## 2 Basic Equations and Problem Formulation
### Dispersion Equations  
A symmetrical geometric model (Figure 1) is constructed to derive the dispersion equation. The media in the model are considered to be homogeneous and isotropic. In this subsection, we provide in detail the process used to derive the dispersion equation for axisymmetric modes with the circumferential number \(n=0\) in a liquid-filled pipe embedded in porous medium. The potential function of this problem is divided into three parts: the pipe, the liquid in the pipe, and the porous medium. For the elastic medium pipe, the displacement field \(\mathbf{u}\) is decomposed into scalar \(\Phi\) and vector potential \(\boldsymbol{\
abla}\) with \(\
abla\cdot\boldsymbol{\
abla}=0\),  
\[\mathbf{u}=\
abla\Phi+\
abla\times\boldsymbol{\
abla} \tag{1}\]  
Because of the axisymmetry of the studied modes, the radial and axial components of the vector potential \(\boldsymbol{\
abla}\) are zero, and only the circumferential component \(\Psi_{\theta}\) is retained [23; 24; 25; 26; 27]. The expressions for \(\Phi\) and \(\Psi_{\theta}\) are given by  
\[\Phi=\left(A_{1}H_{0}^{1}(\alpha_{1}r)+A_{2}H_{0}^{2}(\alpha_{1}r)\right) \exp[i(kz-\omega t)] \tag{2}\]  
\[\Psi_{\theta}=\left(B_{1}H_{1}^{1}(\beta_{1}r)+B_{2}H_{1}^{2}(\beta_{1}r) \right)\exp[i(kz-\omega t)] \tag{3}\]  
where \(\alpha_{1}^{2}=k^{2}-\omega^{2}/c_{EI}^{2}\), \(\beta_{1}^{2}=k^{2}-\omega^{2}/c_{EI}^{2}\), \(k\) is the number of waves, \(w\) is the angular frequency, and \(A_{1}\), \(A_{2}\), \(B_{1}\), and \(B_{2}\) are the undetermined coefficients. \(H_{n}^{(1)}\) and \(H_{n}^{(2)}\) denote the Hankel functions of the first and second order (\(n=0,1\)), which can be expressed as a linear combination of Bessel functions and describe the two-dimensional cylindrical wave solution [28]. The displacement expressions of the pipe are respectively given as  
\[u_{rE}=\frac{\partial\Phi}{\partial r}-\frac{\partial\Psi_{\theta}}{\partial z} \tag{4}\]  
\[u_{zE}=\frac{\partial\Phi}{\partial z}+\frac{1}{r}\frac{\partial(r\Psi_{ \theta})}{\partial r} \tag{5}\]  
Figure 1: Model geometry of a liquid-filled pipe embedded in porous medium. where \(\lambda\) and \(\mu\) are the Lame constants in the elastic medium. For the inviscid liquid in the pipe, the displacement and stress expression are respectively expressed as  
\[u_{rL}=-B_{3}\alpha_{L}I_{1}(\alpha_{L}r)\exp[i(kz-\omega t)] \tag{6}\]  
\[\sigma_{rL}=-B_{3}\lambda_{L}J_{0}(\alpha_{L}r)[k^{2}+(\alpha_{L})^{2}]\exp[i( kz-\omega t)] \tag{7}\]  
where \(\alpha_{L}^{2}=k^{2}-\omega^{2}/c_{L}^{2}\), \(\lambda_{L}=\rho_{L}c_{L}^{2}\), and \(\lambda_{L}\) is the Lame constant in the liquid. \(B_{3}\) is the undetermined coefficient to be determined by boundary conditions. \(J_{0}\) is a modified Bessel function of the first kind. The thickness of the wrapping layer is not included in the study of the influence of the nondestructive testing parameters. However, sometimes, thickness also plays a very important role in the subsequent numerical simulation and analysis. Therefore, it is necessary to discuss the case of the porous medium being finite. In contrast to other media, the potential function of the porous medium includes both solid and liquid phases. The solid phase has three potential functions describing the fast longitudinal, slow longitudinal, and shear waves. Thus, the wave motion equations based on the framework of the Biot theory modified by [PERSON] for the scalar potentials are given as [29]  
\[\Phi_{PSIf}=(D_{1}K_{0}(\alpha_{21}r)+F_{1}I_{0}(\alpha_{21}r))\exp[i(kz- \omega t)] \tag{8}\]  
\[\Phi_{PSIs}=(D_{2}K_{0}(\alpha_{22}r)+F_{2}I_{0}(\alpha_{22}r))\exp[i(kz- \omega t)] \tag{9}\]  
\[\Phi=\left(A_{1}H_{0}^{1}(\alpha_{1}r)+A_{2}H_{0}^{2}(\alpha_{1}r)\right) \exp[i(kz-\omega t)] \tag{2}\]0  
where the longitudinal wave potential function \(\Phi_{PSI}=\Phi_{PSIf}+\Phi_{PSIs}\), \(a_{21}^{2}=k^{2}-k_{p1}^{2}\), \(\alpha_{22}^{2}=k^{2}-k_{p2}^{2}\), and \(\beta_{21}^{2}=k^{2}-k_{t}^{2}\), and \(D_{1}\), \(D_{2}\), \(D_{3}\), \(F_{1}\), \(F_{2}\), \(F_{3}\) are the undetermined coefficients.