## 4 Results Analysis  
### Sensor Signals of Pipeline Structures with Dual Defects at the Same Circumferential Position  
In the positioning study of double defects relative to the axial position, the sensing signals extracted from the piezoelectric element sensors were averaged, to avoid the influence of the angle of the arrangement of the sensors. The averaged sensing signals were then subjected to a wavelet packet decomposition, to calculate the damage index values, which were organized into a two-dimensional damage index matrix. The damage index matrix had the axial distance between the second defect and the piezoelectric element as the x-axis, and the damage index value (referred to as \(l\)) as the y-axis, as shown in Figure 6.  
Figure 5: Defect layout diagram for dual defects in pipeline structures at different circumferential positions. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Case**} & **First Defect** & **Second Defect** & \multirow{2}{*}{**Spacing of**} & \multicolumn{2}{c}{**Double Defect**} \\  & **Axial** & **Axial** & & **Spacing of** & **Relative** \\  & **Distance/mm** & **Distance/mm** & **Damage/mm** & **Circumferential** \\  & & & & **Position Change/\({}^{\circ}\)** \\ \hline case 18 & 750 & 750 & 0 & 90 \\ case 19 & 750 & 900 & 150 & 90 \\ case 20 & 750 & 1050 & 300 & 90 \\ case 21 & 750 & 1200 & 450 & 90 \\ case 22 & 750 & 1350 & 600 & 90 \\ case 23 & 750 & 1500 & 750 & 90 \\ case 24 & 750 & 1650 & 900 & 90 \\ case 25 & 750 & 1800 & 1050 & 90 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Finite element model operating conditions for different circumferential positions. Figure 6 shows that the calculated I showed periodic variation, with an increase in PD for the working conditions, where the first defect was located at 750 mm from the piezoelectric sensor, and the second defect was located at a distance ranging from 750 mm to 1500 mm from the sensor, and both defects were located on the same bus line (case 10 to case 15). The reason for this phenomenon was that the two defects were located on the same bus line, and the axial distance difference between the defects caused a phase difference between the reflected wave 1 and reflected wave 2 of the second defect. By comparing the variation patterns of the damage index values of the double defects with the same circumferential position within the axial distance range of 750 mm to 1800 mm, with those of a single defect, we could locate the relative axial position of the defects. To investigate the effect of the phase difference on the reflected waves of the defects, we compared the time-domain signals of the sensing signals for case 10 with those of case 11 and case 12, and the results are shown in Figure 7.  
From Figure 7, the time at which each wave packet is received by the piezoelectric sensor for case 10, 11, and 12 is shown in Table 6. \begin{table}
\begin{tabular}{l c c c c} \hline
**Receiving Time/s** & **Refection** & **Reflection** & **Refection** & **Reflection** \\
**Echoes 1 of** & **Echoes 2 of** & **Echoes 1 of** & **Echoes 2 of** \\
**Case** & **First Defect/s** & **First Defect/s** & **Second Defect/s** & **Second Defect/s** \\ \hline case 10 & 4.07 \(\times\) 10\({}^{-4}\) & 6.37 \(\times\) 10\({}^{-4}\) & 4.07 \(\times\) 10\({}^{-4}\) & 6.37 \(\times\) 10\({}^{-4}\) \\ case 11 & 4.07 \(\times\) 10\({}^{-4}\) & 6.37 \(\times\) 10\({}^{-4}\) & 4.65 \(\times\) 10\({}^{-4}\) & 7.14 \(\times\) 10\({}^{-4}\) \\ case 12 & 4.07 \(\times\) 10\({}^{-4}\) & 6.37 \(\times\) 10\({}^{-4}\) & 5.22 \(\times\) 10\({}^{-4}\) & 7.72 \(\times\) 10\({}^{-4}\) \\ \hline \end{tabular}
\end{table} 
Table 6: Comparison of the times at which the piezoelectric element sensors received the wave packets 5. Figure 6: Two-dimensional damage index matrix for pipeline structures with dual defects at the same circumferential position 4.