## 3 Results and Discussion
### Results for Time-Frequency-Domain
#### 3.1.2 Water-Filled Pipe in Finite Porous Medium  
In this numerical example, the surrounding medium is changed from an infinite to a finite porous medium, as this generates some different modal waves. Equation (34) is investigated numerically by considering that the porous medium layer is finite, and the thickness is 0.05 m, whereas the material properties of the pipe and the liquid are the same as those in Figure 2.  
The effect of the presence of the finite porous medium outside the water-filled pipe on the phase velocity with respect to frequency is plotted in Figure 3. We find that there are three kinds of guided waves in the water-filled pipe embedded in the finite porous medium. They are denoted as the \(\alpha\) mode, L(0,\(m\))\({}_{E}\), and L(0,\(m\))\({}_{P}\) (\(m\) = 1, 2,... ), respectively. Moreover, the dispersion curve possesses only the modes \(m\) = 1, 2, 3, 4, 5, 6, 7 and 8 at the maximum frequency of 0.5 MHz. An exception is L(0,1)\({}_{E}\), which exhibits a sudden increase in the phase velocity when the frequency rises from 0 to 0.04 MHz, and it subsequently decreases when the frequency increases from 0.04 to 0.5 MHz, while other axisymmetric longitudinal modes in the curves have corresponding cutoff frequencies and decrease with the frequency. In the low-frequency regime, the phase velocity decreases rapidly with an increase in the frequency, especially at the frequencies near the cutoff. Then, L(0,\(m\))\({}_{E}\), L(0,\(m\))\({}_{P}\) (\(m\) = 1, 2, 3,... 8), and the \(\alpha\) mode change slightly with an increase in the frequency at higher frequency bands, and the phase velocity of the L(0,\(m\))\({}_{E}\) modes tends toward the result shown in Figure 2. The velocity of the guided waves L(0,\(m\))\({}_{P}\) approaches that of the shear wave (2871 m/s) in the porous medium, and the velocity of L(0,\(m\))\({}_{E}\) and the \(\alpha\) mode tends toward the longitudinal wave velocity of the fluid in the pipe (1500 m/s). This interesting phenomenon of L(0,\(m\))\({}_{P}\) appearing in a water-filled pipe embedded in a porous medium is discussed further below. A comparison of the dispersion characteristics of the water-filled pipe embedded in infinite and finite porous media is shown in Figures 2 and 3, respectively. The dispersion curves in the infinite porous medium almost coincide with the dispersion curves of the pipe and liquid embedded in finite porous medium, i.e., at the same frequency, the phase velocity is almost equal. Therefore, the change in the medium outside the water-filled  
Figure 3: Dispersion curves of guided waves in water-filled pipe embedded in finite porous medium of porosity 0.1. The thickness of the finite porous medium is d = 0.05 m. Red, black, and blue lines represent three kinds of modes: \(\alpha\) mode, L(0,\(m\))\({}_{E}\), and L(0,\(m\))\({}_{P}\)(\(m\) = 1,2,3,... 8), respectively. pipe does not affect the propagation of the guided wave in the water-filled pipe. The main difference in the velocities in the two cases depends on whether the reflection and transmission of guided waves occurs in a porous medium, and can be used to verify different solutions of potential functions in the finite and the infinite porous medium. This is because, in the water-filled pipe embedded in infinite porous medium, the displacement and stress components at the interface are continuous, which is only consistent with the boundary conditions (22)-(29). Then, Equation (34) indicates that the sound field is not zero at infinity in the infinite porous medium when compared with (35). In general, the dispersion properties of the L(0,\(m\))\({}_{E}\) and \(\alpha\) modes in a water-filled pipe embedded in finite porous medium are consistent with those of the infinite porous medium (e.g., similar cutoff frequencies, phase velocities, and frequency ranges in which the L(0,\(m\))\({}_{E}\) and \(\alpha\) modes exist). These dispersion characteristics agree well with the solution of the displacement potential function in the derivation of the dispersion equation.