## 6 Analysis of the _u-p_ Equation  
Based on the _u-p_ formulation, wave velocities and attenuated coefficients are deduced using the same procedure as the _u-w-p_ formulation. Moreover, the scope of application of the _u-p_ formulation is discussed. Equation (7) is transformed into the frequency domain:  
\[(\lambda+G)\
abla\
abla\cdot\mathbf{u}+G\
abla^{2}\mathbf{u}+\omega^{2}\rho\mathbf{u}- \alpha\
abla p=\mathbf{0} \tag{53}\]  
\[\
abla^{2}p+\frac{\omega^{2}\rho_{m}}{Q_{b}}p+\omega^{2}\alpha p_{m}\
abla \cdot\mathbf{u}=0 \tag{54}\]  
where \(\rho_{m}=-\frac{1}{k}\frac{i}{\omega}\). In the same way, using the Helmholtz decomposition, displacement \(\mathbf{u}\) and pore pressure \(p\), respectively, are substituted into Equations (53) and (54). We can obtain the equations expressed by potential functions:  
\[\left(\lambda+2G+\alpha^{2}Q_{b}\right)\
abla^{2}\varphi_{s}+\alpha Q_{b} \
abla^{2}\varphi_{f}+\omega^{2}\rho\varphi_{s}=0 \tag{55}\]  
\[\alpha Q_{b}\
abla^{2}\varphi_{s}+Q_{b}\
abla^{2}\varphi_{f}+\omega^{2}\rho_{ m}\varphi_{f}=0 \tag{56}\]  
\[G\
abla^{2}\psi_{s}+\omega^{2}\rho\psi_{s}=0 \tag{57}\]  
Figure 2: Relations of permeability, frequency and attenuation coefficients of three waves in four saturated soils. Equations (55)-(57) is written in a matrix form, shown as follows:  
\[\left\{ \begin{array}{l}\left[\mathbf{M}\omega^{2}+\mathbf{K}_{P}\
abla^{2}\right]\mathbf{ \varphi}=\mathbf{0}\\ \left[\rho\omega^{2}+G\
abla^{2}\right]\psi_{s}=0\end{array} \right. \tag{58}\]  
where,  
\[\mathbf{\varphi}=\left[ \begin{array}{c}\varphi_{s}\\ \varphi_{f}\end{array} \right]\mathbf{M}=\left[ \begin{array}{cc}\rho\\ \rho_{m}\end{array} \right]\mathbf{K}_{P}=\left[ \begin{array}{cc}\lambda+2G+ \alpha^{2}Q_{b}&\alpha Q_{b}\\ \alpha Q_{b}&Q_{b}\end{array} \right] \tag{59}\]  
The characteristic equation of Equations (55)-(57) is:  
\[\left|\mathbf{K}_{P}l_{P}^{2}-\mathbf{M}\omega^{2}\right|=0 \tag{60}\]  
\[\left|Gl_{S}^{2}-\rho\omega^{2}\right|=0 \tag{61}\]  
The solutions of Equation (59) are shown as follows:  
\[\left(\frac{l_{P1,2}}{\omega}\right)^{2}=\left(\frac{1}{C_{P1,2}}\right)^{2} =\frac{\rho_{1,2}}{\lambda+2G} \tag{62}\]  
\[\left(\frac{l_{S}}{\omega}\right)^{2}=\left(\frac{1}{C_{S}}\right)^{2}=\frac {\rho}{G} \tag{63}\]  
where,  
\[\rho_{1,2}=\rho\Big{(}b\mp\sqrt{b^{2}-ac}\Big{)}\]  
\[a =\frac{\lambda+2G}{Q_{b}}\]  
\[b =\frac{1}{2\rho}\bigg{[}\rho_{m}\frac{\lambda+2G+\alpha^{2}Q_{b}}{Q_{b}}+ \rho\bigg{]} \tag{64}\] \[c =\frac{\rho_{m}}{\rho}\]  
From Equation (62), we can see that the theoretical velocities of the P1 and P2 waves have the imaginary parts, but the shear wave has a constant velocity. Thus, the two compressional waves are attenuated and the shear wave is not attenuated during the propagation procedure. In addition, Equation (60) is expanded as:  
\[(\lambda+2G)Q_{b}\bigg{(}\frac{l_{P}}{\omega}\bigg{)}^{4}-\Big{[}\Big{(} \lambda+2G+\alpha^{2}Q_{b}\Big{)}\rho_{m}+Q_{b}\rho\Big{]}\bigg{(}\frac{l_{P} }{\omega}\bigg{)}^{2}+\rho\rho_{m}=0 \tag{65}\]  
The Equation (56) is substituted into Equation (55) and the following equation can be obtained  
\[(\lambda+2G)\
abla^{2}\varphi_{s}+\omega^{2}\rho\varphi_{s}-\alpha\omega^{2} \rho_{m}\varphi_{f}=0 \tag{66}\]  
Moreover, the eigenvectors corresponding to the eigenvalue in Equation (62) are expressed as follows:  
Using Equation (67), the potential functions of the solid and fluid displacements can be expressed by the potential function of the P1 and P2 waves. The detailed relations are shown below:  
\[\left\{ \begin{array}{l}\varphi_{s}=\varphi_{P1}+\varphi_{P2}\\ \varphi_{f}=\eta_{1}\varphi_{P1}+\eta_{2}\varphi_{P2}\end{array} \right. \tag{69}\]  
The following two compressional wave equations can be obtained by substituting Equation (68) into Equation (66) and using Equation (65). \[\left\{ \begin{array}{l}C_{P1}^{2}\
abla^{2}\varphi_{P1}=\bar{\varphi}_{P1}\\ C_{P2}^{2}\
abla^{2}\varphi_{P2}=\bar{\varphi}_{P2}\end{array} \right.