## 2 Basic Equations and Problem Formulation
### Dispersion Equations
For the porous medium, the solid displacement and pore fluid displacement are expressed using potential functions as  
\[\Phi=\left(A_{1}H_{0}^{1}(\alpha_{1}r)+A_{2}H_{0}^{2}(\alpha_{1}r)\right) \exp[i(kz-\omega t)] \tag{2}\]8  
\[\Phi=\left(A_{1}H_{0}^{1}(\alpha_{1}r)+A_{2}H_{0}^{2}(\alpha_{1}r)\right) \exp[i(kz-\omega t)] \tag{2}\]9  
\[\Psi_{\theta}=\left(B_{1}H_{1}^{1}(\beta_{1}r)+B_{2}H_{1}^{2}(\beta_{1}r) \right)\exp[i(kz-\omega t)] \tag{3}\]0  
\[\Psi_{\theta}=\left(B_{1}H_{1}^{1}(\beta_{1}r)+B_{2}H_{1}^{2}(\beta_{1}r) \right)\exp[i(kz-\omega t)] \tag{3}\]1  
Thus, the boundary conditions are given as follows. (1) We consider an inviscid liquid, where the liquid-internal tube interface conditions are such that the radial displacement and normal stress components at the boundary are continuous, and the shear stress components are zero at the interface [34]. That is, the boundary conditions at \(r=a\) are  
\[\Psi_{\theta}=\left(B_{1}H_{1}^{1}(\beta_{1}r)+B_{2}H_{1}^{2}(\beta_{1}r) \right)\exp[i(kz-\omega t)] \tag{3}\]2  
\[\Psi_{\theta}=\left(B_{1}H_{1}^{1}(\beta_{1}r)+B_{2}H_{1}^{2}(\beta_{1}r) \right)\exp[i(kz-\omega t)] \tag{3}\]3  
\[\Psi_{\theta}=\left(B_{1}H_{1}^{1}(\beta_{1}r)+B_{2}H_{1}^{2}(\beta_{1}r) \right)\exp[i(kz-\omega t)] \tag{3}\]4  
(2) Owing to the impermeability of the elastic solid, the fluid in the porous medium is confined to flowing out at the boundary, such that the displacement of the fluid is equal to that of the solid phase skeleton [35]. The pipe-porous medium interface conditions state that the displacement and stress components at the boundary must be continuous. That is, the boundary conditions at \(r=b\) are  
\[\Psi_{\theta}=\left(B_{1}H_{1}^{1}(\beta_{1}r)+B_{2}H_{1}^{2}(\beta_{1}r) \right)\exp[i(kz-\omega t)] \tag{3}\]5  
\[\Psi_{\theta}=\left(B_{1}H_{1}^{1}(\beta_{1}r)+B_{2}H_{1}^{2}(\beta_{1}r) \right)\exp[i(kz-\omega t)] \tag{3}\]6  
\[\Psi_{\theta}=\left(B_{1}H_{1}^{1}(\beta_{1}r)+B_{2}H_{1}^{2}(\beta_{1}r) \right)\exp[i(kz-\omega t)] \tag{3}\]7\[\Psi_{\theta}=\left(B_{1}H_{1}^{1}(\beta_{1}r)+B_{2}H_{1}^{2}(\beta_{1}r) \right)\exp[i(kz-\omega t)] \tag{3}\]8 \[\Psi_{\theta}=\left(B_{1}H_{1}^{1}(\beta_{1}r)+B_{2}H_{1}^{2}(\beta_{1}r) \right)\exp[i(kz-\omega t)] \tag{3}\]9  
(3) It is further assumed that the porous medium layer is finite in a vacuum. Thus, all the stress components are equal to zero at the outer surface of the porous medium layer, i.e., the boundary conditions at \(r=b+d\) can be written as  
\[u_{rE}=\frac{\partial\Phi}{\partial r}-\frac{\partial\Psi_{\theta}}{\partial z} \tag{4}\]0 \[u_{rE}=\frac{\partial\Phi}{\partial r}-\frac{\partial\Psi_{\theta}}{\partial z} \tag{4}\]1 (31) \[u_{rE}=\frac{\partial\Phi}{\partial r}-\frac{\partial\Psi_{\theta}}{\partial z} \tag{4}\]2  
The substitution of Equations (2)-(21) into the boundary conditions (22)-(32) results in a system of linear homogeneous equations in the eleven constants \(A_{1}\), \(A_{2}\),..., \(F_{1}\), \(F_{2}\), and \(F_{3}\). \[u_{rE}=\frac{\partial\Phi}{\partial r}-\frac{\partial\Psi_{\theta}}{\partial z} \tag{4}\]3  
where \(\mathbf{M}\) is a \(11\times 11\) matrix, and the elements \(m_{11}\),..., \(m_{1111}\) are listed in the Appendix A. Equation (33) has non-zero solutions, the coefficient determinant of the equation must be zero, and the corresponding dispersion equation can be obtained as  
\[u_{rE}=\frac{\partial\Phi}{\partial r}-\frac{\partial\Psi_{\theta}}{\partial z} \tag{4}\]4  
When the thickness of the porous medium layer \(d\) approaches infinity, the coefficients \(F_{1}\), \(F_{2}\), and \(F_{3}\) would approach zero, as the sound field approaches zero at infinity. The unknown coefficients are reduced to eight in the infinite porous medium. Along similar lines, Equations (2)-(21), along with (22)-(29), yield a system of linear non-homogeneous equations for the axially symmetric vibrations of a liquid-filled pipe embedded in infinite porous medium.