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Influence of initial stress, irregularity and heterogeneity on Love-type wave propagation in double pre-stressed irregular layers lying over a pre-stressed half-space

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Abstract

The present paper deals with the propagation of Love-type wave in an initially stressed irregular vertically heterogeneous layer lying over an initially stressed isotropic layer and an initially stressed isotropic half-space. Two different types of irregularities, viz., rectangular and parabolic, are considered at the interface of uppermost initially stressed heterogeneous layer and intermediate initially stressed isotropic layer. Dispersion equations are obtained in closed form for both cases of irregularities, distinctly. The effect of size and shape of irregularity, horizontal compressive initial stress, horizontal tensile initial stress, heterogeneity of the uppermost layer and width ratio of the layers on phase velocity of Love-type wave are the major highlights of the study. Comparative study has been made to identify the effects of different shapes of irregularity, presence of heterogeneity and initial stresses. Numerical computations have been carried out and depicted by means of graphs for the present study.

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Acknowledgements

The authors convey their sincere thanks to Indian School of Mines, Dhanbad for providing all the necessary facilities for research. The authors also gratefully acknowledge the reviewers for their comprehensive and constructive suggestions for the improvement of quality of the manuscript.

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Authors and Affiliations

  1. Department of Applied Mathematics, Indian School of Mines, Dhanbad, 826 004, Jharkhand, India

    ABHISHEK KUMAR SINGH, AMRITA DAS, ZEENAT PARWEEN & AMARES CHATTOPADHYAY

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  1. ABHISHEK KUMAR SINGH
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  2. AMRITA DAS
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  3. ZEENAT PARWEEN
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  4. AMARES CHATTOPADHYAY
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Correspondence to AMRITA DAS.

Appendices

Appendix I

$$\begin{array}{rcl} &&A_{0} =-B_{0} \frac{({2T+\upsilon \tan TH_{2}} )}{({2T\tan TH_{2} -\upsilon} )},\\ &&A_{1} =-B_{1} \frac{({2T+\upsilon \tan TH_{2}} )}{({2T\tan TH_{2} -\upsilon} )},\\ &&B_{0} =\frac{2p_{2} \mu_{2} \mu_{3} e^{-p_{3} d}({2T\tan TH_{2} -\upsilon} )}{U(k)}, \end{array} $$
$$\begin{array}{rcl} &&B_{1} =\frac{R_{1} {\mu_{1}^{0}} ({-\upsilon \cos TH_{1} +2T\sin TH_{1}} )e^{-\frac{\upsilon} {2}H_{1}} +2R_{2} \cos TH_{1} e^{\frac{\upsilon} {2}H_{1}} }{-2{\mu_{1}^{0}} T}, \end{array} $$
$$\begin{array}{rcl} &&C_{0}=\frac{({2T\cos TH_{1} +\upsilon \tan TH_{2} \cos TH_{1} +2T\tan TH_{2} \sin TH_{1} -\upsilon \sin TH_{1}} )}{({\mu_{2} p_{2} \cos p_{2} H_{1} +\mu_{3} p_{3} \sin p_{2} H_{1}} )U(k)} \\ &&{\kern.4pc}\times ({-2{p_{2}^{2}}\mu_{2}^{2}\mu_{3} e^{-p_{3} d}e^{\frac{\upsilon} {2}H_{1}}} )+\frac{4\mu_{3} \sin p_{2} H_{1} e^{-p_{3} d}}{({\mu_{2} p_{2} \cos p_{2} H_{1} +\mu_{3} p_{3} \sin p_{2} H_{1}} )}, \\ \end{array} $$
$$\begin{array}{rcl} &&D_{0}=\frac{({2T\cos TH_{1} +\upsilon \tan TH_{2} \cos TH_{1} +2T\tan TH_{2} \sin TH_{1} -\upsilon \sin TH_{1}} )}{({\mu_{2} p_{2} \cos p_{2} H_{1} +\mu_{3} p_{3} \sin p_{2} H_{1}} )U(k)} \\ &&{\kern.4pc}\times 2\mu_{2} p_{2} {\mu_{3}^{2}} e^{-p_{3} d}e^{\frac{\upsilon} {2}H_{1}} +\frac{4\mu_{3} \cos p_{2} H_{1} e^{-p_{3} d}}{({\mu_{2} p_{2} \cos p_{2} H_{1} +\mu_{3} p_{3} \sin p_{2} H_{1}} )}, \\ \end{array} $$
$$\begin{array}{rcl} &&E=\frac{({2T\cos TH_{1} +\upsilon \tan TH_{2} \cos TH_{1} +2T\tan TH_{2} \sin TH_{1} -\upsilon \sin TH_{1}} )}{({\mu_{2} p_{2} \cos p_{2} H_{1} +\mu_{3} p_{3} \sin p_{2} H_{1}} )U(k)} \\ &&{\kern.2pc}\times \left( {-2{\mu_{2}^{2}}{p_{2}^{2}}\mu_{3} e^{-p_{3} d}e^{\frac{\upsilon} {2}H_{1}} } \right)\!-\frac{2({\mu_{2} p_{2} \cos p_{2} H_{1} -\mu_{3} p_{3} \sin p_{2} H_{1}} )e^{-p_{3} d}}{({\mu_{2} p_{2} \cos p_{2} H_{1} +\mu_{3} p_{3} \sin p_{2} H_{1}} )p_{3}} . \end{array} $$
$$\begin{array}{rcl} &&Q_{1} =\frac{e^{\frac{\upsilon} {2}H_{1}} }{2}\left( 2\mu_{2} p_{2} T\cos TH_{1} ({p_{2} \mu_{2} \sin TH_{1} -\mu_{3} p_{3} \cos p_{2} H_{1}} )\right.\\ &&{\kern-.2pc}\left.-\upsilon \mu_{2} p_{2} \sin TH_{1} \left( {p_{2} \mu_{2} \sin p_{2} H_{1}} -\mu_{3} p_{3} \cos p_{2} H_{1} \right) \right)\\ &&{\kern-.2pc}-\left( \left( {T^{2}+\,\frac{\upsilon^{2}}{4}}\right)\sin TH_{1}({\mu_{2} p_{2} \cos p_{2} H_{1} +\sin p_{2} H_{1} \mu_{3} p_{3}} ) \right)e^{-\frac{\upsilon} {2}H_{1}} {\mu_{1}^{0}} \end{array} $$
$$\begin{array}{rcl} Q_{2}\!\!\!&=&\!\!\!e^{\frac{\upsilon} {2}H_{1}} \left( \mu_{2} p_{2} ({\mu_{2} p_{2} \sin p_{2} H_{1} -\mu_{3} p_{3} \cos p_{2} H_{1}} ) \left( {\frac{\upsilon} {2}\cos TH_{1} +T\sin TH_{1}} \right) \right)\\ &&{\kern-.6pc}+ \left( \vphantom{\frac{\upsilon^{2}}{4}} \!\!\cos TH_{1} ({\mu_{2} p_{2} \cos p_{2} H_{1} +\mu_{3} p_{3} \sin p_{2} H_{1}} ) \left( \frac{\upsilon^{2}}{4}+T^{2}\cos TH_{1}\right)\right)e^{-\frac{\upsilon} {2}H_{1}} {\mu_{1}^{0}} \end{array} $$
$$\begin{array}{rcl} &&S_{1} =4Q_{2} {\mu_{1}^{0}} \mu_{3} \mu_{2} p_{2} T^{2}(\mu_{2} p_{2} \cos p_{2} h_{1} +\mu_{3} p_{3} \sin p_{2} h_{1} )-Q_{2} (Q_{4} +Q_{6} +Q_{8} ){H}^{\prime},\\ &&S_{2} =2{\mu_{1}^{0}} \mu_{3} p_{2} \mu_{2} T({\mu_{2} p_{2} \cos p_{2} H_{1}\! +\!\mu_{3} p_{3} \sin p_{2} H_{1}} )({2TQ_{1}\! -\!Q_{2} \upsilon} )\,-\,Q_{1} (Q_{4}\! +\!Q_{6}\! +\!Q_{8} ){H}^{\prime} \!\,-\,Q_{2} (Q_{3}\! +\!Q_{5}\! +\!Q_{7} ){H}^{\prime}\!,\\ &&S_{3} =2Q_{1} \upsilon {\mu_{1}^{0}} \mu_{3} \mu_{2} p_{2} T(\mu_{2} p_{2} \cos p_{2} H_{1} \,+\,\mu_{3} p_{3} \sin p_{2} H_{1} ) -Q_{1} (Q_{3} +Q_{5} +Q_{7} ){H}^{\prime}. \end{array} $$
$$\begin{array}{rcl} &&F_{1}=4Q_{2} {\mu_{1}^{0}} \mu_{3} \mu_{2} p_{2} T^{2}({\mu_{2} p_{2} \cos p_{2} h_{1} +\mu_{3} p_{3} \sin p_{2} h_{1}} ) -Q_{2} (Q_{4} +Q_{6} +Q_{8} ){H}^{\prime\prime},\\ &&F_{2}=2{\mu_{1}^{0}} \mu_{3} p_{2} \mu_{2} T({2Q_{1} T\!\,-\,\upsilon Q_{2}} )\!\left( p_{2} \mu_{2} \!\cos p_{2} H_{1}\right. \left.\!\!+\mu_{3} p_{3} \sin p_{2} H_{1}\right)\,-\,Q_{1}\! (Q_{4}\! +\!Q_{6}\! +\!Q_{8} ){H}^{\prime\prime}\!\! -\!Q_{2} (Q_{3} \,+\,Q_{5} \,+\,Q_{7} ){H}^{\prime\prime}, \\ &&F_{3} \,=\,2Q_{1} \upsilon {\mu_{1}^{0}} \mu_{3} \mu_{2} p_{2} T({\mu_{2} p_{2} \cos p_{2} H_{1} \,+\,\mu_{3} p_{3} \sin p_{2} H_{1}} ) -Q_{1} (Q_{3} +Q_{5} +Q_{7} ){H}^{\prime\prime}. \end{array} $$

Appendix II

$$\begin{array}{rcl} &&Q_{3}\!\,=\!\,4\mu_{2} p_{2} T^{2}\mu_{3} {\mu_{1}^{0}} ({T({\mu_{2} p_{2} \cos p_{2} H_{1}\!+\mu_{3} p_{3} \sin p_{2} H_{1}} )} \\ &&{\kern6pt}-\upsilon \cos TH_{1} \sin TH_{1} \left( {\mu_{2} p_{2} \cos p_{2} H_{1}}\right.\\ &&{\kern5pt}\left.\left.{{-\mu_{3} p_{3}}\sin p_{2} H_{1}}\right)\right),\\ &&Q_{4} =\upsilon \mu_{2} p_{2} \mu_{3} {\mu_{1}^{0}} \left\{ ({\mu_{2} p_{2} \cos p_{2} H_{1} +\mu_{3} p_{3} \sin p_{2} H_{1}} )\right.\\ &&{\kern5pt}\times({2T^{2}+\cos^{2}TH_{1} ({\upsilon^{2}+4T^{2}} )} ) +2\upsilon T\cos TH_{1}\\ &&{\kern5pt}\times\left.\sin TH_{1} ({\mu_{3} p_{3} \sin p_{2} H_{1} -\mu_{2} p_{2} \cos p_{2} H_{1}} ) \right\} ,\\ &&Q_{5}=2e^{-\frac{\upsilon} {2}H_{1}} {\mu_{1}^{0}} \mu_{3} p_{2} ({-\upsilon \cos TH_{1} +2T\sin TH_{1}} )Q_{1}\\ &&{\kern4pt}+\!{\mu_{1}^{0}} \!\left( {\kern-.8pt}{2{\kern-.4pt}\upsilon{\kern-.4pt}\mu_{2} {p_{2}^{2}}T\mu_{3}} ({\kern-.4pt}{\mu_{2} p_{2} \sin p_{2} H_{1}\!{\kern-.2pt}+{\kern-.2pt}\!\mu_{3} p_{3} \cos p_{2} H_{1}}{\kern-.8pt}) \right.\\ &&{\kern4pt}-\mu_{2} p_{2}^{\text{2}}\mu_{3} \cos TH_{1} \sin TH_{1} \left( \mu_{2} p_{2} \sin p_{2} H_{1}\right. \\ &&{\kern2pt}\left.\left.-\mu_{3} p_{3} \cos p_{2} H_{1} \right)({4T^{2}-\upsilon^{\text{2}}} )\right), \\ &&Q_{6} =\mu_{2} {p_{2}^{2}}\mu_{3} ({\mu_{2} p_{2} \sin p_{2} H_{1} -\mu_{3} p_{3} \cos p_{2} H_{1}})\\ &&{\kern4pt}\times({\upsilon^{2}\cos^{2}TH_{1} -4T^{2}\sin^{2}TH_{1}} )\,\\ && {\kern4pt}-2\upsilon \mu_{2} {p_{2}^{2}}T{\mu_{3}^{2}} p_{3} \cos p_{2} H_{1} \cos TH_{1} \sin TH_{1} \\ &&{\kern4pt}-2Q_{2} \mu_{3} p_{2}({\upsilon \cos TH_{1} +2T\sin TH_{1}} ),\\ &&Q_{7} =-4e^{\upsilon H_{1}} {\mu_{2}^{2}}{p_{2}^{3}}T\mu_{3} \cos^{2}TH_{1}\\ &&{\kern4pt}\times ({\mu_{2} p_{2} \cos p_{2} H_{1} +\mu_{3} p_{3} \sin p_{2} H_{1}}), \\ &&Q_{8} =-2e^{\upsilon H_{1}} \mu_{2} {p_{2}^{3}}\mu_{3} \cos TH_{1} \\ &&{\kern4pt}\times({\mu_{2} p_{2} \cos p_{2} H_{1} +\mu_{3} p_{3} \sin p_{2} H_{1}})\\ &&{\kern4pt}\times({\upsilon \mu_{2} \cos TH_{1} +2\sin TH_{1}} )\\ &&S_{4} =4L_{2} {\mu_{1}^{0}} \mu_{3} \mu_{2} P_{2} t^{2} ({1+\tan^{2}tkH_{1}} )\\ &&{\kern4pt}\times\!({\kern-.4pt}{\mu_{{\kern-.4pt}2} P_{{\kern-.4pt}2} \,+\,\mu_{3} P_{{\kern-.4pt}3} \tan P_{{\kern-.4pt}2} k{\kern-.4pt}H_{{\kern-.4pt}1}} {\kern-.4pt})\!{\kern-.4pt}-{\kern-.4pt}\!L_{{\kern-.4pt}2} ({\kern-.4pt}{L_{{\kern-.4pt}4} \!{\kern-.4pt}+{\kern-.4pt}\!L_{{\kern-.4pt}6} \!{\kern-.4pt}+{\kern-.4pt}\!L_{{\kern-.4pt}8}}{\kern-.4pt}){H}^{\prime}{\kern-.4pt},\\ &&S_{5} =2{\mu_{1}^{0}} \mu_{3} P_{2} \mu_{2} t({1+\tan^{2}tkH_{1}} )\\ &&{\kern4pt}\times ({P_{2} \mu_{2} +\mu_{3} P_{3} \tan P_{2} kH_{1}} )\\ &&{\kern4pt}\times\left( {2L_{1} t-L_{2} \frac{\upsilon} {k}}\right)-L_{1} (L_{4} +L_{6} +L_{8} ){H}^{\prime},\\ &&S_{6} =2L_{1} \frac{\upsilon} {k}{\mu_{1}^{0}} \mu_{3} \mu_{2} P_{2} t({1+\tan^{2}tkH_{1}} )\\ &&{\kern2pt}\times({\mu_{2} P_{2} +\mu_{3} P_{3}\tan P_{2} kH_{1}} )\\&&{\kern4pt}-L_{1} (L_{3} +L_{5} +L_{7} ){H}^{\prime},\\ &&F_{4} =4L_{2} {\mu_{1}^{0}} \mu_{3} \mu_{2} P_{2} t^{2}({1+\tan^{2}tkH_{1}} )\\ &&{\kern4pt}\times({\mu_{2} P_{2} +\mu_{3} P_{3} \tan P_{2} kH_{1}} )\\&&{\kern4pt}-L_{2} ({L_{4} +L_{6} +L_{8}} ){H}^{\prime\prime},\\ &&F_{5} =2{\mu_{1}^{0}} \mu_{3} \mu_{2} P_{2} t({1+\tan^{2}tkH_{1}} )\\ &&{\kern4pt}\times({\mu_{2} P_{2} +P_{3} \mu_{3} \tan P_{2} kH_{1}} )\left( {2L_{1} t-L_{2} \frac{\upsilon} {k}}\right)\\ &&{\kern4pt}-L_{1} (L_{4} +L_{6} +L_{8} ){H}^{\prime\prime},\\ &&F_{6} =2L_{1} \frac{\upsilon} {k}{\mu_{1}^{0}} \mu_{3} \mu_{2} P_{2} t({1+\tan^{2}tkH_{1}})\\ &&{\kern4pt}\times({\mu_{2} P_{2} +\mu_{3} P_{3} \tan P_{2} kH_{1}} )\\ &&{\kern4pt}-L_{1} (L_{3} +L_{5} +L_{7} ){H}^{\prime\prime}, \end{array} $$
$$\begin{array}{rcl} &&L_{1} =e^{\frac{\upsilon} {2}H_{1}} \mu_{2} P_{2} ({P_{2} \mu_{2} \tan P_{2} kH_{1} +\mu_{3} P_{3}} )\\ &&{\kern4pt}\times\left( {t-\frac{\upsilon} {2k}\tan tkH_{1}}\right )-e^{-\frac{\upsilon} {2}H_{1}} {\mu_{1}^{0}} \tan tkH_{1}\\ && {\kern4pt}\times({\mu_{2} P_{2} +\mu_{3} P_{3} \tan P_{2} kH_{1}} )\left( {t^{2}+\frac{\upsilon^{2}}{4k^{2}}} \right),\\ &&L_{2} =e^{\frac{\upsilon} {2}H_{1}} \mu_{2} P_{2} ({\mu_{2} P_{2} \tan P_{2} kH_{1} -\mu_{3} P_{3}} )\\ &&{\kern4pt}\times\left( {\frac{\upsilon} {2k}+t\tan tkH_{1}}\right)+e^{-\frac{\upsilon} {2}H_{1}} {\mu_{1}^{0}}\\ &&{\kern4pt}\times ({\mu_{2} P_{2} +\mu_{3} P_{3} \tan P_{2} kH_{1}} )\left( {\frac{\upsilon^{2}}{4k^{2}}+t^{2}} \right)\\ &&{\kern-2.2pc}L_{3} =4\mu_{2} P_{2} t^{3}\mu_{3} {\mu_{1}^{0}} ({1+\tan^{2}tkH_{1}} )\\[-1pt] &&\times({\mu_{2} P_{2}{\kern-1.5pt}+{\kern-1.5pt}\mu_{3} P_{3} \tan P_{2} kH_{1}}){\kern-1.5pt}-{\kern-1.5pt}4\frac{\upsilon} {k}\mu_{2} P_{2} t^{2}\mu_{3} {\mu_{1}^{0}} \\ &&\times\tan tkH_{1} ({\mu_{2} P_{2} -\mu_{3} P_{3} \tan P_{2} kH_{1}} ),\\ L_{4}\!\!\!\! &=&\!\!\!\!\frac{\upsilon} {k}\mu_{2} P_{2} \mu_{3} {\mu_{1}^{0}} ({\mu_{3} P_{3} \tan P_{2} kH_{1} +\mu_{2} P_{2}} )\\&&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\times\left( {2t^{2}({1+\tan^{2}tkH_{1}} )+\frac{\upsilon^{2}}{k^{2}}+4t^{2}} \right)\\ &&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!+2\frac{\upsilon^{2}}{k^{2}}\mu_{2} P_{2} t\mu_{3} {\mu_{1}^{0}} \tan tkH_{1}\!({\mu_{3} P_{3} \tan P_{2} kH_{1} -\,\!\mu_{2} P_{2}} ),\\ L_{5} \!\!\!\!&=&\!\!\!\!2e^{-\frac{\upsilon} {2}H_{1}} {\mu_{1}^{0}} \mu_{3} P_{2} \left( {-\frac{\upsilon} {k}+2t\tan tkH_{1}} \right)L_{1}+{\mu_{1}^{0}}\\ &&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\times\!\! \left( \vphantom{\left( {\frac{\upsilon^{2}}{k^{2}}+4t^{2}} \right)}{2\frac{\upsilon} {k}\mu_{2} {P_{2}^{2}}t\mu_{3} \tan^{2}tkH_{1} ({\mu_{3} P_{3} \,+\,\mu_{2} P_{2} \tan P_{2} kH_{1}} )}\right. \\ && \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!+\mu_{2} {P_{2}^{2}}\mu_{3} \left( 2\frac{\upsilon} {k}t-\tan tkH_{1} \left( {\frac{\upsilon^{2}}{k^{2}}+4t^{2}} \right) \right)\\ &&\times\left.\!\!({\mu_{2} P_{2} \tan P_{2} kH_{1} -\mu_{3} P_{3}} ) \vphantom{\left( {\frac{\upsilon^{2}}{k^{2}}+4t^{2}} \right)}\right), \\ L_{6} \!\!\!\!&=&\!\!\!\!\left( {\frac{\upsilon^{2}}{k^{2}}\mu_{2} {P_{2}^{2}}\mu_{3} -4\mu_{2} {P_{2}^{2}}t^{2}\mu_{3} \tan^{2}tkH_{1}} \right)\\ &&\!\!\!\!\times({\mu_{2} P_{2} \tan P_{2} kH_{1} -\mu_{3} P_{3}} ) \\ &&\!\!\!\!-2\mu_{3} P_{2} \left( {\frac{\upsilon} {k}\mu_{2} P_{2} t\mu_{3} P_{3} \tan tkH_{1} }\right.\\&&\left.\!\!\!\!+ \right.\left. {L_{2} \left( {\frac{\upsilon} {k}-4t\tan tkH_{1}}\right)} \right), \\ L_{7}\!\!\!\! &=&\!\!\!\!-4e^{\upsilon H_{1}} {\mu_{2}^{2}}{P_{2}^{3}}t\mu_{3} ({\mu_{2} P_{2} +\mu_{3} P_{3} \tan P_{2} kH_{1}} ), \\ L_{8} \!\!\!\!&=&\!\!\!\!-2e^{\upsilon H_{1}} \mu_{2} {P_{2}^{3}}\mu_{3} ({\mu_{2} P_{2} +P_{3} \mu_{3} \tan P_{2} kH_{1}} )\\ &&\!\!\!\!\times\left( {\frac{\upsilon} {k}\mu_{2} +2\tan tkH_{1}}\right), \\ S_{4}^{(1)} \!\!\!\!&=&\!\!\!\!4L_{2}^{(1)} {\mu_{1}^{0}} \mu_{3} \mu_{2} P_{2} ({t^{(1)}} )^{2}({1+\tan^{2}t^{(1)}kH_{1}} )\\ &&\!\!\!\!\times({\mu_{2} P_{2} +\mu_{3} P_{3} \tan P_{2} kH_{1}} )\\&&\!\!\!\!-L_{2}^{(1)} ({L_{6}^{(1)} +L_{8}^{(1)}} ){H}^{\prime},\\ S_{5}^{(1)} \!\!\!\!&=&\!\!\!\!4{L}^{\prime}_{1} {\mu_{1}^{0}} \mu_{3} P_{2} \mu_{2} ({t^{(1)}} )^{2}({1+\tan^{2}t^{(1)}kH_{1}} )\\ &&\!\!\!\!\times({P_{2} \mu_{2} +\mu_{3} P_{3} \tan P_{2} kH_{1} } )\\&&\!\!\!\!-L_{1}^{(1)} ({L_{6}^{(1)} +L_{8}^{(1)}} ){H}^{\prime},\\ S_{6}^{(1)} \!\!\!\!&=&\!\!\!\!-L_{1}^{(1)} (L_{3}^{(1)} +L_{5}^{(1)} +L_{7}^{(1)} ){H}^{\prime},\\ \end{array} $$
$$\begin{array}{rcl} F_{4}^{(1)} \!\!\!\!&=&\!\!\!\!4{\mu_{1}^{0}} \mu_{3} L_{2}^{(1)} \mu_{2} P_{2} ({t^{(1)}} )^{2}({1+\tan^{2}t^{(1)}kH_{1}} )\\ &&\!\!\!\!\times({\mu_{2} P_{2} +\mu_{3} P_{3} \tan P_{2} kH_{1}} )\\&&\!\!\!\!-L_{2}^{(1)} ({L_{6}^{(1)} +L_{8}^{(1)}} ){H}^{\prime\prime},\\ F_{5}^{(1)}\!\!\!\! &=&\!\!\!\!4{\mu_{1}^{0}} \mu_{3} L_{1}^{(1)} P_{2} \mu_{2} ({t^{(1)}} )^{2}({1+\tan^{2}t^{(1)}kH_{1}} )\\ &&\!\!\!\!\times({P_{2} \mu_{2} +\mu_{3} P_{3} \tan P_{2} kH_{1}} )\\&&\!\!\!\!-L_{1}^{(1)} ({L_{6}^{(1)} +L_{8}^{(1)}} ){H}^{\prime\prime},\\ F_{6}^{(1)} \!\!\!\!&=&\!\!\!\!-L_{1}^{(1)} (L_{3}^{(1)} +L_{5}^{(1)} +L_{7}^{(1)} ){H}^{\prime\prime},\\ L_{1}^{(1)}\!\!\!\! &=&\!\!\!\!\mu_{2} P_{2} t^{(1)}({P_{2} \mu_{2} \tan P_{2} kH_{1} +\mu_{3} P_{3}} )-{\mu_{1}^{0}} ({t^{(1)}} )^{2}\\ &&\!\!\!\!\times \tan t^{(1)}kH_{1} ({\mu_{2} P_{2} +\mu_{3} P_{3} \tan P_{2} kH_{1}} ), \\ L_{2}^{(1)}\!\!\!\! &=&\!\!\!\!\mu_{2} P_{2} t^{(1)}\tan t^{(1)}kH_{1} ({\mu_{2} P_{2} \tan P_{2} kH_{1} -\mu_{3} P_{3}})\\ &&\!\!\!\!+{\mu_{1}^{0}} ({t^{(1)}} )^{2}({\mu_{2} P_{2} +\mu_{3} P_{3} \tan P_{2} kH_{1}} ) \\ L_{3}^{(1)} \!\!\!\!&=&\!\!\!\!4\mu_{2} P_{2} \mu_{3} {\mu_{1}^{0}} ({t^{(1)}} )^{3}({1+\tan^{2}t^{(1)}kH_{1}})\\ &&\!\!\!\!\times({\mu_{2} P_{2} +\mu_{3} P_{3} \tan P_{2} kH_{1}} ),\\ L_{5}^{(1)} \!\!\!\!&=&\!\!\!\!4{\mu_{1}^{0}} \mu_{3} P_{2} t^{(1)}\tan t^{(1)}kH_{1}\\ &&\times ({L_{1} \,-\,\mu_{2} P_{2} t^{(1)}({\mu_{2} P_{2} \tan P_{2} kH_{1} \,-\,\mu_{3} P_{3}} )} ),\\ L_{6}^{(1)} \!\!\!\!&=&\!\!\!\!4\mu_{3} P_{2} t^{(1)}\tan t^{(1)}kH_{1} \,(L_{2}^{(1)} -\mu_{2} P_{2} t^{(1)}\\ &&\!\!\!\!\times\tan t^{(1)}kH_{1} ({\mu_{2} P_{2} \tan P_{2} kH_{1} -\mu_{3} P_{3}} ) ),\\ L_{7}^{(1)} \!\!\!\!&=&\!\!\!\!-4{\mu_{2}^{2}}{P_{2}^{3}}t^{(1)}\mu_{3} ({\mu_{2} P_{2} +\mu_{3} P_{3} \tan P_{2} kH_{1}} ),\\ L_{8}^{(1)} \!\!\!\!&=&\!\!\!\!-4\mu_{2} {P_{2}^{3}}\mu_{3} \tan t^{(1)}kH_{1} \\&&\!\!\!\!\times(\mu_{2} P_{2}+P_{3} \mu_{3} \tan P_{2} kH_{1} ),\\ S_{4}^{(2)} \!\!\!\!&=&\!\!\!\!4L_{2}^{(2)} {\mu_{1}^{0}} \mu_{3} \mu_{2} P_{2}^{(1)} ({t^{(1)}} )^{2}({1+\tan^{2}tkH_{1}} )\\ &&\!\!\!\!\times\left( {\mu_{2} P_{2}^{(1)} +\mu_{3} P_{3}^{(1)} \tan P_{2}^{(1)} kH_{1}} \right)\\ &&\!\!\!\!-L_{2}^{(2)} \left( {L_{6}^{(2)} +L_{8}^{(2)}} \right){H}^{\prime},\\ S_{5}^{(2)}\!\!\!\!&=&\!\!\!\!4L_{1}^{(2)} {\mu_{1}^{0}} \mu_{3} \mu_{2} P_{2}^{(1)} ({t^{(1)}} )^{2}({1+\tan^{2}t^{(1)}kH_{1}} )\\ &&\!\!\!\!\times({P_{2}^{(1)} \mu_{2} +\mu_{3} P_{3}^{(1)} \tan P_{2}^{(1)} kH_{1}})\\ &&\!\!\!\!-L_{1}^{(2)}(L_{6}^{(2)} +L_{8}^{(2)} ){H}^{\prime}, \\ S_{6}^{(2)}\!\!\!\! &=&\!\!\!\!-L_{1}^{(2)}(L_{3}^{(2)}+L_{5}^{(2)} +L_{7}^{(2)} ){H}^{\prime},\\ F_{4}^{(2)}\!\!\!\! &=&\!\!\!\!4L_{2}^{(2)} {\mu_{1}^{0}} \mu_{3} \mu_{2} P_{2}^{(1)}({t^{(1)}} )^{2}({1+\tan^{2}t^{(1)}kH_{1}} )\\ &&\!\!\!\!\times\left( \mu_{2} P_{2}^{(1)} +\mu_{3}P_{3}^{(1)}\tan P_{2}^{(1)} kH_{1} \right)\\ &&\!\!\!\!-L_{2}^{(2)} ({L_{6}^{(2)} +L_{8}^{(2)}} ){H}^{\prime\prime},\\ F_{5}^{(2)}\!\!\!\! &=&\!\!\!\!4L_{1}^{(2)} {\mu_{1}^{0}} \mu_{3} P_{2}^{(1)} \mu_{2} ({t^{(1)}})^{2}({1+\tan^{2}t^{(1)}kH_{1}} )\\ &&\!\!\!\!\times({P_{2}^{(1)} \mu_{2} +\mu_{3} P_{3}^{(1)} \tan P_{2}^{(1)} kH_{1}} )\\ &&\!\!\!\!-L_{1}^{(2)} ({L_{6}^{(2)} +L_{8}^{(2)}} ){H}^{\prime\prime},\\ F_{6}^{(2)} \!\!\!\!&=&\!\!\!\!-L_{1}^{(2)} (L_{3}^{(2)} +L_{5}^{(2)} +L_{7}^{(2)} ){H}^{\prime\prime},\\ L_{1}^{(2)}\!\!\!\! &=&\!\!\!\!\mu_{2} P_{2}^{(1 )}t^{(1)} \left( {\mu_{2} P_{2}^{(1 )}\tan P_{2}^{(1 )}kH_{1} -\mu_{3} P_{3}^{(1 )} } \right)\\ &&\!\!\!\!-{\mu_{1}^{0}} ({t^{(1)}} )^{2}\tan t^{(1)}kH_{1} \\ &&\times\left( \mu_{2} P_{2}^{(1 )}-\mu_{3} P_{3}^{(1 )}\tan P_{2}^{(1 )}kH_{1}\right), \end{array} $$
$$\begin{array}{rcl} L_{2}^{(2)} \!\!\!\!&=&\!\!\!\!\mu_{2} P_{2}^{(1 )}t^{(1)} \tan t^{(1)}kH_{1} \left( \mu_{2} P_{2}^{(1 )}\tan P_{2}^{(1 )}kH_{1}\right.\\ &&\!\!\!\!\left. -\mu_{3} P_{3}^{(1 )} \right)+\mu_{1}^{0}({t^{(1)}} )^{2}(\mu_{2} P_{2}^{(1 )}\\&&\!\!\!\!+\mu_{3} P_{3}{(1 )\tan P_{2}^{(1 )}kH_{1}} )\\ L_{3}^{(2)} \!\!\!\!&=&\!\!\!\!4\mu_{2} P_{2}^{(1 )}({t^{(1)}} )^{3}\mu_{3} {\mu_{1}^{0}} ({1+\tan^{2}t^{(1)}kH_{1}} )\\ &&\!\!\!\!\times({\mu_{2} P_{2}^{(1 )}+\mu_{3} P_{3}^{(1 )}\tan P_{2}^{(1 )}kH_{1}} ),\\ L_{5}^{(2)}\!\!\!\! &=&\!\!\!\!4{\mu_{1}^{0}} \mu_{3} P_{2}^{(1 )}t^{(1)}\tan t^{(1)}kH_{1} \left( L_{1} -P_{2}^{(1 )}\mu_{2} t^{(1)}\right.\\ &&\!\!\!\!\left.\times\left( {\mu_{2} P_{2}^{(1 )}\tan P_{2}^{(1 )}kH_{1} -\mu_{3} P_{3}^{(1)}} \right) \right),\\ L_{6}^{(2)}\!\!\!\! &=&\!\!\!4\mu_{3} P_{2}^{(1 )}t^{(1)}\tan t^{(1)}kH_{1} \left( L_{2}^{(2)} -\mu_{2} P_{2}^{(1 )}t^{(1)}\right.\\ &&\!\!\!\!\left.\times\tan t^{(1)}kH_{1}\left( {\mu_{2} P_{2}^{(1 )}\tan P_{2}^{(1 )}kH_{1}\,-\,\mu_{3} P_{3}^{(1 )}} \right)\right)\ \\ L_{7}^{(2)} \!\!\!\!&=&\!\!\!\!-4{\mu_{2}^{2}}\mu_{3} ({P_{2}^{(1 )}} )^{3}t^{(1)}\\ &&\times\left.\left( \mu_{2} P_{2}^{(1 )}\right.+\mu_{3} P_{3}^{(1 )}\tan P_{2}^{(1 )}kH_{1} \right)\ \end{array} $$
$$\begin{array}{rcl} && L_{8}^{(2)} =-4\mu_{2} ({P_{2}^{(1 )}} )^{3}\mu_{3} \tan t^{(1)}kH_{1} \\ &&\!\!\!\!~\times({\mu_{2} P_{2}^{(1 )}+P_{3}^{(1 )}\mu_{3} \tan P_{2}^{(1 )}kH_{1}} ),\\ t^{(2)}\!\!\!\!&=&\!\!\!\!\sqrt {\frac{c^{2}}{{\beta_{2}^{2}}} -1+\xi} ,\quad t^{(1)}=\sqrt {\frac{c^{2}}{{\beta_{2}^{2}}} -1} ,\\ P_{2}^{(1)}\!\!\!\! &=&\!\!\!\!\sqrt {\frac{c^{2}}{{\beta_{2}^{2}}} -1} , \quad P_{3}^{(1)} =\sqrt {1-\frac{c^{2}}{{\beta_{3}^{2}}} } . \end{array} $$

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SINGH, A.K., DAS, A., PARWEEN, Z. et al. Influence of initial stress, irregularity and heterogeneity on Love-type wave propagation in double pre-stressed irregular layers lying over a pre-stressed half-space. J Earth Syst Sci 124, 1457–1474 (2015). https://doi.org/10.1007/s12040-015-0620-7

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