## 3 Results and Discussion
### Numerical Simulation Results
For the frequency-domain DAS, its ability to examine high-frequency phase variation information allows for higher imaging accuracy and resolution. The imaging results can accurately indicate the distribution of dual damage locations with an accuracy of less than 1 mm. The array performance index (API) is approximately twice that of the single damage case, and the variation with decreasing damage distance is not significant, indicating that when damages are in close proximity, the spatial distribution of imaging peaks for each damage remains stable. Moreover, the portion of the scale above \(-6\) dB is on the same order of magnitude as the wavelength, suggesting that its damage resolution capability is expected to approach the wavelength scale. Although the peak center intensity \(\tau\) difference increases as the damages come closer together, even when \(d/\lambda\) is as low as 0.79, \(\tau=-7.4\) dB remains above the preset resolution threshold of \(-6\) dB, indicating that under these conditions, the presence of dual damage can still be fully resolved, with resolution capability approaching the diffraction limit. This further corroborates the aforementioned conclusions. \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Damage Center** & \multicolumn{2}{c}{**Damage Localization**} & \multicolumn{2}{c}{**Peak-to-Center Intensity**} & \multicolumn{2}{c}{**Array Performance**} \\
**Distance (mm)** & **Deviation (mm)** & \multicolumn{2}{c}{**Difference \(\tau\) (dB)**} & \multicolumn{2}{c}{**Indicator (API)**} \\ \hline  & **f-DAS** & \begin{tabular}{c} **DORT-** \\ **MUSIC** \\ \end{tabular} & **f-DAS** & \begin{tabular}{c} **DORT-** \\ **MUSIC** \\ \end{tabular} & **f-DAS** & \begin{tabular}{c} **DORT-** \\ **MUSIC** \\ \end{tabular} \\ \hline  & \begin{tabular}{c} (0, 1) \\ (0, 1) \\ (0, 1) \\ (0, 1) \\ (0, 1) \\ (0, 1) \\ \end{tabular} & \(-02.0\) & \(-35.4\) & \(0.848\) & \(0.185\) \\  & \begin{tabular}{c} (0, 1) \\ (0, 0.4) \\ (0, 2, -0.6) \\ (0, 1, 1) \\ (0, 1, 0.8) \\ (0, 1) \\ (0, 1) \\ \end{tabular} & \(-17.1\) & \(-24.6\) & \(0.796\) & \(0.21\) \\  & \
\begin{tabular}{c} (0, 1) \\ (0, 2, -0.8) \\ (\(-0.6,-0.4) \\ \end{tabular} & \(-16.1\) & \(-22.8\) & \(0.85\) & \(0.28\) \\  &
\begin{tabular}{c} (0, 1) \\ (0, 1) \\ \end{tabular} & \((-0.6,-0.8)\) & \(-7.4\) & \(-9.2\) & \(0.83\) & \(0.34\) \\ \hline \end{tabular}
\end{table} 
Table 2: Comparison of numerical simulation imaging results for dual damage at different distances. The DORT-MUSIC method demonstrates a more prominent imaging resolution performance. The circumferential deviation of the damage center relative to the array is less than 0.6 mm, and the radial deviation is less than 1.8 mm, indicating that its circumferential positioning accuracy is superior to the radial. At the same time, the peak center intensity difference \(\tau\) increases as the damage approaches; even when the distance to the damage center in the calculation is 6 mm (0.79\(\lambda\), close to the diffraction limit) and the edges of the double damage are only 1 mm apart, the \(\tau\) indicator can still reach \(-\)9.2 dB. Combined with the two-dimensional imaging results, the double damage can still be clearly distinguished at this point. Additionally, the array performance index (API) does not show significant variation, with a maximum change of no more than 0.34, indicating that under the numerical calculation conditions, the peak of the damage imaging can clearly locate the damage area, and the imaging results maintain a high signal-to-noise ratio.