## 4 Results and Discussion
### Detailed Analysis of Leak Position L2  
Figure 6 presents the pressure measurements for leak location L2 obtained from sensors S1 (full line) and S2 (dashed line). The plot is focused at the moment of time-of-arrival of the NPW at the two sensors and is zeroed at the initial pressure level. Vertical line 0 corresponds to the theoretical moment of opening the leak. Vertical lines 1 and 2 then correspond to the moment of time-of-arrival at sensors S1 and S2 of which the times are given below the figure.  
Figure 4: Drawing of the 3/4” tube of the test set-up (dimensions in mm).  
Figure 5: Graphical presentation of the test set-up.  
It must be noted that Figure 6 shows the presence of pressure steps, rather than a continuous increase of the pressure. This is due to the reflections of the NPW at the hard walls of the tube (where the sensors are located). Vertical line 4, for instance, corresponds to the moment of arrival at sensor S1 of the NPW that first went to the side of sensor S2 (and was measured there at the moment corresponding to vertical line 2) and then reflected back to the side of sensor S1.  
The static pressure change can thus be seen as a succession of exponential pressure steps. The measurement of sensor S1 clearly shows a first exponential pressure rise between vertical lines 1 and 4. A second, less clear exponential pressure step starts at vertical line 4 and ends at vertical line 5 where a third exponential pressure step starts. That exponential behavior is the reason why the breakpoint at vertical line 4 is more clear than at vertical line 5: the more the exponential has flattened, the more the slope of the two successive exponentials differ and the clearer the breakpoint becomes. The recognition of a breakpoint in the pressure steps also allows location of the leak, just by using one pressure sensor [15].  
To evaluate the amplitude of the NPW and to compare with the theoretical prediction (Equation (6)), one must must take into account that the sensor measures twice the amplitude of the NPW because of the hard wall at which the wave reflects and the pressure adds up (see section 2). Together with this effect, it is important to note that, due to the contribution of a direct wave and the contribution of a reflected wave that arrives a bit later, multiple NPW might contribute to the pressure rise we are investigating. The pressure rise on sensor S2 between vertical line 2 and 6 flattens out when approaching vertical line 6. We can thus assume that the amplitude of that pressure step can be approximated to the value at vertical line 6, i.e. 530 Pa. Breakpoint 3, however, corresponds to the moment that a second, reflected, wave arrives at the location of sensor S2. The pressure between vertical lines 2 and 6 thus corresponds to the addition of two NPW with amplitude 530 Pa/2/2 = 132.5 Pa. This is in rather good correspondence with the theoretical prediction 106 Pa (Equation (6)) and the simulations 100 Pa (see Section 2).  
Figure 6: Time-of-arrival of the (reflections of) negative pressure waves.  
Given the moments of time-of-arrival of the direct waves below the figure, it is possible to estimate the crack location on basis of Equation (3).  
\[x_{1}=\frac{0.462-343\times 0.00103}{2}\ \text{m}=54.4\pm 0.84\ \text{mm}. \tag{7}\]  
This estimate has to be compared to the real location of the leak, which is 59 mm. We can conclude that the estimation of the leak position is rather good but still larger than the theoretical minimum (related to the finite sampling rate (204.8 kHz)). The remaining error is due to noise present on the measurements, which makes it difficult to select the exact moment of arrival of the NPW.