## 3 Results and Discussion
### Results for Time-Frequency-Domain
#### 3.1.3 Influence of Porosity  
As discussed in the acoustics of the seabed as a poroelastic medium [44], porosity is a key parameter that defines the relative volume of the fluid and solid constituents. To investigate the influence of the porosity on the dispersion curves, a concrete example is presented with the permeability \(k_{0}\) = 10\({}^{-12}\) m\({}^{2}\), where the calculated dispersion curves for \(\beta\) = 0.1, 0.2, and 0.3 are presented in Figure 4. The porosity has no significant effect on the phase velocities' dispersion curve in a liquid-filled pipe embedded in an infinite porous medium, as shown in Figure 4a. When the porosity is 0.2-0.3, the phase velocity is almost the same as that of \(\beta\) = 0.1. In Figure 4b, the effect of porosity on the L(0,\(m\))\({}_{E}\) and \(\alpha\) modes is not evident, and the phase velocity is independent of this effect. The porosity has a significant effect on the phase velocity of the L(0,1)\({}_{P}\) mode in a finite porous medium. As the porosity increases, the cutoff frequency decreases, and the dispersion curves of the L(0,1)\({}_{P}\) mode slope downward, approaching the shear wave velocity in a porous medium at different porosities. A higher porosity corresponds to a lower phase velocity at the same frequency. The reason behind this phenomenon is explained below.  
The potential functions (8)-(10) and boundary conditions (30)-(32) can be used to explain why the porosity has a major impact on the mode L(0,1)\({}_{P}\) in a finite porous medium. Equations (8)-(10) show that the guided modes in a liquid-filled pipe embedded in a finite  
Figure 4: Dispersion curves of guided waves in a water-filled pipe embedded in (**a**) infinite porous medium and (**b**) finite porous medium. The thickness of the finite porous medium is \(d\) = 0.05 m. Black, red, and blue lines represent the porosity \(\beta\) = 0.1, 0.2, and 0.3, respectively.  
porous medium can be viewed as a mode superposition of the guided modes propagating in a liquid-filled pipe embedded in an infinite porous medium and the reflection caused by the surrounding infinite porous medium. That is, the first term of (8)-(10) is infinite when \(r=0\), which represents the direct fields generated by the excitation source; the second term of (8)-(10) is a finite value when \(r=0\), which represents the wave propagating from infinity to the origin of the coordinates, and reflects the field generated by the existence of the interface.  
For a liquid-filled pipe embedded in an infinite porous medium, the second term of (8)-(10) is approximately zero, which means that the effect of reflection is weak. For a liquid-filled pipe embedded in a finite porous medium, the contribution of the second term of (8)-(10) cannot be ignored compared with that of the first term, because a strong reflection field applies. The added boundary conditions state that all stress components are zero on the outer surfaces of the finite porous medium compared with the infinite porous medium, as appears in the matrix elements \(m_{ij}\)(\(i=1\),\(2\),...,\(11\) and \(j=9\),\(10\),\(11\)). This helps to explain the changes in behavior caused by the surrounding infinite porous medium. Therefore, the porosity in the medium outside the liquid-filled pipe can be analyzed by the change in the phase velocity, which provides a certain theoretical support for nondestructive testing. The porosity used in this study may lead to a new aspect of the optimization of the theory of pipeline leak detection.