Seismic ground motion study of free-field earthquakes in unsaturated soils incident with SV wave under thermal effects

Abstract

Based on the theory behind wave propagation in unsaturated porous thermoelastic media and single-phase thermoelastic media, a model has been established to describe the unsaturated soil free-field under plane SV-wave incidence. By employing the Helmholtz vector decomposition principle, the paper has analyzed the wave field in the free field of an unsaturated foundation affected by thermal changes. The paper has derived an analytical solution for the seismic ground motion of the free field of an unsaturated foundation subjected to plane SV-wave incidence under the thermal influence. The influence of thermal physical parameters such as thermal conductivity, medium temperature, and thermal expansion coefficient on the seismic ground motion of free field on unsaturated foundations is analyzed through numerical calculation. The results show that the amplification coefficients for horizontal and vertical displacement obtained under the two theoretical models with and without thermal effects are significantly different. As an increase in the thermal expansion coefficient, the surface vertical displacement amplification coefficient rises, while it has a negligible impact on the horizontal displacement amplification coefficient. As the temperature of the medium rises, the horizontal displacement and vertical displacement amplification coefficients gradually decrease and increase, respectively. On the other hand, the thermal conductivity and phase lag of the heat flux have a tiny impact on the amplification coefficients for surface displacement. Furthermore, with a rise in saturation, the amplification coefficients for horizontal displacement gradually increase, and the amplification coefficients for vertical displacement gradually decrease. A gas phase in pores significantly affects the surface displacement amplification coefficient.

Appendix 1

$$f_{11} = \left[ {\left( {\lambda^{e} + 2\mu^{e} n_{1rp1}^{2} } \right)k_{1rp1}^{2} + 3K_{b}^{e} \beta_{T}^{e} \delta_{TP1}^{e} } \right]\exp (ik_{1rp1} n_{1rp1} H)$$

$f_{12} = \left[ {\left( {\lambda^{e} + 2\mu^{e} n_{1rp2}^{2} } \right)k_{1rp2}^{2} + 3K_{b}^{e} \beta_{T}^{e} \delta_{TP2}^{e} } \right]\exp (ik_{1rp2} n_{1rp2} H)$

$$f_{13} = 2\mu^{e} l_{1rs} n_{1rs} k_{1rs}^{2} \exp (ik_{1rs} n_{1rs} H)$$

$f_{14} = - \left[ {\left( {\overline{\lambda } + 2\mu n_{tp1}^{2} + M_{1} \delta_{fp1} + M_{2} \delta_{ap1} } \right)k_{tp1}^{2} - M_{3} \delta_{Tp1} } \right]\exp ( - ik_{tp1} n_{tp1} H)$

$$f_{15} = - \left[ {\left( {\overline{\lambda } + 2\mu n_{tp2}^{2} + M_{1} \delta_{fp2} + M_{2} \delta_{ap2} } \right)k_{tp2}^{2} - M_{3} \delta_{Tp2} } \right]\exp ( - ik_{tp2} n_{tp2} H)$$

$f_{16} = - \left[ {\left( {\overline{\lambda } + 2\mu n_{tp3}^{2} + M_{1} \delta_{fp3} + M_{2} \delta_{ap3} } \right)k_{tp3}^{2} - M_{3} \delta_{Tp3} } \right]\exp ( - ik_{tp3} n_{tp3} H)$

$$f_{17} = - \left[ {\left( {\overline{\lambda } + 2\mu n_{tp4}^{2} + M_{1} \delta_{fp4} + M_{2} \delta_{ap4} } \right)k_{tp4}^{2} - M_{3} \delta_{Tp4} } \right]\exp ( - ik_{tp4} n_{tp4} H)$$

$f_{18} = 2\mu l_{2ts} n_{2ts} k_{2ts}^{2} \exp ( - ik_{2ts} n_{2ts} H)$

$$f_{19} = - \left[ {\left( {\overline{\lambda } + 2\mu n_{2rp1}^{2} + M_{1} \delta_{2rp1} + M_{2} \delta_{2rp1} } \right)k_{2rp1}^{2} - M_{3} \delta_{Tp1} } \right]\exp (ik_{2rp1} n_{2rp1} H)$$

$$f_{1(10)} = - \left[ {\left( {\overline{\lambda } + 2\mu n_{2rp2}^{2} + M_{1} \delta_{2rp2} + M_{2} \delta_{2rp2} } \right)k_{2rp2}^{2} - M_{3} \delta_{Tp2} } \right]\exp (ik_{2rp2} n_{2rp2} H)$$

$$f_{1(11)} = - \left[ {\left( {\overline{\lambda } + 2\mu n_{2rp3}^{2} + M_{1} \delta_{2rp3} + M_{2} \delta_{2rp3} } \right)k_{2rp3}^{2} - M_{3} \delta_{Tp3} } \right]\exp (ik_{2rp3} n_{2rp3} H)$$

$$f_{1(12)} = - \left[ {\left( {\overline{\lambda } + 2\mu n_{2rp4}^{2} + M_{1} \delta_{2rp4} + M_{2} \delta_{2rp4} } \right)k_{2rp4}^{2} - M_{3} \delta_{Tp4} } \right]\exp (ik_{2rp4} n_{2rp4} H)$$

$f_{1(13)} = - 2\mu l_{2rs} n_{2rs} k_{2rs}^{2} \exp \left( {ik_{2rs} n_{2rs} H} \right)$

$$f_{21} = 2\mu^{e} l_{1rp1} n_{1rp1} k_{1rp1}^{2} \exp (ik_{1rp1} n_{1rp1} H)$$

$f_{22} = 2\mu^{e} l_{1rp2} n_{1rp2} k_{1rp2}^{2} \exp (ik_{1rp2} n_{1rp2} H)$

$$f_{23} = \mu^{e} \left( {l_{1rs}^{2} - n_{1rs}^{2} } \right)k_{1rs}^{2} \exp \left( {ik_{1rs} n_{1rs} H} \right)$$

$f_{24} = 2\mu l_{tp1} n_{tp1} k_{tp1}^{2} \exp \left( { - ik_{tp1} n_{tp1} H} \right)$

$$f_{25} = 2\mu l_{tp2} n_{tp2} k_{tp2}^{2} \exp \left( { - ik_{tp2} n_{tp2} H} \right)$$

$f_{26} = 2\mu l_{tp3} n_{tp3} k_{tp3}^{2} \exp \left( { - ik_{tp3} n_{tp3} H} \right)$

$$f_{27} = 2\mu l_{tp4} n_{tp4} k_{tp4}^{2} \exp \left( { - ik_{tp4} n_{tp4} H} \right)$$

$f_{28} = \mu \left( {n_{2ts}^{2} - l_{2ts}^{2} } \right)k_{2ts}^{2} \exp \left( { - ik_{2ts} n_{2ts} H} \right)$

$$f_{29} = - 2\mu l_{2rp1} n_{2rp1} k_{2rp1}^{2} \exp \left( {ik_{2rp1} n_{2rp1} H} \right)$$

$f_{2(10)} = - 2\mu l_{2rp2} n_{2rp2} k_{2rp2}^{2} \exp \left( {ik_{2rp2} n_{2rp2} H} \right)$

$$f_{2(11)} = - 2\mu l_{2rp3} n_{2rp3} k_{2rp3}^{2} \exp \left( {ik_{2rp3} n_{2rp3} H} \right)$$

$f_{2(12)} = - 2\mu l_{2rp4} n_{2rp4} k_{2rp4}^{2} \exp \left( {ik_{2rp4} n_{2rp4} H} \right)$

$$f_{2(13)} = \mu \left( {n_{2rs}^{2} - l_{2rs}^{2} } \right)k_{2rs}^{2} \exp \left( {ik_{2rs} n_{2rs} H} \right)$$

$f_{31} = - n_{1rp1} k_{1rp1} \exp \left( {ik_{1rp1} n_{1rp1} H} \right)$

$$f_{32} = - n_{1rp2} k_{1rp2} \exp \left( {ik_{1rp2} n_{1rp2} H} \right)$$

$f_{33} = - l_{1rs} k_{1rs} \exp \left( {ik_{1rs} n_{1rs} H} \right)$

$$f_{34} = - n_{tp1} k_{tp1} \exp \left( { - ik_{tp1} n_{tp1} H} \right)$$

$f_{35} = - n_{tp2} k_{tp2} \exp \left( { - ik_{tp2} n_{tp2} H} \right)$

$$f_{36} = - n_{tp3} k_{tp3} \exp \left( { - ik_{tp3} n_{tp3} H} \right)$$

$f_{37} = - n_{tp4} k_{tp4} \exp \left( { - ik_{tp4} n_{tp4} H} \right)$

$$f_{38} = l_{2ts} k_{2ts} \exp \left( { - ik_{2ts} n_{2ts} H} \right)$$

$f_{39} = n_{2rp1} k_{2rp1} \exp \left( {ik_{2rp1} n_{2rp1} H} \right)$

$$f_{3(10)} = n_{2rp2} k_{2rp2} \exp \left( {ik_{2rp2} n_{2rp2} H} \right)$$

$f_{3(11)} = n_{2rp3} k_{2rp3} \exp \left( {ik_{2rp3} n_{2rp3} H} \right)$

$$f_{3(12)} = n_{2rp4} k_{2rp4} \exp \left( {ik_{2rp4} n_{2rp4} H} \right)$$

$f_{3(13)} = l_{2rs} k_{2rs} \exp \left( {ik_{2rs} n_{2rs} H} \right)$

$$f_{41} = - l_{1rp1} k_{1rp1} \exp \left( {ik_{1rp1} n_{1rp1} H} \right)$$

$f_{42} = - l_{1rp2} k_{1rp2} \exp \left( {ik_{1rp2} n_{1rp2} H} \right)$

$$f_{43} = n_{1rs} k_{1rs} \exp \left( {ik_{1rs} n_{1rs} H} \right)$$

$f_{44} = l_{tp1} k_{tp1} \exp \left( { - ik_{tp1} n_{tp1} H} \right)$

$$f_{45} = l_{tp2} k_{tp2} \exp \left( { - ik_{tp2} n_{tp2} H} \right)$$

$f_{46} = l_{tp3} k_{tp3} \exp \left( { - ik_{tp3} n_{tp3} H} \right)$

$$f_{47} = l_{tp4} k_{tp4} \exp \left( { - ik_{tp4} n_{tp4} H} \right)$$

$f_{48} = n_{2ts} k_{2ts} \exp \left( { - ik_{2ts} n_{2ts} H} \right)$

$$f_{49} = l_{2rp1} k_{2rp1} \exp \left( {ik_{2rp1} n_{2rp1} H} \right)$$

$f_{4(10)} = l_{2rp2} k_{2rp2} \exp \left( {ik_{2rp2} n_{2rp2} H} \right)$

$$f_{4(11)} = l_{2rp3} k_{2rp3} \exp \left( {ik_{2rp3} n_{2rp3} H} \right)$$

$f_{4(12)} = l_{2rp4} k_{2rp4} \exp \left( {ik_{2rp4} n_{2rp4} H} \right)$

$$f_{4(13)} = - n_{2rs} k_{2rs} \exp \left( {ik_{2rs} n_{2rs} H} \right)$$

$f_{51} = - \delta_{Tp1}^{e} \exp \left( {ik_{1rp1} n_{1rp1} H} \right)$

$$f_{52} = - \delta_{Tp2}^{e} \exp \left( {ik_{1rp2} n_{1rp2} H} \right)$$

$f_{53} = 0$

$$f_{54} = \delta_{Tp1} \exp \left( { - ik_{tp1} n_{tp1} H} \right)$$

$f_{55} = \delta_{Tp2} \exp \left( { - ik_{tp2} n_{tp2} H} \right)$

$$f_{56} = \delta_{Tp3} \exp \left( { - ik_{tp3} n_{tp3} H} \right)$$

$f_{57} = \delta_{Tp4} \exp \left( { - ik_{tp4} n_{tp4} H} \right)$

$$f_{58} = 0$$

$f_{59} = \delta_{Tp1} \exp \left( {ik_{2rp1} n_{2rp1} H} \right)$

$$f_{5(10)} = \delta_{Tp2} \exp \left( {ik_{2rp2} n_{2rp2} H} \right)$$

$f_{5(11)} = \delta_{Tp3} \exp \left( {ik_{2rp3} n_{2rp3} H} \right)$

$$f_{5(12)} = \delta_{Tp4} \exp \left( {ik_{2rp4} n_{2rp4} H} \right)$$

$f_{5(13)} = 0$

$$f_{61} = K^{e} n_{1rp1} k_{1rp1} \delta_{Tp1}^{e} \exp \left( {ik_{1rp1} n_{1rp1} H} \right)$$

$f_{62} = K^{e} n_{1rp2} k_{1rp2} \delta_{Tp2}^{e} \exp \left( {ik_{1rp2} n_{1rp2} H} \right)$

$$f_{63} = 0$$

$f_{64} = Kn_{tp1} k_{tp1} \delta_{Tp1} \exp \left( { - ik_{tp1} n_{tp1} H} \right)$

$$f_{65} = Kn_{tp2} k_{tp2} \delta_{Tp2} \exp \left( { - ik_{tp2} n_{tp2} H} \right)$$

$f_{66} = Kn_{tp3} k_{tp3} \delta_{Tp3} \exp \left( { - ik_{tp3} n_{tp3} H} \right)$

$$f_{67} = Kn_{tp4} k_{tp4} \delta_{Tp4} \exp \left( { - ik_{tp4} n_{tp4} H} \right)$$

$f_{68} = 0$

$$f_{69} = - Kn_{2rp1} k_{2rp1} \delta_{Tp1} \exp \left( {ik_{2rp1} n_{2rp1} H} \right)$$

$f_{6(10)} = - Kn_{2rp2} k_{2rp2} \delta_{Tp2} \exp \left( {ik_{2rp2} n_{2rp2} H} \right)$

$$f_{6(11)} = - Kn_{2rp3} k_{2rp3} \delta_{Tp3} \exp \left( {ik_{2rp3} n_{2rp3} H} \right)$$

$f_{6(12)} = - Kn_{2rp4} k_{2rp4} \delta_{Tp4} \exp \left( {ik_{2rp4} n_{2rp4} H} \right)$

$$f_{6(13)} = 0$$

$f_{71} = f_{72} = f_{73} = 0$

$$f_{74} = - n_{tp1} k_{tp1} \delta_{fp1} \exp \left( { - ik_{tp1} n_{tp1} H} \right)$$

$f_{75} = - n_{tp2} k_{tp2} \delta_{fp2} \exp \left( { - ik_{tp2} n_{tp2} H} \right)$

$$f_{76} = - n_{tp3} k_{tp3} \delta_{fp3} \exp \left( { - ik_{tp3} n_{tp3} H} \right)$$

$f_{77} = - n_{tp4} k_{tp4} \delta_{fp4} \exp \left( { - ik_{tp4} n_{tp4} H} \right)$

$$f_{78} = l_{2ts} k_{2ts} \delta_{fs} \exp \left( { - ik_{2ts} n_{2ts} H} \right)$$

$f_{79} = n_{2rp1} k_{2rp1} \delta_{fp1} \exp \left( {ik_{2rp1} n_{2rp1} H} \right)$

$$f_{7(10)} = n_{2rp2} k_{2rp2} \delta_{fp2} \exp \left( {ik_{2rp2} n_{2rp2} H} \right)$$

$f_{7(11)} = n_{2rp3} k_{2rp3} \delta_{fp3} \exp \left( {ik_{2rp3} n_{2rp3} H} \right)$

$$f_{7(12)} = n_{2rp4} k_{2rp4} \delta_{fp4} \exp \left( {ik_{2rp4} n_{2rp4} H} \right)$$

$f_{7(13)} = l_{2rs} k_{2rs} \delta_{fs} \exp \left( {ik_{2rs} n_{2rs} H} \right)$

$$f_{81} = f_{82} = f_{83} = 0$$

$f_{84} = - n_{tp1} k_{tp1} \delta_{ap1} \exp \left( { - ik_{tp1} n_{tp1} H} \right)$

$$f_{85} = - n_{tp2} k_{tp2} \delta_{ap2} \exp \left( { - ik_{tp2} n_{tp2} H} \right)$$

$f_{86} = - n_{tp3} k_{tp3} \delta_{ap3} \exp \left( { - ik_{tp3} n_{tp3} H} \right)$

$$f_{87} = - n_{tp4} k_{tp4} \delta_{ap4} \exp \left( { - ik_{tp4} n_{tp4} H} \right)$$

$f_{88} = l_{2ts} k_{2ts} \delta_{as} \exp \left( { - ik_{2ts} n_{2ts} H} \right)$

$$f_{89} = n_{2rp1} k_{2rp1} \delta_{ap1} \exp \left( {ik_{2rp1} n_{2rp1} H} \right)$$

$f_{8(10)} = n_{2rp2} k_{2rp2} \delta_{ap2} \exp \left( {ik_{2rp2} n_{2rp2} H} \right)$

$$f_{8(11)} = n_{2rp3} k_{2rp3} \delta_{ap3} \exp \left( {ik_{2rp3} n_{2rp3} H} \right)$$

$f_{8(12)} = n_{2rp4} k_{2rp4} \delta_{ap4} \exp \left( {ik_{2rp4} n_{2rp4} H} \right)$

$$f_{8(13)} = l_{2rs} k_{2rs} \delta_{as} \exp \left( {ik_{2rs} n_{2rs} H} \right)$$

$f_{91} = f_{92} = f_{93} = 0$

$$f_{94} = \left( {\overline{\lambda } + 2\mu n_{tp1}^{2} + M_{1} \delta_{fp1} + M_{2} \delta_{ap1} } \right)k_{tp1}^{2} - M_{3} \delta_{Tp1}$$

$f_{95} = \left( {\overline{\lambda } + 2\mu n_{tp2}^{2} + M_{1} \delta_{fp2} + M_{2} \delta_{ap2} } \right)k_{tp2}^{2} - M_{3} \delta_{Tp2}$

$$f_{96} = \left( {\overline{\lambda } + 2\mu n_{tp3}^{2} + M_{1} \delta_{fp3} + M_{2} \delta_{ap3} } \right)k_{tp3}^{2} - M_{3} \delta_{Tp3}$$

$f_{97} = \left( {\overline{\lambda } + 2\mu n_{tp4}^{2} + M_{1} \delta_{fp4} + M_{2} \delta_{ap4} } \right)k_{tp4}^{2} - M_{3} \delta_{Tp4}$

$$f_{98} = - 2\mu l_{2ts} n_{2ts} k_{2ts}^{2}$$

$f_{99} = \left( {\overline{\lambda } + 2\mu n_{2rp1}^{2} + M_{1} \delta_{fp1} + M_{2} \delta_{ap1} } \right)k_{2rp1}^{2} - M_{3} \delta_{Tp1}$

$$f_{9(10)} = \left( {\overline{\lambda } + 2\mu n_{2rp2}^{2} + M_{1} \delta_{fp2} + M_{2} \delta_{ap2} } \right)k_{2rp2}^{2} - M_{3} \delta_{Tp2}$$

$f_{9(11)} = \left( {\overline{\lambda } + 2\mu n_{2rp3}^{2} + M_{1} \delta_{fp3} + M_{2} \delta_{ap3} } \right)k_{2rp3}^{2} - M_{3} \delta_{Tp3}$

$$f_{9(12)} = \left( {\overline{\lambda } + 2\mu n_{2rp4}^{2} + M_{1} \delta_{fp4} + M_{2} \delta_{ap4} } \right)k_{2rp4}^{2} - M_{3} \delta_{Tp4}$$

$f_{9(13)} = 2\mu l_{2rs} n_{2rs} k_{2rs}^{2}$

$$f_{10\left( 1 \right)} = f_{10\left( 2 \right)} = f_{10\left( 3 \right)} = 0$$

$f_{10\left( 4 \right)} = - 2\mu l_{tp1} n_{tp1} k_{tp1}^{2}$

$$f_{10\left( 5 \right)} = - 2\mu l_{tp2} n_{tp2} k_{tp2}^{2}$$

$f_{10\left( 6 \right)} = - 2\mu l_{tp3} n_{tp3} k_{tp3}^{2}$

$$f_{10\left( 7 \right)} = - 2\mu l_{tp4} n_{tp4} k_{tp4}^{2}$$

$f_{10\left( 8 \right)} = \mu \left( {1 - 2n_{2ts}^{2} } \right)k_{2ts}^{2}$

$$f_{10\left( 9 \right)} = 2\mu l_{2rp1} n_{2rp1} k_{2rp1}^{2}$$

$f_{{10\left( {10} \right)}} = 2\mu l_{2rp2} n_{2rp2} k_{2rp2}^{2}$

$$f_{{10\left( {11} \right)}} = 2\mu l_{2rp3} n_{2rp3} k_{2rp3}^{2}$$

$f_{{10\left( {12} \right)}} = 2\mu l_{2rp4} n_{2rp4} k_{2rp4}^{2}$

$$f_{{10\left( {13} \right)}} = \mu \left( {1 - 2n_{2rs}^{2} } \right)k_{2rs}^{2}$$

$f_{11\left( 1 \right)} = f_{11\left( 2 \right)} = f_{11\left( 3 \right)} = 0$

$$f_{11\left( 4 \right)} = \left( {N_{1} + N_{2} \delta_{fp1} + N_{3} \delta_{ap1} } \right)k_{tp1}^{2} - N_{4} \delta_{Tp1}$$

$f_{11\left( 5 \right)} = \left( {N_{1} + N_{2} \delta_{fp2} + N_{3} \delta_{ap2} } \right)k_{tp2}^{2} - N_{4} \delta_{Tp2}$

$$f_{11\left( 6 \right)} = \left( {N_{1} + N_{2} \delta_{fp3} + N_{3} \delta_{ap3} } \right)k_{tp1}^{2} - N_{4} \delta_{Tp3}$$

$f_{11\left( 7 \right)} = \left( {N_{1} + N_{2} \delta_{fp4} + N_{3} \delta_{ap4} } \right)k_{tp4}^{2} - N_{4} \delta_{Tp4}$

$$f_{11\left( 8 \right)} = 0$$

$f_{11\left( 9 \right)} = \left( {N_{1} + N_{2} \delta_{fp1} + N_{3} \delta_{ap1} } \right)k_{2rp1}^{2} - N_{4} \delta_{Tp1}$

$$f_{{11\left( {10} \right)}} = \left( {N_{1} + N_{2} \delta_{fp2} + N_{3} \delta_{ap2} } \right)k_{2rp2}^{2} - N_{4} \delta_{Tp2}$$

$f_{{11\left( {12} \right)}} = \left( {N_{1} + N_{2} \delta_{fp4} + N_{3} \delta_{ap4} } \right)k_{2rp4}^{2} - N_{4} \delta_{Tp4}$

$$f_{{11\left( {13} \right)}} = 0$$

$f_{{11\left( {11} \right)}} = \left( {N_{1} + N_{2} \delta_{fp3} + N_{3} \delta_{ap3} } \right)k_{2rp3}^{2} - N_{4} \delta_{Tp3}$

$$f_{12\left( 1 \right)} = f_{12\left( 2 \right)} = f_{12\left( 3 \right)} = 0$$

$f_{12\left( 4 \right)} = \left( {N_{5} + N_{6} \delta_{fp1} + N_{7} \delta_{ap1} } \right)k_{tp1}^{2} - N_{8} \delta_{Tp1}$

$$f_{12\left( 5 \right)} = \left( {N_{5} + N_{6} \delta_{fp2} + N_{7} \delta_{ap2} } \right)k_{tp2}^{2} - N_{8} \delta_{Tp2}$$

$f_{12\left( 6 \right)} = \left( {N_{5} + N_{6} \delta_{fp3} + N_{7} \delta_{ap3} } \right)k_{tp3}^{2} - N_{8} \delta_{Tp3}$

$$f_{12\left( 7 \right)} = \left( {N_{5} + N_{6} \delta_{fp4} + N_{7} \delta_{ap4} } \right)k_{tp4}^{2} - N_{8} \delta_{Tp4}$$

$f_{12\left( 8 \right)} = 0$

$$f_{12\left( 9 \right)} = \left( {N_{5} + N_{6} \delta_{fp1} + N_{7} \delta_{ap1} } \right)k_{2rp1}^{2} - N_{8} \delta_{Tp1}$$

$f_{{12\left( {10} \right)}} = \left( {N_{5} + N_{6} \delta_{fp2} + N_{7} \delta_{ap2} } \right)k_{2rp2}^{2} - N_{8} \delta_{Tp2}$

$$f_{{12\left( {11} \right)}} = \left( {N_{5} + N_{6} \delta_{fp3} + N_{7} \delta_{ap3} } \right)k_{2rp3}^{2} - N_{8} \delta_{Tp3}$$

$f_{{12\left( {12} \right)}} = \left( {N_{5} + N_{6} \delta_{fp4} + N_{7} \delta_{ap4} } \right)k_{2rp4}^{2} - N_{8} \delta_{Tp4}$

$$f_{{12\left( {13} \right)}} = 0$$

$f_{13\left( 1 \right)} = f_{13\left( 2 \right)} = f_{13\left( 3 \right)} = 0$

$$f_{13\left( 4 \right)} = n_{tp1}^{2} k_{tp1}^{2} \delta_{Tp1}$$

$f_{13\left( 5 \right)} = n_{tp2}^{2} k_{tp2}^{2} \delta_{Tp2}$

$$f_{13\left( 6 \right)} = n_{tp3}^{2} k_{tp3}^{2} \delta_{Tp3}$$

$f_{13\left( 7 \right)} = n_{tp4}^{2} k_{tp4}^{2} \delta_{Tp4}$

$$f_{13\left( 8 \right)} = 0$$

$f_{13\left( 9 \right)} = n_{2rp1}^{2} k_{2rp1}^{2} \delta_{Tp1}$

$$f_{{13\left( {10} \right)}} = n_{2rp2}^{2} k_{2rp2}^{2} \delta_{Tp2}$$

$f_{{13\left( {11} \right)}} = n_{2rp3}^{2} k_{2rp3}^{2} \delta_{Tp3}$

$$f_{{13\left( {12} \right)}} = n_{2rp4}^{2} k_{2rp4}^{2} \delta_{Tp4}$$

$f_{{13\left( {13} \right)}} = 0$

$$g_{1} = 2\mu^{e} k_{is}^{2} l_{is} n_{is} \exp ( - ik_{is} n_{is} H)$$

$g_{2} = \mu^{e} (n_{is}^{2} - l_{is}^{2} )k_{is}^{2} \exp ( - ik_{is} n_{is} H)$

$$g_{3} = k_{is} l_{is} \exp ( - ik_{is} n_{is} H)$$

$g_{4} = n_{is} k_{is} \exp \left( { - ik_{is} n_{is} H} \right)$

$$g_{5} = g_{6} = g_{7} = g_{8} = g_{9} = g_{10} = g_{11} = g_{12} = g_{13} = 0$$