## 3 Wave Equations and Theoretical Wave Velocities
### Finite Permeability  
Using Fourier transform, the form of Equation (5) in the frequency domain can be written as:  
\[(\lambda+G)\
abla\
abla\cdot\mathbf{u}+G\
abla^{2}\mathbf{u}+\omega^{2}\left(\rho- \frac{\rho_{f}^{2}}{\rho_{m}}\right)\mathbf{u}-\left(\alpha-\frac{\rho_{f}}{\rho_{ m}}\right)\
abla p=0 \tag{8}\]  
\[\
abla^{2}p+\frac{\omega^{2}\rho_{m}}{Q_{b}}p-\omega^{2}\Big{(}\rho_{f}-\alpha \rho_{m}\Big{)}\
abla\cdot\mathbf{u}=0 \tag{9}\]  
where \(\rho_{m}\) is introduced to make the formulation similar to the wave equations and has the relation of: \(\rho_{m}=\frac{\rho_{f}}{n}-\frac{1}{k}\frac{i}{\omega}\).  
Displacement \(\mathbf{u}\) and pore pressure \(p\) are decomposed by Helmholtz decomposition:  
\[\mathbf{u}=\
abla\varphi_{s}+\
abla\times\psi_{s}\,\mathbf{e} \tag{10}\]  
\[p=-Q_{b}\Big{(}\alpha\
abla^{2}\varphi_{s}+\
abla^{2}\varphi_{f}\Big{)} \tag{11}\]  
where \(\varphi_{s}\) and \(\varphi_{f}\) are the irrotational potential function of the solid and fluid phases, respectively. \(\psi_{s}\) is the rotational potential function of the solid phase. Substituting Equations (10) and (11) into Equations (8) and (9), and using the relation of \(\
abla\cdot\psi_{s}\mathbf{e}=0\), the following equations can be obtained:  
\[\left\{\begin{array}{l}\
abla\Big{[}\Big{(}\lambda+2G+\alpha^{2}Q_{b}-\alpha \frac{\rho_{f}^{2}}{\rho_{m}}Q_{b}\Big{)}\
abla^{2}\varphi_{s}+\Big{(}\alpha- \frac{\rho_{f}^{2}}{\rho_{m}}\Big{)}Q_{b}\
abla^{2}\varphi_{f}+\omega^{2} \Big{(}\rho-\frac{\rho_{f}^{2}}{\rho_{m}}\Big{)}\varphi_{s}\Big{]}\\ +\
abla\times\Big{[}G\
abla^{2}\psi_{s}+\omega^{2}\Big{(}\rho-\frac{\rho_{f} ^{2}}{\rho_{m}}\Big{)}\psi_{s}\Big{]}\mathbf{e}=0\\ \
abla^{2}\Big{[}\alpha Q_{b}\
abla^{2}\varphi_{s}+Q_{b}\
abla^{2}\varphi_{f }+\omega^{2}\Big{(}\rho_{f}\varphi_{s}+\rho_{m}\varphi_{f}\Big{)}\Big{]}=0 \end{array}\right. \tag{12}\]  
\(\varphi_{s}\), \(\varphi_{f}\) and \(\psi_{s}\) in Equation (12) should satisfy the following relations:\[\left(\lambda+2G+\alpha^{2}Q_{b}-\alpha\frac{\rho_{f}}{\rho_{m}}Q_{b}\right)\
abla^ {2}\varphi_{s}+\left(\alpha-\frac{\rho_{f}}{\rho_{m}}\right)Q_{b}\
abla^{2} \varphi_{f}+\omega^{2}\left(\rho-\frac{\rho_{f}{}^{2}}{\rho_{m}}\right)\varphi_ {s}=0 \tag{13}\]  
\[\alpha Q_{b}\
abla^{2}\varphi_{s}+Q_{b}\
abla^{2}\varphi_{f}+\omega^{2}\left( \rho_{f}\varphi_{s}+\rho_{m}\varphi_{f}\right)=0 \tag{14}\]  
\[\omega^{2}\left(\rho-\frac{\rho_{f}{}^{2}}{\rho_{m}}\right)\psi_{s}+G\
abla^{ 2}\psi_{s}=0 \tag{15}\]  
Equations (13)-(15) can be expressed as three scalar equations and is written into the matrix forms shown below:  
\[\left[M\omega^{2}+K_{P}\
abla^{2}\right]\boldsymbol{\varphi}=\boldsymbol{0} \tag{16}\]  
\[\left[\left(\rho-\frac{\rho_{f}{}^{2}}{\rho_{m}}\right)\omega^{2}+G\
abla^{2 }\right]\psi_{s}=0 \tag{17}\]  
where,  
\[\boldsymbol{\varphi}=\left[\begin{array}{c}\varphi_{s}\\ \varphi_{f}\end{array}\right],\boldsymbol{M}=\left[\begin{array}{cc}\rho- \frac{\rho_{f}{}^{2}}{\rho_{m}}&0\\ \rho_{f}&\rho_{m}\end{array}\right],\]  
\[\boldsymbol{K}_{P}=\left[\begin{array}{cc}\lambda+2G+\alpha^{2}Q_{b}-\alpha \frac{\rho_{f}}{\rho_{m}}Q_{b}&\left(\alpha-\frac{\rho_{f}}{\rho_{m}}\right)Q _{b}\\ \alpha Q_{b}&Q_{b}\end{array}\right] \tag{18}\]