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

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Abstract

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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Author information

Authors and Affiliations

  1. Department of Mathematics and Computing, Indian Institute of Technology (Indian School of Mines), Dhanbad, 826004, India

    Shishir Gupta, Snehamoy Pramanik &  Smita

  2. Department of Mathematics, Hampton University, Hampton, USA

    Arun Kumar Verma

Authors
  1. Shishir Gupta
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  2. Snehamoy Pramanik
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  3. Smita
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  4. Arun Kumar Verma
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Corresponding author

Correspondence to Snehamoy Pramanik.

Appendices

Appendix

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). https://doi.org/10.1140/epjp/s13360-020-00767-0

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