## 1 Introduction
### Negative Pressure Waves  
Negative pressure waves (NPW) serve as a basis for the inspection of pipelines. As explained in the paper of [PERSON] _et al._[11], when a leak happens along the pipeline, the fluid density of the leak point declines immediately due to the fluid medium losses and the pressure drops. Then the pressure wave source spreads out from the leak point to both ends of the pipeline. The analysis of the time-of-arrival of these wave fronts at known locations at both sides of the leak then allows for locating the leak with high accuracy. In the paper of [PERSON] _et al._[12], NPW are analyzed to locate leaks in pipelines under operation (with flow). The paper reports that the leak in the pipeline could be located with an accuracy comparable to the theoretical uncertainty related to the finite sampling rate that was used. In this paper, the authors will analyze if NPW can also be used to locate a leak in a closed volume (without flow), as will be the case in the embedded capillaries of the eSHM system. The static pressure inside the volume will continuously change as a consequence of the leak flow, which is not the case in the well known pipeline applications. First, simulations are discussed to understand the nature and amplitude of the negative pressure waves. Secondly, measurements on a test set-up are discussed and the crack localization capability on the basis of negative pressure waves in a closed volume is analyzed. By installing a pressure sensor at each side of the closed tube, one can measure the difference in time-of-arrival of the two waves. The time needed for the NPW to travel to the two sensors is easily deduced from Figure 2. For the latter part of this paper, the sensor with name 'S1' will always be referring to the sensor which is closest to the leak (it receives the signal first), while sensor 'S2' will be the sensor located far from the leak (receiving the signal later). \[\text{Sensor S1}\quad t_{1} = \frac{x_{1}}{c} \tag{1}\] \[\text{Sensor S2}\quad t_{2} = \frac{L-x_{1}}{c} \tag{2}\]  
Figure 1: Embedded capillaries (in red) of the effective Structural Health Monitoring System (eSHM) [7]. Because \(t_{1}\) and \(t_{2}\) are only known relatively, only combination of Equations (1) and (2) leads to a localization formula:  
\[x_{1}=\frac{L-c(t_{2}-t_{1})}{2}. \tag{3}\]  
The theoretical uncertainty on the crack location (\(\Delta x\)) is related to the sampling rate (\(f_{s}=\frac{1}{\Delta T_{s}}\)) of the data acquisition unit through  
\[\Delta x=\frac{c\Delta T_{s}}{2}=\frac{c}{2f_{s}}. \tag{4}\]  
Because of the high sonic speeds (for air at room temperature: \(c=\sqrt{kRT}=343\frac{m}{s}\)), it will be required to use a high sampling rate to reduce the uncertainty on the estimated location and to deduce the localization feasibility. A sampling rate of 204.8 kHz, for instance, results in a theoretical uncertainty on the location of the leak of 0.84 mm. For the ease of practical implementation, the capillaries are represented by 3/4\" tubes (diameter of 16.9 mm) and with a length of 462 mm. The circular leak had a diameter of 1 mm. The initial pressure level is set at approximately 20 kPa (absolute) while the outside environment is at ambient conditions (101.3 kPa). The temperatures are expected to be all in equilibrium at room temperature (293.15 K). The performed simulations and measurements discussed in the following parts of the paper will always be referring to this test set-up.