## 4 Results and Discussion
### Numerical Results: Source Localization Methods  
To introduce failure localization methods, we assume we know neither the failure position nor its occurrence time. In such a case, several simple methods to find the position are available with our current results:  
1. Based on the time difference of arrival (TDOA) of the primary wave between two positions in the main section, one downstream and one upstream from the failure. Source position is estimated with the following expression: \[x_{s}=\frac{(x_{i}+x_{j}-c_{f}\Delta t)}{2}\] (8) with \(x_{i}\) and \(x_{j}\) representing the axial positions of the two points from which pressure histories are extracted, \(\Delta t\) representing the TDOA between the two points, and \(c_{f}\) representing the primary wave speed. 2. Based on the TDOA of the primary wave between P1 and P2. Source position is given by the previous expression. 3. Based on the time delay between precursor waves and the primary wave at a single point (either a point in the main section, or P1, or P2). Assuming that precursor waves and the primary wave are created at the same point and at the same time, the source position is deduced from the following: \[d=(t_{f}-t_{s})\ \frac{c_{f}\ c_{s}}{c_{s}-c_{f}}\] (9) with \(d\) representing the distance between the source and the point from which the pressure signal is extracted, \(c_{f}\) representing the primary wave speed, \(c_{s}\) representing the precursor wave speed, \(t_{f}\) representing the arrival time of the primary wave, and \(t_{s}\) representing the arrival of the precursor waves. Figure 12: Positions of pressure history extraction positions around the outlet transition. These methods require either an assumption about wave velocities (we can use, for instance, the analytical Formulas (3) and (7) or the average values deduced from our numerical results), or an additional point to estimate them by TDOA. In the latter case, the first two methods would require two points on one side of the failure and a third point on the other side. The third method would require two points on the same side of the failure. Here, we applied these methods with assumed velocities (both with analytical ones and with the ones previously deduced from the numerical results). Hence, the first two methods require two sensors; they are referred to as \"multi-sensor methods\", and the third requires a single sensor only and therefore is called a \"single-sensor method\". Results obtained with the different methods are presented below. All positions are relative to the lower end of the rod. **Multi-sensor localization with points in the main section:**  
TDOA is estimated between each of the points shown in Figure 8, which are downstream from the failure, and the point C0, which is upstream from the failure (the failure is quite close from the main section lower end, so only one upstream point is considered), as shown in Figure 9. Then, the source position is estimated with Equation (8). The average value is eventually calculated. Results are shown in Table 3.  
Figure 13: Pressure histories at several positions around the outlet transition. **Multi-sensor localization with P1 and P2:**  
TDOA is estimated between P1 and P2, and the source position is deduced with Equation (8). The results are given in Table 4.  
**Single-sensor localization**  
Localization results with the pressure history at different points in the main section are given in Table 5. C1 is not used here because it is close to the failure, and therefore, the time delay between the precursor and primary waves is too short to yield an accurate result. **Single-sensor localization with points in the main section--results (actual source position: \(135.3\times 10^{-3}\) A.U.L.). **