## 4 Results Analysis
### Sensor Signals of Pipeline Structures with Dual Defects at the Same Circumferential Position
To validate the accuracy of the above wave packet separation theory, an analysis of the propagation mechanism of guided waves in pipe structures, based on the propagation mechanism of piezoelectric guided waves, was conducted, as shown in Figure 8. Furthermore, an analysis of the time-domain signal of the sensing signal at case 17 (the first defect was located 750 mm away from the piezoelectric element sensor, and the second defect was located 1800 mm away from the sensor) was performed, and is shown in Figure 9.  
From Figure 9, it can be observed that the time of the initial wave extracted at the piezoelectric element sensor was \(\mathrm{t}=1.6\times 10^{-4}\) s, the time of the boundary reflection wave 3 was \(\mathrm{t}=11.5\times 10^{-4}\) s, and the time of the first reflection of the second defect was \(\mathrm{t}=8.9\times 10^{-4}\) s. Based on the pulse-echo method and the time-of-flight method, the propagation velocity and the axial position of the simulated wave were calculated, using finite element simulations. The calculated propagation velocity from the finite element simulation showed a low level of error, compared to the theoretical value for the axial distance of the damage, indicating the accuracy of the finite element simulation. The specific values are shown in Table 7.  
Figure 8: The propagation mechanism of guided waves in pipeline structures with dual defects. Sensing Signals of Dual Defects in Pipeline Structures Can Vary with Different Circumferential Positions  
To avoid the influence of the placement angle of the piezoelectric element sensor on the relative axial distance positioning study of pipeline structure defects, the sensing signal extracted at the piezoelectric element sensor was averaged. The averaged sensing signal was subjected to wavelet packet decomposition to calculate the damage index value, which was then used to form a two-dimensional damage index matrix. The damage index matrix had the axial distance between the second defect and the piezoelectric element as the horizontal axis, and the damage index value as the vertical axis, as shown in Figure 10.  
It can be observed from Figure 10 that for case 18 to case 22 (where the second defect was located between 750 mm and 1350 mm away from the piezoelectric element sensor, with a relative circumferential position of 90\({}^{\circ}\) between the two defects), the damage index value showed a trend of first increasing and then decreasing, with the increase in the axial distance between the second defect and the piezoelectric element sensor. This trend was caused not only by the increase in the relative axial distance between the two defects, but also by the phase difference in the damage reflection echo caused by the change in the relative circumferential position of the damages. \begin{table}
\begin{tabular}{c c c c c c} \hline \multirow{2}{*}{**Case**} & \multirow{2}{*}{\begin{tabular}{c} **Time of** \\ **Dissemination/s** \\ \end{tabular} } & \begin{tabular}{c} **Finite Element** \\ **Simulation** \\ **Propagation** \\ **Velocity/m** \(\cdot\)s\({}^{-1}\) \\ \end{tabular} & \
\begin{tabular}{c} **The Second Defect** \\ **Theoretical Damage** \\ **Distance/m** \\ \end{tabular} &
\begin{tabular}{c} **The Second Defect** \\ **Finite Element** \\ **Simulation Damage** \\ **Distance/m** \\ \end{tabular} & **Error/\%** \\ \hline case 17 & 7.3 \(\times\)\(10^{-4}\) & 5191 & 1.8 & 1.895 & 4 \\ \hline \end{tabular}
\end{table} 
Table 7: The comparison between the finite element simulation, and the theoretical calculation, for the second defect. Figure 9: Sensor signal diagram.