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Analysis of Quasi waves in Orthotropic Layer Bonded Between Piezoelectric Half-Spaces with Imperfect and Sliding Interfaces

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Abstract

Purpose

The objective of present work is to investigate seismic wave reflection/transmission in an orthotropic layer bonded between two piezoelectric half-spaces in presence of an imperfect corrugated superstratum and sliding lower interface.

Methods

Closed form analytical expressions for displacement field, electric potential, and velocity profile of quasi seismic waves are deduced for respective geomedia using time harmonic plane wave solution approach with appropriate boundary conditions. Rayleigh’s first-order approximation, generalized Snell’s law and spectrum theorem are used to determine the analytical solution for reflection, transmission (RT) coefficients for regularly and irregularly scattered waves.

Results

Numerical simulations for specific geophysical model have been carried out and results are presented through multiple plots to analyze the impact of incident angle, dimensionless layer width, sliding parameter, and corrugated boundary on resultant velocity profile and RT coefficients. The energy flux ratios of all regularly and irregularly propagating waves are also analyzed and portrayed against angle of incidence (AOI).

Conclusions

Reflection and transmission angle for all the quasi-waves are found monotonically increasing with rise in incident angle. Waves and material parameters are found to have different impact on resultant velocity and RT coefficients. Simulation also reveals the effect of two extrema of sliding parameter corresponding to welded and free contact on RT coefficients. The coefficients obtained by first-order approximation to corrugation are found to have proportional impact of corrugation amplitude.

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Data availability

All data generated or analysed during this study are included in this published article.

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Acknowledgements

The authors gratefully acknowledge the reviewers for their valuable suggestions and insightful comments which leads to enormous improvement in the paper.

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The authors have no relevant financial or non-financial interests to disclose. All authors certify that they have no affiliations with or involvement in any organization or entity with any financial interest or non-financial interest in the subject matter or materials discussed in this manuscript. The authors have no financial or proprietary interests in any material discussed in this article.

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  1. Jaypee Institute of Information Technology, Noida, 201309, India

    Pato Kumari & Rupali Srivastava

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  1. Pato Kumari
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Appendix

Appendix

Appendix 1: The expressions for \({\kappa }_{i}\),\({\kappa }_{jn}\) and \({\kappa }_{jn}^{^{\prime}}\),\(i=0,\dots ,2\) and \(j=3,\dots ,6\) and subexpressions for Eqs. (44), (47) and (48)

$${\kappa }_{0}=\frac{1}{\left({e}_{15}^{^{\prime}}{\mathrm{sin}}^{2}{\theta }_{0}+{e}_{33}^{^{\prime}}{\mathrm{cos}}^{2}{\theta }_{0}\right)}\left[{{\rho }_{3}{c}_{0}^{2}-C}_{44}^{^{\prime}}{\mathrm{sin}}^{2}{\theta }_{0}-{C}_{33}^{^{\prime}}{\mathrm{cos}}^{2}{\theta }_{0}+\left({C}_{13}^{^{\prime}}+{C}_{44}^{^{\prime}}\right){m}_{0}sin{\theta }_{0}cos{\theta }_{0}\right]$$
$${\kappa }_{i}=\frac{1}{\left({e}_{15}^{^{\prime}}{\mathrm{sin}}^{2}{\theta }_{i}+{e}_{33}^{^{\prime}}{\mathrm{cos}}^{2}{\theta }_{i}\right)}\left[{{\rho }_{3}{c}_{i}^{2}-C}_{44}^{^{\prime}}{\mathrm{sin}}^{2}{\theta }_{i}-{C}_{33}^{^{\prime}}{\mathrm{cos}}^{2}{\theta }_{i}-\left({C}_{13}^{^{\prime}}+{C}_{44}^{^{\prime}}\right){m}_{0}sin{\theta }_{i}cos{\theta }_{i}\right] (i=1 ,2)$$
$${\kappa }_{j}=\frac{1}{\left({e}_{15}{\mathrm{sin}}^{2}{\theta }_{i}+{e}_{33}{\mathrm{cos}}^{2}{\theta }_{i}\right)}\left[{\rho }_{1}{c}_{i}^{2}-{C}_{44}{\mathrm{sin}}^{2}{\theta }_{i}-{C}_{33}{\mathrm{cos}}^{2}{\theta }_{i}+\left({C}_{13}+{C}_{44}\right){m}_{0}sin{\theta }_{i}cos{\theta }_{i}\right]$$
$${\kappa }_{jn}=\frac{1}{\left({e}_{15}{\mathrm{sin}}^{2}{\theta }_{in}+{e}_{33}{\mathrm{cos}}^{2}{\theta }_{in}\right)}\left[{\rho }_{1}{c}_{i}^{2}-{C}_{44}{\mathrm{sin}}^{2}{\theta }_{in}-{C}_{33}{\mathrm{cos}}^{2}{\theta }_{in}+\left({C}_{13}+{C}_{44}\right){m}_{0}sin{\theta }_{in}cos{\theta }_{in}\right]$$
$${\kappa }_{jn}^{^{\prime}}=\frac{1}{\left({e}_{15}{\mathrm{sin}}^{2}{\theta }_{in}^{^{\prime}}+{e}_{33}{\mathrm{cos}}^{2}{\theta }_{in}^{^{\prime}}\right)}\left[{\rho }_{1}{c}_{i}^{2}-{C}_{44}{\mathrm{sin}}^{2}{\theta }_{in}^{^{\prime}}-{C}_{33}{\mathrm{cos}}^{2}{\theta }_{in}^{^{\prime}}+\left({C}_{13}+{C}_{44}\right){m}_{0}sin{\theta }_{in}^{^{\prime}}cos{\theta }_{in}^{^{\prime}}\right]$$

\(j=3\, \mathrm{to}\, 6; i=9 \mathrm{to}\) \(12\) respectively. (i.e. for \(j=3\), \(i=\) 9, for \(j=4\), \(i=10,\) and so on).

$${s}_{0}={C}_{11}^{^{\prime}}{m}_{0}-{C}_{13}^{^{\prime}}{R}_{0}+{e}_{31}^{^{\prime}}{\kappa }_{0}{R}_{0}, {s}_{1}={C}_{13}^{^{\prime}}{m}_{0}-{C}_{33}^{^{\prime}}{R}_{0}+{e}_{33}^{^{\prime}}{\kappa }_{0}{R}_{0}$$
$${\gamma }_{1}=\frac{{C}_{13}^{\mathrm{^{\prime}}}{m}_{1}{S}_{0}+{C}_{33}^{\mathrm{^{\prime}}}{R}_{1}+{e}_{33}^{\mathrm{^{\prime}}}{\kappa }_{1}{R}_{1}}{{C}_{13}^{\mathrm{^{\prime}}}{m}_{0}-{C}_{33}^{\mathrm{^{\prime}}}{R}_{0}+{e}_{33}^{\mathrm{^{\prime}}}{\kappa }_{0}{R}_{0}},{\gamma }_{2}=\frac{{C}_{13}^{\mathrm{^{\prime}}}{m}_{2}{S}_{0}+{C}_{33}^{\mathrm{^{\prime}}}{R}_{2}+{e}_{33}^{\mathrm{^{\prime}}}{\kappa }_{2}{R}_{2}}{{C}_{13}^{\mathrm{^{\prime}}}{m}_{0}-{C}_{33}^{\mathrm{^{\prime}}}{R}_{0}+{e}_{33}^{\mathrm{^{\prime}}}{\kappa }_{0}{R}_{0}},$$
$${\gamma }_{3}=\frac{-\sigma ({S}_{31}{m}_{3}{S}_{0}-{S}_{33}{R}_{3})}{{C}_{13}^{\mathrm{^{\prime}}}{m}_{0}-{C}_{33}^{\mathrm{^{\prime}}}{R}_{0}+{e}_{33}^{\mathrm{^{\prime}}}{\kappa }_{0}{R}_{0}}, {\gamma }_{4}=-\frac{\sigma \left({S}_{31}{m}_{4}{S}_{0}-{S}_{33}{R}_{4}\right)}{{C}_{13}^{\mathrm{^{\prime}}}{m}_{0}-{C}_{33}^{\mathrm{^{\prime}}}{R}_{0}+{e}_{33}^{\mathrm{^{\prime}}}{\kappa }_{0}{R}_{0}},$$
$${\gamma }_{5}={C}_{11}^{\mathrm{^{\prime}}}{m}_{1}+{C}_{13}^{\mathrm{^{\prime}}}{R}_{1}+{e}_{31}^{\mathrm{^{\prime}}}{\kappa }_{1}{R}_{1}, {\gamma }_{6}={C}_{11}^{\mathrm{^{\prime}}}{m}_{2}+{C}_{13}^{\mathrm{^{\prime}}}{R}_{2}+{e}_{31}^{\mathrm{^{\prime}}}{\kappa }_{2}{R}_{2}$$
$${\gamma }_{7}={C}_{13}^{\mathrm{^{\prime}}}{m}_{1}+{C}_{33}^{\mathrm{^{\prime}}}{R}_{1}+{e}_{33}^{\mathrm{^{\prime}}}{\kappa }_{1}{R}_{1}, {\gamma }_{8}={C}_{13}^{\mathrm{^{\prime}}}{m}_{2}+{C}_{33}^{\mathrm{^{\prime}}}{R}_{2}+{e}_{33}^{\mathrm{^{\prime}}}{\kappa }_{2}{R}_{2},\sigma =\left(1-\zeta \right)\alpha$$
$${\gamma }_{9}=\frac{{m}_{5}{R}_{5}+{S}_{0}}{{-m}_{3}{R}_{3}+{S}_{0}},{\gamma }_{10}=\frac{{m}_{6}{R}_{6}+{S}_{0}}{{-m}_{3}{R}_{3}+{S}_{0}},{\gamma }_{11}=\frac{{C}_{44}({m}_{9}{R}_{9}-{S}_{0})}{{S}_{55}({-m}_{3}{R}_{3}+{S}_{0})},{\gamma }_{12}=\frac{{C}_{44}({m}_{10}{R}_{10}-{S}_{0})}{{S}_{55}({-m}_{3}{R}_{3}+{S}_{0})}$$
$${\gamma }_{13}=\frac{{S}_{31}{m}_{5}{S}_{0}+{S}_{33}{R}_{5}}{{S}_{31}{m}_{3}{S}_{0}-{S}_{33}{R}_{3}} , {\gamma }_{14}=\frac{{S}_{31}{m}_{6}{S}_{0}+{S}_{33}{R}_{6}}{{S}_{31}{m}_{3}{S}_{0}-{S}_{33}{R}_{3}}, {\gamma }_{15}=\frac{{S}_{31}{m}_{9}{S}_{0}+{S}_{33}{R}_{9}}{{S}_{31}{m}_{3}{S}_{0}-{S}_{33}{R}_{3}}, {\gamma }_{16}=\frac{{S}_{31}{m}_{10}{S}_{0}+{S}_{33}{R}_{10}}{{S}_{31}{m}_{3}{S}_{0}-{S}_{33}{R}_{3}}$$
$${\gamma }_{17}=\frac{{m}_{7}{R}_{7}+{S}_{0}}{{-m}_{4}{R}_{4}+{S}_{0}},{\gamma }_{18}=\frac{{m}_{8}{R}_{8}+{S}_{0}}{{-m}_{4}{R}_{4}+{S}_{0}},{\gamma }_{19}=\frac{{C}_{44}({m}_{11}{R}_{11}-{S}_{0})}{{S}_{55}({-m}_{4}{R}_{4}+{S}_{0})},{\gamma }_{20}=\frac{{C}_{44}({m}_{12}{R}_{12}-{S}_{0})}{{S}_{55}({-m}_{4}{R}_{4}+{S}_{0})}$$
$${\gamma }_{21}=\frac{{S}_{31}{m}_{7}{S}_{0}+{S}_{33}{R}_{7}}{{S}_{31}{m}_{4}{S}_{0}-{S}_{33}{R}_{3}} , {\gamma }_{22}=\frac{{S}_{31}{m}_{8}{S}_{0}+{S}_{33}{R}_{8}}{{S}_{31}{m}_{4}{S}_{0}-{S}_{33}{R}_{3}},{\gamma }_{23}=\frac{{S}_{31}{m}_{11}{S}_{0}+{S}_{33}{R}_{11}}{{S}_{31}{m}_{4}{S}_{0}-{S}_{33}{R}_{4}}, {\gamma }_{24}=\frac{{S}_{31}{m}_{12}{S}_{0}+{S}_{33}{R}_{12}}{{S}_{31}{m}_{4}{S}_{0}-{S}_{33}{R}_{4}}$$

Appendix 2: Subexpressions used in Eqs. (52) and (53)

$${\gamma }_{9}^{\mathrm{^{\prime}}}={S}_{55}\left({m}_{5n}{R}_{5n}+{S}_{0}+nq\right){e}^{-i{R}_{5n}h},{\gamma }_{10}^{\mathrm{^{\prime}}}={S}_{55}({m}_{6n}{R}_{6n}+{S}_{0}+nq){e}^{-i{R}_{6n}h}$$
$${\gamma }_{11}^{\mathrm{^{\prime}}}={C}_{44}\left(-{m}_{9n}{R}_{9n}+{S}_{0}+nq\right){e}^{i{R}_{9n}h},{\gamma }_{12}^{\mathrm{^{\prime}}}={C}_{44}(-{m}_{10n}{R}_{10n}+{S}_{0}+nq){e}^{i{R}_{10n}h}$$
$${\gamma }_{13}^{\mathrm{^{\prime}}}={(S}_{31}{m}_{5n}{S}_{0}+{S}_{31}{m}_{5n}nq+{S}_{33}{R}_{5n}){e}^{-i{R}_{5n}h},{\gamma }_{14}^{\mathrm{^{\prime}}}={{(S}_{31}{m}_{6n}{S}_{0}+{S}_{31}{m}_{6n}nq+{S}_{33}R}_{6n}){e}^{-i{R}_{6n}h}$$
$$\gamma_{15}^{^{\prime}} = (C_{13} m_{9n} S_{0} + C_{13} m_{9n} nq - C_{33} R_{9n} - e_{33} \kappa_{3n} R_{9n} )e^{{iR_{9n} h}}$$
$${\gamma }_{16}^{^{\prime}}={{(C}_{13}{m}_{10n}{S}_{0}+{C}_{13}{m}_{10n}{S}_{0}nq-{C}_{33}R}_{10n}-{e}_{33}{\kappa }_{4n}{R}_{10n}){e}^{i{R}_{10n}h}$$
$${f}_{1}=-i{\xi }_{n}\left({\kappa }_{3}\frac{{I}_{3}}{{C}_{3}}{e}^{i{R}_{9}h}{R}_{9}+{\kappa }_{4}\frac{{J}_{3}}{{C}_{3}}{e}^{i{R}_{10}h}{R}_{10}\right)$$
$${f}_{2}=i{\xi }_{n}\left({m}_{3}{e}^{i{R}_{3}h}{R}_{3}+{m}_{5}\frac{{E}_{3}}{{C}_{3}}{e}^{-i{R}_{5}h}{R}_{5}+{m}_{6}\frac{{F}_{3}}{{C}_{3}}{e}^{-i{R}_{6}h}{R}_{6}-{m}_{9}\frac{{I}_{3}}{{C}_{3}}{e}^{i{R}_{9}h}{R}_{9}- {m}_{10}\frac{{J}_{3}}{{C}_{3}}{e}^{i{R}_{10}h}{R}_{10}\right)$$
$${f}_{3}=i{\xi }_{n}\left[{\alpha }_{1}{e}^{i{R}_{3}h}+{\alpha }_{2}{e}^{-i{R}_{5}h}\frac{{E}_{3}}{{C}_{3}}+{\alpha }_{3}{e}^{-i{R}_{6}h}\frac{{F}_{3}}{{C}_{3}}+{\alpha }_{4}{e}^{i{R}_{9}h}\frac{{I}_{3}}{{C}_{3}}+{\alpha }_{5}{e}^{i{R}_{10}h}\frac{{J}_{3}}{{C}_{3}}\right]$$
$${\alpha }_{1}=-{S}_{55}\left({R}_{3}^{2}{m}_{3}-{R}_{3}{S}_{0}\right)+\left({S}_{31}{-S}_{11}\right){m}_{3}{S}_{0}nq+\left({S}_{33}{-S}_{31}\right){R}_{3}nq$$
$${\alpha }_{2}=-{S}_{55}\left({R}_{5}^{2}{m}_{5}+{R}_{5}{S}_{0}\right)+\left({S}_{31}{-S}_{11}\right){m}_{5}{S}_{0}nq+\left({S}_{33}{-S}_{31}\right){R}_{5}nq$$
$${\alpha }_{3}=-{S}_{55}\left({R}_{6}^{2}{m}_{6}+{R}_{6}{S}_{0}\right)+\left({S}_{31}{-S}_{11}\right){m}_{6}{S}_{0}nq+\left({S}_{33}{-S}_{31}\right){R}_{6}nq$$
$${\alpha }_{4}={C}_{44}\left({R}_{9}^{2}{m}_{9}-{R}_{9}{S}_{0}\right)-{e}_{15}{\kappa }_{3}{S}_{0}{R}_{9}-\left({C}_{31}{-C}_{11}\right){m}_{9}{S}_{0}nq+\left({C}_{33}{-C}_{31}\right){R}_{9}nq+\left({e}_{33}{-e}_{31}\right){R}_{9}nq{\kappa }_{3},{\alpha }_{5}={C}_{44}\left({R}_{10}^{2}{m}_{10}-{R}_{10}{S}_{0}\right)-{e}_{15}{\kappa }_{4}{S}_{0}{R}_{10}-\left({C}_{31}{-C}_{11}\right){m}_{10}{S}_{0}nq+\left({C}_{33}{-C}_{31}\right){R}_{10}nq{\kappa }_{3} +\left({e}_{33}{-e}_{31}\right){R}_{10}nq{\kappa }_{4}$$
$${f}_{4}=i{\xi }_{n}\left[{\alpha }_{6}{e}^{i{R}_{3}h}+{\alpha }_{7}{e}^{-i{R}_{5}h}\frac{{E}_{3}}{{C}_{3}}+{\alpha }_{8}{e}^{-i{R}_{6}h}\frac{{F}_{3}}{{C}_{3}}+{\alpha }_{9}{e}^{i{R}_{9}h}\frac{{I}_{3}}{{C}_{3}}+{\alpha }_{10}{e}^{i{R}_{10}h}\frac{{J}_{3}}{{C}_{3}}\right]$$
$${\alpha }_{6}={S}_{31}{m}_{3}{R}_{3}{S}_{0}-{S}_{33}{R}_{3}^{2}-2{S}_{55}nq\left(-{R}_{3}{m}_{3}+{S}_{0}\right), {\alpha }_{7}={-S}_{31}{m}_{5}{R}_{5}{S}_{0}-{S}_{33}{R}_{5}^{2}-2{S}_{55}nq\left({R}_{5}{m}_{5}+{S}_{0}\right),{\alpha }_{8}={-S}_{31}{m}_{6}{R}_{6}{S}_{0}-{S}_{33}{R}_{6}^{2}-2{S}_{55}nq\left({R}_{6}{m}_{6}+{S}_{0}\right)$$
$${\alpha }_{9}={-C}_{31}{m}_{9}{R}_{9}{S}_{0}+{C}_{33}{R}_{9}^{2}+{e}_{33}{\kappa }_{3}{R}_{9}^{2}+2nq\left\{{C}_{44}\left(-{R}_{9}{m}_{9}+{S}_{0}\right)+{e}_{15}{\kappa }_{3}{S}_{0})\right\}$$
$${\alpha }_{10}={-C}_{31}{m}_{10}{R}_{10}{S}_{0}+{C}_{33}{R}_{10}^{2}+{e}_{33}{\kappa }_{4}{R}_{10}^{2}+2nq\left\{{C}_{44}\left(-{R}_{10}{m}_{10}+{S}_{0}\right)+{e}_{15}{\kappa }_{4}{S}_{0})\right\}$$
$$\gamma_{9}^{{^{\prime\prime}}} = S_{55} \left( {m^{\prime}_{5n} R^{\prime}_{5n} + S_{0} - nq} \right)e^{{ - iR_{5n}^{^{\prime}} h}} ,\gamma_{10}^{{^{\prime\prime}}} = S_{55} \left( {m_{6n}^{^{\prime}} R{^{\prime}}_{6n} + S_{0} - nq} \right)e^{{ - iR_{6n}^{^{\prime}} h}}$$
$$\gamma_{11}^{{^{\prime\prime}}} = C_{44} \left( { - m_{9n}^{^{\prime}} R^{\prime}_{9n} + S_{0} - nq} \right)e^{{iR_{9n}^{^{\prime}} h}} ,\gamma_{12}^{{^{\prime\prime}}} = C_{44} \left( { - m_{10n}^{^{\prime}} R{^{\prime}}_{10n} + S_{0} - nq} \right)e^{{iR_{10n}^{^{\prime}} h}}$$
$$\gamma_{13}^{{^{\prime\prime}}} = (S_{31} m_{5n}^{^{\prime}} S_{0} - S_{31} m_{5n}^{^{\prime}} nq + S_{33} R^{\prime}_{5n} )e^{{ - iR_{5n}^{^{\prime}} h}} ,\gamma_{14}^{{^{\prime\prime}}} = (S_{31} m_{6n}^{^{\prime}} S_{0} - S_{31} m_{6n}^{^{\prime}} nq + S_{33} R{^{\prime}}_{6n} )e^{{ - iR_{6n}^{^{\prime}} h}}$$
$$\gamma_{15}^{^{\prime\prime}} = (C_{13} m_{9n}^{^{\prime}} S_{0} - C_{13} m_{9n}^{^{\prime}} nq - C_{33} R^{\prime}_{9n} - e_{33} \kappa_{3n}^{^{\prime}} R^{\prime}_{9n} )e^{{iR_{9n}^{^{\prime}} h}}$$
$$\gamma_{16}^{^{\prime\prime}} = (C_{13} m_{10n}^{^{\prime}} S_{0} - C_{13} m_{10n}^{^{\prime}} nq - C_{33} R^{\prime}_{10n} - e_{33} \kappa_{4n}^{^{\prime}} R^{\prime}_{10n} )e^{{iR_{10n}^{^{\prime}} h}}$$
$${f}_{5}=i{\xi }_{-n}\left({\kappa }_{3}\frac{{I}_{3}}{{C}_{3}}{e}^{i{R}_{9}h}{R}_{9}+{\kappa }_{4}\frac{{J}_{3}}{{C}_{3}}{e}^{i{R}_{10}h}{R}_{10}\right)$$
$${f}_{6}=i{\xi }_{-n}\left({m}_{3}{e}^{i{R}_{3}h}{R}_{3}+{m}_{5}\frac{{E}_{3}}{{C}_{3}}{e}^{-i{R}_{5}h}{R}_{5}+{m}_{6}\frac{{F}_{3}}{{C}_{3}}{e}^{-i{R}_{6}h}{R}_{6}-{m}_{9}\frac{{I}_{3}}{{C}_{3}}{e}^{i{R}_{9}h}{R}_{9}- {m}_{10}\frac{{J}_{3}^{^{\prime}}}{{C}_{3}}{e}^{i{R}_{10}h}{R}_{10}\right)$$
$${f}_{7}=i{\xi }_{-n}\left[{\alpha }_{1}^{^{\prime}}{e}^{i{R}_{3}h}+{\alpha }_{2}^{^{\prime}}{e}^{-i{R}_{5}h}\frac{{E}_{3}}{{C}_{3}}+{\alpha }_{3}^{^{\prime}}{e}^{-i{R}_{6}h}\frac{{F}_{3}}{{C}_{3}}+{\alpha }_{4}^{^{\prime}}{e}^{i{R}_{9}h}\frac{{I}_{3}}{{C}_{3}}+{\alpha }_{5}^{^{\prime}}{e}^{i{R}_{10}h}\frac{{J}_{3}}{{C}_{3}}\right]$$
$${\alpha }_{1}^{^{\prime}}=-{S}_{55}\left({R}_{3}^{2}{m}_{3}-{R}_{3}{S}_{0}\right)-\left({S}_{31}{-S}_{11}\right){m}_{3}{S}_{0}nq-\left({S}_{33}{-S}_{31}\right){R}_{3}nq$$
$${\alpha }_{2}^{^{\prime}}=-{S}_{55}\left({R}_{5}^{2}{m}_{5}+{R}_{5}{S}_{0}\right)-\left({S}_{31}{-S}_{11}\right){m}_{5}{S}_{0}nq-\left({S}_{33}{-S}_{31}\right){R}_{5}nq$$
$${\alpha }_{3}^{^{\prime}}=-{S}_{55}\left({R}_{6}^{2}{m}_{6}+{R}_{5}{S}_{0}\right)-\left({S}_{31}{-S}_{11}\right){m}_{6}{S}_{0}nq-\left({S}_{33}{-S}_{31}\right){R}_{6}nq$$
$${\alpha }_{4}^{\mathrm{^{\prime}}}={C}_{44}\left({R}_{9}^{2}{m}_{9}-{R}_{9}{S}_{0}\right)-{e}_{15}{\kappa }_{3}{S}_{0}{R}_{9}+\left({C}_{31}{-C}_{11}\right){m}_{9}{S}_{0}nq-\left({C}_{33}{-C}_{31}\right){R}_{9}nq-\left({e}_{33}{-e}_{31}\right){R}_{9}nq{\kappa }_{3},{\alpha }_{5}^{\mathrm{^{\prime}}}={C}_{44}\left({R}_{10}^{2}{m}_{10}-{R}_{10}{S}_{0}\right)-{e}_{15}{\kappa }_{4}{S}_{0}{R}_{10}+\left({C}_{31}{-C}_{11}\right){m}_{10}{S}_{0}nq-\left({C}_{33}{-C}_{31}\right){R}_{10}nq-\left({e}_{33}{-e}_{31}\right){R}_{10}nq{\kappa }_{4}$$
$${f}_{8}=i{\xi }_{-n}\left[{\alpha }_{6}^{^{\prime}}{e}^{i{R}_{3}h}+{\alpha }_{7}^{^{\prime}}{e}^{-i{R}_{5}h}\frac{{E}_{3}}{{C}_{3}}+{\alpha }_{8}^{^{\prime}}{e}^{-i{R}_{6}h}\frac{{F}_{3}}{{C}_{3}}+{\alpha }_{9}^{^{\prime}}{e}^{i{R}_{9}h}\frac{{I}_{3}}{{C}_{3}}+{\alpha }_{10}^{^{\prime}}{e}^{i{R}_{10}h}\frac{{J}_{3}}{{C}_{3}}\right]$$
$${\alpha }_{6}^{\mathrm{^{\prime}}}={S}_{31}{m}_{3}{R}_{3}{S}_{0}-{S}_{33}{R}_{3}^{2}+2{S}_{55}nq\left(-{R}_{3}{m}_{3}+{S}_{0}\right),{\alpha }_{7}^{\mathrm{^{\prime}}}={-S}_{31}{m}_{5}{R}_{5}{S}_{0}-{S}_{33}{R}_{5}^{2}+2{S}_{55}nq\left({R}_{5}{m}_{5}+{S}_{0}\right),{\alpha }_{8}^{\mathrm{^{\prime}}}={-S}_{31}{m}_{6}{R}_{6}{S}_{0}-{S}_{33}{R}_{6}^{2}+2{S}_{55}nq\left({m}_{6}{R}_{6}+{S}_{0}\right)$$
$${\alpha }_{9}^{\mathrm{^{\prime}}}={-C}_{31}{m}_{9}{R}_{9}{S}_{0}+{C}_{33}{R}_{9}^{2}+{e}_{33}{\kappa }_{3}{R}_{9}^{2}-2nq\left\{{C}_{44}\left(-{R}_{9}{m}_{9}+{S}_{0}\right)+{e}_{15}{\kappa }_{3}{S}_{0})\right\},$$
$${\alpha }_{10}^{^{\prime}}={-C}_{31}{m}_{10}{R}_{10}{S}_{0}+{C}_{33}{R}_{10}^{2}+{e}_{33}{\kappa }_{4}{R}_{10}^{2}-2nq\left\{{C}_{44}\left(-{R}_{10}{m}_{10}+{S}_{0}\right)+{e}_{15}{\kappa }_{4}{S}_{0})\right\}$$

Appendix 3: Subexpressions used in Eqs. (54) and (55)

$${\gamma }_{17}^{^{\prime}}={S}_{55}\left({m}_{7n}{R}_{7n}+{S}_{0}+nq\right){e}^{-i{R}_{7n}h}, {\gamma }_{18}^{^{\prime}}={S}_{55}({m}_{8n}{R}_{8n}+{S}_{0}+nq){e}^{-i{R}_{8n}h}$$
$${\gamma }_{19}^{\mathrm{^{\prime}}}={C}_{44}\left(-{m}_{11n}{R}_{11n}+{S}_{0}+nq\right){e}^{i{R}_{11n}h},{\gamma }_{20}^{\mathrm{^{\prime}}}={C}_{44}(-{m}_{12n}{R}_{12n}+{S}_{0}+nq){e}^{i{R}_{12n}h}$$
$${\gamma }_{21}^{\mathrm{^{\prime}}}={(S}_{31}{m}_{7n}{S}_{0}+{S}_{13}{m}_{7n}nq+{S}_{33}{R}_{5n}){e}^{-i{R}_{5n}h},{\gamma }_{22}^{\mathrm{^{\prime}}}={{(S}_{31}{m}_{8n}{S}_{0}+nq{S}_{31}{m}_{8n}+{S}_{33}R}_{8n}){e}^{-i{R}_{8n}h}$$
$${\gamma }_{23}^{^{\prime}}={{(C}_{13}{m}_{11n}{S}_{0}+nq{C}_{13}{m}_{11n}-{C}_{33}R}_{11n}-{e}_{33}{\kappa }_{3n}{R}_{11n}){e}^{i{R}_{11n}h}$$
$${\gamma }_{24}^{^{\prime}}={{(C}_{13}{m}_{12n}{S}_{0}+nq{C}_{13}{m}_{12n}-{C}_{33}R}_{12n}-{e}_{33}{\kappa }_{4n}{R}_{12n}){e}^{i{R}_{12n}h}$$
$${g}_{1}=-i{\xi }_{-n}\left({\kappa }_{5}\frac{{L}_{3}}{{D}_{3}}{e}^{i{R}_{11}h}{R}_{11}+{\kappa }_{6}\frac{{M}_{3}}{{D}_{3}}{e}^{i{R}_{12}h}{R}_{12}\right)$$
$${g}_{2}=i{\xi }_{-n}\left({m}_{4}{e}^{i{R}_{4}h}{R}_{4}+{m}_{7}\frac{{G}_{3}}{{D}_{3}}{e}^{-i{R}_{7}h}{R}_{7}+{m}_{8}\frac{{H}_{3}}{{D}_{3}}{e}^{-i{R}_{8}h}{R}_{8}-{m}_{11}\frac{{L}_{3}}{{D}_{3}}{e}^{i{R}_{11}h}{R}_{11}- {m}_{12}\frac{{M}_{3}}{{D}_{3}}{e}^{i{R}_{12}h}{R}_{12}\right)$$
$${g}_{3}=i{\xi }_{-n}\left[{\beta }_{1}{e}^{i{R}_{4}h}+{\beta }_{2}{e}^{-i{R}_{7}h}\frac{{G}_{3}}{{D}_{3}}+{\beta }_{3}{e}^{-i{R}_{8}h}\frac{{H}_{3}}{{D}_{3}}+{\beta }_{4}{e}^{i{R}_{11}h}\frac{{L}_{3}}{{D}_{3}}+{\beta }_{5}{e}^{i{R}_{12}h}\frac{{M}_{3}}{{D}_{3}}\right]$$
$${\beta }_{1}=-{S}_{55}\left({R}_{4}^{2}{m}_{4}-{R}_{4}{S}_{0}\right)+\left({S}_{31}{-S}_{11}\right){m}_{4}{S}_{0}nq+\left({S}_{33}{-S}_{31}\right){R}_{4}nq$$
$${\beta }_{2}=-{S}_{55}\left({R}_{7}^{2}{m}_{7}+{R}_{7}{S}_{0}\right)+\left({S}_{31}{-S}_{11}\right){m}_{7}{S}_{0}nq+\left({S}_{33}{-S}_{31}\right){R}_{7}nq$$
$${\beta }_{3}=-{S}_{55}\left({R}_{8}^{2}{m}_{8}+{R}_{8}{S}_{0}\right)+\left({S}_{31}{-S}_{11}\right){m}_{8}{S}_{0}nq+\left({S}_{33}{-S}_{31}\right){R}_{8}nq$$
$${\beta }_{4}={C}_{44}\left({R}_{11}^{2}{m}_{11}-{R}_{11}{S}_{0}\right)-{e}_{15}{\kappa }_{5}{S}_{0}{R}_{11}-\left({C}_{31}{-C}_{11}\right){m}_{11}{S}_{0}nq+\left({C}_{33}{-C}_{31}\right){R}_{11}nq+\left({e}_{33}{-e}_{31}\right){R}_{11}nq{\kappa }_{5},{\beta }_{5}={C}_{44}\left({R}_{12}^{2}{m}_{12}-{R}_{12}{S}_{0}\right)-{e}_{15}{\kappa }_{6}{S}_{0}{R}_{12}-\left({C}_{31}{-C}_{11}\right){m}_{12}{S}_{0}nq+\left({C}_{33}{-C}_{31}\right){R}_{12}nq{\kappa }_{6} +\left({e}_{33}{-e}_{31}\right){R}_{10}nq{\kappa }_{4}$$
$${g}_{4}=i{\xi }_{-n}\left[{\beta }_{6}{e}^{i{R}_{4}h}+{\beta }_{7}{e}^{-i{R}_{7}h}\frac{{G}_{3}}{{D}_{3}}+{\beta }_{8}{e}^{-i{R}_{8}h}\frac{{H}_{3}}{{D}_{3}}+{\beta }_{9}{e}^{i{R}_{11}h}\frac{{L}_{3}}{{D}_{3}}+{\beta }_{10}{e}^{i{R}_{12}h}\frac{{M}_{3}}{{D}_{3}}\right]$$
$${\beta }_{6}={S}_{31}{m}_{4}{R}_{4}{S}_{0}-{S}_{33}{R}_{4}^{2}-2nq{S}_{55}\left(-{R}_{4}{m}_{4}+{S}_{0}\right),{\beta }_{7}={-S}_{31}{m}_{7}{R}_{5}{S}_{0}-{S}_{33}{R}_{7}^{2}-2nq{S}_{55}\left({R}_{5}{m}_{7}+{S}_{0}\right), {\beta }_{8}={-S}_{31}{m}_{8}{R}_{6}{S}_{0}-{S}_{33}{R}_{8}^{2}-2{nqS}_{55}\left({R}_{6}{m}_{8}+{S}_{0}\right)$$
$${\beta }_{9}={-C}_{31}{m}_{11}{R}_{11}{S}_{0}+{C}_{33}{R}_{11}^{2}+{e}_{33}{\kappa }_{5}{R}_{11}^{2}+2nq\left\{{C}_{44}\left(-{R}_{11}{m}_{11}+{S}_{0}\right)+{e}_{15}{\kappa }_{5}{S}_{0})\right\},$$
$${\beta }_{10}={-C}_{31}{m}_{12}{R}_{12}{S}_{0}+{C}_{33}{R}_{12}^{2}+{e}_{33}{\kappa }_{6}{R}_{12}^{2}+2nq\left\{{C}_{44}\left(-{R}_{12}{m}_{12}+{S}_{0}\right)+{e}_{15}{\kappa }_{6}{S}_{0})\right\}.$$
$$\gamma_{17}^{{^{\prime\prime}}} = S_{55} \left( {m^{\prime}_{7n} R^{\prime}_{7n} + S_{0} - nq} \right)e^{{ - iR_{7n}^{^{\prime}} h}} ,\gamma_{18}^{{^{\prime\prime}}} = S_{55} \left( {m_{8n}^{^{\prime}} R{^{\prime}}_{8n} + S_{0} - nq} \right)e^{{ - iR_{8n}^{^{\prime}} h}}$$
$$\gamma_{19}^{{^{\prime\prime}}} = C_{44} \left( { - m_{11n}^{^{\prime}} R^{\prime}_{11n} + S_{0} - nq} \right)e^{{iR_{11n}^{^{\prime}} h}} ,\gamma_{20}^{{^{\prime\prime}}} = C_{44} \left( { - m_{12n}^{^{\prime}} R{^{\prime}}_{12n} + S_{0} - nq} \right)e^{{iR_{12n}^{^{\prime}} h}}$$
$$\gamma_{21}^{{^{\prime\prime}}} = (S_{31} m{^{\prime}}_{7n} S_{0} - S_{31} m^{\prime}_{7n} nq + S_{33} R^{\prime}_{5n} )e^{{ - iR_{7n}^{^{\prime}} h}} ,\gamma_{22}^{{^{\prime\prime}}} = (S_{31} m_{8n}^{^{\prime}} S_{0} - S_{31} m{^{\prime}}_{8n} nq + S_{33} R{^{\prime}}_{6n} )e^{{ - iR_{8n}^{^{\prime}} h}}$$
$$\gamma_{23}^{^{\prime\prime}} = (C_{13} m_{11n}^{^{\prime}} S_{0} - C_{13} m_{11n}^{^{\prime}} nq - C_{33} R^{\prime}_{11n} - e_{33} \kappa_{5n}^{^{\prime}} R^{\prime}_{11n} )e^{{iR_{11n}^{^{\prime}} h}}$$
$$\gamma_{24}^{^{\prime\prime}} = (C_{13} m_{12n}^{^{\prime}} S_{0} - C_{13} m_{12n}^{^{\prime}} nq - C_{33} R^{\prime}_{12n} - e_{33} \kappa_{6n}^{^{\prime}} R^{\prime}_{12n} )e^{{iR_{12n}^{^{\prime}} h}}$$
$${g}_{5}=i{\xi }_{-n}\left({\kappa }_{5}\frac{{L}_{3}}{{D}_{3}}{e}^{i{R}_{11}h}{R}_{11}+{\kappa }_{6}\frac{{M}_{3}}{{D}_{3}}{e}^{i{R}_{12}h}{R}_{12}\right)$$
$${g}_{6}=i{\xi }_{-n}\left({m}_{4}{e}^{i{R}_{4}h}{R}_{4}+{m}_{7}\frac{{G}_{3}}{{D}_{3}}{e}^{-i{R}_{7}h}{R}_{7}+{m}_{8}\frac{{H}_{3}}{{D}_{3}}{e}^{-i{R}_{8}h}{R}_{8}-{m}_{11}\frac{{L}_{3}}{{D}_{3}}{e}^{i{R}_{11}h}{R}_{11}- {m}_{12}\frac{{M}_{3}}{{D}_{3}}{e}^{i{R}_{12}h}{R}_{12}\right)$$
$${g}_{7}=i{\xi }_{-n}\left[{\beta }_{1}^{^{\prime}}{e}^{i{R}_{4}h}+{\beta }_{2}^{^{\prime}}{e}^{-i{R}_{7}h}\frac{{G}_{3}}{{D}_{3}}+{\beta }_{3}^{^{\prime}}{e}^{-i{R}_{8}h}\frac{{H}_{3}}{{D}_{3}}+{\beta }_{4}^{^{\prime}}{e}^{i{R}_{11}h}\frac{{L}_{3}}{{D}_{3}}+{\beta }_{5}^{^{\prime}}{e}^{i{R}_{12}h}\frac{{M}_{3}}{{D}_{3}}\right]$$
$${\beta }_{1}^{^{\prime}}=-{S}_{55}\left({R}_{4}^{2}{m}_{4}-{R}_{4}{S}_{0}\right)-\left({S}_{31}{-S}_{11}\right){m}_{4}{S}_{0}nq-\left({S}_{33}{-S}_{31}\right){R}_{4}nq$$
$${\beta }_{2}^{^{\prime}}=-{S}_{55}\left({R}_{7}^{2}{m}_{7}+{R}_{7}{S}_{0}\right)-\left({S}_{31}{-S}_{11}\right){m}_{7}{S}_{0}nq-\left({S}_{33}{-S}_{31}\right){R}_{7}nq$$
$${\beta }_{3}^{^{\prime}}=-{S}_{55}\left({R}_{8}^{2}{m}_{8}+{R}_{8}{S}_{0}\right)-\left({S}_{31}{-S}_{11}\right){m}_{8}{S}_{0}nq-\left({S}_{33}{-S}_{31}\right){R}_{8}nq$$
$${\beta }_{4}^{\mathrm{^{\prime}}}={C}_{44}\left({R}_{11}^{2}{m}_{11}-{R}_{11}\right)-{e}_{15}{\kappa }_{5}{S}_{0}{R}_{11}+\left({C}_{31}{-C}_{11}\right){m}_{11}{S}_{0}nq-\left({C}_{33}{-C}_{31}\right){R}_{11}nq-\left({e}_{33}{-e}_{31}\right){R}_{11}nq{\kappa }_{3}^{\mathrm{^{\prime}}},{\beta }_{5}^{\mathrm{^{\prime}}}={C}_{44}\left({R}_{12}^{2}{m}_{12}-{R}_{12}{S}_{0}\right)-{e}_{15}{\kappa }_{6}{S}_{0}{R}_{12}+\left({C}_{31}{-C}_{11}\right){m}_{12}{S}_{0}nq-\left({C}_{33}{-C}_{31}\right){R}_{12}nq-\left({e}_{33}{-e}_{31}\right){R}_{12}nq{\kappa }_{6}$$
$${g}_{8}=i{\xi }_{-n}\left[{\beta }_{6}^{^{\prime}}{e}^{i{R}_{4}h}+{\beta }_{7}^{^{\prime}}{e}^{-i{R}_{7}h}\frac{{G}_{3}}{{D}_{3}}+{\beta }_{8}^{^{\prime}}{e}^{-i{R}_{8}h}\frac{{H}_{3}}{{D}_{3}}+{\beta }_{9}^{^{\prime}}{e}^{i{R}_{11}h}\frac{{L}_{3}}{{D}_{3}}+{\beta }_{10}^{^{\prime}}{e}^{i{R}_{12}h}\frac{{M}_{3}}{{D}_{3}}\right]$$
$${\beta }_{6}^{\mathrm{^{\prime}}}={S}_{31}{m}_{4}{R}_{4}{S}_{0}-{S}_{33}{R}_{4}^{2}+2nq{S}_{55}\left(-{m}_{4}{R}_{4}+{S}_{0}\right), {\beta }_{7}^{\mathrm{^{\prime}}}={-S}_{31}{m}_{7}{R}_{7}{S}_{0}-{S}_{33}{R}_{7}^{2}+2{nqS}_{55}\left({m}_{7}{R}_{7}+{S}_{0}\right),{\beta }_{8}^{\mathrm{^{\prime}}}={-S}_{31}{m}_{8}{R}_{8}{S}_{0}-{S}_{33}{R}_{8}^{2}+2{nqS}_{55}\left({m}_{8}{R}_{8}+{S}_{0}\right)$$
$${\beta }_{9}^{^{\prime}}={-C}_{31}{m}_{11}{R}_{11}{S}_{0}+{C}_{33}{R}_{11}^{2}+{e}_{33}{\kappa }_{5}{R}_{11}^{2}-2nq\left\{{C}_{44}\left(-{R}_{11}{m}_{11}+{S}_{0}\right)+{e}_{15}{\kappa }_{5}{S}_{0})\right\}$$
$${\beta }_{10}^{^{\prime}}={-C}_{31}{m}_{12}{R}_{12}{S}_{0}+{C}_{33}{R}_{12}^{2}+{e}_{33}{\kappa }_{6}{R}_{12}^{2}-2nq\left\{{C}_{44}\left(-{R}_{12}{m}_{12}+{S}_{0}\right)+{e}_{15}{\kappa }_{6}{S}_{0})\right\}$$

Appendix 4: Expressions used for Energy Ratios \((j=\mathrm{1,2},\dots 12)\)

$$E_{i} = \left| {R^{2} } \right|, E_{2} = \left| {T^{2} } \right|\frac{{\rho_{1} c_{t} cos\theta_{j} }}{{\rho_{2} c_{i} cos\theta_{0} }}, E_{3} = \frac{{cos\theta_{j1} }}{{cos\theta_{0} }}\left| {R^{{1^{2} }} } \right|,E_{4} = \frac{{\rho_{1} c_{t} cos\theta_{j1} }}{{\rho_{2} c_{i} cos\theta_{0} }}\left| {T^{{1^{2} }} } \right|, E_{5} = \frac{{cos\theta_{j1} }}{{cos\theta_{0} }}|R^{{1^{{\prime\prime}{2}} }} \left| {, E_{6} = \frac{{\rho_{1} c_{t} cos\theta_{j1} }}{{\rho_{2} c_{i} cos\theta_{0} }}|T^{{1^{{\prime\prime}{2}} }} } \right|$$

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Kumari, P., Srivastava, R. Analysis of Quasi waves in Orthotropic Layer Bonded Between Piezoelectric Half-Spaces with Imperfect and Sliding Interfaces. J. Vib. Eng. Technol. 12, 1577–1602 (2024). https://doi.org/10.1007/s42417-023-00927-3

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