Appendix A
In the case of the couple-stress elastic slab sandwiched between the couple-stress elastic half-spaces, the non-zero elements of matrix \(\varvec{P}=\left[ {a_{i,j} } \right] _{1{2}\times 1{2}} \) are listed as follows
$$\begin{aligned}&a_{1,1} =\mu _1 \left( {2\xi _1^2 -{\omega ^{2}}/{c_{2\left( 1 \right) }^2 }} \right) ,\quad a_{1,2} =2\mu _1 \zeta _1 \beta _1 ,\\&a_{1,3} = 2\text {i}\mu _1 \gamma _1 \zeta _1 ,\quad a_{1,{4}} =-\mu _2 \left( {2\xi _2^2 -{\omega ^{2}}/{c_{2\left( 2 \right) }^2 }} \right) ,\\&a_{1,{5}} = 2\mu _2 \zeta _2 \beta _2 , \quad a_{1,{6}} =2\text {i}\mu _2 \gamma _2 \zeta _2 ,\\&a_{1,{7}} =-\mu _2 \left( {2\xi _2^2 -{\omega ^{2}}/{c_{2\left( 2 \right) }^2 }} \right) \exp \left( {\text {i}\alpha _2 d} \right) ,\\&a_{1,{8}} =-2\mu _2 \zeta _2 \beta _2 \exp \left( {\text {i}\beta _2 d}\right) ;\\&a_{2,1} =-2\mu _1 \xi _1 \alpha _1 ,\quad a_{{2,}2} =\mu _1 \left( {2\zeta _{1}^{2} -{\omega ^{2}}/{c_{2\left( 1 \right) }^2 }} \right) ,\\&a_{{2,3}} =\mu _1 \left( {2\zeta _{1}^{2} -{\omega ^{2}}/{c_{2\left( 1 \right) }^2 }} \right) , \quad a_{2,{4}} =-2\mu _2 \xi _2 \alpha _2 , \\&a_{2,{5}} =-\mu _2 \left( {2\zeta _2^2 -{\omega ^{2}}/{c_{2\left( 2 \right) }^2 }} \right) ,\\&a_{2,{6}} =-\mu _2 \left( {2\zeta _2^2 -{\omega ^{2}}/{c_{2\left( 2 \right) }^2 }} \right) , \\&a_{2,{7}} = 2\mu _2 \xi _2 \alpha _2 \exp \left( {\text {i}\alpha _2 d} \right) ,\\ \end{aligned}$$
$$\begin{aligned}&a_{2,{8}} =-\mu _2 \left( {2\zeta _2^2 -{\omega ^{2}}/{c_{2\left( 2 \right) }^2 }} \right) \exp \left( {\text {i}\beta _2 d} \right) ;\\&a_{3,2} = 2\text {i}\eta _1 \beta _1 \beta _{1\left( 1 \right) }^2 , a_{3,3}=2\eta _1 \gamma _1 \beta _{2\left( 1 \right) }^2 ,\, a_{3,5} =2\text {i}\eta _2 \beta _2 \beta _{{1}\left( {2} \right) }^{2},\\&a_{3,6} =2\eta _2 \gamma _2 \beta _{2\left( 2 \right) }^2 ,\quad a_{3,8} =-2\text {i}\eta _2 \beta _2 \beta _{1\left( 2 \right) }^2 \exp \left( {\text {i}\beta _2 d} \right) ;\\&a_{4,1} = \text {i}\alpha _1 ,\quad a_{4,2} =-\text {i}\zeta _1 , a_{4,3} =-\text {i}\zeta _1 , \quad a_{4,{4}} =\text {i}\alpha _2 ,\\&a_{4,{5}} =\text {i}\zeta _2 , \quad a_{4,{6}} =\text {i}\zeta _2,\quad a_{4,{7}} =-\text {i}\alpha _2 \exp \left( {\text {i}\alpha _2 d} \right) ,\\&a_{4,{8}} = \text {i}\zeta _2 \exp \left( {\text {i}\beta _2 d} \right) ; \quad a_{5,1} =\text {i}\xi _1 , \quad a_{5,2} =\text {i}\beta _1 ,\\&a_{5,3} =-\gamma _1 , \quad a_{5,{4}} =-\text {i}\xi _2 , \quad a_{5,{5}} =\text {i}\beta _2 ,\quad a_{5,{6}} =-\gamma _2 , \\&a_{5,{7}} =-\text {i}\xi _2 \exp \left( {\text {i}\alpha _2 d} \right) , \quad a_{{5,8}} =-\text {i}\beta _2 \exp \left( {\text {i}\beta _2 d} \right) ;\\&a_{6,2} = \beta _{1\left( 1 \right) }^2 ,\quad a_{6,3} =-\beta _{2\left( 1 \right) }^2 ,\quad a_{6,5} =-\beta _{1\left( 2 \right) }^2 ,\\&a_{6,6} = \beta _{2\left( 2 \right) }^2 ,\quad a_{6,8} = -\beta _{1\left( 2 \right) }^2 \exp \left( {\text {i}\beta _2 d} \right) ;\\&a_{{7,4}} =\mu _2 \left( {2\xi _2^2 -{\omega ^{2}}/{c_{2\left( 2 \right) }^2 }} \right) \exp \left( {\text {i}\alpha _2 d} \right) ,\\&a_{{7,5}} =-2\mu _2 \zeta _2 \beta _2 \exp \left( {\text {i}\beta _2 d} \right) ,\\&a_{{7,7}} =\mu _2 \left( {2\xi _2^2 -{\omega ^{2}}/{c_{2\left( 2 \right) }^2 }} \right) , \\ \end{aligned}$$
$$\begin{aligned}&a_{{7,8}} =2\mu _2 \zeta _2 \beta _2 , a_{{7,9}} =2\text {i}\mu _2 \gamma _2 \zeta _2 ,\\&a_{{7,10}} =-\mu _3 \left( {2\xi _3^2 -{\omega ^{2}}/{c_{2\left( 3 \right) }^2 }} \right) , \quad a_{{7,}1{1}} =2\mu _3 \zeta _3 \beta _3 ,\\&a_{7,12} = 2\text {i}\mu _3 \gamma _3 \zeta _3 ;\quad a_{{8,4}} =2\mu _2 \xi _2 \alpha _2 \exp \left( {\text {i}\alpha _2 d} \right) , \\&a_{{8,5}} =\mu _2 \left( {2\zeta _2^2 -{\omega ^{2}}/{c_{2\left( 2 \right) }^2 }} \right) \exp \left( {\text {i}\beta _2 d} \right) ,\\&a_{{8,7}} =-2\mu _2 \xi _2 \alpha _2 ,\quad a_{{8,8}} =\mu _2 \left( {2\zeta _2^2 -{\omega ^{2}}/{c_{2\left( 2 \right) }^2 }} \right) ,\\&a_{{8,9}} =\mu _2 \left( {2\zeta _2^2 -{\omega ^{2}}/{c_{2\left( 2 \right) }^2 }} \right) ,\quad a_{{8,10}} =-2\mu _3 \xi _3 \alpha _3 ,\\&a_{8,11} =-\mu _3 \left( {2\zeta _{3}^{2} -{\omega ^{2}}/{c_{{2}\left( {3} \right) }^{2} }} \right) ,\\&a_{8,1{2}} = -\mu _3 \left( {2\zeta _{3}^{2} -{\omega ^{2}}/{c_{{2}\left( {3} \right) }^{2} }} \right) ;\\&a_{{9,5}} = -2\text {i}\eta _2 \beta _2 \beta _{1\left( 2 \right) }^2 \exp \left( {\text {i}\beta _2 d} \right) , \\&a_{{9,8}} =2\text {i}\eta _2 \beta _2 \beta _{1\left( 2 \right) }^2 ,\quad a_{{9,9}} =2\eta _2 \gamma _2 \beta _{2\left( 2 \right) }^2 ,\\&a_{9,11} =2\text {i}\eta _3 \beta _3 \beta _{{1}\left( {3} \right) }^{2} ,\quad a_{9,12} = 2\eta _3 \gamma _3 \beta _{2\left( 3 \right) }^2 ;\\ \end{aligned}$$
$$\begin{aligned}&a_{10,4} =-\text {i}\alpha _2 \exp \left( {\text {i}\alpha _2 d} \right) ,\quad a_{10,5}=-\text {i}\zeta _2 \exp \left( {\text {i}\beta _2 d} \right) ,\\&a_{10,7} =\text {i}\alpha _2 ,\quad a_{10,8} =-\text {i}\zeta _2 , \quad a_{10,9} =-\text {i}\zeta _2 , \\&a_{10,10} = \text {i}\alpha _{3} ,\quad a_{10,11} =\text {i}\zeta _3 , \quad a_{10,12} =\text {i}\zeta _3 ;\\&a_{{11,4}} =\text {i}\xi _2 \exp \left( {\text {i}\alpha _2 d} \right) ,\, a_{{11,5}} =-\text {i}\beta _2 \exp \left( {\text {i}\beta _2 d} \right) ,\\&a_{11,7} =\text {i}\xi _2 , \quad a_{11,8} =\text {i}\beta _2 , \quad a_{11,9} =-\gamma _2 ,\\&a_{11,10} =-\text {i}\xi _3 , \quad a_{11,11} =\text {i}\beta _3 , \quad a_{11,12} =-\gamma _3 ;\quad \\&a_{12,5} =\beta _{1\left( 2 \right) }^2 \exp \left( {\text {i}\beta _2 d} \right) , \\&a_{12,8} =\beta _{1\left( 2 \right) }^2 ,\quad a_{12,9} =-\beta _{2\left( 2 \right) }^2 , \\&a_{12,11} =-\beta _{1\left( 3 \right) }^2 , \quad a_{12,12} =\beta _{2\left( 3 \right) }^2 . \end{aligned}$$
In the case of incident P wave, the non-zero elements of matrix \(\varvec{Q}=\left[ {q_i } \right] _{12\times {1}} \) are
$$\begin{aligned} q_1= & {} -\mu _1 \left( {2\xi _1^2 -{\omega ^{2}}/{c_{2\left( 1 \right) }^2 }} \right) , \quad q_2 =-2\mu _1 \xi _1 \alpha _1, \quad q_4 =\text {i}\alpha _1, \\ q_5= & {} -\text {i}\xi _1. \end{aligned}$$
In the case of incident SV wave, the non-zero elements of matrix \({\varvec{Q}}'=\left[ {{q}'_i } \right] _{12\times {1}} \) are
$$\begin{aligned}&{q}'_1 =2\mu _1 \zeta _1 \beta _1, \quad {q}'_2 =-\mu _1 \left( {2\zeta _{1}^{2} -{\omega ^{2}}/{c_{2\left( 1 \right) }^2 }} \right) ,\\&{q}'_3 =2\text {i}\eta _1 \beta _1 \beta _{1\left( 1 \right) }^2,\, {q}'_4 =\text {i}\zeta _1, \quad {q}'_5 =\text {i}\beta _1, \quad {q}'_6 =-\beta _{1\left( 1 \right) }^2. \end{aligned}$$
Appendix B
In the case of the couple-stress elastic slab sandwiched by the classical elastic half-spaces, there are two kinds of interface conditions. For the couple-free interface, the non-zero elements of matrix \(\varvec{P}=\left[ {a_{i,j} } \right] _{1{0}\times 1{0}} \) are
$$\begin{aligned}&a_{1,1} = \mu _1 \left( {2\xi _1^2 -{\omega ^{2}}/{c_{2\left( 1 \right) }^2 }} \right) , \quad a_{1,2} =2\mu _1 \zeta _1 \beta _1 ,\\&a_{1,{3}} = -\mu _2 \left( {2\xi _2^2 -{\omega ^{2}}/{c_{2\left( 2 \right) }^2 }} \right) ,\quad a_{1,{4}} =2\mu _2 \zeta _2 \beta _2 ,\\&a_{1,{5}} = 2\text {i}\mu _2 \gamma _2 \zeta _2 , \quad a_{1,{6}} =-\mu _2 \left( {2\xi _2^2 -{\omega ^{2}}/{c_{2\left( 2 \right) }^2 }} \right) \exp \left( {\text {i}\alpha _2 d} \right) ,\\&a_{1,{7}} = -2\mu _2 \zeta _2 \beta _2 \exp \left( {\text {i}\beta _2 d} \right) ;\\&a_{2,1} =-2\mu _1 \xi _1 \alpha _1 , \\&a_{2,2} =\mu _1 \left( {-\beta _1^2 +\zeta _1^2 } \right) ,\quad a_{2,3} = -2\mu _2 \xi _2 \alpha _2 , \\&a_{2,4} =-\mu _2 \left( {2\zeta _2^2 -{\omega ^{2}}/{c_{2\left( 2 \right) }^2 }} \right) ,\\&a_{2,5} = -\mu _2 \left( {2\zeta _2^2 -{\omega ^{2}}/{c_{2\left( 2 \right) }^2 }} \right) , \\ \end{aligned}$$
$$\begin{aligned}&a_{2,6} =2\mu _2 \xi _2 \alpha _2 \exp \left( {\text {i}\alpha _2 d} \right) ,\\&a_{2,7} = -\mu _2 \left( {2\zeta _2^2 -{\omega ^{2}}/{c_{2\left( 2 \right) }^2 }} \right) \exp \left( {\text {i}\beta _2 d} \right) ; \\&a_{3,4} =2\text {i}\eta _2 \beta _2 \beta _{1\left( 2 \right) }^2 ,\quad a_{3,5} = 2\eta _2 \gamma _2 \beta _{2\left( 2 \right) }^2 ,\\&a_{3,7} = -2\text {i}\eta _2 \beta _2 \beta _{1\left( 2 \right) }^2 \exp \left( {\text {i}\beta _2 d} \right) ;\, a_{4,1} = \text {i}\alpha _1 , \quad a_{4,2} =-\text {i}\zeta _1 ,\\&a_{4,3} = \text {i}\alpha _2 , \quad a_{4,4} =\text {i}\zeta _2 ,\quad a_{4,5} = \text {i}\zeta _2 , \\&a_{4,6} =-\text {i}\alpha _2 \exp \left( {\text {i}\alpha _2 d} \right) ,\quad a_{4,7} = \text {i}\zeta _2 \exp \left( {\text {i}\beta _2 d} \right) ;\\&a_{5,1} =\text {i}\xi _1 , \quad a_{5,2} =\text {i}\beta _1 ,\quad a_{5,3} = -\text {i}\xi _2 , \quad \\&a_{5,4} =\text {i}\beta _2 , \quad a_{5,5} =-\gamma _2 ,\\&a_{5,6} = -\text {i}\xi _2 \exp \left( {\text {i}\alpha _2 d} \right) , \quad a_{{5,}7} =-\text {i}\beta _2 \exp \left( {\text {i}\beta _2 d} \right) ;\\&a_{6,3} = \mu _2 \left( {2\xi _2^2 -{\omega ^{2}}/{c_{2\left( 2 \right) }^2 }} \right) \exp \left( {\text {i}\alpha _2 d} \right) ,\\&a_{6,4} = -2\mu _2 \zeta _2 \beta _2 \exp \left( {\text {i}\beta _2 d} \right) , \\&a_{6,6} = \mu _2 \left( {2\xi _2^2 -{\omega ^{2}}/{c_{2\left( 2 \right) }^2 }} \right) , \\ \end{aligned}$$
$$\begin{aligned}&a_{6,7} =2\mu _2 \zeta _2 \beta _2 , \quad a_{6,8} =2\text {i}\mu _2 \gamma _2 \zeta _2 ,\\&a_{6,9} = -\mu _3 \left( {2\xi _3^2 -{\omega ^{2}}/{c_{2\left( 3 \right) }^2 }} \right) , \quad a_{6,10} =2\mu _3 \zeta _3 \beta _3 ;\\&a_{7,3} = 2\mu _2 \xi _2 \alpha _2 \exp \left( {\text {i}\alpha _2 d} \right) ,\\&a_{7,4} = \mu _2 \left( {2\zeta _2^2 -{\omega ^{2}}/{c_{2\left( 2 \right) }^2 }} \right) \exp \left( {\text {i}\beta _2 d} \right) , \\&a_{7,6} =-2\mu _2 \xi _2 \alpha _2 ,\quad a_{7,7} = \mu _2 \left( {2\zeta _2^2 -{\omega ^{2}}/{c_{2\left( 2 \right) }^2 }} \right) ,\\&a_{7,8} = \mu _2 \left( {2\zeta _2^2 -{\omega ^{2}}/{c_{2\left( 2 \right) }^2 }} \right) , \quad a_{7,9} =-2\mu _3 \xi _3 \alpha _3 ,\\&a_{7,10} = -\mu _3 \left( {-\beta _3^2 +\zeta _3^2 } \right) ;\\&a_{8,4} = -2\text {i}\eta _2 \beta _2 \beta _{1\left( 2 \right) }^2 \exp \left( {\text {i}\beta _2 d} \right) , \\ \end{aligned}$$
$$\begin{aligned}&a_{8,7} =2\text {i}\eta _2 \beta _2 \beta _{1\left( 2 \right) }^2 ,\quad a_{8,8} = 2\eta _2 \gamma _2 \beta _{2\left( 2 \right) }^2 ;\\&a_{9,3} = -\text {i}\alpha _2 \exp \left( {\text {i}\alpha _2 d} \right) , \quad a_{9,4} =-\text {i}\zeta _2 \exp \left( {\text {i}\beta _2 d} \right) ,\\&a_{9,6} = \text {i}\alpha _2 , \quad a_{9,7} =-\text {i}\zeta _2 , \quad a_{9,8} =-\text {i}\zeta _2 ,\\&a_{9,9} = \text {i}\alpha _{3} , \quad a_{9,10} =\text {i}\zeta _3 ; \quad \\&a_{10,3} =\text {i}\xi _2 \exp \left( {\text {i}\alpha _2 d} \right) ,\\&a_{10,4} = -\text {i}\beta _2 \exp \left( {\text {i}\beta _2 d} \right) , \quad a_{10,6} =\text {i}\xi _2 ,\quad a_{10,7} = \text {i}\beta _2 , \\&a_{10,8} =-\gamma _2 , \quad a_{10,9} =-\text {i}\xi _3 , \quad a_{10,10} =\text {i}\beta _3. \end{aligned}$$
In the case of incident P wave, the non-zero elements of matrix \(\varvec{Q}=\left[ {q_i } \right] _{1{0}\times {1}} \) are
$$\begin{aligned}&q_1 =-\mu _1 \left( {2\xi _1^2 -{\omega ^{2}}/{c_{2\left( 1 \right) }^2 }} \right) ,\quad q_2 =-2\mu _1 \xi _1 \alpha _1,\\&q_4 =\text {i}\alpha _1,\quad q_5 =-\text {i}\xi _1. \end{aligned}$$
In the case of incident SV wave, the non-zero elements of matrix \({\varvec{Q}}'=\left[ {{q}'_i } \right] _{1{0}\times {1}} \) are
$$\begin{aligned}&{q}'_1 =2\mu _1 \zeta _1 \beta _1, \quad {q}'_2 =-\mu _1 \left( {-\beta _1^2 +\zeta _1^2 } \right) , \\&{q}'_4 =\text {i}\zeta _1, \quad {q}'_5 =\text {i}\beta _1. \end{aligned}$$
For the rotation-free interface, compared with the couple-free interface, only following elements of matrix \(\varvec{P}=\left[ {a_{i,j} } \right] _{1{0}\times 1{0}} \) are replaced by
$$\begin{aligned} a_{3,4}= & {} -\beta _{1\left( 2 \right) }^2 , \quad a_{3,5} =\beta _{2\left( 2 \right) }^2 , \quad a_{3,7} =-\beta _{1\left( 2 \right) }^2 \exp \left( {\text {i}\beta _2 d} \right) ;\\ a_{8,4}= & {} \beta _{1\left( 2 \right) }^2 \exp \left( {\text {i}\beta _2 d} \right) , \quad a_{8,7} =\beta _{1\left( 2 \right) }^2 , \quad a_{8,8} =-\beta _{2\left( 2 \right) }^2 . \end{aligned}$$
Appendix C
In the case of the classical elastic slab sandwiched between the classical elastic half-spaces, the non-zero elements of matrix \(\varvec{P}=\left[ {a_{i,j} } \right] _{{8}\times {8}} \) are
$$\begin{aligned} a_{1,1}= & {} \mu _1 \left( {2\xi _1^2 -{\omega ^{2}}/{c_{2\left( 1 \right) }^2 }} \right) , \quad a_{1,2} =2\mu _1 \zeta _1 \beta _1 ,\\ a_{1,{3}}= & {} -\mu _2 \left( {2\xi _2^2 -{\omega ^{2}}/{c_{2\left( 2 \right) }^2 }} \right) , \quad a_{1,{4}} =2\mu _2 \zeta _2 \beta _2 ,\\ a_{1,{5}}= & {} -\mu _2 \left( {2\xi _2^2 -{\omega ^{2}}/{c_{2\left( 2 \right) }^2 }} \right) \exp \left( {\text {i}\alpha _2 d} \right) , \\ a_{1,{6}}= & {} -2\mu _2 \zeta _2 \beta _2 \exp \left( {\text {i}\beta _2 d} \right) ;\\ a_{2,1}= & {} -2\mu _1 \xi _1 \alpha _1 ,\quad a_{2,2} =\mu _1 \left( {-\beta _1^2 +\zeta _1^2 } \right) , \quad \\ a_{2,3}= & {} -2\mu _2 \xi _2 \alpha _2 ,\quad a_{2,4} =-\mu _2 \left( {-\beta _2^2 +\zeta _2^2 } \right) ,\quad \\ a_{2,{5}}= & {} 2\mu _2 \xi _2 \alpha _2 \exp \left( {\text {i}\alpha _2 d} \right) ,\\ a_{2,{6}}= & {} -\mu _2 \left( {-\beta _2^2 +\zeta _2^2 } \right) \exp \left( {\text {i}\beta _2 d} \right) ;\quad \\ \end{aligned}$$
$$\begin{aligned} a_{{3,}1}= & {} \text {i}\alpha _1 ,\quad a_{{3,}2} =-\text {i}\zeta _1 ,\quad a_{{3,}3} =\text {i}\alpha _2, \quad a_{{3,}4} =\text {i}\zeta _2 ,\\ a_{{3,5}}= & {} -\text {i}\alpha _2 \exp \left( {\text {i}\alpha _2 d} \right) ,\quad a_{{3,6}} =\text {i}\zeta _2 \exp \left( {\text {i}\beta _2 d} \right) ; \\ a_{{4,}1}= & {} \text {i}\xi _1 ,\quad a_{{4,}2} =\text {i}\beta _1 , \quad a_{{4,}3} =-\text {i}\xi _2 ,\\ a_{{4,}4}= & {} \text {i}\beta _2 ,\quad a_{{4,5}} =-\text {i}\xi _2 \exp \left( {\text {i}\alpha _2 d} \right) , \\ a_{{4,6}}= & {} -\text {i}\beta _2 \exp \left( {\text {i}\beta _2 d} \right) ;\\ a_{{5,}3}= & {} \mu _2 \left( {2\xi _2^2 -{\omega ^{2}}/{c_{2\left( 2 \right) }^2 }} \right) \exp \left( {\text {i}\alpha _2 d} \right) ,\\ a_{{5,}4}= & {} -2\mu _2 \zeta _2 \beta _2 \exp \left( {\text {i}\beta _2 d} \right) ,\\ a_{{5,5}}= & {} \mu _2 \left( {2\xi _2^2 -{\omega ^{2}}/{c_{2\left( 2 \right) }^2 }} \right) , \quad a_{{5,6}} =2\mu _2 \zeta _2 \beta _2 ,\\ a_{{5,7}}= & {} -\mu _3 \left( {2\xi _3^2 -{\omega ^{2}}/{c_{2\left( 3 \right) }^2 }} \right) , \quad a_{{5,8}} =2\mu _3 \zeta _3 \beta _3 ;\\ a_{{6,}3}= & {} 2\mu _2 \xi _2 \alpha _2 \exp \left( {\text {i}\alpha _2 d} \right) ,\\ \end{aligned}$$
$$\begin{aligned} a_{{6,}4}= & {} \mu _2 \left( {-\beta _2^2 +\zeta _2^2 } \right) \exp \left( {\text {i}\beta _2 d} \right) ,\\ a_{{6,5}}= & {} -2\mu _2 \xi _2 \alpha _2 ,\quad a_{{6,6}} =\mu _2 \left( {-\beta _2^2 +\zeta _2^2 } \right) ,\\ a_{{6,7}}= & {} -2\mu _3 \xi _3 \alpha _3 ,\quad a_{{6,8}} =-\mu _3 \left( {-\beta _3^2 +\zeta _3^2 } \right) ;\\ a_{{7,}3}= & {} -\text {i}\alpha _2 \exp \left( {\text {i}\alpha _2 d} \right) ,\quad a_{{7,4}} =-\text {i}\zeta _2 \exp \left( {\text {i}\beta _2 d} \right) ,\\ a_{{7,5}}= & {} \text {i}\alpha _2 ,\quad a_{{7,6}} =-\text {i}\zeta _2 ,\quad a_{{7,7}} =\text {i}\alpha _{3} ,\quad a_{{7,8}} =\text {i}\zeta _3 ;\quad \\ a_{{8,}3}= & {} \text {i}\xi _2 \exp \left( {\text {i}\alpha _2 d} \right) ,\\ a_{{8,}4}= & {} -\text {i}\beta _2 \exp \left( {\text {i}\beta _2 d} \right) , a_{{8,5}} =\text {i}\xi _2 ,\quad a_{{8,6}} =\text {i}\beta _2 ,\\ a_{{8,7}}= & {} -\text {i}\xi _3 ,\quad a_{{8,8}} =\text {i}\beta _3. \end{aligned}$$
In the case of incident P wave, the non-zero elements of matrix \(\varvec{Q}=\left[ {q_i } \right] _{8\times {1}} \) are
$$\begin{aligned} q_1= & {} -\mu _1 \left( {2\xi _1^2 -{\omega ^{2}}/{c_{2\left( 1 \right) }^2 }} \right) , \quad q_2 =-2\mu _1 \xi _1 \alpha _1, \\ q_3= & {} \text {i}\alpha _1, \quad q_{4} =-\text {i}\xi _1. \end{aligned}$$
In the case of incident SV wave, the non-zero elements of matrix \({\varvec{Q}}'=\left[ {{q}'_i } \right] _{8\times {1}} \) are
$$\begin{aligned} {q}'_1= & {} 2\mu _1 \zeta _1 \beta _1, \quad {q}'_2 =-\mu _1 \left( {-\beta _1^2 +\zeta _1^2 } \right) ,\\ {q}'_3= & {} \text {i}\zeta _1, \quad {q}'_{4} =\text {i}\beta _1. \end{aligned}$$
