# Bulk and Rayleigh Waves Propagation in Three-Phase Soil with Flow-Independent Viscosity
## 3 Wavefield Solution for Bulk and Rayleigh Waves
### Rayleigh Wave  
As portrayed in Figure 2, the Rayleigh wave is generated by superimposing the longitudinal and transverse waves at the soil boundary \(z=0\). Accordingly, the displacement potential in Equations (29) and (30) for the Rayleigh wave can be further rewritten as  
\[ \begin{vmatrix}\phi^{s}=\left[A_{(1)}^{s}e^{-i\gamma_{1}z}+A_{(2)}^{s}e^{-i \gamma_{1}z}+A_{(3)}^{s}e^{-i\gamma_{3}z}\right]e^{i(\omega t-k_{g}x)}\\ \phi^{w}=\left[p_{(1)}^{\text{ms}}A_{(1)}^{s}e^{-i\gamma_{1}z}+p_{(2)}^{\text{ ms}}A_{(1)}^{s}e^{-i\gamma_{2}z}+p_{(3)}^{\text{ms}}A_{(3)}^{s}e^{-i\gamma_{2}z} \right]e^{i(\omega t-k_{g}x)}\\ \phi^{g}=\left[p_{(1)}^{\text{gs}}A_{(1)}^{s}e^{-i\gamma_{1}z}+p_{(2)}^{\text{ gs}}A_{(2)}^{s}e^{-i\gamma_{2}z}+p_{(3)}^{\text{gs}}A_{(3)}^{s}e^{-i\gamma_{2}z} \right]e^{i(\omega t-k_{g}x)}\end{vmatrix}  \tag{41}\]  
\[ \begin{vmatrix}\mathbf{\psi}^{s}=\mathbf{B}^{s}e^{-i\gamma_{4}z}e^{i(\omega t-k_{g}x)} \\ \mathbf{\psi}^{w}=s^{\text{ss}}\mathbf{B}^{s}e^{-i\gamma_{2}z}e^{i(\omega t-k_{g}x)} \\ \mathbf{\psi}^{g}=s^{\text{ss}}\mathbf{B}^{s}e^{-i\gamma_{4}z}e^{i(\omega t-k_{g}x)} \end{vmatrix}  \tag{42}\]where  
\[\gamma_{1}=\sqrt{k_{p1}^{2}-k_{R}^{2}},\ \gamma_{2}=\sqrt{k_{p2}^{2}-k_{R}^{2}},\ \gamma_{3}=\sqrt{k_{p3}^{2}-k_{R}^{2}},\ \gamma_{4}=\sqrt{k_{3}^{2}-k_{R}^{2}} \tag{43}\]  
As discussed earlier, the Rayleigh wave in unsaturated soil is a kind of superimposed wave at the soil boundary. In this work, considering the stress-free boundary, the stresses (\(\sigma_{zz}\) and \(\sigma_{xz}\)) and fluid pressures (\(p_{l}\) and \(p_{a}\)) disappear at the soil boundary. Accordingly, the following formulas can be obtained as  
\[\sigma_{zz}=\bigg{(}C_{11}+C_{14}\frac{\partial}{\partial t} \bigg{)}\bigg{(}\frac{\partial u_{x}^{s}}{\partial x}+\frac{\partial u_{x}^{s }}{\partial z}\bigg{)}+2\bigg{(}\mu_{e}+\mu_{v}\frac{\partial}{\partial t} \bigg{)}\frac{\partial u_{x}^{s}}{\partial z}+C_{12}\bigg{(}\frac{\partial u_{ x}^{w}}{\partial x}+\frac{\partial u_{z}^{w}}{\partial z}\bigg{)}+C_{13} \bigg{(}\frac{\partial u_{x}^{s}}{\partial x}+\frac{\partial u_{z}^{s}}{ \partial z}\bigg{)}=0 \tag{44}\] \[\sigma_{xz}=\bigg{(}\mu_{e}+\mu_{v}\frac{\partial}{\partial t} \bigg{)}\bigg{(}\frac{\partial u_{x}^{s}}{\partial z}+\frac{\partial u_{z}^{s}} {\partial x}\bigg{)}=0\] (45) \[p_{l}=-\bigg{(}C_{21}+C_{24}\frac{\partial}{\partial t}\bigg{)} \bigg{(}\frac{\partial u_{x}^{s}}{\partial x}+\frac{\partial u_{z}^{s}}{ \partial z}\bigg{)}-C_{22}\bigg{(}\frac{\partial u_{x}^{w}}{\partial x}+\frac{ \partial u_{z}^{w}}{\partial z}\bigg{)}-C_{23}\bigg{(}\frac{\partial u_{x}^{s }}{\partial x}+\frac{\partial u_{z}^{s}}{\partial z}\bigg{)}=0\] (46) \[p_{a}=-\bigg{(}C_{31}+C_{34}\frac{\partial}{\partial t}\bigg{)} \bigg{(}\frac{\partial u_{x}^{s}}{\partial x}+\frac{\partial u_{z}^{s}}{ \partial z}\bigg{)}-C_{32}\bigg{(}\frac{\partial u_{x}^{w}}{\partial x}+\frac{ \partial u_{z}^{w}}{\partial z}\bigg{)}-C_{33}\bigg{(}\frac{\partial u_{x}^{s }}{\partial x}+\frac{\partial u_{z}^{s}}{\partial z}\bigg{)}=0 \tag{47}\]  
Substituting Equations (41) and (42) into Equations (44)-(47) and making algebraic operations, the following formula can be derived as  
\[ \begin{pmatrix}\Re_{11}&\Re_{12}&\Re_{13}&\Re_{14}\\ \Re_{21}&\Re_{22}&\Re_{23}&\Re_{24}\\ \Re_{31}&\Re_{32}&\Re_{33}&\Re_{34}\\ \Re_{41}&\Re_{42}&\Re_{43}&\Re_{44}\end{pmatrix}\begin{pmatrix}A_{(1)}^{s}\\ A_{(2)}^{s}\\ A_{(3)}^{s}\\ B^{s}\end{pmatrix} = \begin{pmatrix}0\\ 0\\ 0\\ 0\end{pmatrix}