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Coupled dilatational waves at a plane interface between two dissimilar magneto-elastic half-spaces containing voids

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Abstract

Reflection and transmission phenomenon due to a set of coupled dilatational waves incident obliquely at a plane interface between two different magneto-elastic half-spaces containing voids has been investigated. Using suitable boundary conditions, the equations providing the reflection and transmission coefficients corresponding to various reflected and transmitted waves are presented. Some new/earlier known results have been deduced from the present formulation. The effect of induced electric field due to applied magnetic field has been studied on the reflection and transmission coefficients. For a specific material, the amplitude ratios and corresponding energy ratios have been computed numerically and depicted graphically against the angle of incidence and frequency. The effects of various parameters have been noticed.

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Notes

  1. When both the half-spaces are of low conductivity, the potential form for the same can be obtained by replacing \(\alpha \) in Eq.(4.12) with \(\mathcal {A}.\)

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Authors and Affiliations

  1. Smt. Aruna Asaf Ali Government PG College, Kalka, Haryana, 133302, India

    Ashish Kumar

  2. Department of Mathematics, Panjab University, Chandigarh, 160014, India

    S. K. Tomar

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  1. Ashish Kumar
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  2. S. K. Tomar
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Correspondence to S. K. Tomar.

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Appendix

Appendix

Using Cramer’s rule, the reflection and refraction coefficients from the system of linear Eqs. (4.13) are obtained as

$$\begin{aligned} X_{i}=\frac{\Delta _{i}}{\Delta }, \quad (i=1,2,...,6) \end{aligned}$$
(4.1)

where

$$\begin{aligned} \Delta= & {} m_{31}[m_{31}\{(m_{23} + m_{26})\{(R_{1}L_{6} - R_{6}L_{1}) - (R_{2}L_{5} + R_{5}L_{2}) + (R_{3}L_{4} + R_{4}L_{3})\}\\{} & {} +(m_{13}+m_{16})\{(S_{4}L'_{3} +S_{3}L'_{4})-(S_{5}L'_{2}+S_{2}L'_{5})-(S_{6}L'_{1}+S_{1}L'_{6})\}\}\\{} & {} + (m_{33}+m_{36})\{(R_{2}S_{5} + R_{5}S_{2}) - (R_{3}S_{4} +R_{4}S_{3}) - ( R_{1}S_{6} + R_{6}S_{1})\}\\{} & {} +(m_{13}m_{26} - m_{16}m_{23})\{(Q_{2}L_{5} + Q_{5}L_{2}) -(Q_{1}L_{6} -Q_{6}L_{1}) - (Q_{4}L_{3} + Q_{3}L_{4})\}]\\{} & {} +(m_{23}m_{36} - m_{26}m_{33})\{(T_{2}Q_{5} + T_{5}Q_{2}) -(T_{1}Q_{6} - T_{6}Q_{1}) - (T_{3}Q_{4} + T_{4}Q_{3})\}\\{} & {} +(m_{13}m_{36} + m_{16}m_{33})\{S_{1} - S_{2} - S_{3} - S_{4} -S_{5} - S_{6}\},\\ \Delta _{1}= & {} m_{31}[m_{31}\{(m_{23} + m_{26})\{(R_{6}L_{1} +R_{1}L_{6}) + (R_{5}L_{2} - R_{2}L_{5}) - (R_{4}L_{3} -R_{3}L_{4})\}\\{} & {} +(m_{13} + m_{16})\{(S_{4}L''_{3} - S_{3}L'_{4}) + (S_{2}L'_{5} -S_{5}L''_{2}) - (S_{6}L''_{1} + S_{1}L'_{6})\}\}\\{} & {} +(m_{33} + m_{36})\{(R_{2}S_{5} - R_{5}S_{2}) + (R_{4}S_{3} -R_{3}S_{4}) - (R_{1}S_{6} - R_{6}S_{1})\}]\\{} & {} +(m_{13}m_{26} - m_{16}m_{23})\{(Q_{2}L_{5} - Q_{5}L_{2}) -(Q_{1}L_{6} + Q_{6}L_{1})+ (Q_{4}L_{3} - Q_{3}L_{4})\}\\{} & {} +(m_{23}m_{36} - m_{26}m_{33})\{(T'_{2}Q_{5} + T_{5}Q_{2}) -(T'_{3}Q_{4} + T_{4}Q_{3}) - (T'_{1}Q_{6} + T_{6}Q_{1})\}\\{} & {} +(m_{13}m_{36} + m_{16}m_{31})[S_{1} - S_{2} - S_{3} - S_{4} -S_{5} - S_{6}],\\ \Delta _{2}= & {} 2m_{31}[m_{31}\{(m_{23} + m_{26})\{R_{2}L_{3} -R_{3}L_{2}\} + m_{61}(m_{13} + m_{16})\{- S_{2} + S_{3} +S_{6}\}\}\\{} & {} +(m_{33} + m_{36})\{R_{3}S_{2} - R_{2}S_{3}\}+ (m_{13}m_{26} +m_{16}m_{23})\{Q_{3}L_{2} - Q_{2}L_{3}\}]\\{} & {} + 2[m_{51}(m_{23}m_{36} - m_{26}m_{33})\{m_{11}Q_{6} - m_{14}Q_{3}+ m_{15}Q_{2}\}\\{} & {} -(m_{13}m_{36}+ m_{16}m_{33})\{Q_{2}S_{3} + Q_{3}S_{2}\}],\\ \Delta _{3}= & {} 2m_{31}[(m_{16} + m_{31})\{(R_{2} + Q_{2})\{S_{1} +S_{3} - S_{5}\} + (R_{1} + Q_{1})\{- S_{2} + S_{3} + S_{6}\}\\{} & {} +(R_{3} + Q_{3})\{- S_{1} - S_{2} + S_{4}\}\}+m_{26}\{m_{11}\{-Q_{4}L_{3} + Q_{5}L_{2} + Q_{6}L_{1}\}\\{} & {} + m_{12}\{Q_{2}L_{3} - Q_{3}L_{2}\} - m_{14}\{Q_{1}L_{3} +Q_{3}L_{1}\} + m_{15}\{Q_{1}L_{2} + Q_{2}L_{1}\}\}]\\{} & {} + 2m_{36}[m_{15}\{Q_{1}S_{2} - Q_{2}S_{1}\} -m_{11}\{-Q_{4}S_{3} +Q_{5}S_{2} - Q_{6}S_{1}\} + m_{12}\{Q_{2}S_{3} - Q_{3}S_{2}\}\\{} & {} - m_{14}\{Q_{1}S_{3} - Q_{3}S_{1}\}],\\ \Delta _{4}= & {} -2m_{31}[m_{31}\{(m_{23} + m_{26})\{R_{3}L_{1} + R_{1}S_{3}\} + m_{61}(m_{13} + m_{16})\{S_{1} + S_{3} - S_{5}\}\}\\{} & {} +(m_{33} + m_{36})\{R_{1}S_{3} - R_{3}S_{1}\}+ (m_{13}m_{26} - m_{16}m_{23}) \{Q_{1}L_{3} + Q_{3}L_{1}\}]\\{} & {} +2[m_{51}(m_{23}m_{36} - m_{26}m_{33})\{- m_{11}Q_{4} + m_{12}Q_{3} - m_{15}Q_{1}\}\\{} & {} + (m_{16}m_{33} - m_{13}m_{36})\{Q_{1}S_{3} - Q_{3}S_{1}\}],\\ \Delta _{5}= & {} 2m_{31}[m_{31}\{(m_{23} + m_{26})\{R_{1}L_{2} + R_{2}L_{1}\} + m_{61}(m_{13} + m_{16})\{- S_{1} - S_{2} + S_{4}\}\}\\{} & {} +(m_{33} + m_{36})\{R_{2}S_{1} - R_{1}S_{2}\} +(m_{16}m_{23} - m_{13}m_{26})\{Q_{1}L_{2} + Q_{2}L_{1}\}]\\{} & {} +2[m_{51}(m_{23}m_{36} - m_{26}m_{33})\{m_{11}Q_{4} - m_{12}Q_{2} + m_{14}Q_{1}\}\\{} & {} - (m_{16}m_{33} - m_{13}m_{36})\{Q_{1}S_{2} - Q_{2}S_{1}\}],\\ \Delta _{6}= & {} 2 m_{31}[(m_{13} + m_{31})\{(R_{2} - Q_{2})(S_{1} - S_{5} + S_{3}) - (R_{1} - Q_{1})(S_{2} - S_{3} - S_{6})\\{} & {} -(R_{3} - Q_{3})(S_{1} + S_{2} - S_{4})\}+ m_{23}\{m_{11}(Q_{4}L_{3} - Q_{5}L_{2} - Q_{6}L_{1})\\{} & {} + m_{12}(Q_{3}L_{2} - Q_{2}L_{3}) +m_{14}(Q_{1}L_{3} + Q_{3}L_{1}) - m_{15}(Q_{1}L_{2} + Q_{2}L_{1})\}]\\{} & {} - 2m_{33}[m_{11}(Q_{4}S_{3} - Q_{5}S_{2} + Q_{6}S_{1}) +m_{12}(Q_{3}S_{2} - Q_{2}S_{3})\\{} & {} + m_{14}(Q_{1}S_{3} + Q_{3}S_{1}) - m_{15}(Q_{1}S_{2} + Q_{2}S_{1})], \\{} & {} S_{1}=m_{21}m_{52} - m_{22}m_{51}, \quad S_{2}=m_{21}m_{54} - m_{24}m_{51}, \quad S_{3}=m_{21}m_{55} - m_{25}m_{51},\\{} & {} S_{4}=m_{22}m_{54} - m_{24}m_{52}, \quad S_{5}=m_{22}m_{55} - m_{25}m_{52}, \quad S_{6}=m_{24}m_{55} - m_{25}m_{54},\\{} & {} R_{1}=m_{11}m_{62} - m_{12}m_{61}, \quad R_{2}=m_{11}m_{64} - m_{14}m_{61}, \quad R_{3}=m_{11}m_{65} - m_{15}m_{61},\\{} & {} R_{4}=m_{12}m_{64} - m_{14}m_{62}, \quad R_{5}=m_{12}m_{65} - m_{15}m_{62}, \quad R_{6}=m_{14}m_{65} - m_{15}m_{64}\\{} & {} T_{1}=m_{11}m_{52} - m_{12}m_{51}, \quad T_{2}=m_{11}m_{54} - m_{14}m_{51}, \quad T_{3}=m_{11}m_{55} - m_{15}m_{51},\\{} & {} T_{4}=m_{12}m_{54} - m_{14}m_{52}, \quad T_{5}=m_{12}m_{55} - m_{15}m_{52}, \quad T_{6}=m_{14}m_{55} - m_{15}m_{54},\\{} & {} Q_{1}=m_{41}m_{62} - m_{42}m_{61}, \quad Q_{2}=m_{41}m_{64} - m_{44}m_{61}, \quad Q_{3}=m_{41}m_{65} - m_{45}m_{61},\\{} & {} Q_{4}=m_{42}m_{64} - m_{44}m_{62}, \quad Q_{5}=m_{42}m_{65} - m_{45}m_{62}, \quad Q_{6}=m_{44}m_{65} - m_{45}m_{64},\\{} & {} L_{1}=m_{51} - m_{52}, \quad L_{2}=m_{51} + m_{54}, \quad L_{3}=m_{51} + m_{55}, \quad L_{4}=m_{52} + m_{54},\\{} & {} L_{5}=m_{52} + m_{55}, \quad L_{6}=m_{54} - m_{55}, \quad L'_{1}=m_{61} - m_{62}, \quad L'_{2}=m_{61} + m_{64},\\{} & {} L'_{3}=m_{61} + m_{65}, \quad L'_{4}=m_{62} + m_{64}, \quad L'_{5}=m_{62} + m_{65}, \quad L'_{6}=m_{64} - m_{65},\\{} & {} L''_{1}=m_{61} + m_{62}, \qquad L''_{2}=m_{61} - m_{64}, \qquad L''_{3}=m_{61} - m_{65}. \end{aligned}$$

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Kumar, A., Tomar, S.K. Coupled dilatational waves at a plane interface between two dissimilar magneto-elastic half-spaces containing voids. Acta Mech 233, 5061–5087 (2022). https://doi.org/10.1007/s00707-022-03353-w

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