# Scattering of Plane Waves by Cylindrical Cavity in Unsaturated Poroelastic Medium +
## 2 Model and Solution
### Solution of Dynamic Equation of Unsaturated Porous Media  
Based on the [PERSON] theorem, the displacement vector \(\mathbf{u}\) may be expressed as:  
\[\mathbf{u}^{\alpha}=\
abla\varphi^{\alpha}+\
abla\times\mathbf{\Psi}^{\alpha}(\alpha =S,W,N) \tag{5}\]  
where, \(\varphi^{\alpha}\), \(\mathbf{\Psi}^{\alpha}(\alpha=S,W,N)\) are scalar potentials and vector potentials of \(\alpha\) phase in unsaturated porous media, respectively. Figure 1: Cylindrical cavity in unsaturated medium. By introducing Equation (5), Equation (1) is transformed into:  
\[\left\{ \begin{array}{l}n_{0}^{S}\rho_{0}^{S}\frac{\partial^{2}\varphi^{S}}{ \partial\hat{t}^{2}}=\left(M_{SS}+2n_{0}^{S}I_{S}\right)\
abla^{2}\varphi^{S}+M_ {SW}\
abla^{2}\varphi^{W}+M_{SN}\
abla^{2}\varphi^{W}\\ \hskip 14.226378 pt+\hat{\mu}^{W}\left(\frac{\partial\varphi^{W}}{\partial t}- \frac{\partial\varphi^{S}}{\partial t}\right)+\hat{\mu}^{N}\left(\frac{ \partial\varphi^{N}}{\partial t}-\frac{\partial\varphi^{S}}{\partial t}\right) \\ n_{0}^{W}\rho_{0}^{W}\frac{\partial^{2}\varphi^{W}}{\partial\hat{t}^{2}}=M_{ SW}\
abla^{2}\varphi^{S}+M_{WW}\
abla^{2}\varphi^{W}+M_{WN}\
abla^{2}\varphi^{W}- \hat{\mu}^{W}\left(\frac{\partial\varphi^{W}}{\partial t}-\frac{\partial \varphi^{S}}{\partial t}\right)\\ n_{0}^{N}\rho_{0}^{N}\frac{\partial^{2}\varphi^{W}}{\partial\hat{t}^{2}}=M_{ SN}\
abla^{2}\varphi^{S}+M_{WN}\
abla^{2}\varphi^{W}+M_{NN}\
abla^{2} \varphi^{N}-\hat{\mu}^{N}\left(\frac{\partial\varphi^{N}}{\partial t}-\frac{ \partial\varphi^{S}}{\partial t}\right)\end{array} \right. \tag{6}\]  
\[\left\{ \begin{array}{l}n_{0}^{S}\rho_{0}^{S}\frac{\partial^{2}\textbf{\Psi}^{ S}}{\partial\hat{t}^{2}}=n_{0}^{S}n_{S}\
abla^{2}\textbf{\Psi}^{S}+\hat{\mu}^{W} \left(\frac{\partial\textbf{\Psi}^{W}}{\partial t}-\frac{\partial\textbf{ \Psi}^{S}}{\partial t}\right)+\hat{\mu}^{N}\left(\frac{\partial\textbf{\Psi} ^{N}}{\partial t}-\frac{\partial\textbf{\Psi}^{S}}{\partial t}\right)\\ n_{0}^{W}\rho_{0}^{W}\frac{\partial^{2}\textbf{\Psi}^{W}}{\partial\hat{t}^{2}} =-\hat{\mu}^{W}\left(\frac{\partial\textbf{\Psi}^{W}}{\partial t}-\frac{ \partial\textbf{\Psi}^{S}}{\partial t}\right)\\ n_{0}^{N}\rho_{0}^{N}\frac{\partial^{2}\textbf{\Psi}^{W}}{\partial\hat{t}^{2}} =-\hat{\mu}^{N}\left(\frac{\partial\textbf{\Psi}^{N}}{\partial t}-\frac{ \partial\textbf{\Psi}^{S}}{\partial t}\right)\end{array} \right. \tag{7}\]  
Assuming that the incident wave is a harmonic with circular frequency \(\omega\), the expressions of scalar potentials and vector potentials can be expressed as:  
\[\varphi^{a}=\varphi^{a}(x,y,z)e^{-i\omega t} \tag{8a}\] \[\textbf{\Psi}^{a}=\Psi^{a}(x,y,z)e^{-i\omega t} \tag{8b}\]  
By applying Equations (8a) and (8b), Equations (6) and (7) can be transformed into:  
\[\left\{ \begin{array}{l}(M_{SS}^{s}\
abla^{2}+\gamma_{SS}\omega^{2})\varphi^ {S}+(M_{SW}\
abla^{2}-\gamma_{SW}\omega^{2})\varphi^{W}+(M_{SN}\
abla^{2}- \gamma_{SN}\omega^{2})\varphi^{N}=0\\ (M_{SW}\
abla^{2}-\gamma_{SW}\omega^{2})\varphi^{S}+(M_{WW}\
abla^{2}+\gamma _{WW}\omega^{2})\varphi^{W}+M_{WN}\
abla^{2}\varphi^{N}=0\\ (M_{SN}\
abla^{2}-\gamma_{SN}\omega^{2})\varphi^{S}+M_{WN}\
abla^{2}\varphi^ {W}+(M_{NN}\
abla^{2}+\gamma_{NN}\omega^{2})\varphi^{N}=0\end{array} \right. \tag{9}\]  
\[\left\{ \begin{array}{l}-\omega^{2}n_{0}^{S}\rho_{0}^{S}\textbf{\[PERSON]}^{S}=n_ {0}^{S}\mu_{S}\
abla^{2}\[PERSON]^{S}-i\omega\hat{\mu}^{W}\left(\[PERSON]^{W}-\[PERSON]^{S} \right)-i\omega\hat{\mu}^{N}\left(\[PERSON]^{N}-\[PERSON]^{S}\right)\\ -\omega^{2}n_{0}^{W}\rho_{0}^{W}\textbf{\[PERSON]}^{W}=i\omega\hat{\mu}^{W}\left( \[PERSON]^{W}-\[PERSON]^{S}\right)\\ -\omega^{2}n_{0}^{N}\rho_{0}^{N}\textbf{\[PERSON]}^{N}=i\omega\hat{\mu}^{N}\left( \[PERSON]^{N}-\[PERSON]^{S}\right)\end{array} \right.