## 3 Results and Discussion
### Results for Time-Frequency-Domain
#### 3.1.1 Water-Filled Pipe in Infinite Porous Medium  
Using the parameters listed in Tables 1 and 2, the expressions in (35) are numerically calculated to obtain roots that correspond to the propagating modes. The water-filled \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{1}{c}{\multirow{2}{*}{**Material**}} & \multicolumn{1}{c}{**Density**} & \multicolumn{1}{c}{**Longitudinal**} & \multirow{2}{*}{**Shear Velocity**} \\  & & **(kg/m\({}^{3}\))** & & **Velocity** \\  & & & **(m/s)** & \\ \hline \multirow{4}{*}{
\begin{tabular}{c} Porous \\ medium \\ \end{tabular} } & Pipe & 7800 & 4100 & 2100 \\  & solid skeleton & 2700 & 5370 & 3100 \\  & Pore fluid & 998 & 1483 & \\ \cline{1-1}  & Water & 998 & 1483 & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material properties and geometrical parameters for a case study of a liquid-filled pipe embedded in a porous medium. The pipe fluid and pore fluids are water, and the internal and external pipe radii are \(a\) = 14 mm and \(b\) = 16 mm. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Simulation Value** \\ \hline Pore curvature \(\alpha_{\infty}\) & 5.5 \\ Static permeability \(\kappa_{0}\) & \(1.0\times 10^{-12}\) m\({}^{2}\) \\ Viscous coefficient \(n\) & 0.001 kg/(s-m) \\ Bulk modulus of fluid \(k_{PL}\) & 2.19 GPa \\ Bulk modulus of solid skeleton \(K_{\text{PB}}\) & 33.70 GPa \\ Shear modulus of solid skeleton \(N\) & 20.86 GPa \\ Bulk modulus of solid matrix \(K_{\text{PS}}\) & 43.33 GPa \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Material properties of saturated porous medium at porosity \(\beta=0.1\). pipe's internal and external radii are \(a=14\) mm and \(b=16\) mm, respectively. The dispersion curves of the longitudinal modes in the water-filled pipe embedded in an infinite porous medium are presented in Figure 2.  
This study follows the mode classification of [PERSON] [41]. The longitudinal modes of the liquid-filled pipe's forward propagation in an infinite porous medium are denoted as L(0,_m_), where L represents the axisymmetric longitudinal modes with circumferential order \(n=0\) and mode order \(m=1\), 2, 3,.... The fundamental modes (i.e., those propagating at zero frequency) are given the value of \(m=1\), and the higher order modes are numbered consecutively. For clarity, in Figure 2, we compare the dispersion curves of the water-filled pipe embedded in elastic and porous media. Blue and red lines represent the propagating modes in the water-filled pipe embedded in infinite elastic and porous media, respectively. No porous medium is present outside the pipe, but it is used to validate the dispersion characteristics of guided waves in a water-filled pipe embedded in an infinite porous medium (\(\beta=0\)), and the choice of parameters. The longitudinal and shear velocities, and the density of the elastic medium, are taken to be 5370, 3100 m/s, and 2700 kg/m\({}^{3}\), respectively; for the water-filled pipe, the longitudinal and shear velocities and the density are 4100, 2100 m/s, and 7800 kg/m\({}^{3}\), respectively, and its internal and external radii are 14 mm and 16 mm, respectively. Figure 2 shows that in the frequency range \(f<0.5\) MHz, the longitudinal modes of the water-filled pipe in infinite porous medium comprise eight modes L(0,_m_) (\(m=1\), 2, 3,..., 8) and one \(\alpha\) mode. The bottom red line represents the \(\alpha\) mode in the liquid of the pipe, whose phase velocity is similar to the Scholte wave, and which increases with frequency and gradually tends to a value of 1500 m/s [12]. The remaining red lines depict the longitudinal guided waves of the water-filled pipe in elastic medium. Each exists in a wide frequency range. More specifically, the axisymmetric longitudinal guided waves exist in numerous modes. There is more than one mode at any frequency, and the characteristics of each mode are unique.