Study on vibration isolation performance of composite multilayer wave impeding block based on wave impedance ratio under an underground dynamic load

Abstract

Based on the theory of single-phase elastic medium and unsaturated porous medium, the vibration isolation effect of composite multilayer wave impeding block (WIB) in the unsaturated ground under an underground dynamic load is investigated. The results show that the best vibration isolation effect can be obtained by designing the wave impedance ratio between the composite multilayer WIB and unsaturated ground. The composite multilayer WIB improves the vibration-damping bandwidth compared with the homogeneous WIB. The vibration isolation effect is better the closer the embedded depth of the composite multilayer WIB is to the vibration source, and its vibration isolation effect increases significantly with the increase of thickness, but when the thickness of the WIB exceeds a certain critical thickness, its vibration isolation effect decreases with the increase of thickness. Soil saturation has a significant effect on the vibration isolation effect of composite multilayer WIB in the unsaturated ground, and the composite multilayer WIB can achieve a better vibration isolation effect at low saturation.

Keywords
Underground dynamic load; unsaturated soil; composite multilayer WIB; wave impedance ratio; vibration isolation design

Graphical Abstract

1 INTRODUCTION

Internationally, vibration pollution has been listed as one of the seven environmental hazards and has begun to study the laws of vibration pollution, the causes of its generation, the ways of its propagation, and the methods of its control. As the environmental vibration generated during the work of underground rail transit and other engineering activities has many adverse effects on various precision instruments as well as people's living and working environments. Therefore, it is of great practical significance to find effective vibration damping and vibration isolation measures for ground vibration caused by underground dynamic loads, and ground vibration and its control have become one of the important topics that need to be urgently solved in the field of geotechnical engineering research.

The propagation process and attenuation law of ground vibration vary in different sites, and a series of important research results have been conducted by domestic and foreign scholars from various aspects such as load types and ground characteristics (Zienkiewicz et al., 1980; Zhou et al., 2015; Shi et al., 2021; Huang et al., 2022). The establishment of a barrier between the existing vibration source and the protected area to cut off the propagation path of elastic waves, attenuate the vibration energy and reduce the vibration amplitude is a commonly adopted vibration isolation measure internationally. At present, scholars at home and abroad have carried out a lot of research work on the vibration reduction performance of various vibration isolation barriers (Woods., 1968; Xu et al., 2015; Schevenels and Lombaert, 2017; Gao, 1998; Gao et al., 2018; Liu et al., 2019; Li et al., 2021; Ba et al., 2022). Among them, according to the form of vibration isolation barriers, Gao (1998) further classified barriers into continuous barriers and discontinuous barriers according to their geometric characteristics.

Chouw et al. (1991) were the first to analyze the active and passive vibration isolation effects of WIB in the elastic ground, and the results showed that the vibration isolation effect of WIB was better than that of filled trenches. Takemiya and Fujiwara (1994) studied the vibration isolation effect of group pile foundation excitation in a single soil layer on bedrock, and the results showed that the WIB is an effective vibration isolation measure. Gao and Li (2005); Gao et al. (2007) analyzed the vibration isolation effect of WIB in two-dimensional elastic foundations as well as two-dimensional laminated elastic foundations, and the study showed that the vibration isolation effect of WIB is significantly affected by the elastic modulus. Most of the materials of WIB in the above studies are relatively single, and the materials are mainly concentrated in single-phase solid media, and there is less research on the composite mode of WIB and non-uniform materials. Ma (2018), Ma et al. (2017) established a ground vibration isolation system with a functional gradient WIB in an elastic ground, and the study showed that the gradient WIB can effectively reduce the vibration amplitude. Gao et al. (2020, 2021) proposed a joint vibration isolation method combining Duxseal industrial filler and WIB (referred to as DXWIB), and found that the DXWIB barrier enhanced the bandwidth of traditional homogeneous WIB vibration reduction through experimental studies. Zhou et al. (2020) proposed a new type of combined air trench-WIB vibration isolation barrier and analyzed its vibration isolation performance in elastic foundations, which showed that the combined air trench-WIB vibration isolation barrier combined the respective advantages of air trench and WIB and could effectively control the foundation vibration caused by different frequency vibration sources. Due to the complexity of the soil, simplifying to a single elastic foundation does not fully reflect the true state of the soil. Gao et al. (2014) studied the vibration isolation performance of WIB in the saturated ground under the effect of rail traffic load for the saturated ground model. Gao et al. (2017) studied the vibration isolation effect of WIB in saturated soil and analyzed the saturated soil-foundation-WIB interaction. Based on the Biot theory of saturated porous media, Ma et al. (2019) investigated the vibration isolation effect of liquid-containing saturated porous gradient WIB in the saturated ground under a moving load. Subsequently, Shu et al. (2022) extended the soil type to unsaturated soils and investigated the propagation characteristics of P₁ waves through single-layer homogeneous WIB in unsaturated soils, and the results showed that the selection of shear modulus and density of WIB in the appropriate range can achieve better vibration isolation effects in the unsaturated ground.

As can be seen above, most of the current research on WIB vibration barriers is focused on single-phase elastic ground and saturated ground. However, natural soil in nature is generally a porous multiphase medium composed of solid-liquid-gas phases, and unsaturated soil is the more common state of existence in nature. Therefore, it is of more general significance to study the vibration damping and isolation effect of WIB in unsaturated soil ground. In addition, most of the previous studies on WIB vibration isolation barriers were limited to homogeneous WIB, and it was found that (Chouw et al., 1991; Takemiya and Fujiwara, 1994; Gao and Li, 2005; Gao et al., 2007) single-layer homogeneous WIB vibration isolation barriers have a good vibration isolation effect for a low-frequency vibration, but the deficiency of vibration amplification will occur for medium and high-frequency vibration.

Therefore, this paper proposes a new vibration isolation system with a composite multilayer WIB as a vibration isolation barrier based on the property that the greater the interfacial difference of thin layer media the more significant the vibration wave transmission and reflection effect, and studies the vibration isolation performance of the composite multilayer WIB in the unsaturated ground. Based on the governing equations of unsaturated porous media, the mathematical model of composite multilayer WIB in the unsaturated ground under an underground dynamic load is established, and the general solution of displacement and stress of soil body in the domain of Fourier transform is derived by using Fourier transform and Helmholtz vector decomposition. The influence of the wave impedance between the material layers on the vibration isolation effect of the composite multilayer WIB is analyzed by numerical calculation, and the vibration isolation effect of the composite multilayer WIB and the homogeneous WIB in the unsaturated soil foundation is compared.

2 Mathematical model of the problem

The unsaturated ground is internally considered to be subjected to a vertical strip harmonic load is q₀e^iωt with an angular frequency is ω and amplitude is q₀ as shown in Figure 1. A composite multilayer WIB with thicknesses h_w1, h_w2, and h_w3 is set at a depth of H from the surface of the unsaturated ground, and the composite multilayer WIB are located in the areas of WIB₁, WIB₂, WIB₃, respectively. The distance of the vibration source from the WIB is h. The composite multilayer WIB and the vibration source divide the unsaturated ground into three parts, I, II, and III. A schematic diagram of the wave propagation and the wave amplitude is given in the figure.

Figure 1
Wave propagation diagram of WIB in unsaturated soil under an underground dynamic load.

3 Governing equations

3.1 Single-phase solid medium

Governing equations for an isotropic linear elastic single-phase continuous solid medium are as follows:

$$\left({\lambda }_e+{\mu }_e\right)\nabla \left(\nabla \cdot u^e\right)+{\mu }_e{\nabla }^2{u}^e={\rho }_e{\ddot{u}}^e$$ (1)

where ${\lambda }_{\text{e}}$and ${\mu }_{\text{e}}$denote the Lame elastic constants of the solid material, ${\rho }_e$is the density of the solid material, $u^e$denotes the displacement vector.

The corresponding stress-displacement relation is as follows:

$${\sigma }^e={\lambda }_e\left(\nabla \cdot u^e\right)I+{\mu }_e\left(\nabla u^e+u^e\nabla \right)I$$ (2)

where ${\sigma }^e$denotes the stress tensor in a single-phase elastic solid, $I$is the unit matrix.

According to the Helmholtz vector decomposition, the displacement vector can be expressed as a potential function as follows:

$$u^e=\nabla {\phi }_e+\nabla \times {\psi }_e$$ (3)

where ${\phi }_e$and ${\psi }_e$are the scalar potential function and vector potential function of the single-phase solid, respectively.

Substituting Eq. (3) into Eq. (1), the single-phase solid wave equation is as follows:

$${\nabla }^2{\phi }_e=\frac{1}{v_p^2}\frac{{\partial }^2{\phi }_e}{\partial t^2}$$, $${\nabla }^2{\psi }_e=\frac{1}{v_s^2}\frac{{\partial }^2{\psi }_e}{\partial t^2}$$(4)

where $v_p=\sqrt{\left({\lambda }_e+2{\mu }_e\right)/{\rho }_e}$and$v_s=\sqrt{{\mu }_e/{\rho }_e}$ are the propagation velocity of the longitudinal and transverse waves, respectively.

3.2 Unsaturated porous medium

Based on the fact that the model built in this paper is considered to be subjected to simple harmonic loading, all variables in the steady-state case can be written as follows:

$$f\left(x_1,x_3,t\right)=f{\left(x_1,x_3\right)}^{*}{\text{e}}^{\text{i}\omega t}$$ (5)

where $f\left(x_1,x_3,t\right)$is the functional form of all variables in a steady state, and ω is the circular frequency.

For convenience, the asterisk is omitted in the following derivation process.

Based on the theory of continuous medium mechanics and Bishop's effective stress formula, the dynamic governing equations of unsaturated soil were proposed by Xu and Wei (2011) as follows:

$${\mu }_p{\nabla }^2{u}^s+\left({\lambda }_p+{\mu }_p\right)\nabla \nabla \cdot u^s-a\gamma \nabla p^l-a\left(1-\gamma \right)\nabla p^g=-{\omega }^2{\overline{\rho }}_s{u}^s-{\omega }^2{\overline{\rho }}_l{u}^l-{\omega }^2{\overline{\rho }}_g{u}^g$$ (6a)

$$-\nabla p^l=b^l\text{i}\omega \left(u^l-u^s\right)-{\omega }^2{\rho }_l{u}^l$$ (6b)

$$-\nabla p^g=b^g\text{i}\omega \left(u^g-u^s\right)-{\omega }^2{\rho }_g{u}^g$$ (6c)

$$-p^l=a_{11}\nabla \cdot u^s+a_{12}\nabla \cdot u^l+a_{13}\nabla \cdot u^g$$ (6d)

$$-p^g=a_{21}\nabla \cdot u^s+a_{22}\nabla \cdot u^l+a_{23}\nabla \cdot u^g$$ (6e)

where the coefficients (Xu and Wei, 2011) of $a_{11}\text{~}{a}_{23}$,$A_{11}\text{~}{A}_{23}$, $b^l$,$b^g$,$a$,$\gamma$, etc. are the corresponding coefficients in the governing equations for unsaturated porous media. $u^s$, $u^l$,$u^g$,${\overline{\rho }}_s=\left(1-n\right){\rho }_s$, ${\overline{\rho }}_l=n{S}_r{\rho }_l$and ${\overline{\rho }}_g=n\left(1-S_r\right){\rho }_g$are the displacements and relative densities of the solid, liquid, and gas phases of the unsaturated soil, respectively. ${\rho }_s$, ${\rho }_l$and ${\rho }_g$are the densities of the solid, liquid, and gas phases, respectively. $\omega =2\pi f$ is the angular frequency, $f$ is the frequency, n is the porosity, S_r is the saturation, ${\lambda }_p$and ${\mu }_p$are the Lame coefficients of the unsaturated soil. $p^l$and $p^g$are the pore water and pore gas pressures, respectively.

The stress-strain relationship of unsaturated soil is as follows:

$${\sigma }_{ij}={\lambda }_{p}e{\delta }_{ij}+2{\mu }_p{\epsilon }_{ij}-{\delta }_{ij}ap$$ (7)

where ${\sigma }_{ij}$is the total stress component of the unsaturated soil medium (i=j=1,3), $e=\nabla \cdot u^s$is the volumetric strain of the soil skeleton, ${\delta }_{ij}$is the Kronecker symbol, ${\epsilon }_{ij}$is the strain of the soil skeleton, and $p=\gamma p^l+\left(1-\gamma \right)p^g$is the equivalent pore fluid pressure.

Substituting Eqs. (6d) and (6e) into Eqs. (6a), (6b), and (6c) yields

$${\mu }_p{\nabla }^2{u}^s+B_1\nabla \left(\nabla \cdot u^s\right)+B_2\nabla \left(\nabla \cdot u^l\right)+B_3\nabla \left(\nabla \cdot u^g\right)=-{\omega }^2{\overline{\rho }}_s{u}^s-{\omega }^2{\overline{\rho }}_l{u}^l-{\omega }^2{\overline{\rho }}_g{u}^g$$ (8a)

$$a_{11}\nabla \left(\nabla \cdot u^s\right)+a_{12}\nabla \left(\nabla \cdot u^l\right)+a_{13}\nabla \left(\nabla \cdot u^g\right)=C_1{u}^s+C_2{u}^l$$ (8b)

$$a_{21}\nabla \left(\nabla \cdot u^s\right)+a_{22}\nabla \left(\nabla \cdot u^l\right)+a_{23}\nabla \left(\nabla \cdot u^g\right)=C_3{u}^s+C_4{u}^g$$ (8c)

where $B_1={\lambda }_p+{\mu }_p+a\gamma a_{11}+a\left(1-\gamma \right)a_{21}$,$B_2=a\gamma a_{12}+a\left(1-\gamma \right)a_{22}$,$B_3=a\gamma a_{13}+a\left(1-\gamma \right)a_{23}$, $C_1=-b^l\text{i}\omega$,$C_2=b^{1}i\omega -{\omega }^2{\rho }_1$,$C_3=-b^g\text{i}\omega$,$C_4=b^g\text{i}\omega -{\omega }^2{\rho }_g$.

According to the Helmholtz vector decomposition, the displacement vectors$u^s$, $u^l$, and $u^g$can be expressed in terms of potential functions as follows:

$$u^z=\nabla {\phi }_z+\nabla \times {\psi }_z$$ (9)

where (z=s, l, g). ${\phi }_s$, ${\phi }_l$, ${\phi }_g$are the scalar potential functions of solid, liquid, and gas phases, respectively. ${\psi }_s$, ${\psi }_l$, ${\psi }_g$are the vector potential functions of solid, liquid, and gas phases, respectively.

Substituting Eq. (9) into Eqs. (8a~8c) for divergence and curl operations yields:

$$\left({\mu }_p+B_1\right){\nabla }^2{\phi }_s+B_2{\nabla }^2{\phi }_l+B_3{\nabla }^2{\phi }_g=-{\omega }^2{\overline{\rho }}_s{\phi }_s-{\omega }^2{\overline{\rho }}_l{\phi }_l-{\omega }^2{\overline{\rho }}_g{\phi }_g$$ (10a)

$$a_{11}{\nabla }^2{\phi }_s+a_{12}{\nabla }^2{\phi }_l+a_{13}{\nabla }^2{\phi }_g=C_1{\phi }_s+C_2{\phi }_l$$ (10b)

$$a_{21}{\nabla }^2{\phi }_s+a_{22}{\nabla }^2{\phi }_l+a_{23}{\nabla }^2{\phi }_g=C_3{\phi }_s+C_4{\phi }_g$$ (10c)

$${\mu }_p{\nabla }^2{\psi }_s=-{\omega }^2{\overline{\rho }}_s{\psi }_s-{\omega }^2{\overline{\rho }}_l{\psi }_l-{\omega }^2{\overline{\rho }}_g{\psi }_g$$ (10d)

$$C_1{\psi }_s+C_2{\psi }_l=0$$ (10e)

$$C_3{\psi }_s+C_4{\psi }_g=0$$ (10f)

4 Solution of the displacement potential function

4.1 Solution of single-phase solid medium

The Fourier transform with respect to $x_1$is defined as follows:

$$\tilde{f}\left(\xi ,x_3\right)={\displaystyle {\int }_{-\infty }^{\infty }f\left(x_1,x_3\right){\text{e}}^{-\text{i}\xi x_1}\text{d}}{x}_1$$ (11)

where $\xi$ are the wave numbers in the $x_1$ direction.

Substituting Eqs. (5) and (11) into Eq. (4), after Fourier transforms yields:

$$\frac{{\text{d}}^2{\tilde{\phi }}_e}{\text{d}{x}_3^2}+{\alpha }_e^2{\tilde{\phi }}_e=0$$, $$\frac{{\text{d}}^2{\tilde{\psi }}_e}{\text{d}{x}_3^2}+{\beta }_e^2{\tilde{\psi }}_e=0$$(12)

where ${\alpha }_e=\sqrt{{\omega }^2/v_p^2-{\xi }^2}$, ${\beta }_e=\sqrt{{\omega }^2/v_s^2-{\xi }^2}$.

Therefore, the general solution of displacement potential function in a single-phase elastic solid medium can be obtained as follows:

$${\tilde{\phi }}_e=A_{te}{\text{e}}^{\text{-i}{\alpha }_e{x}_3}+A_{re}{\text{e}}^{\text{i}{\alpha }_e{x}_3}$$, $${\tilde{\psi }}_e=B_{te}{\text{e}}^{\text{-i}{\beta }_e{x}_3}+B_{re}{\text{e}}^{\text{i}{\beta }_e{x}_3}$$(13)

where $A_{te}$and $A_{re}$are the amplitudes of the transmitted P wave and reflected P wave in a single-phase solid medium respectively. $B_{te}$and $B_{re}$are the amplitudes of transmitted S wave and reflected S wave in a single-phase solid medium, respectively.

4.2 Solution of the unsaturated porous medium

Substituting Eq. (11) into Eqs. (10a~10c), after Fourier transforms yields:

$$\left({\mu }_p+B_1\right)\frac{{\text{d}}^2{\tilde{\phi }}_s}{\text{d}{x}_3^2}+b_{11}{\tilde{\phi }}_s+B_2\frac{{\text{d}}^2{\tilde{\phi }}_l}{\text{d}{x}_3^2}+b_{12}{\tilde{\phi }}_l+B_3\frac{{\text{d}}^2{\tilde{\phi }}_g}{\text{d}{x}_3^2}+b_{13}{\tilde{\phi }}_g=0$$ (14a)

$$a_{11}\frac{{\text{d}}^2{\tilde{\phi }}_s}{\text{d}{x}_3^2}+b_{21}{\tilde{\phi }}_s+a_{12}\frac{{\text{d}}^2{\tilde{\phi }}_l}{\text{d}{x}_3^2}+b_{22}{\tilde{\phi }}_l+a_{13}\frac{{\text{d}}^2{\tilde{\phi }}_g}{\text{d}{x}_3^2}+b_{23}{\tilde{\phi }}_g=0$$ (14b)

$$a_{21}\frac{{\text{d}}^2{\tilde{\phi }}_s}{\text{d}{x}_3^2}+b_{31}{\tilde{\phi }}_s+a_{22}\frac{{\text{d}}^2{\tilde{\phi }}_l}{\text{d}{x}_3^2}+b_{32}{\tilde{\phi }}_l+a_{23}\frac{{\text{d}}^2{\tilde{\phi }}_g}{\text{d}{x}_3^2}+b_{33}{\tilde{\phi }}_g=0$$ (14c)

where $b_{11}={\overline{\rho }}_s{\omega }^2-\left({\mu }_p+B_1\right){\xi }^2$, $b_{12}={\overline{\rho }}_l{\omega }^2-B_2{\xi }^2$, $b_{13}={\overline{\rho }}_g{\omega }^2-B_3{\xi }^2$, $b_{21}=-C_1-a_{11}{\xi }^2$, $b_{22}=-C_2-a_{12}{\xi }^2$, $b_{23}=-a_{13}{\xi }^2$, $b_{31}=-C_3-a_{21}{\xi }^2$, $b_{32}=-a_{22}{\xi }^2$, $b_{33}=-C_4-a_{23}{\xi }^2$.

Assuming that the solution of the Eqs. (14) as follows:

$${\left[\begin{array}{ccc}{\tilde{\phi }}_s& {\tilde{\phi }}_l& {\tilde{\phi }}_g\end{array}\right]}^{\text{T}}={\left[\begin{array}{ccc}{c}^s& c^l& c^g\end{array}\right]}^{\text{T}}\exp \left(\lambda x_3\right)$$ (15)

Substituting Eq. (15) into Eqs. (14) yields the linear equations as follows:

$$\left[\begin{array}{ccc}{\lambda }^2{B}_3+b_{13}& {\lambda }^2{B}_2+b_{12}& {\lambda }^2({\mu }_p+B_1)+b_{11}\\ {\lambda }^2{a}_{13}+b_{23}& {\lambda }^2{a}_{12}+b_{22}& {\lambda }^2{a}_{11}+b_{21}\\ {\lambda }^2{a}_{23}+b_{33}& {\lambda }^2{a}_{22}+b_{32}& {\lambda }^2{a}_{21}+b_{31}\end{array}\right]\left[\begin{array}{c}{c}^g\\ c^l\\ c^s\end{array}\right]=0$$ (16)

The condition that Eq. (16) has a non-zero solution is that the determinant of its coefficient matrix is 0, namely:

$${\beta }_1{\lambda }^6+{\beta }_2{\lambda }^4+{\beta }_3{\lambda }^2+{\beta }_4=0$$ (17)

where $${\beta }_n\left(n=\mathrm{1,2,3,4}\right)$$ are detailed in Appendix A.

The roots of Eq. (17) are$\pm {\lambda }_n\left(n=\mathrm{1,2,3}\right)$ , ${\lambda }_n$ are given by the following equation:

$${\lambda }_n=\sqrt{r_n}\left(\mathrm{Re}\left[{\lambda }_n\right]\ge \mathrm{0,}n=\mathrm{1,2,3}\right)$$ (18)

where $r_n$ are determined by the equation${\beta }_1{r}_n^3+{\beta }_2{r}_n^2+{\beta }_3{r}_n+{\beta }_4=0$.

The solution of ordinary differential Eqs. (14) can be obtained as follows:

$${\tilde{\phi }}_s={\displaystyle \sum _{n=1}^3{A}_{tpn}^{}{\text{e}}^{-{\lambda }_n{x}_3}+A_{rpn}^{}{\text{e}}^{{\lambda }_n{x}_3}}$$ (19a)

$${\tilde{\phi }}_l={\displaystyle \sum _{n=1}^3{\delta }_{pn}^l}\left(A_{tpn}^{}{\text{e}}^{-{\lambda }_n{x}_3}+A_{rpn}^{}{\text{e}}^{{\lambda }_n{x}_3}\right)$$ (19b)

$${\tilde{\phi }}_g={\displaystyle \sum _{n=1}^3{\delta }_{pn}^g}\left(A_{tpn}^{}{\text{e}}^{-{\lambda }_n{x}_3}+A_{rpn}^{}{\text{e}}^{{\lambda }_n{x}_3}\right)$$ (19c)

where ${\delta }_{pn}^l=-\frac{\left(B_1{a}_{13}-B_3{a}_{11}+a_{13}{\mu }_p\right)r_n^2+\left(B_1{b}_{23}-B_3{b}_{21}-a_{11}{b}_{13}+a_{13}{b}_{11}+b_{23}{\mu }_p\right)r_n+b_{11}{b}_{23}-b_{13}{b}_{21}}{\left(B_2{a}_{13}-B_3{a}_{12}\right)r_n^2+\left(B_2{b}_{23}-B_3{b}_{22}-a_{12}{b}_{13}+a_{13}{b}_{12}\right)r_n+b_{12}{b}_{23}-b_{13}{b}_{22}}$,${\delta }_{pn}^g=\frac{\left(B_1{a}_{12}-B_2{a}_{11}+a_{12}{\mu }_p\right)r_n^2+\left(B_1{b}_{22}-B_2{b}_{21}-a_{11}{b}_{12}+a_{12}{b}_{11}+b_{22}{\mu }_p\right)r_n+b_{11}{b}_{22}-b_{12}{b}_{21}}{\left(B_2{a}_{13}-B_3{a}_{12}\right)r_n^2+\left(B_2{b}_{23}-B_3{b}_{22}-a_{12}{b}_{13}+a_{13}{b}_{12}\right)r_n+b_{12}{b}_{23}-b_{13}{b}_{22}}$, $A_{tpn}^{}$ and $A_{rpn}^{}$ (n=1,2,3) are the amplitudes of transmitted P₁ wave, P₂ wave, P₃ wave and reflected P₁ wave, P₂ wave, and P₃ wave in unsaturated soil porous medium, respectively.

Substituting Eq. (11) into Eqs. (10d~10f), then performing the Fourier transform on them yields:

$${\mu }_p\frac{{\text{d}}^2{\tilde{\psi }}_s}{\text{d}{x}_3^2}+d_{11}{\tilde{\psi }}_s+d_{12}{\tilde{\psi }}_l+d_{13}{\tilde{\psi }}_g=0$$ (20a)

$$d_{21}{\tilde{\psi }}_s+d_{22}{\tilde{\psi }}_l=0$$ (20b)

$$d_{31}{\tilde{\psi }}_s+d_{32}{\tilde{\psi }}_l=0$$ (20c)

where $d_{11}={\overline{\rho }}_s{\omega }^2-{\mu }_p{\xi }^2$, $d_{12}={\overline{\rho }}_l{\omega }^2$, $d_{13}={\overline{\rho }}_g{\omega }^2$, $d_{21}=C_1$, $d_{22}=C_2$, $d_{31}=C_3$, $d_{32}=C_4$.

Assuming that the solution of the Eqs. (20) as follows:

$${\left[\begin{array}{ccc}{\tilde{\psi }}_s& {\tilde{\psi }}_l& {\tilde{\psi }}_g\end{array}\right]}^{\text{T}}={\left[\begin{array}{ccc}{d}^s& d^l& d^g\end{array}\right]}^{\text{T}}\exp \left(r{x}_3\right)$$ (21)

Substituting Eq. (21) into Eqs. (20) yields the linear equations:

$$\left[\begin{array}{ccc}{d}_{13}& d_{12}& {\mu }_p{r}^2+d_{11}\\ 0& d_{22}& d_{21}\\ d_{33}& 0& d_{31}\end{array}\right]\left[\begin{array}{c}{d}^g\\ d^l\\ d^s\end{array}\right]=0$$ (22)

The condition that Eq. (22) has a non-zero solution is that the determinant of its coefficient matrix is 0, namely:

$${\beta }_5{r}^2+{\beta }_6=0$$ (23)

The roots of Eq. (23) are $\pm r$, $r$ is then given by the following equation:

$$r=\sqrt{-{\beta }_6/{\beta }_5}\left(\mathrm{Re}\left[r\right]\ge 0\right)$$ (24)

where ${\beta }_n\left(n=\mathrm{5,6}\right)$ are detailed in ^{Appendix A}.

The solution of ordinary differential Eqs. (20) can be obtained as follows:

$${\tilde{\psi }}_s=B_{ts}^{}{\text{e}}^{-r{x}_3}+B_{rs}^{}{\text{e}}^{r{x}_3}$$ (25a)

$${\tilde{\psi }}_l={\delta }_s^l\left(B_{ts}^{}{\text{e}}^{-r{x}_3}+B_{rs}^{}{\text{e}}^{r{x}_3}\right)$$ (25b)

$${\tilde{\psi }}_g={\delta }_s^g\left(B_{ts}^{}{\text{e}}^{-r{x}_3}+B_{rs}^{}{\text{e}}^{r{x}_3}\right)$$ (25c)

where ${\delta }_s^l=-\frac{d_{21}}{d_{22}}$, ${\delta }_s^g=\frac{d_{12}{d}_{21}-({\mu }_p{r}^2+d_{11})d_{22}}{d_{13}{d}_{22}}$. $B_{ts}^{}$and $B_{rs}^{}$are the amplitudes of transmission and reflection waves of unsaturated porous media, respectively.

5 Solving the dynamic response problem of unsaturated ground

5.1 Dynamic response of unsaturated ground

In the coordinate system $\left(x_1,x_3\right)$, each displacement component can be expressed by the displacement potential functions $\phi$ and $\psi$ as follows:

$$u_1=\frac{\partial \phi }{\partial x_1}-\frac{\partial \psi }{\partial x_3}$$, $$u_3=\frac{\partial \phi }{\partial x_3}+\frac{\partial \psi }{\partial x_1}$$(26)

Substituting Eqs. (13) and (26) into Eqs. (2) and (3) and substituting Eqs. (19), (25), (26) into Eqs. (6d), (6e), (7), and (9), the displacement and stress expressions in the Fourier transform domain can be obtained for the elastic medium and the unsaturated porous medium, respectively.

In the region $0\le x_3\le H$:

$${\tilde{u}}_1^{s\text{I}}=\text{i}\xi \left({\displaystyle \sum _{n=1}^3{A}_{tpn}^{\text{I}}}{\text{e}}^{-{\lambda }_n{x}_3}+A_{rpn}^{\text{I}}{\text{e}}^{{\lambda }_n{x}_3}\right)+r\left(B_{ts}^{\text{I}}{\text{e}}^{-r{x}_3}-B_{rs}^{\text{I}}{\text{e}}^{r{x}_3}\right)$$ (27a)

$${\tilde{u}}_3^{s\text{I}}=\left[{\displaystyle \sum _{n=1}^3-{\lambda }_n\left(A_{tpn}^{\text{I}}{\text{e}}^{-{\lambda }_n{x}_3}-A_{rpn}^{\text{I}}{\text{e}}^{{\lambda }_n{x}_3}\right)}\right]+\text{i}\xi \left(B_{ts}^{\text{I}}{\text{e}}^{-r{x}_3}+B_{rs}^{\text{I}}{\text{e}}^{r{x}_3}\right)$$ (27b)

$${\tilde{u}}_1^{l\text{I}}=\text{i}\xi \left[{\displaystyle \sum _{n=1}^3{\delta }_{pn}^l\left(A_{tpn}^{\text{I}}{\text{e}}^{-{\lambda }_n{x}_3}+A_{rpn}^{\text{I}}{\text{e}}^{{\lambda }_n{x}_3}\right)}\right]+r{\delta }_s^l\left(B_{ts}^{\text{I}}{\text{e}}^{-r{x}_3}-B_{rs}^{\text{I}}{\text{e}}^{r{x}_3}\right)$$ (27c)

$${\tilde{u}}_3^{l\text{I}}=\left[{\displaystyle \sum _{n=1}^3-{\lambda }_n{\delta }_{pn}^l\left(A_{tpn}^{\text{I}}{\text{e}}^{-{\lambda }_n{x}_3}-A_{rpn}^{\text{I}}{\text{e}}^{{\lambda }_n{x}_3}\right)}\right]+\text{i}\xi {\delta }_s^l\left(B_{ts}^{\text{I}}{\text{e}}^{-r{x}_3}+B_{rs}^{\text{I}}{\text{e}}^{r{x}_3}\right)$$ (27d)

$${\tilde{u}}_1^{g\text{I}}=\text{i}\xi \left[{\displaystyle \sum _{n=1}^3{\delta }_{pn}^g\left(A_{tpn}^{\text{I}}{\text{e}}^{-{\lambda }_n{x}_3}+A_{rpn}^{\text{I}}{\text{e}}^{{\lambda }_n{x}_3}\right)}\right]+r{\delta }_s^g\left(B_{ts}^{\text{I}}{\text{e}}^{-r{x}_3}-B_{rs}^{\text{I}}{\text{e}}^{r{x}_3}\right)$$ (27e)

$${\tilde{u}}_3^{g\text{I}}=\left[{\displaystyle \sum _{n=1}^3-{\lambda }_n{\delta }_{pn}^g\left(A_{tpn}^{\text{I}}{\text{e}}^{-{\lambda }_n{x}_3}-A_{rpn}^{\text{I}}{\text{e}}^{{\lambda }_n{x}_3}\right)}\right]+\text{i}\xi {\delta }_s^g\left(B_{ts}^{\text{I}}{\text{e}}^{-r{x}_3}+B_{rs}^{\text{I}}{\text{e}}^{r{x}_3}\right)$$ (27f)

$${\tilde{\sigma }}_{13}^{\text{I}}=2{\mu }_p\text{i}\xi \left[{\displaystyle \sum _{n=1}^3-{\lambda }_n\left(A_{tpn}^{\text{I}}{\text{e}}^{-{\lambda }_n{x}_3}-A_{rpn}^{\text{I}}{\text{e}}^{{\lambda }_n{x}_3}\right)}\right]-{\mu }_p\left(r^2+{\xi }^2\right)\left(B_{ts}^{\text{I}}{\text{e}}^{-r{x}_3}+B_{rs}^{\text{I}}{\text{e}}^{r{x}_3}\right)$$ (27g)

$${\tilde{\sigma }}_{33}^{\text{I}}=\left[{\displaystyle \sum _{n=1}^3{\chi }_n}\left(A_{tpn}^{\text{I}}{\text{e}}^{-{\lambda }_n{x}_3}+A_{rpn}^{\text{I}}{\text{e}}^{{\lambda }_n{x}_3}\right)\right]-2\text{i}\xi r{\mu }_p\left(B_{ts}^{\text{I}}{\text{e}}^{-r{x}_3}-B_{rs}^{\text{I}}{\text{e}}^{r{x}_3}\right)$$ (27h)

$${\tilde{p}}^{l\text{I}}={\displaystyle \sum _{n=1}^3\left(a_{11}+a_{12}{\delta }_{pn}^l+a_{13}{\delta }_{pn}^g\right)\left({\xi }^2-{\lambda }_n^2\right)\left(A_{tpn}^{\text{I}}{\text{e}}^{-{\lambda }_n{x}_3}+A_{rpn}^{\text{I}}{\text{e}}^{{\lambda }_n{x}_3}\right)}$$ (27i)

$${\tilde{p}}^{\text{gI}}={\displaystyle \sum _{n=1}^3\left(a_{21}+a_{22}{\delta }_{pn}^l+a_{23}{\delta }_{pn}^g\right)\left({\xi }^2-{\lambda }_n^2\right)\left(A_{tpn}^{\text{I}}{\text{e}}^{-{\lambda }_n{x}_3}+A_{rpn}^{\text{I}}{\text{e}}^{{\lambda }_n{x}_3}\right)}$$ (27j)

where ${\chi }_n\text{=}\left[a\gamma \left(a_{11}+a_{12}{\delta }_{pn}^l+a_{13}{\delta }_{pn}^g\right)+a\left(1-\gamma \right)\left(a_{21}+a_{22}{\delta }_{pn}^l+a_{23}{\delta }_{pn}^g\right)\right]\left({\lambda }_n^2-{\xi }^2\right)+\left[\left({\lambda }_p+2{\mu }_p\right){\lambda }_n^2-{\lambda }_p{\xi }^2\right]$.

In the region $H\le x_3\le H\text{+}{h}_{w1}$:

$${\tilde{u}}_1^{1e}=\text{i}\xi \left(A_{1te}{\text{e}}^{-\text{i}{\alpha }_{1e}{x}_3}+A_{1re}{\text{e}}^{\text{i}{\alpha }_{1e}{x}_3}\right)\text{+i}{\beta }_{1e}\left(B_{1te}{\text{e}}^{-\text{i}{\beta }_{1e}{x}_3}-B_{1re}{\text{e}}^{\text{i}{\beta }_{1e}{x}_3}\right)$$ (28a)

$${\tilde{u}}_3^{1e}=-\text{i}{\alpha }_{1e}\left(A_{1te}{\text{e}}^{-\text{i}{\alpha }_{1e}{x}_3}-A_{1re}{\text{e}}^{\text{i}{\alpha }_{1e}{x}_3}\right)+\text{i}\xi \left(B_{1te}{\text{e}}^{-\text{i}{\beta }_{1e}{x}_3}+B_{1re}{\text{e}}^{\text{i}{\beta }_{1e}{x}_3}\right)$$ (28b)

$${\tilde{\sigma }}_{13}^{1e}=2{\mu }_{1e}\xi {\alpha }_{1e}\left(A_{1te}{\text{e}}^{-\text{i}{\alpha }_{1e}{x}_3}-A_{1re}{\text{e}}^{\text{i}{\alpha }_{1e}{x}_3}\right)\text{+}{\mu }_{1e}\left({\beta }_{1e}{}^2\text{-}{\xi }^2\right)\left(B_{1te}{\text{e}}^{-\text{i}{\beta }_{1e}{x}_3}+B_{1re}{\text{e}}^{\text{i}{\beta }_{1e}{x}_3}\right)$$ (28c)

$${\tilde{\sigma }}_{33}^{1e}=\left[-{\lambda }_{1e}{\xi }^2-{\alpha }_{1e}{}^2\left({\lambda }_{1e}+2{\mu }_{1e}\right)\right]\left(A_{1te}{\text{e}}^{-\text{i}{\alpha }_{1e}{x}_3}+A_{1re}{\text{e}}^{\text{i}{\alpha }_{1e}{x}_3}\right)+2{\mu }_{1e}{\beta }_{1e}\xi \left(B_{1te}{\text{e}}^{-\text{i}{\beta }_{1e}{x}_3}-B_{1re}{\text{e}}^{\text{i}{\beta }_{1e}{x}_3}\right)$$ (28d)

In the region $H\text{+}{h}_{w1}\le x_3\le H\text{+}{h}_{w1}+h_{w2}$:

$${\tilde{u}}_1^{2e}=\text{i}\xi \left(A_{2te}{\text{e}}^{-\text{i}{\alpha }_{2e}{x}_3}+A_{2re}{\text{e}}^{\text{i}{\alpha }_{2e}{x}_3}\right)\text{+i}{\beta }_{2e}\left(B_{2te}{\text{e}}^{-\text{i}{\beta }_{2e}{x}_3}-B_{2re}{\text{e}}^{\text{i}{\beta }_{2e}{x}_3}\right)$$ (29a)

$${\tilde{u}}_3^{2e}=-\text{i}{\alpha }_{2e}\left(A_{2te}{\text{e}}^{-\text{i}{\alpha }_{2e}{x}_3}-A_{2re}{\text{e}}^{\text{i}{\alpha }_{2e}{x}_3}\right)+\text{i}\xi \left(B_{2te}{\text{e}}^{-\text{i}{\beta }_{2e}{x}_3}+B_{2re}{\text{e}}^{\text{i}{\beta }_{2e}{x}_3}\right)$$ (29b)

$${\tilde{\sigma }}_{13}^{2e}=2{\mu }_{2e}\xi {\alpha }_{2e}\left(A_{2te}{\text{e}}^{-\text{i}{\alpha }_{2e}{x}_3}-A_{2re}{\text{e}}^{\text{i}{\alpha }_{2e}{x}_3}\right)\text{+}{\mu }_{2e}\left({\beta }_{2e}{}^2\text{-}{\xi }^2\right)\left(B_{2te}{\text{e}}^{-\text{i}{\beta }_{2e}{x}_3}+B_{2re}{\text{e}}^{\text{i}{\beta }_{2e}{x}_3}\right)$$ (29c)

$${\tilde{\sigma }}_{33}^{2e}=\left[-{\lambda }_{2e}{\xi }^2-{\alpha }_{2e}^2\left({\lambda }_{2e}+2{\mu }_{2e}\right)\right]\left(A_{2te}{\text{e}}^{-\text{i}{\alpha }_{2e}{x}_3}+A_{2re}{\text{e}}^{\text{i}{\alpha }_{2e}{x}_3}\right)+2{\mu }_{2e}{\beta }_{2e}\xi \left(B_{2te}{\text{e}}^{-\text{i}{\beta }_{2e}{x}_3}-B_{2re}{\text{e}}^{\text{i}{\beta }_{2e}{x}_3}\right)$$ (29d)

In the region $H\text{+}{h}_{w1}+h_{w2}\le x_3\le H\text{+}{h}_{w1}+h_{w2}+h_{w3}$:

$${\tilde{u}}_1^{3e}=\text{i}\xi \left(A_{3te}{\text{e}}^{-\text{i}{\alpha }_{3e}{x}_3}+A_{3re}{\text{e}}^{\text{i}{\alpha }_{3e}{x}_3}\right)\text{+i}{\beta }_{3e}\left(B_{3te}{\text{e}}^{-\text{i}{\beta }_{3e}{x}_3}-B_{3re}{\text{e}}^{\text{i}{\beta }_{3e}{x}_3}\right)$$ (30a)

$${\tilde{u}}_3^{3e}=-\text{i}{\alpha }_{3e}\left(A_{3te}{\text{e}}^{-\text{i}{\alpha }_{3e}{x}_3}-A_{3re}{\text{e}}^{\text{i}{\alpha }_{3e}{x}_3}\right)+\text{i}\xi \left(B_{3te}{\text{e}}^{-\text{i}{\beta }_{3e}{x}_3}+B_{3re}{\text{e}}^{\text{i}{\beta }_{3e}{x}_3}\right)$$ (30b)

$${\tilde{\sigma }}_{13}^{3e}=2{\mu }_{3e}\xi {\alpha }_{3e}\left(A_{3te}{\text{e}}^{-\text{i}{\alpha }_{3e}{x}_3}-A_{3re}{\text{e}}^{\text{i}{\alpha }_{3e}{x}_3}\right)\text{+}{\mu }_{3e}\left({\beta }_{3e}^2\text{-}{\xi }^2\right)\left(B_{3te}{\text{e}}^{-\text{i}{\beta }_{3e}{x}_3}+B_{3re}{\text{e}}^{\text{i}{\beta }_{3e}{x}_3}\right)$$ (30c)

$${\tilde{\sigma }}_{33}^{3e}=\left[-{\lambda }_{3e}{\xi }^2-{\alpha }_{3e}{}^2({\lambda }_{3e}+2{\mu }_{3e})\right]\left(A_{3te}{\text{e}}^{-\text{i}{\alpha }_{3e}{x}_3}+A_{3re}{\text{e}}^{\text{i}{\alpha }_{3e}{x}_3}\right)+2{\mu }_{3e}{\beta }_{3e}\xi \left(B_{3te}{\text{e}}^{-\text{i}{\beta }_{3e}{x}_3}-B_{3re}{\text{e}}^{\text{i}{\beta }_{3e}{x}_3}\right)$$ (30d)

In the region $H\text{+}{h}_{w1}+h_{w2}+h_{w3}\le x_3\le H\text{+}{h}_{w1}+h_{w2}+h_{w3}+h$:

$${\tilde{u}}_1^{s\text{II}}=\text{i}\xi \left({\displaystyle \sum _{n=1}^3{A}_{tpn}^{\text{II}}}{\text{e}}^{-{\lambda }_n{x}_3}+A_{rpn}^{\text{II}}{\text{e}}^{{\lambda }_n{x}_3}\right)+r\left(B_{ts}^{\text{II}}{\text{e}}^{-r{x}_3}-B_{rs}^{\text{II}}{\text{e}}^{r{x}_3}\right)$$ (31a)

$${\tilde{u}}_3^{s\text{II}}=\left[{\displaystyle \sum _{n=1}^3-{\lambda }_n\left(A_{tpn}^{\text{II}}{\text{e}}^{-{\lambda }_n{x}_3}-A_{rpn}^{\text{II}}{\text{e}}^{{\lambda }_n{x}_3}\right)}\right]+\text{i}\xi \left(B_{ts}^{\text{II}}{\text{e}}^{-r{x}_3}+B_{rs}^{\text{II}}{\text{e}}^{r{x}_3}\right)$$ (31b)

$${\tilde{u}}_1^{l\text{II}}=\text{i}\xi \left[{\displaystyle \sum _{n=1}^3{\delta }_{pn}^l\left(A_{tpn}^{\text{II}}{\text{e}}^{-{\lambda }_n{x}_3}+A_{rpn}^{\text{II}}{\text{e}}^{{\lambda }_n{x}_3}\right)}\right]+r{\delta }_s^l\left(B_{ts}^{\text{II}}{\text{e}}^{-r{x}_3}-B_{rs}^{\text{II}}{\text{e}}^{r{x}_3}\right)$$ (31c)

$${\tilde{u}}_3^{l\text{II}}=\left[{\displaystyle \sum _{n=1}^3-{\lambda }_n{\delta }_{pn}^l\left(A_{tpn}^{\text{II}}{\text{e}}^{-{\lambda }_n{x}_3}-A_{rpn}^{\text{II}}{\text{e}}^{{\lambda }_n{x}_3}\right)}\right]+\text{i}\xi {\delta }_s^l\left(B_{ts}^{\text{II}}{\text{e}}^{-r{x}_3}+B_{rs}^{\text{II}}{\text{e}}^{r{x}_3}\right)$$ (31d)

$${\tilde{u}}_1^{g\text{II}}=\text{i}\xi \left[{\displaystyle \sum _{n=1}^3{\delta }_{pn}^g\left(A_{tpn}^{\text{II}}{\text{e}}^{-{\lambda }_n{x}_3}+A_{rpn}^{\text{II}}{\text{e}}^{{\lambda }_n{x}_3}\right)}\right]+r{\delta }_s^g\left(B_{ts}^{\text{II}}{\text{e}}^{-r{x}_3}-B_{rs}^{\text{II}}{\text{e}}^{r{x}_3}\right)$$ (31e)

$${\tilde{u}}_3^{g\text{II}}=\left[{\displaystyle \sum _{n=1}^3-{\lambda }_n{\delta }_{pn}^g\left(A_{tpn}^{\text{II}}{\text{e}}^{-{\lambda }_n{x}_3}-A_{rpn}^{\text{II}}{\text{e}}^{{\lambda }_n{x}_3}\right)}\right]+\text{i}\xi {\delta }_s^g\left(B_{ts}^{\text{II}}{\text{e}}^{-r{x}_3}+B_{rs}^{\text{II}}{\delta }_s^g{\text{e}}^{r{x}_3}\right)$$ (31f)

$${\tilde{\sigma }}_{13}^{\text{II}}=2{\mu }_p\text{i}\xi \left[{\displaystyle \sum _{n=1}^3-{\lambda }_n\left(A_{tpn}^{\text{II}}{\text{e}}^{-{\lambda }_n{x}_3}-A_{rpn}^{\text{II}}{\text{e}}^{{\lambda }_n{x}_3}\right)}\right]-{\mu }_p\left(r^2+{\xi }^2\right)\left(B_{ts}^{\text{II}}{\text{e}}^{-r{x}_3}+B_{rs}^{\text{II}}{\text{e}}^{r{x}_3}\right)$$ (31g)

$${\tilde{\sigma }}_{33}^{\text{II}}=\left[{\displaystyle \sum _{n=1}^3{\chi }_n}\left(A_{tpn}^{\text{II}}{\text{e}}^{-{\lambda }_n{x}_3}+A_{rpn}^{\text{II}}{\text{e}}^{{\lambda }_n{x}_3}\right)\right]-2\text{i}\xi r{\mu }_p\left(B_{ts}^{\text{II}}{\text{e}}^{-r{x}_3}-B_{rs}^{\text{II}}{\text{e}}^{r{x}_3}\right)$$ (31h)

In the region $x_3\ge H\text{+}{h}_{w1}+h_{w2}+h_{w3}+h$:

$${\tilde{u}}_1^{s\text{III}}=\text{i}\xi \left({\displaystyle \sum _{n=1}^3{A}_{tpn}^{\text{III}}}{\text{e}}^{-{\lambda }_n{x}_3}\right)+r{B}_{ts}^{\text{III}}{\text{e}}^{-r{x}_3}$$ (32a)

$${\tilde{u}}_3^{s\text{III}}=\left({\displaystyle \sum _{n=1}^3-{\lambda }_n{A}_{tpn}^{\text{III}}}{\text{e}}^{-{\lambda }_n{x}_3}\right)+\text{i}\xi B_{ts}^{\text{III}}{\text{e}}^{-r{x}_3}$$ (32b)

$${\tilde{\sigma }}_{13}^{\text{III}}=2{\mu }_p\text{i}\xi \left({\displaystyle \sum _{n=1}^3-{\lambda }_n{A}_{tpn}^{\text{III}}{\text{e}}^{-{\lambda }_n{x}_3}}\right)-{\mu }_p\left(r^2+{\xi }^2\right)B_{ts}^{\text{III}}{\text{e}}^{-r{x}_3}$$ (32c)

$${\tilde{\sigma }}_{33}^{\text{III}}=\left({\displaystyle \sum _{n=1}^3{\chi }_n{A}_{tpn}^{\text{III}}{\text{e}}^{-{\lambda }_n{x}_3}}\right)-2\text{i}\xi r{\mu }_p{B}_{ts}^{\text{III}}{\text{e}}^{-r{x}_3}$$ (32d)

5.2 Boundary conditions and solution

For the case where the load acts in the interior of the half-plane, the boundary conditions and the continuity conditions at the interfaces of the layers are as follows:

At $x_3=0$:

$${\tilde{\sigma }}_{33}^{\text{I}}=0$$, $${\tilde{\sigma }}_{13}^{\text{I}}=0$$, $${\tilde{p}}^{lI}=0$$, $${\tilde{p}}^{gI}=0$$(33)

At $x_3=H$:

$${\tilde{\sigma }}_{33}^{\text{I}}={\tilde{\sigma }}_{33}^{1e}$$,$${\tilde{\sigma }}_{13}^{\text{I}}={\tilde{\sigma }}_{13}^{1e}$$, $${\tilde{u}}_3^{\text{sI}}={\tilde{u}}_3^{1e}$$, $${\tilde{u}}_1^{s\text{I}}={\tilde{u}}_1^{1e}$$, $${\tilde{u}}_3^{l\text{I}}={\tilde{u}}_3^{s\text{I}}$$, $${\tilde{u}}_3^{g\text{I}}={\tilde{u}}_3^{s\text{I}}$$, $${\tilde{u}}_1^{l\text{I}}={\tilde{u}}_1^{s\text{I}}$$, $${\tilde{u}}_1^{g\text{I}}={\tilde{u}}_1^{s\text{I}}$$(34)

At $x_3=H+h_{w1}$:

$${\tilde{\sigma }}_{33}^{1e}={\tilde{\sigma }}_{33}^{2e}$$, $${\tilde{\sigma }}_{13}^{1e}={\tilde{\sigma }}_{13}^{2e}$$, $$u_3^{1e}=u_3^{2e}$$, $$u_1^{1e}=u_1^{2e}$$(35)

At $x_3=H+h_{w1}+h_{w2}$:

$${\tilde{\sigma }}_{33}^{2e}={\tilde{\sigma }}_{33}^{3e}$$, $${\tilde{\sigma }}_{13}^{2e}={\tilde{\sigma }}_{13}^{3e}$$, $$u_3^{2e}=u_3^{3e}$$, $$u_1^{2e}=u_1^{3e}$$(36)

At $x_3=H\text{+}{h}_{w1}+h_{w2}\text{+}{h}_{w3}$:

$${\tilde{\sigma }}_{33}^{3e}={\tilde{\sigma }}_{33}^{\text{II}}$$, $${\tilde{\sigma }}_{13}^{3e}={\tilde{\sigma }}_{13}^{\text{II}}$$, $${\tilde{u}}_3^{3e}={\tilde{u}}_3^{\text{sII}}$$, $${\tilde{u}}_1^{3e}={\tilde{u}}_1^{\text{sII}}$$, $${\tilde{u}}_3^{s\text{II}}={\tilde{u}}_3^{l\text{II}}$$, $${\tilde{u}}_3^{s\text{II}}={\tilde{u}}_3^{\text{gII}}$$, $${\tilde{u}}_1^{\text{sII}}={\tilde{u}}_1^{l\text{II}}$$, $${\tilde{u}}_1^{\text{sII}}={\tilde{u}}_1^{g\text{II}}$$(37)

At $x_3=H\text{+}{h}_w+h_{w2}+h_{w3}\text{+}h$:

$${\tilde{\sigma }}_{33}^{\text{II}}-{\tilde{\sigma }}_{33}^{\text{III}}=q_0\frac{\sin \left(\xi l\right)}{\xi l}$$, $${\tilde{\sigma }}_{13}^{\text{II}}={\tilde{\sigma }}_{13}^{\text{III}}$$, $${\tilde{u}}_3^{\text{sII}}={\tilde{u}}_3^{\text{sIII}}$$, $${\tilde{u}}_1^{\text{sII}}={\tilde{u}}_1^{\text{sIII}}$$(38)

Substituting Eqs. (27-32) into the boundary conditions (33-38), the following matrix equations yields:

$$\left[T\right]\left\{\left.x\right\}\right.=\left\{\left.f\right\}\right.$$ (39)

where the elements in matrix$\left[T\right]$and vectors$\left\{\left.x\right\}\right.$and$\left\{\left.f\right\}\right.$are detailed in ^{Appendix B}.

By solving the Eqs. (39), the wave amplitudes of various types of waves in$\left\{\left.x\right\}\right.$are obtained, and the stress and displacement responses at any point in the unsaturated soil and the WIB can be obtained by combining Eqs. (27-32).

6 Numerical examples

For unsaturated soil, due to the change of saturation lead to a change in the shear modulus in the soil (Lu et al., 2018; Xu and Wei, 2011), then the modified dynamic shear modulus is used in this paper:

$${\mu }_p\text{=}{\mu }_{\text{s}}\text{+}\frac{2050}{\alpha }\ln \left[\sqrt{{\left(S_{\text{e}}\right)}^{\text{-}2}\text{-}1}\text{+}{\left(S_{\text{e}}\right)}^{\text{-}1}\right]\tan {\varphi }^{\text{'}}$$ (40)

In order to study the effect of WIB on the vibration control of unsaturated ground, a set of physical and mechanical parameters (Xu and Wei, 2011) of unsaturated ground are selected in this paper as shown in Table 1. Taking the load amplitude is q₀=1kPa and the distribution length is l=1m. Since the complexity of the integrand function expression, it is difficult to obtain the closed-form solution of the Fourier inverse transform. In this paper, the FFT method is used to complete the Fourier inverse transform. The discrete point of the wave number is 1024, and the spatial calculation interval is 100 m.

Table 1
Physical and mechanical parameters of unsaturated soils

Parameter	Unit	Value of a quantity	Parameter	Unit	Value of a quantity	Parameter	Value of a quantity
Kg	kPa	100	Ks	GPa	36	Sr	0.1-1
ρg	kg/m3	1.29	ρs	kg/m3	0.2	d	2
ηg	Ns/m2	1.5×10-5	φ’	。	21	Sw0	0.05
Kl	GPa	2.1	α	Pa-1	1×10-4	k	1×10-12
ρl	kg/m3	1000	μs	MPa	19.4	υ	0.2
ηl	Ns/m2	0.001	n		0.6	m	0.5

6.1 Validation of the present solution

In order to verify the correctness of the method in this paper, Eq. (39) degenerates to the case without WIB. When H+h_w1+h_w2+h_w3+h=0, the load degenerates to the surface strip harmonic load. Shi et al. (2021) studied the dynamic response of unsaturated soil foundations under strip harmonic load. Saturation is S_r=0.8, circular frequency is ω=1rad/s, and the rest of the unsaturated soil physical and mechanical parameters are shown in Table 1. Figure 2 shows the variation curve of the vertical displacement of the surface along the horizontal direction. It can be seen from the figure that the calculation results of this paper are in good agreement with the results of Shi.

Figure 2
Variation curve of vertical displacement of the ground surface along the horizontal direction.

In order to further verify the correctness of the method in this paper, when S_r→S_w0, n→0, the unsaturated soil foundation degenerates into an elastic foundation. When the material parameters of the local foundation and the composite layer WIB are the same, the model in this paper degenerates into the Lamb problem where the vibration source is located in the semi-infinite homogeneous medium. Taking ω=0.01rad/s, the elastic modulus of the material is E=1×10⁷Pa, Poisson's ratio is ν=0.3, density is ρ=1884kg/m³. Figure 3 shows the positive stress diagram of strip load acting on the edge of load H+h_w1+h_w2+h_w3+h=4m from the foundation surface, and compared with the analytical solution obtained by Yuan and Zhao (1995). It can be seen from Figure 3 that when the frequency is very small, it is basically consistent with the static solution.

Figure 3
The normal stress under the corner point of internal load.

6.2 Example analysis

It is known that the wave impedance (Xiang., 2011) is defined as the product of velocity and density, i.e., Z=ρv. The ratio of the wave impedance of the first medium ρ₁v₁ and the wave impedance of the second medium ρ₂v₂ on the dividing surface of different media is the wave impedance ratio, which can be expressed as: Z=ρ₁v₁/ρ₂v₂. Let the wave impedance of unsaturated soil be Z₀, and the wave impedances of WIB₁, WIB₂, and WIB₃ be Z₁, Z₂, and Z₃, respectively. Considering the saturation of soil S_r=0.8 and the frequency is f=5Hz, then the wave speed of P₁ wave in unsaturated soil (Xu and Wei, 2009) can be obtained as v_p1=208.15m/s, the density of unsaturated soil is $\rho ={\overline{\rho }}_s+{\overline{\rho }}_l+{\overline{\rho }}_g$, then the wave impedance ratio at the intersection of WIB₁ and the unsaturated ground is ${\gamma }_1=Z_1/Z_0=\sqrt{\frac{{\rho }_{1e}{E}_{1e}\left(1-{\upsilon }_{1e}\right)}{(1+{\upsilon }_{1e})(1-2{\upsilon }_{1e})}}/\rho v_{p1}$, the wave impedance ratio at the intersection of WIB₂ and WIB₁ layers is ${\gamma }_2=Z_2/Z_1=\sqrt{\frac{{\rho }_{2e}{E}_{2e}\left(1-{\upsilon }_{2e}\right)(1+{\upsilon }_{1e})(1-2{\upsilon }_{1e})}{{\rho }_{1e}{E}_{1e}(1-{\upsilon }_{1e})(1+{\upsilon }_{2e})(1-2{\upsilon }_{2e})}}$, the wave impedance ratio at the intersection of WIB₃ and WIB₂ layers is ${\gamma }_3=Z_3/Z_2=\sqrt{\frac{{\rho }_{3e}{E}_{3e}\left(1-{\upsilon }_{3e}\right)(1+{\upsilon }_{2e})(1-2{\upsilon }_{2e})}{{\rho }_{2e}{E}_{2e}(1-{\upsilon }_{2e})(1+{\upsilon }_{3e})(1-2{\upsilon }_{3e})}}$.

The greater the difference at the interface of the media intersection the more significant the amplitude attenuation of the elastic wave, so the greater the difference in wave impedance at the interface, the more significant the amplitude attenuation of the elastic wave. From the definition of wave impedance, it can be seen that the material parameters affecting the medium wave impedance are mainly density and elastic modulus. Therefore, this paper by analyzing the influence of the wave impedance ratio at the interface between the composite multilayer WIB and the unsaturated ground layer on the vibration isolation performance of the composite multilayer WIB, the corresponding interlayer wave impedance ratio with the best vibration isolation effect is selected, and then back calculates the elastic modulus and density of the interlayer material of the composite multilayer WIB, and finally obtains the design guidelines of the interlayer material parameters of the composite multilayer WIB. The physical and mechanical parameters of the composite multilayer WIB are selected as follows: Poisson's ratio is υ_1e=υ_2e=υ_3e=0.3, the thickness of the composite multilayer WIB is h_w1=h_w2=h_w3=0.5m respectively, and the density of the composite multilayer WIB is discussed in the following six cases:

• Case1: ${\rho }_{1e}\text{=}3700kg/m^3>{\rho }_{2e}\text{=2}700kg/m^3>{\rho }_{3e}\text{=1}700kg/m^3$

• Case2: ${\rho }_{1e}\text{=}3700kg/m^3>{\rho }_{3e}\text{=2}700kg/m^3>{\rho }_{2e}\text{=1}700kg/m^3$

• Case3: ${\rho }_{2e}\text{=}3700kg/m^3>{\rho }_{1e}\text{=2}700kg/m^3>{\rho }_{3e}\text{=1}700kg/m^3$

• Case4: ${\rho }_{2e}\text{=}3700kg/m^3>{\rho }_{3e}\text{=2}700kg/m^3>{\rho }_{1e}\text{=1}700kg/m^3$

• Case5: ${\rho }_{3e}\text{=}3700kg/m^3>{\rho }_{1e}\text{=2}700kg/m^3>{\rho }_{2e}\text{=1}700kg/m^3$

• Case6: ${\rho }_{3e}\text{=}3700kg/m^3>{\rho }_{2e}\text{=2}700kg/m^3>{\rho }_{1e}\text{=1}700kg/m^3$

Considering the soil saturation of S_r=0.8, the load frequency of f=5Hz, the embedded depth of WIB is H=1m, and the distance of the vibration source from WIB is h=3-H-h_w1-h_w2-h_w3. Figure 4 gives the four-dimensional plots of the simultaneous variation of the surface vertical displacement maxima with the wave impedance ratios${\gamma }_1$, ${\gamma }_2$, and${\gamma }_3$ under an underground dynamic load in six cases, respectively, with the independent variables being the wave impedance ratios ${\gamma }_1$ , ${\gamma }_2$, and ${\gamma }_3$, and the stress variable being the surface vertical displacement maxima |u₃|_max, and the color changes represent the variation of the surface vertical displacement maxima. By comparing the minimum value of the maximum value of the vertical displacement of the ground under the simultaneous change of the interlayer wave impedance ratio, the value of the interlayer wave impedance ratio of the composite multilayer WIB with the best vibration isolation effect can be selected. From Figure 4(a~f), it can be seen that when the wave impedance ratio is small, with the increase of the wave impedance ratio, the maximum value of vertical displacement of the ground surface decreases sharply, but when the wave impedance ratio increases to a certain degree, the maximum value of vertical displacement of the surface reaches the minimum. In the range of the wave impedance ratio discussed in this paper, there is the interlayer wave impedance ratio that corresponds to the maximum value of vertical displacement of the ground surface when it reaches the minimum, which is the best vibration isolation effect of the composite multilayer WIB. From Figure 4(a~f), we can get the interlayer wave impedance ratio corresponding to the optimum vibration isolation effect of composite multilayer WIB in six cases, i.e., when the maximum value of vertical displacement of the ground surface reaches the minimum, respectively:

Figure 4
Four-dimensional plot of the variation of the vertical displacement maximum of the ground surface with the wave impedance ratio.

• Case1: ${\gamma }_1$=4.5, ${\gamma }_2$=20.5, ${\gamma }_3$=0.5, |u₃|_max=1.22×10^-8m

• Case2: ${\gamma }_1$=4.5, ${\gamma }_2$=16.5, ${\gamma }_3$=1.0, |u₃|_max=1.08×10^-8m

• Case3: ${\gamma }_1$=4.5, ${\gamma }_2$=22.0, ${\gamma }_3$=0.5, |u₃|_max=1.44×10^-8m

• Case4: ${\gamma }_1$=4.0, ${\gamma }_2$=21.5, ${\gamma }_3$=0.5, |u₃|_max=2.38×10^-8m

• Case5: ${\gamma }_1$=3.5, ${\gamma }_2$=21.0, ${\gamma }_3$=1.0, |u₃|_max=1.38×10^-8m

• Case6: ${\gamma }_1$=3.5, ${\gamma }_2$=22.0, ${\gamma }_3$=1.0, |u₃|_max=1.82×10^-8m

From the above six cases and in the range of wave impedance ratio considered in this paper, it can be seen that when the density case is Case2, the composite multilayer WIB can achieve the optimal vibration isolation effect. At this time, the elastic modulus of the composite multilayer WIB is E_1e=4.62×10⁸Pa, E_2e=2.74×10¹¹Pa, and E_3e=1.72×10¹¹Pa. Therefore, the optimal vibration isolation effect of the composite multilayer WIB can be obtained by designing the wave impedance ratio between the layers of the composite multilayer WIB. In addition, due to the transmission effect of the barrier, directly affects the vibration isolation efficiency (Gao et al., 2021), when the impedance of the barrier is relatively large, it can reduce the transmission of vibration waves from the soil surface, and the larger the impedance ratio is the better vibration isolation efficiency can be obtained. However, it is obviously unrealistic to take the infinite wave impedance ratio in practical engineering. Therefore, the composite multilayer WIB vibration isolation barrier system proposed in this paper is superior and designable.

In the following analysis and comparison, the density corresponding to Case2 and the inter-layer wave impedance ratio corresponding to the optimal isolation effect of the composite multilayer WIB is taken to analyze the vibration isolation performance law of the composite multilayer WIB.

In order to compare the advantages and disadvantages of the vibration isolation effect of composite multilayer WIB and homogeneous WIB under the same conditions, consider the embedded depth of WIB is H =1m, the soil saturation is S_r=0.8, the load frequency is f=5Hz, the distance of vibration source from WIB is h=3-H-h_w1-h_w2-h_w3m, the Poisson's ratio of composite multilayer WIB and homogeneous WIB are υ_1e=υ_2e=υ_3e=0.3, the thickness of composite multilayer WIB are h_w1=h_w2=h_w3=0.5m , the thickness of the homogeneous WIB is h_w=1.5m, the density of the composite multilayer WIB are ρ_1e=3700kg/m³, ρ_2e=1700kg/m³, ρ_3e=2700kg/m³, the elastic modulus of the composite multilayer WIB are E_1e=4.62×10⁸Pa, E_2e=2.74×10¹¹Pa, E_3e=1.72×10¹¹Pa. Because the vibration isolation effect of the homogeneous WIB increases with the increase of elastic modulus (Gao and Li, 2005; Gao et al., 2007), in order to compare the vibration isolation effect of the composite multilayer WIB and the homogeneous WIB, the density and elastic modulus of the homogeneous WIB are taken as their maximum values, namely: ρ_e=3700kg/m³, E_e=2.74×10¹¹Pa. For the evaluation of the vibration isolation effect of the composite multilayer WIB, the amplitude attenuation coefficient A_R proposed by Woods (1968) is used to measure :

$$A_R=\frac{\overline{u}}{u}$$ (41)

where $\overline{u}$ is the surface displacement after setting the WIB barrier, and $u$ is the surface displacement without the WIB barrier.

Figure 5 plots the change curve of the amplitude attenuation coefficient of the vertical displacement of the ground surface with the distance when the composite multilayer WIB and homogeneous WIB are installed in the unsaturated ground. According to Figure 5, it can be seen that the surface vertical displacement amplitude attenuation coefficient of the composite multilayer WIB is significantly smaller than that of the homogeneous WIB, and the corresponding surface vertical displacement amplitude attenuation coefficient of the composite multilayer WIB is much smaller than 1. Therefore, under the same conditions, the design of composite multilayer WIB compared to homogeneous WIB can not only reduce the use of high-strength materials, and improve the vibration isolation performance, composite multilayer WIB than homogeneous WIB has a superior vibration isolation effect.

Figure 5
Variation curve of amplitude attenuation coefficient A_R of the ground surface with distance for vertical displacement.

In order to analyze the effect of the saturation of the soil on the vibration isolation effect of the composite multilayer WIB, Figure 6 plots the change curve of the amplitude attenuation coefficient of the vertical displacement of the ground surface with the distance when the saturation increases gradually from S_r=0.3, 0.5, 0.7, 0.8, 0.9 at H=1m, h_w1=h_w2=h_w3=0.5m, f=5Hz, h=3-H-h_w1-h_w2- h_w3m. It can be seen from Figure 6 that the amplitude attenuation coefficient of vertical displacement of the ground surface increases with the increase of the saturation degree, which indicates that the vibration isolation effect of the composite multilayer WIB is decreasing with the increase of the saturation degree. When S_r=0.9, the surface vertical displacement amplitude attenuation coefficient increases sharply, and the value is greater than 1, indicating that the vibration isolation failure occurs at this time. When S_r≤0.8, the surface vertical displacement amplitude attenuation coefficient is much less than 1, which indicates that the composite multilayer WIB can play a good vibration isolation effect in the unsaturated ground in the range of S_r≤0.8. From the perspective of the vibration isolation mechanism of the wave impeding block, the good vibration isolation performance of the wave impeding block is mainly due to its reflection and scattering of waves (Song et al., 2022). When the saturation of the unsaturated soil foundation is less than 0.8, the total compression modulus of the soil is controlled by the soil skeleton. When the saturation increases to 0.9, the gas phase in the soil decreases, the liquid phase increases, and the equivalent compression modulus of the pore fluid increases rapidly. The total compression modulus of the soil is controlled by the compression modulus of the soil skeleton and the equivalent compression modulus of the pore fluid, which makes the longitudinal wave velocity in the foundation increase rapidly under the action of underground dynamic load, which leads to the increase of the saturation of unsaturated foundation. The complex change of the vertical displacement amplitude attenuation coefficient of the wave impeding block. Therefore, the saturation of the site soil has a significant effect on the vibration isolation effect of the WIB, which indicates that the design of vibration isolation of WIB in actual engineering should choose a ground model that matches the actual site, and simplify the site soil as a single-phase elastic ground or a saturated ground without considering the effect of the saturation of the soil on the vibration isolation effect, which is very different from the actual phenomenon. It can be seen from Figure 6 that it is desirable to set the composite multilayer WIB in the range of S_r≤0.8 in the unsaturated ground to obtain the ideal vibration isolation effect.

Figure 6
Variation curve of amplitude attenuation coefficient A_R of the ground surface under different saturation with distance for vertical displacement.

In order to analyze the effect of load frequency on the vibration isolation effect of the composite multilayer WIB, Figure 7 shows the change curve of the amplitude attenuation coefficient of the vertical displacement of the ground surface with the distance when the load frequency increases gradually from f=5, 20, 40, 80, 100, and 120 Hz at H=1m, S_r=0.8, h_w1=h_w2=h_w3=0.5m, and h=3-H-h_w1-h_w2-h_w3m. It can be seen from Figure 7 that when the load frequency f ≤20Hz, with the increase of load frequency, the attenuation coefficient of surface vertical displacement amplitude decreases, and the vibration isolation effect of composite multilayer WIB is gradually enhanced. When the load frequency is in the range of 20Hz≤f≤100Hz, with the increase of load frequency, the surface vertical displacement amplitude attenuation coefficient increases, and the surface vertical displacement amplitude attenuation coefficient is less than1, only when f = 120Hz, the corresponding surface vertical displacement attenuation coefficient is greater than1. In summary, the composite multilayer WIB has a good damping effect on the high-frequency vibration in the load frequency range of 5Hz≤f≤100Hz. In contrast, the traditional homogeneous WIB only has a good vibration isolation effect on low-frequency vibration, and there is a phenomenon of far-field amplification failure in the vibration isolation of medium and high- frequency vibration. The composite multilayer WIB vibration isolation barrier proposed in this paper improves the frequency width of the homogeneous WIB vibration isolation. The composite multilayer WIB also has a good vibration isolation effect on low, medium, and high-frequency vibration (5Hz≤f≤100Hz), which again illustrates the superiority of the composite multilayer WIB vibration isolation system.

Figure 7
Variation curve of amplitude attenuation coefficient A_R of the ground surface under different loading frequencies with distance for vertical displacement.

In order to analyze the effect of the thickness of the composite multilayer WIB on the vibration isolation effect of the ground, Figure 8 plots the variation curve of the amplitude attenuation coefficient of the surface vertical displacement with distance when the thickness of the WIB increases gradually from h_w1=h_w2=h_w3=0.3, 0.4, 0.5, 0.6m at H=1m, S_r=0.8, f=5Hz, h=3-H-h_w1-h_w2-h_w3m. From Figure 8, it can be seen that when 0.3m≤h_w1=h_w2=h_w3≤0.5m, the attenuation coefficient of vertical displacement amplitude decreases with the increase of the thickness of the WIB, indicating that the vibration isolation effect of the composite multilayer WIB is gradually enhanced. When the thickness of the WIB h_w1=h_w2=h_w3 ≥0.5m, the attenuation coefficient of the vertical displacement amplitude increases with the increase of the thickness of the WIB, indicating that the vibration isolation effect is no longer obvious when the thickness of the WIB continues to increase, and the composite multilayer WIB has a critical thickness. This phenomenon is consistent with the law of the WIB-Duxseal joint vibration isolation test study (Tian et al., 2020, Gao et al., 2021 ) on the effect of thickness under vertical excitation force. Therefore, when designing the vibration isolation of composite multilayer WIB in practical engineering, the thickness of composite multilayer WIB should not be set too thick considering the appropriate economic cost and better vibration isolation effect. From Figure 8, it can be seen that the thickness of the composite multilayer WIB should be h_w1=h_w2=h_w3=0.5m, which can play a better vibration control effect in unsaturated soil foundations.

Figure 8
Variation curve of amplitude attenuation coefficient A_R of the ground surface under different thicknesses with distance for vertical displacement.

In order to analyze the effect of the embedded depth of the composite multilayer WIB on the vibration isolation effect of the ground, Figure 9 plots the variation curve of the amplitude attenuation coefficient of the surface vertical displacement with distance when the embedded depth of the WIB increases gradually from H=1, 2, 3m at h_w1=h_w2=h_w3=0.5m, S_r=0.8, f=5Hz, h=5.5-H-h_w1-h_w2-h_w3m. It can be seen from Figure 9 that when the embedded depth is larger, the attenuation coefficient of the vertical displacement amplitude of the surface is smaller, that is, when the WIB is closer to the vibration source, the vibration isolation effect is better. This is because the WIB uses the principle of the cut-off frequency of the foundation to isolate the vibration. Only when the WIB is close to the vibration source, even if there is an effective cut-off frequency of the foundation, can the vibration isolation effect on the environmental vibration be achieved. Therefore, when the position of the WIB is set, the composite multilayer WIB is set closer to the vibration source so that there is an effective cut-off frequency in the foundation, and the vibration isolation effect of the WIB is better.

Figure 9
Variation curve of amplitude attenuation coefficient A_R of the ground surface under different embedded depths with distance for vertical displacement.

7 Conclusions

Based on single-phase elastic medium theory and unsaturated porous medium theory, the vibration isolation performance of composite multilayer WIB vibration isolation system in the unsaturated ground under an underground dynamic load is studied in this paper, and the advantages and disadvantages of the vibration isolation effect of composite multilayer WIB and homogeneous WIB under the same conditions are compared. The influence of soil saturation, load frequency, thickness, and embedded depth of the WIB on the vibration isolation performance of composite multilayer WIB was analyzed. The results of the study show that:

(1) The effect of the interlayer wave impedance ratio of composite multilayer WIB and unsaturated soil foundation on the vibration isolation effect of composite multilayer WIB is very obvious. By designing the wave impedance ratio between the composite multilayer WIB and the unsaturated soil foundation, the optimal vibration isolation effect of the composite multilayer WIB vibration isolation barrier can be obtained.

(2) Under the same thickness condition, the vibration isolation effect of the designed composite multilayer WIB is much better than that of the homogeneous WIB. Compared with the homogeneous WIB, the composite multilayer WIB improves the vibration isolation bandwidth and has a good vibration isolation effect in the range of 5≤f≤100 Hz.

(3) The soil saturation has a significant influence on the vibration isolation effect of the composite multilayer WIB, and the composite multilayer WIB can have a good vibration isolation effect in the range of S_r≤0.8. The foundation model that is consistent with the actual site should be selected in the design of WIB vibration isolation in practical engineering.

(4) The closer the embedded depth of the composite multilayer WIB is to the vibration source, the better the vibration isolation effect is. There is a critical thickness of the composite multilayer WIB. When the thickness of the WIB exceeds the critical thickness, the vibration isolation effect of the WIB decreases with the increase of thickness.

Appendix A - The coefficient of the root of the potential function

$${\beta }_1=-B_1{a}_{12}{a}_{23}+B_1{a}_{13}{a}_{22}+B_2{a}_{11}{a}_{23}-B_2{a}_{13}{a}_{21}-B_3{a}_{11}{a}_{22}+B_3{a}_{12}{a}_{21}-{\mu }_p{a}_{12}{a}_{23}+{\mu }_p{a}_{13}{a}_{22}$$ $$\begin{array}{l}{\beta }_2=-B_1{a}_{12}{b}_{33}+B_1{a}_{13}{b}_{32}+B_1{a}_{22}{b}_{23}-B_1{a}_{23}{b}_{22}+B_2{a}_{11}{b}_{33}-B_2{a}_{13}{b}_{31}-B_2{a}_{21}{b}_{23}+B_2{a}_{23}{b}_{21}-B_3{a}_{11}{b}_{32}\\ +B_3{a}_{12}{b}_{31}+B_3{a}_{21}{b}_{22}-B_3{a}_{22}{b}_{21}-a_{11}{a}_{22}{b}_{13}+a_{11}{a}_{23}{b}_{12}+a_{12}{a}_{21}{b}_{13}-a_{12}{a}_{23}{b}_{11}-a_{13}{a}_{21}{b}_{12}+a_{13}{a}_{22}{b}_{11}\\ -{\mu }_p{a}_{12}{b}_{33}+{\mu }_p{a}_{13}{b}_{32}+{\mu }_p{a}_{22}{b}_{23}-{\mu }_p{a}_{23}{b}_{22}\end{array}$$ $$\begin{array}{l}{\beta }_3=-B_1{b}_{22}{b}_{33}+B_1{b}_{23}{b}_{32}+B_2{b}_{21}{b}_{33}-B_2{b}_{23}{b}_{31}-B_3{b}_{21}{b}_{32}+B_3{b}_{22}{b}_{31}+a_{11}{b}_{12}{b}_{33}-a_{11}{b}_{13}{b}_{32}-a_{12}{b}_{11}{b}_{33}\\ +a_{12}{b}_{13}{b}_{31}+a_{13}{b}_{11}{b}_{32}-a_{13}{b}_{12}{b}_{31}-a_{21}{b}_{12}{b}_{23}+a_{21}{b}_{13}{b}_{22}+a_{22}{b}_{11}{b}_{23}-a_{22}{b}_{13}{b}_{21}-a_{23}{b}_{11}{b}_{22}+a_{23}{b}_{12}{b}_{21}\\ -{\mu }_p{b}_{22}{b}_{33}+{\mu }_p{b}_{23}{b}_{32}\end{array}$$ $${\beta }_4=-b_{11}{b}_{22}{b}_{33}+b_{11}{b}_{23}{b}_{32}+b_{12}{b}_{21}{b}_{33}-b_{12}{b}_{23}{b}_{31}-b_{13}{b}_{21}{b}_{32}+b_{13}{b}_{22}{b}_{31}$$

$${\beta }_5=-{\mu }_p{d}_{22}{d}_{33}$$,$${\beta }_6=-d_{11}{d}_{22}{d}_{33}+d_{21}{d}_{12}{d}_{33}+d_{13}{d}_{22}{d}_{31}$$

Appendix B - Specific representations of the wave amplitude coefficient matrix

The expression of the system of Eqs. (39) are as follows:

$$\begin{array}{l}{T}_{32\times 32}[\begin{array}{cc}\begin{array}{cccccccc}{A}_{tp1}^{\text{I}}& A_{tp2}^{\text{I}}& A_{tp3}^{\text{I}}& A_{rp1}^{\text{I}}& A_{rp2}^{\text{I}}& A_{rp3}^{\text{I}}& B_{ts}^{\text{I}}& B_{rs}^{\text{I}}\end{array}& \begin{array}{cccccccc}{A}_{1te}& A_{1re}& B_{1te}& B_{1re}& A_{2te}& A_{2re}& B_{2te}& B_{2re}\end{array}\end{array}\\ \begin{array}{cc}\begin{array}{cccccccc}{A}_{3te}& A_{3re}& B_{3te}& B_{3re}& A_{tp1}^{\text{II}}& A_{tp2}^{\text{II}}& A_{tp3}^{\text{II}}& A_{rp1}^{\text{II}}\end{array}& \begin{array}{cccccccc}{A}_{rp2}^{\text{II}}& A_{rp3}^{\text{II}}& B_{ts}^{\text{II}}& B_{rs}^{\text{II}}& A_{tp1}^{\text{III}}& A_{tp2}^{\text{III}}& A_{tp3}^{\text{III}}& B_{ts}^{\text{III}}\end{array}\end{array}{]}^{\text{T}}=\\ [\begin{array}{cc}\begin{array}{cccccccccccccc}0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}& \begin{array}{cccccccccccccc}0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\end{array}\\ \begin{array}{c}\begin{array}{cccc}{q}_0\frac{\sin (\xi l)}{\xi l}& 0& 0& 0\end{array}\end{array}{]}^{\text{T}}\end{array}$$

The elements not 0 in matrix T are as follows:

$$T_{0101}=T_{0104}={\chi }_1$$,$$T_{0102}=T_{0105}={\chi }_2$$,$$T_{0103}=T_{0106}={\chi }_3$$,$$T_{0107}=\text{-}2{\mu }_p\text{i}\xi r$$,$$T_{0108}=2{\mu }_p\text{i}\xi r$$,$$T_{0201}=\text{-}2{\mu }_p\text{i}\xi {\lambda }_1$$,$$T_{0202}=\text{-}2{\mu }_p\text{i}\xi {\lambda }_2$$,$$T_{0203}=\text{-}2{\mu }_p\text{i}\xi {\lambda }_3$$,$$T_{0204}=2{\mu }_p\text{i}\xi {\lambda }_1$$,$$T_{0205}=2{\mu }_p\text{i}\xi {\lambda }_2$$,$$T_{0206}=2{\mu }_p\text{i}\xi {\lambda }_3$$, $$T_{0207}=T_{0208}\text{=-}{\mu }_p�r^2+{\xi }^2)$$,$$T_{0301}=T_{0304}\text{=}\left(a_{11}+a_{12}{\delta }_{p1}^l+a_{13}{\delta }_{p1}^g\right)({\xi }^2-{\lambda }_1^2)$$,$$T_{0302}=T_{0305}\text{=}\left(a_{11}+a_{12}{\delta }_{p2}^l+a_{13}{\delta }_{p2}^g\right)({\xi }^2-{\lambda }_2^2)$$,$$T_{0303}=T_{0306}\text{=}\left(a_{11}+a_{12}{\delta }_{p3}^l+a_{13}{\delta }_{p3}^g\right)({\xi }^2-{\lambda }_3^2)$$,$$T_{0401}=T_{0304}\text{=}\left(a_{21}+a_{22}{\delta }_{p1}^l+a_{23}{\delta }_{p1}^g\right)({\xi }^2-{\lambda }_1^2)$$,$$T_{0402}=T_{0405}\text{=}\left(a_{21}+a_{22}{\delta }_{p2}^l+a_{23}{\delta }_{p2}^g\right)({\xi }^2-{\lambda }_2^2)$$,$$T_{0403}=T_{0406}\text{=}\left(a_{21}+a_{22}{\delta }_{p3}^l+a_{23}{\delta }_{p3}^g\right)({\xi }^2-{\lambda }_3^2)$$,$$T_{0501}\text{=}{\chi }_1{\text{e}}^{\text{-}{\lambda }_{1}H}$$,$$T_{0502}\text{=}{\chi }_2{\text{e}}^{\text{-}{\lambda }_{2}H}$$,$$T_{0503}\text{=}{\chi }_3{\text{e}}^{\text{-}{\lambda }_{3}H}$$,$$T_{0504}\text{=}{\chi }_1{\text{e}}^{{\lambda }_{1}H}$$,$$T_{0505}\text{=}{\chi }_2{\text{e}}^{{\lambda }_{2}H}$$,$$T_{0506}\text{=}{\chi }_3{\text{e}}^{{\lambda }_{3}H}$$,$$T_{0507}=\text{-}2{\mu }_p\text{i}\xi r{\text{e}}^{-rH}$$,$$T_{0508}=2{\mu }_p\text{i}\xi r{\text{e}}^{rH}$$,$$T_{0509}=\left[{\lambda }_{1e}{\xi }^2+{\alpha }_{1e}^2({\lambda }_{1e}+2{\mu }_{1e})\right]{\text{e}}^{-\text{i}{\alpha }_{1e}H}$$,$$T_{0510}=\left[{\lambda }_{1e}{\xi }^2+{\alpha }_{1e}^2({\lambda }_{1e}+2{\mu }_{1e})\right]{\text{e}}^{\text{i}{\alpha }_{1e}H}$$,$$T_{0511}=\text{-2}{\mu }_{1e}\xi {\beta }_{1e}{\text{e}}^{-\text{i}{\beta }_{1e}H}$$,$$T_{0512}=\text{2}{\mu }_{1e}\xi {\beta }_{1e}{\text{e}}^{\text{i}{\beta }_{1e}H}$$,$$T_{0601}=\text{-}2{\mu }_p\text{i}\xi {\lambda }_1{\text{e}}^{-{\lambda }_{1}H}$$,$$T_{0602}=\text{-}2{\mu }_p\text{i}\xi {\lambda }_2{\text{e}}^{-{\lambda }_{2}H}$$,$$T_{0603}=\text{-}2{\mu }_p\text{i}\xi {\lambda }_3{\text{e}}^{-{\lambda }_{3}H}$$,$$T_{0604}=2{\mu }_p\text{i}\xi {\lambda }_1{\text{e}}^{{\lambda }_{1}H}$$,$$T_{0605}=2{\mu }_p\text{i}\xi {\lambda }_2{\text{e}}^{{\lambda }_{2}H}$$,$$T_{0606}=2{\mu }_p\text{i}\xi {\lambda }_3{\text{e}}^{{\lambda }_{3}H}$$,$$T_{0607}=\text{-}{\mu }_p�r^2+{\xi }^2){\text{e}}^{\text{-}rH}$$,$$T_{0608}=\text{-}{\mu }_p�r^2+{\xi }^2){\text{e}}^{rH}$$, $$T_{0609}=\text{-2}{\mu }_{1e}\xi {\alpha }_{1e}{\text{e}}^{\text{-i}{\alpha }_{1e}H}$$,$$T_{0610}=\text{2}{\mu }_{1e}\xi {\alpha }_{1e}{\text{e}}^{\text{i}{\alpha }_{1e}H}$$,$$T_{0611}={\mu }_{1e}�{\xi }^2\text{-}{\beta }_{1e}^2){\text{e}}^{\text{-i}{\beta }_{1e}H}$$,$$T_{0612}={\mu }_{1e}�{\xi }^2\text{-}{\beta }_{1e}^2){\text{e}}^{\text{i}{\beta }_{1e}H}$$,$$T_{0701}=\text{-}{\lambda }_1{\text{e}}^{-{\lambda }_{1}H}$$,$$T_{0701}=\text{-}{\lambda }_2{\text{e}}^{-{\lambda }_{2}H}$$,$$T_{0703}=\text{-}{\lambda }_3{\text{e}}^{-{\lambda }_{3}H}$$,$$T_{0704}={\lambda }_1{\text{e}}^{{\lambda }_{1}H}$$,$$T_{0705}={\lambda }_2{\text{e}}^{{\lambda }_{2}H}$$,$$T_{0706}={\lambda }_3{\text{e}}^{{\lambda }_{3}H}$$,$$T_{0707}=\text{i}\xi {\text{e}}^{-rH}$$,$$T_{0708}=\text{i}\xi {\text{e}}^{rH}$$,$$T_{0709}=\text{i}{\alpha }_{1e}{\text{e}}^{-\text{i}{\alpha }_{1e}H}$$,$$T_{0710}=\text{-i}{\alpha }_{1e}{\text{e}}^{\text{i}{\alpha }_{1e}H}$$,$$T_{0711}=\text{-i}\xi {\text{e}}^{\text{-i}{\beta }_{1e}H}$$,$$T_{0712}=\text{-i}\xi {\text{e}}^{\text{i}{\beta }_{1e}H}$$,$$T_{0801}=\text{i}\xi {\text{e}}^{-{\lambda }_{1}H}$$,$$T_{0802}=\text{i}\xi {\text{e}}^{-{\lambda }_{2}H}$$,$$T_{0803}=\text{i}\xi {\text{e}}^{-{\lambda }_{3}H}$$,$$T_{0804}=\text{i}\xi {\text{e}}^{{\lambda }_{1}H}$$,$$T_{0805}=\text{i}\xi {\text{e}}^{{\lambda }_{2}H}$$,$$T_{0806}=\text{i}\xi {\text{e}}^{{\lambda }_{3}H}$$,$$T_{0807}=r{\text{e}}^{-rH}$$,$$T_{0808}=\text{-}r{\text{e}}^{rH}$$,$$T_{0809}=\text{-i}\xi {\text{e}}^{\text{-i}{\alpha }_{1e}H}$$,$$T_{0810}=\text{-i}\xi {\text{e}}^{\text{i}{\alpha }_{1e}H}$$, $$T_{0811}=\text{-i}{\beta }_{1e}{\text{e}}^{-\text{i}{\beta }_{1e}H}$$,$$T_{0812}=\text{i}{\beta }_{1e}{\text{e}}^{\text{i}{\beta }_{1e}H}$$,$$T_{0901}={\lambda }_1(1-{\delta }_{p1}^l){\text{e}}^{-{\lambda }_{1}H}$$,$$T_{0902}={\lambda }_2(1-{\delta }_{p2}^l){\text{e}}^{-{\lambda }_{2}H}$$,$$T_{0903}={\lambda }_3(1-{\delta }_{p3}^l){\text{e}}^{-{\lambda }_{3}H}$$,$$T_{0904}={\lambda }_1({\delta }_{p1}^l-1){\text{e}}^{{\lambda }_{1}H}$$,$$T_{0905}={\lambda }_2({\delta }_{p2}^l-1){\text{e}}^{{\lambda }_{2}H}$$,$$T_{0906}={\lambda }_3({\delta }_{p3}^l-1){\text{e}}^{{\lambda }_{3}H}$$,$$T_{0907}=\text{i}\xi �{\delta }_s^l-1){\text{e}}^{-rH}$$,$$T_{0908}=\text{i}\xi �{\delta }_s^l-1){\text{e}}^{rH}$$,$$T_{1001}={\lambda }_1(1-{\delta }_{p1}^g){\text{e}}^{-{\lambda }_{1}H}$$,$$T_{1002}={\lambda }_2(1-{\delta }_{p2}^g){\text{e}}^{-{\lambda }_{2}H}$$,$$T_{1003}={\lambda }_3(1-{\delta }_{p3}^g){\text{e}}^{-{\lambda }_{3}H}$$,$$T_{1004}={\lambda }_1({\delta }_{p1}^g-1){\text{e}}^{{\lambda }_{1}H}$$,$$T_{1005}={\lambda }_2({\delta }_{p2}^g-1){\text{e}}^{{\lambda }_{2}H}$$,$$T_{1006}={\lambda }_3({\delta }_{p3}^g-1){\text{e}}^{{\lambda }_{3}H}$$,$$T_{1007}=\text{i}\xi �{\delta }_s^g-1){\text{e}}^{-rH}$$,$$T_{1008}=\text{i}\xi �{\delta }_s^g-1){\text{e}}^{rH}$$,$$T_{1101}=\text{i}\xi \left({\delta }_{p1}^l-1\right){\text{e}}^{-{\lambda }_{1}H}$$,$$T_{1102}=\text{i}\xi \left({\delta }_{p2}^l-1\right){\text{e}}^{-{\lambda }_{2}H}$$,$$T_{1103}=\text{i}\xi \left({\delta }_{p3}^l-1\right){\text{e}}^{-{\lambda }_{3}H}$$,$$T_{1104}=\text{i}\xi \left({\delta }_{p1}^l-1\right){\text{e}}^{{\lambda }_{1}H}$$,$$T_{1105}=\text{i}\xi \left({\delta }_{p2}^l-1\right){\text{e}}^{{\lambda }_{2}H}$$,$$T_{1106}=\text{i}\xi \left({\delta }_{p3}^l-1\right){\text{e}}^{{\lambda }_{3}H}$$,$$T_{1107}=r\left({\delta }_s^l-1\right){\text{e}}^{-rH}$$,$$T_{1108}=r\left(1-{\delta }_s^l\right){\text{e}}^{rH}$$,$$T_{1201}=\text{i}\xi \left({\delta }_{p1}^g-1\right){\text{e}}^{-{\lambda }_{1}H}$$,$$T_{1202}=\text{i}\xi \left({\delta }_{p2}^g-1\right){\text{e}}^{-{\lambda }_{2}H}$$,$$T_{1203}=\text{i}\xi \left({\delta }_{p3}^g-1\right){\text{e}}^{-{\lambda }_{3}H}$$,$$T_{1204}=\text{i}\xi \left({\delta }_{p1}^g-1\right){\text{e}}^{{\lambda }_{1}H}$$,$$T_{1205}=\text{i}\xi \left({\delta }_{p2}^g-1\right){\text{e}}^{{\lambda }_{2}H}$$,$$T_{1206}=\text{i}\xi \left({\delta }_{p3}^g-1\right){\text{e}}^{{\lambda }_{3}H}$$,$$T_{1207}=r\left({\delta }_s^g-1\right){\text{e}}^{-rH}$$,$$T_{1208}=r\left(1-{\delta }_s^g\right){\text{e}}^{rH}$$,$$T_{1309}=-\left[{\lambda }_{1e}{\xi }^2+{\alpha }_{1e}^2({\lambda }_{1e}+2{\mu }_{1e})\right]{\text{e}}^{-\text{i}{\alpha }_{1e}\left(H+h_{w1}\right)}$$,$$T_{1310}=-\left[{\lambda }_{1e}{\xi }^2+{\alpha }_{1e}^2({\lambda }_{1e}+2{\mu }_{1e})\right]{\text{e}}^{\text{i}{\alpha }_{1e}\left(H+h_{w1}\right)}$$,$$T_{1311}=2{\mu }_{1e}\xi {\beta }_{1e}{\text{e}}^{-\text{i}{\beta }_{1e}\left(H+h_{w1}\right)}$$,$$T_{1312}=-2{\mu }_{1e}\xi {\beta }_{1e}{\text{e}}^{-\text{i}{\beta }_{1e}\left(H+h_{w1}\right)}$$,$$T_{1313}=\left[{\lambda }_{2e}{\xi }^2+{\alpha }_{2e}^2({\lambda }_{2e}+2{\mu }_{2e})\right]{\text{e}}^{-\text{i}{\alpha }_{2e}\left(H+h_{w1}\right)}$$,$$T_{1314}=\left[{\lambda }_{2e}{\xi }^2+{\alpha }_{2e}^2({\lambda }_{2e}+2{\mu }_{2e})\right]{\text{e}}^{\text{i}{\alpha }_{2e}\left(H+h_{w1}\right)}$$,$$T_{1315}=-2{\mu }_{2e}\xi {\beta }_{2e}{\text{e}}^{-\text{i}{\beta }_{2e}\left(H+h_{w1}\right)}$$,$$T_{1316}=2{\mu }_{2e}\xi {\beta }_{2e}{\text{e}}^{\text{i}{\beta }_{2e}\left(H+h_{w1}\right)}$$,$$T_{1409}=2{\mu }_{1e}\xi {\alpha }_{1e}{\text{e}}^{-\text{i}{\alpha }_{1e}\left(H+h_{w1}\right)}$$,$$T_{1410}=-2{\mu }_{1e}\xi {\alpha }_{1e}{\text{e}}^{\text{i}{\alpha }_{1e}\left(H+h_{w1}\right)}$$,$$T_{1411}={\mu }_{1e}\left({\beta }_{1e}^2-{\xi }^2\right){\text{e}}^{\text{-i}{\beta }_{1e}\left(H+h_{w1}\right)}$$,$$T_{1412}={\mu }_{1e}\left({\beta }_{1e}^2-{\xi }^2\right){\text{e}}^{\text{i}{\beta }_{1e}\left(H+h_{w1}\right)}$$,$$T_{1413}=-2{\mu }_{2e}\xi {\alpha }_{2e}{\text{e}}^{-\text{i}{\alpha }_{2e}\left(H+h_{w1}\right)}$$,$$T_{1414}=2{\mu }_{2e}\xi {\alpha }_{2e}{\text{e}}^{\text{i}{\alpha }_{2e}\left(H+h_{w1}\right)}$$,$$T_{1415}={\mu }_{2e}\left({\xi }^2-{\beta }_{2e}^2\right){\text{e}}^{\text{-i}{\beta }_{2e}\left(H+h_{w1}\right)}$$,$$T_{1416}={\mu }_{2e}\left({\xi }^2-{\beta }_{2e}^2\right){\text{e}}^{\text{i}{\beta }_{2e}\left(H+h_{w1}\right)}$$,$$T_{1509}=-\text{i}{\alpha }_{1e}{\text{e}}^{-\text{i}{\alpha }_{1e}\left(H+h_{w1}\right)}$$,$$T_{1510}=\text{i}{\alpha }_{1e}{\text{e}}^{\text{i}{\alpha }_{1e}\left(H+h_{w1}\right)}$$,$$T_{1511}=\text{i}\xi {\text{e}}^{-\text{i}{\beta }_{1e}\left(H+h_{w1}\right)}$$,$$T_{1512}=\text{i}\xi {\text{e}}^{\text{i}{\beta }_{1e}\left(H+h_{w1}\right)}$$,$$T_{1513}=\text{i}{\alpha }_{2e}{\text{e}}^{-\text{i}{\alpha }_{2e}\left(H+h_{w1}\right)}$$,$$T_{1514}=-\text{i}{\alpha }_{2e}{\text{e}}^{\text{i}{\alpha }_{2e}\left(H+h_{w1}\right)}$$,$$T_{1515}=-\text{i}\xi {\text{e}}^{-\text{i}{\beta }_{2e}\left(H+h_{w1}\right)}$$,$$T_{1516}=-\text{i}\xi {\text{e}}^{\text{i}{\beta }_{2e}\left(H+h_{w1}\right)}$$,$$T_{1609}=\text{i}\xi {\text{e}}^{-\text{i}{\alpha }_{1e}\left(H+h_{w1}\right)}$$,$$T_{1610}=\text{i}\xi {\text{e}}^{\text{i}{\alpha }_{1e}\left(H+h_{w1}\right)}$$,$$T_{1611}=\text{i}{\beta }_{1e}{\text{e}}^{-\text{i}{\beta }_{1e}\left(H+h_{w1}\right)}$$,$$T_{1612}=-\text{i}{\beta }_{1e}{\text{e}}^{\text{i}{\beta }_{1e}\left(H+h_{w1}\right)}$$,$$T_{1613}=-\text{i}\xi {\text{e}}^{\text{-i}{\alpha }_{2e}\left(H+h_{w1}\right)}$$,$$T_{1614}=-\text{i}\xi {\text{e}}^{\text{i}{\alpha }_{2e}\left(H+h_{w1}\right)}$$,$$T_{1615}=-\text{i}{\beta }_{2e}{\text{e}}^{-\text{i}{\beta }_{2e}\left(H+h_{w1}\right)}$$, $$T_{1616}=\text{i}{\beta }_{2e}{\text{e}}^{\text{i}{\beta }_{2e}\left(H+h_{w1}\right)}$$,$$T_{1713}=-\left[{\lambda }_{2e}{\xi }^2+{\alpha }_{2e}^2({\lambda }_{2e}+2{\mu }_{2e})\right]{\text{e}}^{-\text{i}{\alpha }_{2e}\left(H+h_{w1}\text{+}{h}_{w2}\right)}$$,$$T_{1714}=-\left[{\lambda }_{2e}{\xi }^2+{\alpha }_{2e}^2({\lambda }_{2e}+2{\mu }_{2e})\right]{\text{e}}^{\text{i}{\alpha }_{2e}\left(H+h_{w1}\text{+}{h}_{w2}\right)}$$,$$T_{1715}=2{\mu }_{2e}\xi {\beta }_{2e}{\text{e}}^{-\text{i}{\beta }_{2e}\left(H+h_{w1}\text{+}{h}_{w2}\right)}$$,$$T_{1716}=-2{\mu }_{2e}\xi {\beta }_{2e}{\text{e}}^{-\text{i}{\beta }_{2e}\left(H+h_{w1}\text{+}{h}_{w2}\right)}$$,$$T_{1717}=\left[{\lambda }_{3e}{\xi }^2+{\alpha }_{3e}^2({\lambda }_{3e}+2{\mu }_{3e})\right]{\text{e}}^{-\text{i}{\alpha }_{3e}\left(H+h_{w1}\text{+}{h}_{w2}\right)}$$,$$T_{1718}=\left[{\lambda }_{3e}{\xi }^2+{\alpha }_{3e}^2({\lambda }_{3e}+2{\mu }_{3e})\right]{\text{e}}^{\text{i}{\alpha }_{3e}\left(H+h_{w1}\text{+}{h}_{w2}\right)}$$,$$T_{1719}=-2{\mu }_{3e}\xi {\beta }_{3e}{\text{e}}^{-\text{i}{\beta }_{3e}\left(H+h_{w1}\text{+}{h}_{w2}\right)}$$,$$T_{1720}=2{\mu }_{3e}\xi {\beta }_{3e}{\text{e}}^{\text{i}{\beta }_{3e}\left(H+h_{w1}\text{+}{h}_{w2}\right)}$$,$$T_{1813}=2{\mu }_{2e}\xi {\alpha }_{2e}{\text{e}}^{-\text{i}{\alpha }_{2e}\left(H+h_{w1}\text{+}{h}_{w2}\right)}$$,$$T_{1814}=-2{\mu }_{2e}\xi {\alpha }_{2e}{\text{e}}^{\text{i}{\alpha }_{2e}\left(H+h_{w1}\text{+}{h}_{w2}\right)}$$,$$T_{1815}={\mu }_{2e}\left({\beta }_{2e}^2-{\xi }^2\right){\text{e}}^{\text{-i}{\beta }_{2e}\left(H+h_{w1}\text{+}{h}_{w2}\right)}$$,$$T_{1816}={\mu }_{2e}\left({\beta }_{2e}^2-{\xi }^2\right){\text{e}}^{\text{i}{\beta }_{2e}\left(H+h_{w1}\text{+}{h}_{w2}\right)}$$,$$T_{1817}=-2{\mu }_{3e}\xi {\alpha }_{3e}{\text{e}}^{-\text{i}{\alpha }_{3e}\left(H+h_{w1}\text{+}{h}_{w2}\right)}$$,$$T_{1818}=2{\mu }_{3e}\xi {\alpha }_{3e}{\text{e}}^{\text{i}{\alpha }_{3e}\left(H+h_{w1}\text{+}{h}_{w2}\right)}$$,$$T_{1819}={\mu }_{3e}\left({\xi }^2-{\beta }_{3e}^2\right){\text{e}}^{\text{-i}{\beta }_{3e}\left(H+h_{w1}\text{+}{h}_{w2}\right)}$$,$$T_{1820}={\mu }_{3e}\left({\xi }^2-{\beta }_{3e}^2\right){\text{e}}^{\text{i}{\beta }_{3e}\left(H+h_{w1}\text{+}{h}_{w2}\right)}$$,$$T_{1913}=-\text{i}{\alpha }_{2e}{\text{e}}^{-\text{i}{\alpha }_{2e}\left(H+h_{w1}\text{+}{h}_{w2}\right)}$$,$$T_{1914}=\text{i}{\alpha }_{2e}{\text{e}}^{\text{i}{\alpha }_{2e}\left(H+h_{w1}\text{+}{h}_{w2}\right)}$$,$$T_{1915}=\text{i}\xi {\text{e}}^{-\text{i}{\beta }_{2e}\left(H+h_{w1}\text{+}{h}_{w2}\right)}$$,$$T_{1916}=\text{i}\xi {\text{e}}^{\text{i}{\beta }_{2e}\left(H+h_{w1}\text{+}{h}_{w2}\right)}$$,$$T_{1917}=\text{i}{\alpha }_{3e}{\text{e}}^{-\text{i}{\alpha }_{3e}\left(H+h_{w1}\text{+}{h}_{w2}\right)}$$,$$T_{1918}=-\text{i}{\alpha }_{3e}{\text{e}}^{\text{i}{\alpha }_{3e}\left(H+h_{w1}\text{+}{h}_{w2}\right)}$$,$$T_{1919}=-\text{i}\xi {\text{e}}^{-\text{i}{\beta }_{3e}\left(H+h_{w1}\text{+}{h}_{w2}\right)}$$,$$T_{1920}=-\text{i}\xi {\text{e}}^{\text{i}{\beta }_{3e}\left(H+h_{w1}\text{+}{h}_{w2}\right)}$$,$$T_{2013}=\text{i}\xi {\text{e}}^{-\text{i}{\alpha }_{2e}\left(H+h_{w1}\text{+}{h}_{w2}\right)}$$,$$T_{2014}=\text{i}\xi {\text{e}}^{\text{i}{\alpha }_{2e}\left(H+h_{w1}\text{+}{h}_{w2}\right)}$$,$$T_{2015}=\text{i}{\beta }_{2e}{\text{e}}^{-\text{i}{\beta }_{2e}\left(H+h_{w1}\text{+}{h}_{w2}\right)}$$,$$T_{2016}=-\text{i}{\beta }_{2e}{\text{e}}^{\text{i}{\beta }_{2e}\left(H+h_{w1}\text{+}{h}_{w2}\right)}$$,$$T_{2017}=-\text{i}\xi {\text{e}}^{\text{-i}{\alpha }_{3e}\left(H+h_{w1}\text{+}{h}_{w2}\right)}$$,$$T_{2018}=-\text{i}\xi {\text{e}}^{\text{i}{\alpha }_{3e}\left(H+h_{w1}\text{+}{h}_{w2}\right)}$$,$$T_{2019}=-\text{i}{\beta }_{3e}{\text{e}}^{-\text{i}{\beta }_{3e}\left(H+h_{w1}\text{+}{h}_{w2}\right)}$$, $$T_{2020}=\text{i}{\beta }_{3e}{\text{e}}^{\text{i}{\beta }_{3e}\left(H+h_{w1}\text{+}{h}_{w2}\right)}$$,$$T_{2117}=\text{-}\left[{\lambda }_{3e}{\xi }^2\text{+}{\alpha }_{3e}^2({\lambda }_{3e}+2{\mu }_{3e})\right]{\text{e}}^{-\text{i}{\alpha }_{3e}(H\text{+}{h}_{w1}\text{+}{h}_{w2}+h_{w3})}$$,$$T_{2118}=\text{-}\left[{\lambda }_{3e}{\xi }^2\text{+}{\alpha }_{3e}^2({\lambda }_{3e}+2{\mu }_{3e})\right]{\text{e}}^{\text{i}{\alpha }_{3e}(H\text{+}{h}_{w1}\text{+}{h}_{w2}+h_{w3})}$$,$$T_{2119}=\text{2}{\mu }_{3e}\xi {\beta }_{3e}{\text{e}}^{-\text{i}{\beta }_{3e}(H\text{+}{h}_{w1}\text{+}{h}_{w2}+h_{w3})}$$,$$T_{2120}=-\text{2}{\mu }_{3e}\xi {\beta }_{3e}{\text{e}}^{\text{i}{\beta }_{3e}(H\text{+}{h}_{w1}\text{+}{h}_{w2}+h_{w3})}$$,$$T_{2121}=-{\chi }_1{\text{e}}^{\text{-}{\lambda }_1(H\text{+}{h}_{w1}\text{+}{h}_{w2}+h_{w3})}$$,$$T_{2122}=-{\chi }_2{\text{e}}^{\text{-}{\lambda }_2(H\text{+}{h}_{w1}\text{+}{h}_{w2}+h_{w3})}$$,$$T_{2123}=-{\chi }_3{\text{e}}^{\text{-}{\lambda }_3(H\text{+}{h}_{w1}\text{+}{h}_{w2}+h_{w3})}$$,$$T_{2124}=-{\chi }_1{\text{e}}^{{\lambda }_1(H\text{+}{h}_{w1}\text{+}{h}_{w2}+h_{w3})}$$,$$T_{2125}=-{\chi }_2{\text{e}}^{{\lambda }_2(H\text{+}{h}_{w1}\text{+}{h}_{w2}+h_{w3})}$$,$$T_{2126}=-{\chi }_3{\text{e}}^{{\lambda }_3(H\text{+}{h}_{w1}\text{+}{h}_{w2}+h_{w3})}$$,$$T_{2127}=2{\mu }_p\text{i}\xi r{\text{e}}^{-r(HH\text{+}{h}_{w1}\text{+}{h}_{w2}+h_{w3})}$$,$$T_{2128}=-2{\mu }_p\text{i}\xi r{\text{e}}^{r(H\text{+}{h}_{w1}\text{+}{h}_{w2}+h_{w3})}$$,$$T_{2217}=\text{2}{\mu }_{3e}\xi {\alpha }_{3e}{\text{e}}^{\text{-i}{\alpha }_{3e}(H\text{+}{h}_{w1}\text{+}{h}_{w2}+h_{w3})}$$,$$T_{2218}=-\text{2}{\mu }_{3e}\xi {\alpha }_{3e}{\text{e}}^{\text{i}{\alpha }_{3e}(H\text{+}{h}_{w1}\text{+}{h}_{w2}+h_{w3})}$$,$$T_{2219}={\mu }_{3e}({\beta }_{3e}^2\text{-}{\xi }^2){\text{e}}^{\text{-i}{\beta }_{3e}(H\text{+}{h}_{w1}\text{+}{h}_{w2}+h_{w3})}$$,$$T_{2220}={\mu }_{3e}({\beta }_{3e}^2\text{-}{\xi }^2){\text{e}}^{\text{i}{\beta }_{3e}(H\text{+}{h}_{w1}\text{+}{h}_{w2}+h_{w3})}$$,$$T_{2221}=2{\mu }_p\text{i}\xi {\lambda }_1{\text{e}}^{-{\lambda }_1(H\text{+}{h}_{w1}\text{+}{h}_{w2}+h_{w3})}$$,$$T_{2222}=2{\mu }_p\text{i}\xi {\lambda }_2{\text{e}}^{-{\lambda }_2(H\text{+}{h}_{w1}\text{+}{h}_{w2}+h_{w3})}$$,$$T_{2223}=2{\mu }_p\text{i}\xi {\lambda }_3{\text{e}}^{-{\lambda }_3(H\text{+}{h}_{w1}\text{+}{h}_{w2}+h_{w3})}$$,$$T_{2224}=-2{\mu }_p\text{i}\xi {\lambda }_1{\text{e}}^{{\lambda }_1(H\text{+}{h}_{w1}\text{+}{h}_{w2}+h_{w3})}$$,$$T_{2225}=-2{\mu }_p\text{i}\xi {\lambda }_2{\text{e}}^{{\lambda }_2(H\text{+}{h}_{w1}\text{+}{h}_{w2}+h_{w3})}$$,$$T_{2226}=-2{\mu }_p\text{i}\xi {\lambda }_3{\text{e}}^{{\lambda }_3(H\text{+}{h}_{w1}\text{+}{h}_{w2}+h_{w3})}$$,$$T_{2227}={\mu }_p(r^2+{\xi }^2){\text{e}}^{\text{-}r(H\text{+}{h}_{w1}\text{+}{h}_{w2}+h_{w3})}$$$$T_{2228}={\mu }_p(r^2+{\xi }^2){\text{e}}^{r(H\text{+}{h}_{w1}\text{+}{h}_{w2}+h_{w3})}$$$$T_{2317}=\text{-i}{\alpha }_{3e}{\text{e}}^{-\text{i}{\alpha }_{3e}(H\text{+}{h}_{w1}\text{+}{h}_{w2}+h_{w3})}$$,$$T_{2318}=\text{i}{\alpha }_{3e}{\text{e}}^{\text{i}{\alpha }_{3e}(H\text{+}{h}_{w1}\text{+}{h}_{w2}+h_{w3})}$$,$$T_{2319}=\text{i}\xi {\text{e}}^{\text{-i}{\beta }_{3e}�H\text{+}{h}_{w1}\text{+}{h}_{w2}+h_{w3})}$$,$$T_{2320}=\text{i}\xi {\text{e}}^{\text{i}{\beta }_{3e}(H\text{+}{h}_{w1}\text{+}{h}_{w2}+h_{w3})}$$,$$T_{2321}={\lambda }_1{\text{e}}^{-{\lambda }_1�H\text{+}{h}_{w1}\text{+}{h}_{w2}+h_{w3})}$$,$$T_{2322}={\lambda }_2{\text{e}}^{-{\lambda }_2(H\text{+}{h}_{w1}\text{+}{h}_{w2}+h_{w3})}$$,$$T_{2323}={\lambda }_3{\text{e}}^{-{\lambda }_3(H\text{+}{h}_{w1}\text{+}{h}_{w2}+h_{w3})}$$,$$T_{2324}=-{\lambda }_1{\text{e}}^{{\lambda }_1(H\text{+}{h}_{w1}\text{+}{h}_{w2}+h_{w3})}$$,$$T_{2325}=-{\lambda }_2{\text{e}}^{{\lambda }_2(H\text{+}{h}_{w1}\text{+}{h}_{w2}+h_{w3})}$$,$$T_{2326}=-{\lambda }_3{\text{e}}^{{\lambda }_3(H\text{+}{h}_{w1}\text{+}{h}_{w2}+h_{w3})}$$,$$T_{2327}=-\text{i}\xi {\text{e}}^{-r(H\text{+}{h}_{w1}\text{+}{h}_{w2}+h_{w3})}$$,$$T_{2328}=-\text{i}\xi {\text{e}}^{r(H\text{+}{h}_{w1}\text{+}{h}_{w2}+h_{w3})}$$,$$T_{2417}=\text{i}\xi {\text{e}}^{\text{-i}{\alpha }_{3e}(H\text{+}{h}_{w1}\text{+}{h}_{w2}+h_{w3})}$$,$$T_{2418}=\text{i}\xi {\text{e}}^{\text{i}{\alpha }_{3e}(H\text{+}{h}_{w1}\text{+}{h}_{w2}+h_{w3})}$$,$$T_{2419}=\text{i}{\beta }_{3e}{\text{e}}^{-\text{i}{\beta }_{3e}(H\text{+}{h}_{w1}\text{+}{h}_{w2}+h_{w3})}$$,$$T_{2420}=-\text{i}{\beta }_{3e}{\text{e}}^{\text{i}{\beta }_{3e}(H\text{+}{h}_{w1}\text{+}{h}_{w2}+h_{w3})}$$,$$T_{2421}=\text{-i}\xi {\text{e}}^{-{\lambda }_1(H\text{+}{h}_{w1}\text{+}{h}_{w2}+h_{w3})}$$,$$T_{2422}=\text{-i}\xi {\text{e}}^{-{\lambda }_2(H\text{+}{h}_{w1}\text{+}{h}_{w2}+h_{w3})}$$,$$T_{2423}=\text{-i}\xi {\text{e}}^{-{\lambda }_3(H\text{+}{h}_{w1}\text{+}{h}_{w2}+h_{w3})}$$,$$T_{2424}=\text{-i}\xi {\text{e}}^{{\lambda }_1(H\text{+}{h}_{w1}\text{+}{h}_{w2}+h_{w3})}$$,$$T_{2425}=\text{-i}\xi {\text{e}}^{{\lambda }_2(H\text{+}{h}_{w1}\text{+}{h}_{w2}+h_{w3})}$$,$$T_{2426}=\text{-i}\xi {\text{e}}^{{\lambda }_3(H\text{+}{h}_{w1}\text{+}{h}_{w2}+h_{w3})}$$,$$T_{2427}=\text{-}r{\text{e}}^{-r(H\text{+}{h}_{w1}\text{+}{h}_{w2}+h_{w3})}$$,$$T_{2428}=r{\text{e}}^{r(H\text{+}{h}_{w1}\text{+}{h}_{w2}+h_{w3})}$$,$$T_{2521}={\lambda }_1({\delta }_{p1}^l\text{-}1){\text{e}}^{-{\lambda }_1(H\text{+}{h}_{w1}\text{+}{h}_{w2}+h_{w3})}$$,$$T_{2522}={\lambda }_2({\delta }_{p2}^l\text{-}1){\text{e}}^{-{\lambda }_2(H\text{+}{h}_{w1}\text{+}{h}_{w2}+h_{w3})}$$,$$T_{2523}={\lambda }_3({\delta }_{p3}^l\text{-}1){\text{e}}^{-{\lambda }_3(H\text{+}{h}_{w1}\text{+}{h}_{w2}+h_{w3})}$$,$$T_{2524}={\lambda }_1(1-{\delta }_{p1}^l){\text{e}}^{{\lambda }_1(H\text{+}{h}_{w1}\text{+}{h}_{w2}+h_{w3})}$$,$$T_{2525}={\lambda }_2(1-{\delta }_{p2}^l){\text{e}}^{{\lambda }_2(H\text{+}{h}_{w1}\text{+}{h}_{w2}+h_{w3})}$$,$$T_{2526}={\lambda }_3(1-{\delta }_{p3}^l){\text{e}}^{{\lambda }_3(H\text{+}{h}_{w1}\text{+}{h}_{w2}+h_{w3})}$$,$$T_{2527}=\text{i}\xi (1\text{-}{\delta }_s^l){\text{e}}^{-r(H\text{+}{h}_{w1}\text{+}{h}_{w2}+h_{w3})}$$,$$T_{2528}=\text{i}\xi (1\text{-}{\delta }_s^l){\text{e}}^{r(H\text{+}{h}_{w1}\text{+}{h}_{w2}+h_{w3})}$$,$$T_{2621}={\lambda }_1({\delta }_{p1}^g\text{-}1){\text{e}}^{-{\lambda }_1(H\text{+}{h}_{w1}\text{+}{h}_{w2}+h_{w3})}$$,$$T_{2622}={\lambda }_2({\delta }_{p2}^g\text{-}1){\text{e}}^{-{\lambda }_2(H\text{+}{h}_{w1}\text{+}{h}_{w2}+h_{w3})}$$,$$T_{2623}={\lambda }_3({\delta }_{p3}^g\text{-}1){\text{e}}^{-{\lambda }_3(H\text{+}{h}_{w1}\text{+}{h}_{w2}+h_{w3})}$$,$$T_{2624}={\lambda }_1(1-{\delta }_{p1}^g){\text{e}}^{{\lambda }_1(H\text{+}{h}_{w1}\text{+}{h}_{w2}+h_{w3})}$$,$$T_{2625}={\lambda }_2(1-{\delta }_{p2}^g){\text{e}}^{{\lambda }_2(H\text{+}{h}_{w1}\text{+}{h}_{w2}+h_{w3})}$$,$$T_{2626}={\lambda }_3(1-{\delta }_{p3}^g){\text{e}}^{{\lambda }_3(H\text{+}{h}_{w1}\text{+}{h}_{w2}+h_{w3})}$$,$$T_{2627}=\text{i}\xi (1\text{-}{\delta }_s^g){\text{e}}^{-r(H\text{+}{h}_{w1}\text{+}{h}_{w2}+h_{w3})}$$,$$T_{2628}=\text{i}\xi (1\text{-}{\delta }_s^g){\text{e}}^{r(H\text{+}{h}_{w1}\text{+}{h}_{w2}+h_{w3})}$$,

$$T_{2721}=\text{i}\xi \left({\delta }_{p1}^l-1\right){\text{e}}^{-{\lambda }_1\left(H\text{+}{h}_{w1}+h_{w2}+h_{w3}\right)}$$,$$T_{2722}=\text{i}\xi \left({\delta }_{p2}^l-1\right){\text{e}}^{-{\lambda }_2\left(H\text{+}{h}_{w1}+h_{w2}+h_{w3}\right)}$$,$$T_{2723}=\text{i}\xi \left({\delta }_{p3}^l-1\right){\text{e}}^{-{\lambda }_3\left(H\text{+}{h}_{w1}+h_{w2}+h_{w3}\right)}$$,$$T_{2724}=\text{i}\xi \left({\delta }_{p1}^l-1\right){\text{e}}^{{\lambda }_1\left(H\text{+}{h}_{w1}+h_{w2}+h_{w3}\right)}$$,$$T_{2725}=\text{i}\xi \left({\delta }_{p2}^l-1\right){\text{e}}^{{\lambda }_2\left(H\text{+}{h}_{w1}+h_{w2}+h_{w3}\right)}$$,$$T_{2726}=\text{i}\xi \left({\delta }_{p3}^l-1\right){\text{e}}^{{\lambda }_3\left(H\text{+}{h}_{w1}+h_{w2}+h_{w3}\right)}$$,$$T_{2727}=r\left({\delta }_s^l-1\right){\text{e}}^{-r\left(H\text{+}{h}_{w1}+h_{w2}+h_{w3}\right)}$$,$$T_{2728}=r\left(1-{\delta }_s^l\right){\text{e}}^{r\left(H\text{+}{h}_{w1}+h_{w2}+h_{w3}\right)}$$,$$T_{2821}=\text{i}\xi \left({\delta }_{p1}^g-1\right){\text{e}}^{-{\lambda }_1\left(H\text{+}{h}_{w1}+h_{w2}+h_{w3}\right)}$$,$$T_{2822}=\text{i}\xi \left({\delta }_{p2}^g-1\right){\text{e}}^{-{\lambda }_2\left(H\text{+}{h}_{w1}+h_{w2}+h_{w3}\right)}$$,$$T_{2823}=\text{i}\xi \left({\delta }_{p3}^g-1\right){\text{e}}^{-{\lambda }_3\left(H\text{+}{h}_{w1}+h_{w2}+h_{w3}\right)}$$,$$T_{2824}=\text{i}\xi \left({\delta }_{p1}^g-1\right){\text{e}}^{{\lambda }_1\left(H\text{+}{h}_{w1}+h_{w2}+h_{w3}\right)}$$,$$T_{2825}=\text{i}\xi \left({\delta }_{p2}^g-1\right){\text{e}}^{{\lambda }_2\left(H\text{+}{h}_{w1}+h_{w2}+h_{w3}\right)}$$,$$T_{2826}=\text{i}\xi \left({\delta }_{p3}^g-1\right){\text{e}}^{{\lambda }_3\left(H\text{+}{h}_{w1}+h_{w2}+h_{w3}\right)}$$,$$T_{2827}=r\left({\delta }_s^g-1\right){\text{e}}^{-r\left(H\text{+}{h}_{w1}+h_{w2}+h_{w3}\right)}$$,$$T_{2828}=r\left(1-{\delta }_s^g\right){\text{e}}^{r\left(H\text{+}{h}_{w1}+h_{w2}+h_{w3}\right)}$$,$$T_{2921}\text{=}{\chi }_1{\text{e}}^{\text{-}{\lambda }_1\left(H+h_{w1}+h_{w2}\text{+}{h}_{w3}\text{+}h\right)}$$,$$T_{2922}\text{=}{\chi }_2{\text{e}}^{\text{-}{\lambda }_2\left(H+h_{w1}+h_{w2}\text{+}{h}_{w3}\text{+}h\right)}$$,$$T_{2923}\text{=}{\chi }_3{\text{e}}^{\text{-}{\lambda }_3\left(H+h_{w1}+h_{w2}\text{+}{h}_{w3}\text{+}h\right)}$$,$$T_{2924}\text{=}{\chi }_1{\text{e}}^{{\lambda }_1\left(H+h_{w1}+h_{w2}\text{+}{h}_{w3}\text{+}h\right)}$$,$$T_{2925}\text{=}{\chi }_2{\text{e}}^{{\lambda }_2\left(H+h_{w1}+h_{w2}\text{+}{h}_{w3}\text{+}h\right)}$$,$$T_{2926}\text{=}{\chi }_3{\text{e}}^{{\lambda }_3\left(H+h_{w1}+h_{w2}\text{+}{h}_{w3}\text{+}h\right)}$$,$$T_{2927}=\text{-}2{\mu }_p\text{i}\xi r{\text{e}}^{-r\left(H+h_{w1}+h_{w2}\text{+}{h}_{w3}\text{+}h\right)}$$,$$T_{2928}=2{\mu }_p\text{i}\xi r{\text{e}}^{r\left(H+h_{w1}+h_{w2}\text{+}{h}_{w3}\text{+}h\right)}$$,$$T_{2929}\text{=-}{\chi }_1{\text{e}}^{\text{-}{\lambda }_1\left(H+h_{w1}+h_{w2}\text{+}{h}_{w3}\text{+}h\right)}$$,$$T_{2930}\text{=-}{\chi }_2{\text{e}}^{\text{-}{\lambda }_2\left(H+h_{w1}+h_{w2}\text{+}{h}_{w3}\text{+}h\right)}$$,$$T_{2931}\text{=-}{\chi }_3{\text{e}}^{\text{-}{\lambda }_3\left(H+h_{w1}+h_{w2}\text{+}{h}_{w3}\text{+}h\right)}$$,$$T_{2932}=2{\mu }_p\text{i}\xi r{\text{e}}^{-r\left(H+h_{w1}+h_{w2}\text{+}{h}_{w3}\text{+}h\right)}$$,$$T_{3021}=\text{-}2{\mu }_p\text{i}\xi {\lambda }_1{\text{e}}^{-{\lambda }_1\left(H+h_{w1}+h_{w2}\text{+}{h}_{w3}\text{+}h\right)}$$$$T_{3022}=\text{-}2{\mu }_p\text{i}\xi {\lambda }_2{\text{e}}^{-{\lambda }_2\left(H+h_{w1}+h_{w2}\text{+}{h}_{w3}\text{+}h\right)}$$,$$T_{3023}=\text{-}2{\mu }_p\text{i}\xi {\lambda }_3{\text{e}}^{-{\lambda }_3\left(H+h_{w1}+h_{w2}\text{+}{h}_{w3}\text{+}h\right)}$$,$$T_{3024}=2{\mu }_p\text{i}\xi {\lambda }_1{\text{e}}^{{\lambda }_1\left(H+h_{w1}+h_{w2}\text{+}{h}_{w3}\text{+}h\right)}$$,$$T_{3025}=2{\mu }_p\text{i}\xi {\lambda }_2{\text{e}}^{{\lambda }_2\left(H+h_{w1}+h_{w2}\text{+}{h}_{w3}\text{+}h\right)}$$,$$T_{3026}=2{\mu }_p\text{i}\xi {\lambda }_3{\text{e}}^{{\lambda }_3\left(H+h_{w1}+h_{w2}\text{+}{h}_{w3}\text{+}h\right)}$$,$$T_{3027}=\text{-}{\mu }_p(r^2+{\xi }^2){\text{e}}^{\text{-}r\left(H+h_{w1}+h_{w2}\text{+}{h}_{w3}\text{+}h\right)}$$,$$T_{3028}=\text{-}{\mu }_p(r^2+{\xi }^2){\text{e}}^{r\left(H+h_{w1}+h_{w2}\text{+}{h}_{w3}\text{+}h\right)}$$,$$T_{3029}=2{\mu }_p\text{i}\xi {\lambda }_1{\text{e}}^{-{\lambda }_1\left(H+h_{w1}+h_{w2}\text{+}{h}_{w3}\text{+}h\right)}$$,$$T_{3030}=2{\mu }_p\text{i}\xi {\lambda }_2{\text{e}}^{-{\lambda }_2\left(H+h_{w1}+h_{w2}\text{+}{h}_{w3}\text{+}h\right)}$$,$$T_{3031}=2{\mu }_p\text{i}\xi {\lambda }_3{\text{e}}^{-{\lambda }_3\left(H+h_{w1}+h_{w2}\text{+}{h}_{w3}\text{+}h\right)}$$,$$T_{3032}={\mu }_p(r^2+{\xi }^2){\text{e}}^{\text{-}r\left(H+h_{w1}+h_{w2}\text{+}{h}_{w3}\text{+}h\right)}$$,$$T_{3121}=\text{-}{\lambda }_1{\text{e}}^{-{\lambda }_1\left(H+h_{w1}+h_{w2}\text{+}{h}_{w3}\text{+}h\right)}$$,$$T_{3122}=\text{-}{\lambda }_2{\text{e}}^{-{\lambda }_2\left(H+h_{w1}+h_{w2}\text{+}{h}_{w3}\text{+}h\right)}$$,$$T_{3123}=\text{-}{\lambda }_3{\text{e}}^{-{\lambda }_3\left(H+h_{w1}+h_{w2}\text{+}{h}_{w3}\text{+}h\right)}$$,$$T_{3124}={\lambda }_1{\text{e}}^{{\lambda }_1\left(H+h_{w1}+h_{w2}\text{+}{h}_{w3}\text{+}h\right)}$$,$$T_{3125}={\lambda }_2{\text{e}}^{{\lambda }_2\left(H+h_{w1}+h_{w2}\text{+}{h}_{w3}\text{+}h\right)}$$,$$T_{3126}={\lambda }_3{\text{e}}^{{\lambda }_3\left(H+h_{w1}+h_{w2}\text{+}{h}_{w3}\text{+}h\right)}$$,$$T_{3127}=\text{i}\xi {\text{e}}^{-r\left(H+h_{w1}+h_{w2}\text{+}{h}_{w3}\text{+}h\right)}$$,$$T_{3128}=\text{i}\xi {\text{e}}^{r\left(H+h_{w1}+h_{w2}\text{+}{h}_{w3}\text{+}h\right)}$$,$$T_{3129}={\lambda }_1{\text{e}}^{-{\lambda }_1\left(H+h_{w1}+h_{w2}\text{+}{h}_{w3}\text{+}h\right)}$$,$$T_{3130}={\lambda }_2{\text{e}}^{-{\lambda }_2\left(H+h_{w1}+h_{w2}\text{+}{h}_{w3}\text{+}h\right)}$$,$$T_{3131}={\lambda }_3{\text{e}}^{-{\lambda }_3\left(H+h_{w1}+h_{w2}\text{+}{h}_{w3}\text{+}h\right)}$$,$$T_{3132}=-\text{i}\xi {\text{e}}^{-r\left(H+h_{w1}+h_{w2}\text{+}{h}_{w3}\text{+}h\right)}$$,$$T_{3221}=\text{i}\xi {\text{e}}^{-{\lambda }_1\left(H+h_{w1}+h_{w2}\text{+}{h}_{w3}\text{+}h\right)}$$,$$T_{3222}=\text{i}\xi {\text{e}}^{-{\lambda }_2\left(H+h_{w1}+h_{w2}\text{+}{h}_{w3}\text{+}h\right)}$$,$$T_{3223}=\text{i}\xi {\text{e}}^{-{\lambda }_3\left(H+h_{w1}+h_{w2}\text{+}{h}_{w3}\text{+}h\right)}$$,$$T_{3224}=\text{i}\xi {\text{e}}^{{\lambda }_1\left(H+h_{w1}+h_{w2}\text{+}{h}_{w3}\text{+}h\right)}$$,$$T_{3225}=\text{i}\xi {\text{e}}^{{\lambda }_2\left(H+h_{w1}+h_{w2}\text{+}{h}_{w3}\text{+}h\right)}$$,$$T_{3226}=\text{i}\xi {\text{e}}^{{\lambda }_3\left(H+h_{w1}+h_{w2}\text{+}{h}_{w3}\text{+}h\right)}$$,$$T_{3227}=r{\text{e}}^{-r\left(H+h_{w1}+h_{w2}\text{+}{h}_{w3}\text{+}h\right)}$$,$$T_{3228}=-r{\text{e}}^{r\left(H+h_{w1}+h_{w2}\text{+}{h}_{w3}\text{+}h\right)}$$,$$T_{3229}=-\text{i}\xi {\text{e}}^{-{\lambda }_1\left(H+h_{w1}+h_{w2}\text{+}{h}_{w3}\text{+}h\right)}$$,$$T_{3230}=-\text{i}\xi {\text{e}}^{-{\lambda }_2\left(H+h_{w1}+h_{w2}\text{+}{h}_{w3}\text{+}h\right)}$$,$$T_{3231}=-\text{i}\xi {\text{e}}^{-{\lambda }_3\left(H+h_{w1}+h_{w2}\text{+}{h}_{w3}\text{+}h\right)}$$,$$T_{3232}=-r{\text{e}}^{-r\left(H+h_{w1}+h_{w2}\text{+}{h}_{w3}\text{+}h\right)}$$.

Acknowledgments

The authors gratefully acknowledge the financial support of the Chinese Natural Science Foundation (Grant No. 52168053), the authors are also grateful to the reviewers for their helpful advice and comments.

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Edited by

Editor: Pablo Andrés Muñoz Rojas

Publication Dates

• Publication in this collection
  13 Mar 2023
• Date of issue
  2023

History

• Received
  10 Jan 2023
• Reviewed
  28 Jan 2023
• Accepted
  08 Feb 2023