated piezomagnetic layer and elastic substrate. J. Sandwich Struct. Mater. 21(8), 2921–2948 (2019)

    Article  Google Scholar 

  11. 11.

    A.J.M. Spencer, (1972). Deformations of fibre-reinforced materials

  12. 12.

    R. Ergun, A. Ercengiz, Propagation of harmonic waves in prestressed fiber viscoelastic thick tubes filled with a viscous dusty fluid. Appl. Math. Model. 34(9), 2597–2614 (2010)

    MathSciNet  Article  Google Scholar 

  13. 13.

    S.D. Akbarov, T. Kepceler, On the torsional wave dispersion in a hollow sandwich circular cylinder made from viscoelastic materials. Appl. Math. Model. 39(13), 3569–3587 (2015)

    MathSciNet  Article  Google Scholar 

  14. 14.

    J. Ding, B. Wu, C. He, Reflection and transmission coefficients of the SH mode in the adhesive structures with imperfect interface. Ultrasonics 70, 248–257 (2016)

    Article  Google Scholar 

  15. 15.

    J. Zhang, N. Hu, J. Zhang, Independent scattering model for evaluating antiplane shear wave attenuation in fiber-reinforced composite materials. Ultrasonics 78, 185–194 (2017)

    Article  Google Scholar 

  16. 16.

    S. Saha, A.K. Singh, A. Chattopadhyay, Analysis of reflection and refraction of plane wave at the separating interface of two functionally graded incompressible monoclinic media under initial stress and gravity. Eur. Phys. J. Plus 135(2), 173 (2020)

    Article  Google Scholar 

  17. 17.

    R. Sato, The reflection of elastic waves on corrugated surfaces. J. Seismol. Soc. Jpn. 8, 8–22 (1955)

    Google Scholar 

  18. 18.

    S.K. Tomar, S.L. Saini, Reflection and refraction of SH-waves at a corrugated interface between two-dimensional transversely isotropic half-spaces. J. Phys. Earth 45(5), 347–362 (1997)

    Article  Google Scholar 

  19. 19.

    J. Kaur, S.K. Tomar, V.P. Kaushik, Reflection and refraction of SH-waves at a corrugated interface between two laterally and vertically heterogeneous viscoelastic solid half-spaces. Int. J. Solids Struct. 42(13), 3621–3643 (2005)

    Article  Google Scholar 

  20. 20.

    A.J. Belfield, T.G. Rogers, A.J.M. Spencer, Stress in elastic plates reinforced by fibres lying in concentric circles. J. Mech. Phys. Solids 31(1), 25–54 (1983)

    ADS  Article  Google Scholar 

  21. 21.

    M.A. Biot, D.C. Drucker, (1965). Mechanics of incremental deformation

  22. 22.

    M. Chakraborty, Reflection and transmission of SH waves from an inhomogeneous half space. Proc. Indian Natl. Sci. Acad. 51(4), 716–723 (1985)

    MATH  Google Scholar 

  23. 23.

    A. Chattopadhyay, S. Gupta, S.A. Sahu, A.K. Singh, Dispersion equation of magnetoelastic shear waves in irregular monoclinic layer. Appl. Math. Mech. 32(5), 571–586 (2011)

    MathSciNet  Article  Google Scholar 

  24. 24.

    S. Asano, Reflection and refraction of elastic waves at a corrugated interface. Bull. Seismol. Soc. Am. 56(1), 201–221 (1966)

    Google Scholar 

  25. 25.

    I. Abubakar (1962). Scattering of plane elastic waves at rough surfaces. I. In: Mathematical Proceedings of the Cambridge Philosophical Society (Vol. 58, No. 1, pp. 136-157). Cambridge University Press

  26. 26.

    S. Kumar, P.C. Pal, S. Majhi, Reflection and transmission of SH-waves at a corrugated interface between two semi-infinite anisotropic magnetoelastic half-spaces. Waves Random Compl. Media 27(2), 339–358 (2017)

    ADS  MathSciNet  Article  Google Scholar 

  27. 27.

    S. Kumar, P.C. Pal, S. Majhi, 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(10), 2621–2634 (2015)

    ADS  Article  Google Scholar 

  28. 28.

    A. Chattopadhyay, S. Gupta, A.K. Singh, S.A. Sahu, G-type seismic wave in magnetoelastic monoclinic layer. Appl. Math. 2(02), 145 (2011)

    MathSciNet  Article  Google Scholar 

  29. 29.

    D. Gubbins, Seismol. Plate Tectonics (Cambridge University Press, Oxford, 1990)

    Google Scholar 

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Correspondence to Snehamoy Pramanik.



Appendix I

$$\begin{aligned} M_1= & {} -T_2\left( 1+E_1\right) ,~ M_2= T_3\left( 1+E_1\right) ,~ M_3=(T_1-T_2)E_1-T_2D_2+T_1D_2,~\\ M_4= & {} E_1T_3+D_1 T_3, M_5=-(E_1T_3+D_2 T_3), M_{6}=T_1(1+E_1),~ M_{7}=-T_3(1+E_1),~\\ M_{8}= & {} T_1(1+E_1), ~ M_{9}=-T_2(1+E_1), M_{10}=(T_1-T_2)+D_1T_2-T_1D_2, \\ M_{11}= & {} T_3-D_1 T_3, M_{12}=D_2T_3-T_3. \end{aligned}$$

Appendix II

$$\begin{aligned} \begin{aligned} \Theta _1=\,&T_9\left( D_4+D_5\right) ,~\Theta _2=-T_9\left( X_3D_5+X_4\right) ,~\Theta _3=X_2\left( D_3+D_4\right) -T_6\left( X_3D_5+X_4\right) ,~ \\ \Theta _4=\,&-T_9\left( D_3+D_4\right) ,~\Theta _5=T_9\left( D_3+D_5\right) ,~\Theta _6=T_7\left( D_4+D_5\right) -T_6\left( D_3+D_5\right) , \\ \Theta _7=\,&-T_9\left( X_3D_5+X_4\right) ,~\Theta _8=T_9\left( X_3D_3-X_4\right) ,~\Theta _9=-X_1T_9\left( D_3+D_5\right) , \\ \Theta _{10}=\,&T_9\left( \left( X_4-X_3D_5\right) +X_2\left( X_4-X_3D_4\right) \right) ,~\Theta _{11}=T_7\left( X_3D_5-X_4\right) -X_2\left( X_4-X_3D_3\right) , \\ \Theta _{12}=\,&X_1\left( T_7\left( D_4+D_5\right) -T_6\left( D_4-D_3\right) \right) ,~\Theta _{13}=T_9\left( X_4-X_3D_4\right) ,~\\ \Theta _{14}=\,&T_9\left( X_4-X_3D_3\right) ,~\Theta _{15}=T_7\left( X_4-X_3D_4\right) -T_6\left( X_4-X_3D_3\right) ,\\ \Theta _{16} =\,&X_1T_9\left( T_9-D_3\right) , \\ \Theta '_1=\,&T'_9\left( D'_4+D'_5\right) ,~\Theta '_2=-T'_9\left( X'_3D'_5+X'_4\right) ,~\Theta '_3=X'_2\left( D'_3+D'_4\right) -T'_6\left( X'_3D'_5+X'_4\right) ,~ \\ \Theta '_4=\,&-T'_9\left( D'_3+D'_4\right) ,~\Theta '_5=T'_9\left( D'_3+D'_5\right) ,~\Theta '_6=T'_7\left( D'_4+D'_5\right) -T'_6\left( D'_3+D'_5\right) , \\ \Theta '_7=\,&-T'_9\left( X'_3D'_5+X'_4\right) ,~\Theta '_8=T'_9\left( X'_3D'_3-X'_4\right) ,~\Theta '_9=-X'_1T'_9\left( D'_3+D'_5\right) , \\ \Theta '_{10}=\,&T'_9\left( \left( X'_4-X'_3D'_5\right) +X'_2\left( X'_4-X'_3D'_4\right) \right) ,~\Theta '_{11}=T'_7\left( X'_3D'_5-X'_4\right) -X'_2\left( X'_4-X'_3D'_3\right) , \\ \Theta '_{12}=\,&X'_1\left( T'_7\left( D'_4+D'_5\right) -T'_6\left( D'_4-D'_3\right) \right) ,~\Theta '_{13}=T'_9\left( X'_4-X'_3D'_4\right) ,~\\ \Theta '_{14}=\,&T'_9\left( X'_4-X'_3D'_3\right) ,~\Theta '_{15}=T'_7\left( X'_4-X'_3D'_4\right) -T'_6\left( X'_4-X'_3D'_3\right) ,\\ \Theta '_{16} =\,&X'_1T'_9\left( T'_9-D'_3\right) . \end{aligned} \end{aligned}$$

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Gupta, S., Pramanik, S., Smita et al. Reflection and refraction phenomena of shear horizontal waves at the interfaces of sandwiched anisotropic magnetoelastic medium with corrugated boundaries. Eur. Phys. J. Plus 135, 737 (2020).

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