## 4 Results and Discussion
### Numerical Results: Pressure History at Different Positions  
The signals used in this part are the average pressure over the cross-section at defined axial positions. As the pressure field shows in Figures 5-7, pressure waves can reasonably be considered as plane waves. It allows the use of the cross-section average pressure instead of the value at a point with specific angular and radial coordinates. Figure 4: Simulated pressure at P1 and P2 for different simulated failure lengths. Figure 5: Representation of the fluid pressure field in the main section at different time steps. Approximate distances are indicated to show the figure scale. For the sake of clarity, the structure is not shown. Figure 6: Representation of the fluid pressure field around the outlet transition at different time steps when the main pressure wavefront crosses the transition. For the sake of clarity, the structure is not shown. Figure 7: Representation of the fluid pressure field \(9\times 10^{-3}\) A.U.L. downstream from the center of the outlet transition, when the main wavefront reaches the section (at 0.88 ms). Figure 8 shows the pressure history at six different axial positions in the main channel. Figure 9 shows these positions. Each signal in Figure 8 can be divided in four parts:  
1. An empty part before signal arrival;
2. Low-amplitude waves (hardly visible on Figure 8; see Figure 10);
3. Main wave front followed by slow pressure decrease;
4. First reflection followed by additional resonance. The main wave front is assumed to be the primary wave (propagation of the pressure wave in water), and low-amplitude waves appearing before the main wavefront are assumed to be precursor waves. Precursor waves are hardly visible in Figure 8, but we can clearly see them with a magnification such as the example in Figure 10. They are also slightly noticeable on some frames in Figures 5 and 6. Figure 8: Simulated pressure at different positions in the main section. Figure 9: Positions of pressure history extraction points in the main section. To confirm that interpretation, we estimate the velocities of the observed low-amplitude waves and main wavefront to compare them with the theoretical values of precursor and primary waves. Velocities are estimated by the time difference of arrival (TDOA) between the different signals. TDOA is estimated for each couple of signals (such as the example in Figure 11); then, the average value is computed. We obtain the following average values (with a 95% confidence interval):  
* For the early low-amplitude waves (assumed precursor waves): \(3550\pm 132\) m \(\cdot\) s\({}^{-1}\);
* For the main wave front (assumed primary wave): \(1052\pm 9\) m \(\cdot\) s\({}^{-1}\). These values are quite close to the values given by the simplified theoretical model (Section 4.1) for precursor and primary waves. Therefore, our interpretation can be confirmed. The channel in the main section has a quasi-constant cross-section (there is only a small and gradual increase of 40% of the cross-section, upstream from the rod, due to a change in upper extension diameter). Therefore, in that area, the distortion of propagating waves is very low, so two pressure signals extracted at two different positions in that section look very similar. In Figure 8, if we look at the first peak only, all the signals look quite similar and differ almost exclusively by time shifts (after the first peak, more significant differences arise because of reflections on channel ends). One may especially notice that  
Figure 11: Example of TDOA estimation between two signals. Figure 10: Example of precursor waves: Magnification around the precursor waves on the pressure history at point C5 (\(535.8\times 10^{-3}\) A.U.L. from the failure). the amplitude of that first peak is nearly constant over the six positions. However, at the transitions between the main section and the inlet or outlet sections, there are strong and steep cross-section reductions.