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The study of reflection/transmission phenomena in a corrugated interface between two magnetoelastic transversely isotropic media

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Abstract

In this paper, an integrated approach has been carried out with an intent to study the reflection and transmission phenomena of plane SH-type wave on a corrugated interface separating two magnetoelastic transversely isotropic half-space. Closed form expressions for reflection and transmission coefficients have been derived for both plane and corrugated surface. Rayleigh’s method of approximation have been incorporated to deduce equations for the first- and second-order approximation of corrugation. Analytical solutions for both the half-spaces have been worked out. All possible cases have been apprehended specially for anisotropic and an isotropic medium. The effects of magnetoelastic coupling parameter, angle at which the wave crosses the magnetic field, corrugation amplitude, frequency factor and wave number have been explained by collaborating with graphical analysis.

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Authors and Affiliations

  1. Department of Applied Mathematics, Indian Institute of Technology (ISM), Dhanbad, Jharkhand, 826004, India

    Shishir Gupta, Neelima Bhengra & Mostaid Ahmed

Authors
  1. Shishir Gupta
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  2. Neelima Bhengra
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  3. Mostaid Ahmed
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Corresponding author

Correspondence to Neelima Bhengra.

Appendix

Appendix

$$ {\displaystyle \begin{array}{l}{\varPi}_{R_0}=\left[-L\left({\varOmega}_0{S}^{\prime }-\frac{Q^{\prime}\omega \sin e}{2{\beta}_1}\right)+{L}^{\prime}\left(\eta {S}^{{\prime\prime} }-\frac{Q^{{\prime\prime}}\omega \sin f}{2{\beta}_2}\right)\right],\\ {}{\varPi}_{T_0}=\left[L\left(-\varOmega {S}^{\prime }-\frac{Q^{\prime}\omega \sin e}{2{\beta}_1}\right)-L\left({\varOmega}_0{S}^{\prime }-\frac{Q^{\prime}\omega \sin e}{2{\beta}_1}\right)\right],\\ {}{\varPi}_0=\left[L\left(-\varOmega {S}^{\prime }-\frac{Q^{\prime}\omega \sin e}{2{\beta}_1}\right)-{L}^{\prime}\left(\eta {S}^{{\prime\prime} }-\frac{Q^{{\prime\prime}}\omega \sin f}{2{\beta}_2}\right)\right],\\ {}{d}_n=L\left[-{S}^{\prime }{\varOmega}_n-\left(\frac{\omega \sin e}{\beta_1}+ n\lambda \right)\frac{Q^{\prime }}{2}\right],{a}_n={L}^{\prime}\left[{\eta}_n{S}^{{\prime\prime} }-\left(\frac{\omega \sin f}{\beta_2}+ n\lambda \right)\frac{Q^{{\prime\prime} }}{2}\right],\\ {}{q}_n=L\left[ n\lambda \left(\frac{Q^{\prime }{\varOmega}_0}{2}-{P}^{\prime}\frac{\omega \sin e}{\beta_1}\right)+{\varOmega}_0\left({S}^{\prime }{\varOmega}_0-{Q}^{\prime}\frac{\omega \sin e}{2{\beta}_1}\right)\right],\\ {}{b}_n=L\left[- n\lambda \left(\frac{Q^{\prime}\varOmega }{2}+{P}^{\prime}\frac{\omega \sin e}{\beta_1}\right)+\varOmega \left({S}^{\prime}\varOmega +{Q}^{\prime}\frac{\omega \sin e}{2{\beta}_1}\right)\right],\\ {}{g}_n={L}^{\prime}\left[ n\lambda \left(\frac{\eta {Q}^{{\prime\prime} }}{2}-{P}^{{\prime\prime}}\frac{\omega \sin f}{\beta_2}\right)+\eta \left(\eta {S}^{{\prime\prime} }-{Q}^{{\prime\prime}}\frac{\omega \sin f}{2{\beta}_2}\right)\right\}\\ {}{d}_n^{\prime }=L\left[-{S}^{\prime }{\varOmega}_n^{\prime }+\left( n\lambda -\frac{\sin e\omega}{\beta_1}\right)\frac{Q^{\prime }}{2}\right],{a}_n^{\prime }={L}^{\prime}\left[{\eta}_n^{\prime }{S}^{{\prime\prime} }+\left( n\lambda -\frac{\omega \sin f}{\beta_2}\right)\frac{Q^{{\prime\prime} }}{2}\right],\\ {}{q}_n^{\prime }=L\left[- n\lambda \left(\frac{Q^{\prime }{\varOmega}_0}{2}-{P}^{\prime}\frac{\omega \sin e}{\beta_1}\right)+{\varOmega}_0\left({S}^{\prime }{\varOmega}_0-{Q}^{\prime}\frac{\omega \sin e}{2{\beta}_1}\right)\right],\\ {}{b}_n^{\prime }=L\left[ n\lambda \left(\frac{Q^{\prime}\varOmega }{2}+\frac{P^{\prime}\omega \sin e}{\beta_1}\right)+\varOmega \left({S}^{\prime}\varOmega +{Q}^{\prime}\frac{\omega \sin e}{2{\beta}_1}\right)\right],\\ {}{g}_n^{\prime }={L}^{\prime}\left[- n\lambda \left(\frac{\eta {Q}^{{\prime\prime} }}{2}-{P}^{{\prime\prime}}\frac{\omega \sin f}{\beta_2}\right)+\eta \left(\eta {S}^{{\prime\prime} }-{Q}^{{\prime\prime}}\frac{\omega \sin f}{2{\beta}_2}\right)\right].\\ {}{d}_1=L\left[-{S}^{\prime }{\varOmega}_1+\left(\lambda -\frac{\sin e\omega}{\beta_1}\right)\frac{Q^{\prime }}{2}\right],{a}_1={L}^{\prime}\left[\eta {S}^{{\prime\prime} }-\left(\frac{\omega \sin f}{\beta_2}+\lambda \right)\frac{Q^{{\prime\prime} }}{2}\right],\\ {}{q}_1=L\left[\lambda \left(\frac{Q^{\prime }{\varOmega}_0}{2}-\frac{\omega \sin e{P}^{\prime }}{\beta_1}\right)+{\varOmega}_0\left({S}^{\prime }{\varOmega}_0-\frac{\omega \sin e{Q}^{\prime }}{2{\beta}_1}\right)\right],\\ {}{b}_1=L\left\{-\lambda \left(\frac{Q^{\prime}\varOmega }{2}+{P}^{\prime}\frac{\omega \sin e}{\beta_1}\right)+\varOmega \left({S}^{\prime}\varOmega +{Q}^{\prime}\frac{\omega \sin e}{2{\beta}_1}\right)\right\},\\ {}{g}_1={L}^{\prime}\left[\lambda \left(\frac{Q^{{\prime\prime}}\eta }{2}-{P}^{{\prime\prime}}\frac{\omega \sin f}{\beta_2}\right)+\eta \left(\eta {S}^{{\prime\prime} }-\frac{\omega \sin f{Q}^{{\prime\prime} }}{2{\beta}_2}\right)\right],\\ {}{d}_1^{\prime }=L\left[-{S}^{\prime }{\varOmega}_1^{\prime }+\left(-\frac{\sin e\omega}{\beta_1}+\lambda \right)\frac{Q^{\prime }}{2}\right],{a}_1^{\prime }={L}^{\prime}\left[{\eta}_1^{\prime }{S}^{{\prime\prime} }+\left(-\frac{\omega \sin f}{\beta_2}+\lambda \right)\frac{Q^{{\prime\prime} }}{2}\right],\\ {}{q}_1^{\prime }=L\left[-\lambda \left(\frac{Q^{\prime }{\varOmega}_0}{2}-{P}^{\prime}\frac{\omega \sin e}{\beta_1}\right)+{\varOmega}_0\left({S}^{\prime }{\varOmega}_0-{Q}^{\prime}\frac{\omega \sin e}{2{\beta}_1}\right)\right],\\ {}{b}_1^{\prime }=L\left[\lambda \left(\frac{Q^{\prime}\varOmega }{2}+{P}^{\prime}\frac{\omega \sin e}{\beta_1}\right)+\varOmega \left({S}^{\prime}\varOmega +\frac{\omega \sin e{Q}^{\prime }}{2{\beta}_1}\right)\right],\\ {}{g}_1^{\prime }={L}^{\prime}\left[-\lambda \left(\frac{Q^{{\prime\prime}}\eta }{2}-\frac{\omega \sin f{P}^{{\prime\prime} }}{\beta_2}\right)+\eta \left(\eta {S}^{{\prime\prime} }-\frac{\omega \sin f{Q}^{{\prime\prime} }}{2{\beta}_2}\right)\right]\\ {}{\varOmega}_1=\frac{\omega \sin {e}_1}{\beta_1}\left(-\frac{Q^{\prime }}{2{S}^{\prime }}+\frac{1}{2}\sqrt{\frac{{Q^{\prime}}^2}{{S^{\prime}}^2}+\frac{4{P}^{\prime }}{S^{\prime }}\left\{\frac{1}{P^{\prime }{\sin}^2{e}_1}-1\right\}}\right)\\ {}{\varOmega}_1^{\prime }=\frac{\omega \sin {e}_1^{\prime }}{\beta_1}\left(-\frac{Q^{\prime }}{2{S}^{\prime }}+\frac{1}{2}\sqrt{\frac{{Q^{\prime}}^2}{{S^{\prime}}^2}+\frac{4{P}^{\prime }}{S^{\prime }}\left\{\frac{1}{P^{\prime }{\sin}^2{e}_1^{\prime }}-1\right\}}\right)\\ {}{\eta}_1^{\prime }=\frac{\omega \sin {f}_1^{\prime }}{\beta_2}\left(\frac{Q^{{\prime\prime} }}{2{S}^{{\prime\prime} }}+\frac{1}{2}\sqrt{\frac{{Q^{{\prime\prime}}}^2}{{S^{{\prime\prime}}}^2}+\frac{4{P}^{{\prime\prime} }}{S^{{\prime\prime} }}\left\{\frac{1}{P^{{\prime\prime} }{\sin}^2{f}_1^{\prime }}-1\right\}}\right)\end{array}} $$

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Gupta, S., Bhengra, N. & Ahmed, M. The study of reflection/transmission phenomena in a corrugated interface between two magnetoelastic transversely isotropic media. Arab J Geosci 11, 526 (2018). https://doi.org/10.1007/s12517-018-3883-x

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