Appendix I
$$ \begin{aligned} R_{1} & = J_{{p_{1} }}^{\prime } (\delta_{1} e^{{ - n_{\,1} h_{\,1} }} )Y_{{p_{1} }} (\delta_{1} ) - Y_{{p_{1} }}^{\prime } (\delta_{1} e^{{ - n_{\,1} h_{\,1} }} )J_{{p_{1} }} (\delta_{1} ),R_{2} = J_{{p_{1} }}^{\prime } (\delta_{1} e^{{ - n_{\,1} h_{\,1} }} )Y_{{p_{1} }}^{\prime } \left( {\delta_{1} } \right) - Y_{{p_{1} }}^{\prime } (\delta_{1} e^{{ - n_{\,1} h_{\,1} }} )J_{{p_{1} }}^{\prime } (\delta_{1} ), \\ R_{3} & = J_{{p_{2} }}^{\prime } (\delta_{2} e^{{n_{\,2} h_{\,2} }} )Y_{{p_{2} }} (\delta_{2} ) - Y_{{p_{2} }}^{\prime } (\delta_{2} e^{{n_{\,2} h_{\,2} }} )J_{{p_{2} }} (\delta_{2} ),R_{4} = J_{{p_{2} }} (\delta_{2} e^{{n_{\,2} h_{\,2} }} )Y_{{p_{2} }} (\delta_{2} ) - Y_{{p_{2} }} (\delta_{2} e^{{n_{\,2} h_{\,2} }} )J_{{p_{2} }} (\delta_{2} ), \\ R_{5} & = J_{{p_{2} }}^{\prime } (\delta_{2} e^{{n_{2} h_{2} }} )Y_{{p_{2} }} '(\delta_{2} ) - Y_{{p_{2} }}^{\prime } (\delta_{2} e^{{n_{2} h_{2} }} )J_{{p_{2} }} '(\delta_{2} ),R_{6} = J_{{p_{2} }} (\delta_{2} e^{{n_{\,2} h_{2} }} )Y_{{p_{2} }}^{\prime } (\delta_{2} ) - Y_{{p_{2} }} (\delta_{2} e^{{n_{\,2} h_{2} }} )J_{{p_{2} }}^{\prime } (\delta_{2} ), \\ \end{aligned} $$
$$ p_{1} \sec \phi_{1} = \frac{kc}{{n_{\,1} \beta_{1}^{(1)} }}e^{{ - n_{1} h_{1} }} ,\quad p_{2} \sec \phi_{2} = \frac{kc}{{n_{\,1} \beta_{1}^{(1)} }},\quad p_{1} \tan \phi_{1} = \frac{k}{{n_{1} }}\sqrt {\frac{{c^{2} }}{{(\beta_{1}^{(1)} )^{2} }} - 1 + \zeta_{1}^{(1)} } \left[ {1 - \frac{{c^{2} n_{1} h_{1} }}{{c^{2} - (\beta_{1}^{(1)} )^{2} }}} \right], $$
$$ p_{1} \tan \phi_{2} = \frac{k}{{n_{1} }}\sqrt {\frac{{c^{2} }}{{(\beta_{1}^{(1)} )^{2} }} - 1 + \zeta_{1}^{(1)} } ,(\phi_{1} - \phi_{2} )\sim n_{1} h_{1} \frac{{\sqrt {1 - \zeta_{1}^{(1)} } }}{{\sqrt {\frac{{c^{2} }}{{(\beta_{1}^{(1)} )^{2} }} - 1 + \zeta_{1}^{(1)} } }}, $$
$$ p_{1} (\tan \phi_{1} - \tan \phi_{2} ) - p_{1} (\phi_{1} - \phi_{2} )\sim - kh_{1} \sqrt {\frac{{c^{2} }}{{(\beta_{1}^{(1)} )^{2} }} - 1 + \zeta_{1}^{(1)} } , $$
$$ \sin \phi_{1} \sim \frac{{\beta_{1}^{(1)} }}{c}\sqrt {\frac{{c^{2} }}{{(\beta_{1}^{(1)} )^{2} }} - 1 + \zeta_{1}^{(1)} } \sim \sin \phi_{2} ,\quad p_{2} \sec \phi_{3} = \frac{kc}{{n_{2} \beta_{2}^{(1)} }}e^{{n_{2} h_{2} }} ,\quad p_{2} \sec \phi_{4} = \frac{kc}{{n_{2} \beta_{2}^{(1)} }}, $$
$$ p_{2} \tan \phi_{3} = \frac{k}{{n_{2} }}\sqrt {\frac{{c^{2} }}{{(\beta_{2}^{(1)} )^{2} }} - 1 + \zeta_{2}^{(1)} } ,\quad p_{2} \tan \phi_{4} = \frac{k}{{n_{2} }}\sqrt {\frac{{c^{2} }}{{(\beta_{2}^{(1)} )^{2} }} - 1 + \zeta_{2}^{(1)} } \left[ {1 + \frac{{c^{2} n_{2} h_{2} }}{{c^{2} - (\beta_{2}^{(1)} )^{2} }}} \right], $$
$$ (\phi_{3} - \phi_{4} )\sim n_{2} h_{2} \frac{{\sqrt {1 - \zeta_{2}^{(1)} } }}{{\sqrt {\frac{{c^{2} }}{{(\beta_{2}^{(1)} )^{2} }} - 1 + \zeta_{2}^{\left( 1 \right)} } }},\quad p_{2} (\tan \phi_{3} - \tan \phi_{4} ) - p_{2} (\phi_{3} - \phi_{4} )\sim kh_{2} \sqrt {\frac{{c^{2} }}{{(\beta_{2}^{(1)} )^{2} }} - 1 + \zeta_{2}^{(1)} } , $$
$$ {\text{and}}\;\sin \phi_{3} \sim \frac{{\beta_{2}^{(1)} }}{c}\sqrt {\frac{{c^{2} }}{{\left( {\beta_{2}^{(1)} } \right)^{2} }} - 1 + \zeta_{2}^{(1)} } \sim \sin \phi_{4} . $$
Appendix II
$$ \begin{aligned} R_{7} = J_{{q_{1} }} (\delta_{1} e^{{l_{1} h_{1} }} )Y_{{q_{1} }} (\delta_{1} ) - J_{{q_{1} }} (\delta_{1} )Y_{{q_{1} }} (\delta_{1} e^{{l_{1} h_{1} }} ),\quad R_{8} = J_{{q_{1} }}^{\prime } (\delta_{1} e^{{l_{1} h_{1} }} )Y_{{q_{1} }} (\delta_{1} ) - J_{{q_{1} }} (\delta_{1} )Y_{{q_{1} }}^{\prime } (\delta_{1} e^{{l_{1} h_{1} }} ), \hfill \\ R_{9} = J_{{q_{1} }} (\delta_{1} e^{{l_{1} h_{1} }} )Y_{{q_{1} }}^{\prime } (\delta_{1} ) - J_{{q_{1} }}^{\prime } (\delta_{1} )Y_{{q_{1} }} (\delta_{1} e^{{l_{1} h_{1} }} ),\quad R_{10} = J_{{q_{1} }}^{\prime } (\delta_{1} e^{{l_{1} h_{1} }} )Y_{{q_{1} }}^{\prime } (\delta_{1} ) - J_{{q_{1} }}^{\prime } (\delta_{1} )Y_{{q_{1} }}^{\prime } (\delta_{1} e^{{l_{1} h_{1} }} ), \hfill \\ \end{aligned} $$
$$ \begin{aligned} R_{11} = J_{{q_{2} }} (\delta_{2} e^{{ - l_{2} h_{2} }} )Y_{{q_{2} }} (\delta_{2} ) - J_{{q_{2} }} (\delta_{2} )Y_{{q_{2} }} (\delta_{2} e^{{ - l_{2} h_{2} }} ),\quad R_{12} = J_{{q_{2} }} (\delta_{2} e^{{ - l_{2} h_{2} }} )Y_{{q_{2} }}^{\prime } (\delta_{2} ) - J_{{q_{2} }}^{\prime } (\delta_{2} )Y_{{q_{2} }} (\delta_{2} e^{{ - l_{2} h_{2} }} ), \hfill \\ R_{13} = J_{{q_{2} }}^{\prime } (\delta_{2} e^{{ - l_{2} h_{2} }} )Y_{{q_{2} }} (\delta_{2} ) - J_{{q_{2} }} (\delta_{2} )Y_{{q_{2} }}^{\prime } (\delta_{2} e^{{ - l_{2} h_{2} }} ),\quad R_{14} = J_{{q_{2} }}^{\prime } (\delta_{2} e^{{ - l_{2} h_{2} }} )Y_{{q_{2} }}^{\prime } (\delta_{2} ) - J_{{q_{2} }}^{\prime } (\delta_{2} )Y_{{q_{2} }}^{\prime } (\delta_{2} e^{{ - l_{2} h_{2} }} ). \hfill \\ \end{aligned} $$
$$ \left[ {J_{{q_{1} }} (\gamma_{1} e^{{l_{1} h_{1} }} )Y_{{q_{1} }} (\gamma_{1} ) - Y_{{q_{1\,} }} (\delta \gamma_{1} e^{{l_{1} h_{1} }} )J_{{q_{1} }} (\gamma_{1} )} \right] \approx \frac{{ - 2\sin [q_{1} (\tan \psi_{1} - \tan \psi_{2} ) - q_{1} (\psi_{1} - \psi_{2} )]}}{{\pi q_{1} \sqrt {\tan \psi_{1} \tan \psi_{2} } }}, $$
$$ \left[ {J_{{q_{2} }} (\gamma_{2} e^{{ - l_{2} h_{2} }} )Y_{{q_{2} }} (\delta_{2} ) - Y_{{q_{2} }} (\gamma_{2} e^{{ - l_{\,2} h_{2} }} )J_{{q_{2} }} (\gamma_{2} )} \right] \approx \frac{{ - 2\sin [q_{2} (\tan \psi_{3} - \tan \psi_{4} ) - q_{2} (\psi_{3} - \psi_{4} )]}}{{\pi q_{2} \sqrt {\tan \psi_{3} \tan \psi_{4} } }}, $$
$$ \left[ {J_{{q_{1} }} (\gamma_{1} e^{{l_{1} h_{1} }} )Y_{{q_{1} }}^{\prime } (\gamma_{1} ) - Y_{{q_{1\,} }} (\gamma_{1} e^{{l_{1} h_{1} }} )J_{{q_{1} }}^{\prime } (\gamma_{1} )} \right] \approx \frac{{2\sin \psi_{2} \cos [q_{1} (\tan \psi_{1} - \tan \psi_{2} ) - q_{1} (\psi_{1} - \psi_{2} )]}}{{\pi q_{1} \sqrt {\tan \psi_{1} \tan \psi_{2} } }}, $$
$$ \left[ {J_{{q_{2} }} (\gamma_{2} e^{{ - l_{2} h_{2} }} )Y_{{q_{2} }}^{\prime } (\gamma_{2} ) - Y_{{q_{2} }} (\gamma_{2} e^{{ - l_{2} h_{2} }} )J_{{q_{2} }}^{\prime } (\gamma_{2} )} \right] \approx \frac{{2\sin \psi_{4} \cos [q_{2} (\tan \psi_{3} - \tan \psi_{4} ) - q_{2} (\psi_{3} - \psi_{4} )]}}{{\pi q_{2} \sqrt {\tan \psi_{3} \tan \psi_{4} } }}, $$
$$ \left[ {J_{{q_{1} }}^{\prime } (\gamma_{1} e^{{l_{1} h_{1} }} )Y_{{q_{1} }} (\gamma_{1} ) - Y_{{q_{1\,} }}^{\prime } (\gamma_{1} e^{{l_{1} h_{1} }} )J_{{q_{1} }} (\gamma_{1} )} \right] \approx \frac{{ - 2sin\psi_{1} \cos [q_{1} (\tan \psi_{1} - \tan \psi_{2} ) - q_{1} (\psi_{1} - \psi_{2} )]}}{{\pi q_{1} \sqrt {\tan \psi_{1} \tan \psi_{2} } }}, $$
$$ \left[ {J_{{q_{2} }}^{\prime } (\gamma_{2} e^{{ - l_{2} h_{2} }} )Y_{{q_{2} }} (\gamma_{2} ) - Y_{{q_{2} }} '(\gamma_{2} e^{{ - l_{2} h_{2} }} )J_{{q_{2} }} (\gamma_{2} )} \right] \approx \frac{{ - 2\sin \psi_{3} \cos [q_{2} (\tan \psi_{3} - \tan \psi_{4} ) - q_{2} (\psi_{3} - \psi_{4} )]}}{{\pi q_{2} \sqrt {\tan \psi_{3} \tan \psi_{4} } }}, $$
$$ \left[ {J_{{q_{1} }}^{\prime } (\gamma_{1} e^{{l_{1} h_{1} }} )Y_{{q_{1} }}^{\prime } (\gamma_{1} ) - Y_{{q_{1\,} }}^{\prime } (\gamma_{1} e^{{l_{1} h_{1} }} )J_{{q_{1} }}^{\prime } (\gamma_{1} )} \right] \approx \frac{{ - 2\sin \psi_{1} \sin \psi_{2} \sin [q_{1} (\tan \psi_{1} - \tan \psi_{2} ) - q_{1} (\psi_{1} - \psi_{2} )]}}{{\pi q_{1} \sqrt {\tan \psi_{1} \tan \psi_{2} } }}, $$
$$ \left[ {J_{{q_{2} }}^{\prime } (\gamma_{2} e^{{ - l_{2} h_{2} }} )Y_{{q_{2} }}^{\prime } (\gamma_{2} ) - Y_{{q_{2} }}^{\prime } (\gamma_{2} e^{{\, - l_{2} \,h_{2} }} )J_{{q_{2} }}^{\prime } (\gamma_{2} )} \right] \approx \frac{{ - 2\sin \psi_{3} \sin \psi_{4} \sin [q_{2} (\tan \psi_{3} - \tan \psi_{4} ) - q_{2} (\psi_{3} - \psi_{4} )]}}{{\pi q_{2} \sqrt {\tan \psi_{3} \tan \psi_{4} } }}, $$
$$ q_{1} \sec \psi_{1} = \frac{kc}{{l_{1} \beta_{1}^{\left( 2 \right)} }}e^{{l_{1} h_{1} }} ,\quad q_{1} \sec \psi_{2} = \frac{kc}{{l_{1} \beta_{1}^{(2)} }},\quad q_{1} \tan \psi_{1} = \frac{k}{{l_{1} }}\sqrt {\frac{{c^{2} }}{{(\beta_{1}^{(2)} )^{2} }} - 1 + \zeta_{1}^{\left( 1 \right)} } \left[ {1 + \frac{{c^{2} l_{1} h_{1} }}{{c^{2} - (\beta_{1}^{(2)} )^{2} }}} \right], $$
$$ q_{1} \tan \psi_{2} = \frac{k}{{l_{1} }}\sqrt {\frac{{c^{2} }}{{\left( {\beta_{1}^{(2)} } \right)^{2} }} - 1 + \zeta_{1}^{(2)} } ,\quad \psi_{1} - \psi_{2} \sim l_{1} h_{1} \frac{{\sqrt {1 - \zeta_{1}^{(2)} } }}{{\sqrt {\frac{{c^{2} }}{{\left( {\beta_{1}^{(2)} } \right)^{2} }} - 1 + \zeta_{1}^{(2)} } }}, $$
$$ q_{1} (\tan \psi_{1} - \tan \psi_{2} ) - q_{1} (\psi_{1} - \psi_{2} )\sim kh_{1} \sqrt {\frac{{c^{2} }}{{\left( {\beta_{1}^{(2)} } \right)^{2} }} - 1 + \zeta_{1}^{(2)} } ,\quad \sin \psi_{1} \sim \frac{{\beta_{1}^{(2)} }}{c}\sqrt {\frac{{c^{2} }}{{\left( {\beta_{1}^{(2)} } \right)^{2} }} - 1 + \zeta_{1}^{(2)} } \sim \sin \psi_{2} . $$
$$ q_{2} \sec \psi_{3} = \frac{kc}{{l_{2} \beta_{2}^{(1)} }}e^{{ - l_{2} h_{2} }} ,\quad q_{2} \sec \psi_{4} = \frac{kc}{{l_{2} \beta_{2}^{(2)} }},\quad q_{2} \tan \psi_{3} = \frac{k}{{l_{2} }}\sqrt {\frac{{c^{2} }}{{(\beta_{2}^{(2)} )^{2} }} - 1 + \zeta_{2}^{(2)} } \left[ {1 - \frac{{c^{2} l_{2} h_{2} }}{{c^{2} - (\beta_{2}^{(2)} )^{2} }}} \right], $$
$$ q_{2} \tan \psi_{4} = \frac{k}{{l_{2} }}\sqrt {\frac{{c^{2} }}{{(\beta_{2}^{(2)} )^{2} }} - 1 + \zeta_{2}^{(2)} } ,\quad q_{2} (\tan \psi_{3} - \tan \psi_{4} ) - q_{2} (\psi_{3} - \psi_{4} )\sim - kh_{2} \sqrt {\frac{{c^{2} }}{{(\beta_{2}^{(2)} )^{2} }} - 1 + \zeta_{2}^{(2)} } , $$
$$ \psi_{3} - \psi_{4} \sim l_{2} h_{2} \frac{{\sqrt {1 - \zeta_{2}^{\left( 2 \right)} } }}{{\sqrt {\frac{{c^{2} }}{{\left( {\beta_{2}^{(2)} } \right)^{2} }} - 1 + \zeta_{2}^{(2)} } }}\quad {\text{and}}\quad \sin \psi_{3} \sim \frac{{\beta_{2}^{(2)} }}{c}\sqrt {\frac{{c^{2} }}{{\left( {\beta_{2}^{(2)} } \right)^{2} }} - 1 + \zeta_{2}^{(2)} } \sim \sin \psi_{4} . $$
Appendix III
$$ \begin{aligned} R_{15} = J_{{s_{3} }} (\varOmega_{1} e^{{\hbar_{1} h_{1} }} )Y_{{s_{1} }} (\varOmega_{1} ) - J_{{s_{3} }} (\varOmega_{1} )Y_{{s_{3} }} (\varOmega_{1} e^{{\hbar_{1} h_{1} }} ),\quad R_{16} = J_{{s_{3} }}^{'} (\varOmega_{1} e^{{\hbar_{1} h_{1} }} )Y_{{s_{3} }} (\varOmega_{1} ) - J_{{s_{3} }} (\varOmega_{1} )Y_{{s_{3} }}^{'} (\varOmega_{1} e^{{\hbar_{1} h_{1} }} ), \hfill \\ R_{17} = J_{{s_{3} }}^{{}} (\varOmega_{1} e^{{\hbar_{1} h_{1} }} )Y_{{s_{3} }}^{'} (\varOmega_{1} ) - J_{{s_{3} }}^{'} (\varOmega_{1} )Y_{{s_{3} }}^{{}} (\varOmega_{1} e^{{\hbar_{1} h_{1} }} ),\quad R_{18} = J_{{s_{3} }}^{'} (\varOmega_{1} e^{{\hbar_{1} h_{1} }} )Y_{{s_{3} }}^{'} (\varOmega_{1} ) - J_{{s_{3} }}^{'} (\varOmega_{1} )Y_{{s_{3} }}^{'} (\varOmega_{1} e^{{\hbar_{1} h_{1} }} ), \hfill \\ \end{aligned} $$
$$ \begin{aligned} R_{19} = J_{{s_{4} }} (\varOmega_{2} e^{{ - \hbar_{2} h_{2} }} )Y_{{s_{4} }} (\varOmega_{2} ) - J_{{s_{4} }} (\varOmega_{2} )Y_{{s_{4} }} (\varOmega_{2} e^{{ - \hbar_{2} h_{2} }} ),\quad R_{20} = J_{{s_{4} }}^{'} (\varOmega_{2} e^{{ - \hbar_{2} h_{2} }} )Y_{{s_{4} }} (\varOmega_{2} ) - J_{{s_{4} }} (\varOmega_{2} )Y_{{s_{4} }}^{'} (\varOmega_{2} e^{{ - \hbar_{2} h_{2} }} ), \hfill \\ R_{21} = J_{{s_{4} }}^{{}} (\varOmega_{2} e^{{ - \hbar_{2} h_{2} }} )Y_{{s_{4} }}^{'} (\varOmega_{2} ) - J_{{s_{4} }}^{'} (\varOmega_{2} )Y_{{s_{4} }}^{{}} (\varOmega_{2} e^{{ - \hbar_{2} h_{2} }} ),\quad R_{22} = J_{{s_{4} }}^{'} (\varOmega_{2} e^{{ - \hbar_{2} h_{2} }} )Y_{{s_{4} }}^{'} (\varOmega_{2} ) - J_{{s_{4} }}^{'} (\varOmega_{2} )Y_{{s_{4} }}^{'} (\varOmega_{2} e^{{ - \hbar_{2} h_{2} }} ), \hfill \\ \end{aligned} $$
$$ \left[ {J_{{s_{3} }} (\varOmega_{1} e^{{\hbar_{1} h_{\,1} }} )Y_{{s_{3} }} (\varOmega_{1} ) - Y_{{s_{3} }} (\varOmega_{1} e^{{\hbar_{1} h_{\,1} }} )J_{{s_{3} }} (\varOmega_{1} )} \right] \approx \frac{{ - \,2\sin [s_{3} (\tan \phi_{21} - \tan \phi_{22} ) - s_{3} (\phi_{21} - \phi_{22} )]}}{{\pi s_{3} \sqrt {\tan \phi_{21} \tan \phi_{22} } }}, $$
$$ \left[ {J_{{s_{4} }} (\varOmega_{2} e^{{ - \hbar_{2} h_{2} }} )Y_{{s_{4} }} (\varOmega_{2} ) - Y_{{s_{4} }} (\varOmega_{2} e^{{ - \hbar_{2} h_{2} }} )J_{{s_{2} }} (\varOmega_{2} )} \right] \approx \frac{{ - \,2\sin [s_{4} (\tan \phi_{23} - \tan \phi_{24} ) - s_{4} (\phi_{23} - \phi_{24} )]}}{{\pi s_{4} \sqrt {\tan \phi_{23} \tan \phi_{24} } }}, $$
$$ \left[ {J_{{s_{3} }} (\varOmega_{1} e^{{\hbar_{1} h_{\,1} }} )Y_{{s_{3} }} '(\varOmega_{1} ) - Y_{{s_{3} }} (\varOmega_{1} e^{{\hbar_{1} h_{\,1} }} )J_{{s_{3} }} '(\varOmega_{1} )} \right] \approx \frac{{2\sin \phi_{21} \sin [s_{3} (\tan \phi_{21} - \tan \phi_{22} ) - s_{3} (\phi_{21} - \phi_{22} )]}}{{\pi s_{3} \sqrt {\tan \phi_{21} \tan \phi_{22} } }}, $$
$$ \left[ {J_{{s_{4} }} (\varOmega_{2} e^{{ - \hbar_{2} h_{2} }} )Y_{{s_{4} }} '(\varOmega_{2} ) - Y_{{s_{4} }} (\varOmega_{2} e^{{ - \hbar_{2} h_{2} }} )J_{{s_{4} }} '(\varOmega_{2} )} \right] \approx \frac{{2\sin \phi_{24} \cos [s_{4} (\tan \phi_{23} - \tan \phi_{24} ) - s_{4} (\phi_{23} - \phi_{24} )]}}{{\pi s_{4} \sqrt {\tan \phi_{23} \tan \phi_{24} } }}, $$
$$ \left[ {J_{{s_{3} }} '(\varOmega_{1} e^{{\hbar_{1} h_{\,1} }} )Y_{{s_{3} }} (\varOmega_{1} ) - Y_{{s_{3} }} '(\varOmega_{1} e^{{\hbar_{1} h_{\,1} }} )J_{{s_{3} }} (\varOmega_{1} )} \right] \approx \frac{{ - \,2\sin \phi_{21} \cos [s_{3} (\tan \phi_{21} - \tan \phi_{22} ) - s_{3} (\phi_{21} - \phi_{22} )]}}{{\pi s_{3} \sqrt {\tan \phi_{21} \tan \phi_{22} } }}, $$
$$ \left[ {J_{{s_{4} }} '(\varOmega_{2} e^{{ - \hbar_{2} h_{2} }} )Y_{{s_{4} }} (\varOmega_{2} ) - Y_{{s_{4} }} '(\varOmega_{2} e^{{ - \hbar_{2} h_{2} }} )J_{{s_{4} }} (\varOmega_{2} )} \right] \approx \frac{{ - \,2\sin \phi_{23} \cos [s_{4} (\tan \phi_{23} - \tan \phi_{24} ) - s_{4} (\phi_{23} - \phi_{24} )]}}{{\pi s_{4} \sqrt {\tan \phi_{23} \tan \phi_{24} } }}, $$
$$ \left[ {J_{{s_{3} }} '(\varOmega_{1} e^{{\hbar_{1} h_{\,1} }} )Y_{{s_{3} }} '(\varOmega_{1} ) - Y_{{s_{3} }} '(\varOmega_{1} e^{{\hbar_{1} h_{\,1} }} )J_{{s_{3} }} '(\varOmega_{1} )} \right] \approx \frac{{ - \,2\sin \phi_{21} \sin \phi_{22} \sin [s_{3} (\tan \phi_{21} - \tan \phi_{22} ) - s_{3} (\phi_{21} - \phi_{22} )]}}{{\pi s_{3} \sqrt {\tan \phi_{21} \tan \phi_{22} } }}, $$
$$ \left[ {J_{{s_{4} }} '(\varOmega_{2} e^{{ - \hbar_{2} h_{2} }} )Y_{{s_{4} }} '(\varOmega_{2} ) - Y_{{s_{4} }} '(\varOmega_{2} e^{{ - \hbar_{2} h_{2} }} )J_{{s_{4} }} '(\varOmega_{2} )} \right] \approx \frac{{ - \,2\sin \phi_{23} \cos [s_{4} (\tan \phi_{23} - \tan \phi_{24} ) - s_{4} (\phi_{23} - \phi_{24} )]}}{{\pi s_{4} \sqrt {\tan \phi_{23} \tan \phi_{24} } }}, $$
$$ s_{3} \sec \varphi_{21} = \frac{kc}{{\hbar_{1} \beta_{1}^{(3)} }}e^{{\hbar_{1} h_{1} }} ,\quad s_{3} \sec \varphi_{22} = \frac{kc}{{\hbar_{1} \beta_{1}^{(3)} }},\quad s_{3} \tan \varphi_{21} = \frac{k}{{\hbar_{1} }}\sqrt {\frac{{c^{2} }}{{(\beta_{1}^{(3)} )^{2} }} - 1 + \zeta_{1}^{(3)} } \left[ {1 - \frac{{c^{2} \hbar_{1} h_{1} }}{{c^{2} - (\beta_{1}^{(3)} )^{2} }}} \right], $$
$$ s_{3} \tan \phi_{22} = \frac{k}{{\hbar_{1} }}\sqrt {\frac{{c^{2} }}{{\left( {\beta_{\,1}^{(3)} } \right)^{2} }} - 1 + \zeta_{1}^{(3)} } ,\quad \left( {\phi_{21} - \phi_{22} } \right)\sim \hbar_{1} h_{1} \frac{{\sqrt {1 - \zeta_{1}^{(3)} } }}{{\sqrt {\frac{{c^{2} }}{{\left( {\beta_{\,1}^{(3)} } \right)^{2} }} - 1 + \zeta_{1}^{(3)} } }}, $$
$$ s_{3} (\tan \phi_{21} - \tan \phi_{22} ) - s_{3} (\phi_{21} - \phi_{22} )\sim kh_{1} \sqrt {\frac{{c^{2} }}{{(\beta_{1}^{(3)} )^{2} }} - 1 + \zeta_{1}^{(3)} } , $$
$$ \sin \phi_{21} \sim \frac{{\beta_{1}^{(3)} }}{c}\sqrt {\frac{{c^{2} }}{{(\beta_{1}^{(3)} )^{2} }} - 1 + \zeta_{1}^{(3)} } \sim \sin \phi_{22} ,\quad s_{4} \sec \phi_{23} = \frac{kc}{{\hbar_{2} \beta_{2}^{(3)} }}e^{{\hbar_{2} h_{2} }} ,\quad s_{4} \sec \phi_{24} = \frac{kc}{{\hbar_{2} \beta_{2}^{(3)} }}, $$
$$ s_{4} \tan \phi_{23} = \frac{k}{{\hbar_{2} }}\sqrt {\frac{{c^{2} }}{{(\beta_{2}^{(3)} )^{2} }} - 1 + \zeta_{2}^{(3)} } ,\quad s_{4} \tan \phi_{24} = \frac{k}{{\hbar_{2} }}\sqrt {\frac{{c^{2} }}{{(\beta_{2}^{(3)} )^{2} }} - 1 + \zeta_{2}^{(3)} } \left[ {1 + \frac{{c^{2} \hbar_{2} h_{2} }}{{c^{2} - (\beta_{2}^{(3)} )^{2} }}} \right], $$
$$ \left( {\phi_{23} - \phi_{24} } \right)\sim \hbar_{2} h_{2} \frac{{\sqrt {1 - \zeta_{2}^{\left( 3 \right)} } }}{{\sqrt {\frac{{c^{2} }}{{\left( {\beta_{2}^{\left( 3 \right)} } \right)^{2} }} - 1 + \zeta_{2}^{\left( 3 \right)} } }},s_{4} \left( {tan\phi_{23} - tan\phi_{24} } \right) - s_{4} \left( {\phi_{23} - \phi_{24} } \right)\sim \hbar_{2} h_{2} \sqrt {\frac{{c^{2} }}{{\left( {\beta_{2}^{\left( 3 \right)} } \right)^{2} }} - 1 + \zeta_{2}^{\left( 3 \right)} } \,, $$
$$ {\text{and}}\;\sin \phi_{23} \sim \frac{{\beta_{2}^{(3)} }}{c}\sqrt {\frac{{c^{2} }}{{(\beta_{2}^{(3)} )^{2} }} - 1 + \zeta_{2}^{(3)} } \sim \sin \phi_{24} . $$
Appendix IV
$$ \begin{aligned} R_{23} & = I_{0}^{'} \left\{ {{{t_{1} (1 - \varepsilon_{1} h_{1} )} \mathord{\left/ {\vphantom {{t_{1} (1 - \varepsilon_{1} h_{1} )} {i\varepsilon_{1} }}} \right. \kern-0pt} {i\varepsilon_{1} }}} \right\}K_{0}^{'} ({{t_{1} } \mathord{\left/ {\vphantom {{t_{1} } {i\varepsilon_{1} }}} \right. \kern-0pt} {i\varepsilon_{1} }}) - K_{0}^{'} \left\{ {{{t_{1} (1 - \varepsilon_{1} h_{1} )} \mathord{\left/ {\vphantom {{t_{1} (1 - \varepsilon_{1} h_{1} )} {i\varepsilon_{1} }}} \right. \kern-0pt} {i\varepsilon_{1} }}} \right\}I_{0}^{'} ({{t_{1} } \mathord{\left/ {\vphantom {{t_{1} } {i\varepsilon_{1} }}} \right. \kern-0pt} {i\varepsilon_{1} }}), \\ R_{24} & = I_{0}^{'} \left\{ {{{t_{1} (1 - \varepsilon_{1} h_{1} )} \mathord{\left/ {\vphantom {{t_{1} (1 - \varepsilon_{1} h_{1} )} {i\varepsilon_{1} }}} \right. \kern-0pt} {i\varepsilon_{1} }}} \right\}K_{0}^{{}} ({{t_{1} } \mathord{\left/ {\vphantom {{t_{1} } {i\varepsilon_{1} }}} \right. \kern-0pt} {i\varepsilon_{1} }}) - K_{0}^{'} \left\{ {{{t_{1} (1 - \varepsilon_{1} h_{1} )} \mathord{\left/ {\vphantom {{t_{1} (1 - \varepsilon_{1} h_{1} )} {i\varepsilon_{1} }}} \right. \kern-0pt} {i\varepsilon_{1} }}} \right\}I_{0}^{{}} ({{t_{1} } \mathord{\left/ {\vphantom {{t_{1} } {i\varepsilon_{1} }}} \right. \kern-0pt} {i\varepsilon_{1} }}), \\ R_{25} & = I_{0}^{'} \left\{ {{{t_{2} (1 + \varepsilon_{2} h_{2} )} \mathord{\left/ {\vphantom {{t_{2} (1 + \varepsilon_{2} h_{2} )} {i\varepsilon_{2} }}} \right. \kern-0pt} {i\varepsilon_{2} }}} \right\}K_{0}^{'} ({{t_{2} } \mathord{\left/ {\vphantom {{t_{2} } {i\varepsilon_{2} }}} \right. \kern-0pt} {i\varepsilon_{2} }}) - K_{0}^{'} \left\{ {{{t_{2} (1 + \varepsilon_{2} h_{2} )} \mathord{\left/ {\vphantom {{t_{2} (1 + \varepsilon_{2} h_{2} )} {i\varepsilon_{2} }}} \right. \kern-0pt} {i\varepsilon_{2} }}} \right\}I_{0}^{'} ({{t_{2} } \mathord{\left/ {\vphantom {{t_{2} } {i\varepsilon_{2} }}} \right. \kern-0pt} {i\varepsilon_{2} }}), \\ R_{26} & = I_{0}^{{}} \left\{ {{{t_{2} (1 + \varepsilon_{2} h_{2} )} \mathord{\left/ {\vphantom {{t_{2} (1 + \varepsilon_{2} h_{2} )} {i\varepsilon_{2} }}} \right. \kern-0pt} {i\varepsilon_{2} }}} \right\}K_{0}^{'} ({{t_{2} } \mathord{\left/ {\vphantom {{t_{2} } {i\varepsilon_{2} }}} \right. \kern-0pt} {i\varepsilon_{2} }}) - K_{0}^{{}} \left\{ {{{t_{2} (1 + \varepsilon_{2} h_{2} )} \mathord{\left/ {\vphantom {{t_{2} (1 + \varepsilon_{2} h_{2} )} {i\varepsilon_{2} }}} \right. \kern-0pt} {i\varepsilon_{2} }}} \right\}I_{0}^{'} ({{t_{2} } \mathord{\left/ {\vphantom {{t_{2} } {i\varepsilon_{2} }}} \right. \kern-0pt} {i\varepsilon_{2} }}), \\ R_{27} & = I_{0}^{'} \left\{ {{{t_{2} (1 + \varepsilon_{2} h_{2} )} \mathord{\left/ {\vphantom {{t_{2} (1 + \varepsilon_{2} h_{2} )} {i\varepsilon_{2} }}} \right. \kern-0pt} {i\varepsilon_{2} }}} \right\}K_{0}^{{}} ({{t_{2} } \mathord{\left/ {\vphantom {{t_{2} } {i\varepsilon_{2} }}} \right. \kern-0pt} {i\varepsilon_{2} }}) - K_{0}^{'} \left\{ {{{t_{2} (1 + \varepsilon_{2} h_{2} )} \mathord{\left/ {\vphantom {{t_{2} (1 + \varepsilon_{2} h_{2} )} {i\varepsilon_{2} }}} \right. \kern-0pt} {i\varepsilon_{2} }}} \right\}I_{0}^{{}} ({{t_{2} } \mathord{\left/ {\vphantom {{t_{2} } {i\varepsilon_{2} }}} \right. \kern-0pt} {i\varepsilon_{2} }}), \\ R_{28} & = I_{0}^{{}} \left\{ {{{t_{2} (1 + \varepsilon_{2} h_{2} )} \mathord{\left/ {\vphantom {{t_{2} (1 + \varepsilon_{2} h_{2} )} {i\varepsilon_{2} }}} \right. \kern-0pt} {i\varepsilon_{2} }}} \right\}K_{0}^{{}} ({{t_{2} } \mathord{\left/ {\vphantom {{t_{2} } {i\varepsilon_{2} }}} \right. \kern-0pt} {i\varepsilon_{2} }}) - K_{0}^{{}} \left\{ {{{t_{2} (1 + \varepsilon_{2} h_{2} )} \mathord{\left/ {\vphantom {{t_{2} (1 + \varepsilon_{2} h_{2} )} {i\varepsilon_{2} }}} \right. \kern-0pt} {i\varepsilon_{2} }}} \right\}I_{0}^{{}} ({{t_{2} } \mathord{\left/ {\vphantom {{t_{2} } {i\varepsilon_{2} }}} \right. \kern-0pt} {i\varepsilon_{2} }}), \\ R_{29} & = J_{0}^{'} ({{t_{1} } \mathord{\left/ {\vphantom {{t_{1} } {\varepsilon_{1} }}} \right. \kern-0pt} {\varepsilon_{1} }})Y_{0}^{'} \left\{ {{{t_{1} (1 - \varepsilon_{1} h_{1} )} \mathord{\left/ {\vphantom {{t_{1} (1 - \varepsilon_{1} h_{1} )} {\varepsilon_{1} }}} \right. \kern-0pt} {\varepsilon_{1} }}} \right\} - J_{0}^{'} \left\{ {{{t_{1} (1 - \varepsilon_{1} h_{1} )} \mathord{\left/ {\vphantom {{t_{1} (1 - \varepsilon_{1} h_{1} )} {\varepsilon_{1} }}} \right. \kern-0pt} {\varepsilon_{1} }}} \right\}Y_{0}^{'} ({{t_{1} } \mathord{\left/ {\vphantom {{t_{1} } {\varepsilon_{1} }}} \right. \kern-0pt} {\varepsilon_{1} }}), \\ R_{30} & = J_{0}^{'} \left\{ {{{t_{1} (1 - \varepsilon_{1} h_{1} )} \mathord{\left/ {\vphantom {{t_{1} (1 - \varepsilon_{1} h_{1} )} {\varepsilon_{1} }}} \right. \kern-0pt} {\varepsilon_{1} }}} \right\}Y_{0}^{{}} ({{t_{1} } \mathord{\left/ {\vphantom {{t_{1} } {\varepsilon_{1} }}} \right. \kern-0pt} {\varepsilon_{1} }}) - J_{0}^{{}} ({{t_{1} } \mathord{\left/ {\vphantom {{t_{1} } {\varepsilon_{1} }}} \right. \kern-0pt} {\varepsilon_{1} }})Y_{0}^{'} \left\{ {{{t_{1} (1 - \varepsilon_{1} h_{1} )} \mathord{\left/ {\vphantom {{t_{1} (1 - \varepsilon_{1} h_{1} )} {\varepsilon_{1} }}} \right. \kern-0pt} {\varepsilon_{1} }}} \right\}, \\ R_{31} & = J_{0}^{'} ({{t_{2} } \mathord{\left/ {\vphantom {{t_{2} } {\varepsilon_{2} }}} \right. \kern-0pt} {\varepsilon_{2} }})Y_{0}^{'} \left\{ {{{t_{2} (1 + \varepsilon_{2} h_{2} )} \mathord{\left/ {\vphantom {{t_{2} (1 + \varepsilon_{2} h_{2} )} {\varepsilon_{2} }}} \right. \kern-0pt} {\varepsilon_{2} }}} \right\} - J_{0}^{'} \left\{ {{{t_{2} (1 + \varepsilon_{2} h_{2} )} \mathord{\left/ {\vphantom {{t_{2} (1 + \varepsilon_{2} h_{2} )} {\varepsilon_{2} }}} \right. \kern-0pt} {\varepsilon_{2} }}} \right\}Y_{0}^{'} ({{t_{2} } \mathord{\left/ {\vphantom {{t_{2} } {\varepsilon_{2} }}} \right. \kern-0pt} {\varepsilon_{2} }}), \\ R_{32} & = J_{0}^{{}} \left\{ {{{t_{2} (1 + \varepsilon_{2} h_{2} )} \mathord{\left/ {\vphantom {{t_{2} (1 + \varepsilon_{2} h_{2} )} {\varepsilon_{2} }}} \right. \kern-0pt} {\varepsilon_{2} }}} \right\}Y_{0}^{'} ({{t_{2} } \mathord{\left/ {\vphantom {{t_{2} } {\varepsilon_{2} }}} \right. \kern-0pt} {\varepsilon_{2} }}) - J_{0}^{'} ({{t_{2} } \mathord{\left/ {\vphantom {{t_{2} } {\varepsilon_{2} }}} \right. \kern-0pt} {\varepsilon_{2} }})Y_{0}^{{}} \left\{ {{{t_{2} (1 + \varepsilon_{2} h_{2} )} \mathord{\left/ {\vphantom {{t_{2} (1 + \varepsilon_{2} h_{2} )} {\varepsilon_{2} }}} \right. \kern-0pt} {\varepsilon_{2} }}} \right\}, \\ R_{33} & = J_{0}^{'} \left\{ {{{t_{2} (1 + \varepsilon_{2} h_{2} )} \mathord{\left/ {\vphantom {{t_{2} (1 + \varepsilon_{2} h_{2} )} {\varepsilon_{2} }}} \right. \kern-0pt} {\varepsilon_{2} }}} \right\}Y_{0}^{{}} ({{t_{2} } \mathord{\left/ {\vphantom {{t_{2} } {\varepsilon_{2} }}} \right. \kern-0pt} {\varepsilon_{2} }}) - J_{0}^{{}} ({{t_{2} } \mathord{\left/ {\vphantom {{t_{2} } {\varepsilon_{2} }}} \right. \kern-0pt} {\varepsilon_{2} }})Y_{0}^{'} \left\{ {{{t_{2} (1 + \varepsilon_{2} h_{2} )} \mathord{\left/ {\vphantom {{t_{2} (1 + \varepsilon_{2} h_{2} )} {\varepsilon_{2} }}} \right. \kern-0pt} {\varepsilon_{2} }}} \right\}, \\ R_{34} & = J_{0}^{{}} \left\{ {{{t_{2} (1 + \varepsilon_{2} h_{2} )} \mathord{\left/ {\vphantom {{t_{2} (1 + \varepsilon_{2} h_{2} )} {\varepsilon_{2} }}} \right. \kern-0pt} {\varepsilon_{2} }}} \right\}Y_{0}^{{}} ({{t_{2} } \mathord{\left/ {\vphantom {{t_{2} } {\varepsilon_{2} }}} \right. \kern-0pt} {\varepsilon_{2} }}) - J_{0}^{{}} ({{t_{2} } \mathord{\left/ {\vphantom {{t_{2} } {\varepsilon_{2} }}} \right. \kern-0pt} {\varepsilon_{2} }})Y_{0}^{{}} \left\{ {{{t_{2} (1 + \varepsilon_{2} h_{2} )} \mathord{\left/ {\vphantom {{t_{2} (1 + \varepsilon_{2} h_{2} )} {\varepsilon_{2} }}} \right. \kern-0pt} {\varepsilon_{2} }}} \right\}, \\ \end{aligned} $$
