## 1 Introduction: Crack Localization Based on Pressure Wave Propagation
### Nature of Pressure Wave Propagation  
The exact nature of the propagating waves does not seem to be completely identified in the available literature. The pressure difference measured by the sensors can be of a hydrodynamic,  
Figure 1: **(Left)** Cogwheel with integrated capillaries. **(Right)** Schematics of the effective SHM (eSHM) system. acoustic, or hydro-acoustic nature depending on the moment of acquisition and measurement conditions. The suddenness of the opening, as well as the pressure difference between atmospheric conditions (in the crack) and low-pressure conditions (in the capillary) seem to be determinant. Depending on their nature, these pressure waves could be defined as acoustic pressure waves (acoustic phenomenon) [11] or shock waves (fluid dynamics phenomenon) [12]. Acoustic pressure waves (also called negative pressure waves in some references) have been deeply studied in the past in both gaseous and liquid media to retrieve the location of cracks in large-scale pipelines [13; 14]. Pipelines are one of the most secure methods to deliver oil and gas. The potential risk of leaks in the pipeline should be lowered as much as possible for environmental and safety reasons. In some particular cases, the losses can even become detrimental for the performance of the installation. Here again, the localization principle relies on the constant propagation speed characteristic of the propagating wave. The only difference w.r.t. the eSHM-system is the scale of the pipelines in terms of length and diameter. Analytically, the acoustic pressure waves are described by the isentropic linearized Navier-Stokes equations. These waves are associated with small pressure changes and result in a zero net mass flow. Their propagation speed corresponds to the local speed of sound, and after their passage, the state of the medium stays unchanged. This process is adiabatic, reversible, and keeps the entropy constant. The acoustic linearized shock tube model [15] can be used as a reference to described the phenomenon. On the other hand, considering different initial conditions, shock waves can travel through the capillary at a speed higher than the local speed of sound in the medium. These waves are defined as a sudden irreversible discontinuity in pressure, density, and temperature of the fluid and are associated with violent pressure changes. Shock waves give a change in entropy and are described by the non-linear Navier-Stokes equations. Their passage is non-adiabatic and abruptly changes the state of the medium in an irreversible way. The main difference with the acoustic pressure waves is their non-constant propagation speed and the net mass flow behind the shock after its passage. Propagating shock waves are not a common topic in the literature. In the case of detached bowed and oblique shocks around moving bodies, as well as in converging-diverging nozzles, the shock is considered in a fixed position and the fluid is moving. However, when a balloon is bursting, an atomic bomb is exploding, or in turbocharged internal combustion engines, the shock wave propagates through a stationary medium. The theoretical shock tube model [16] used in the field of gas dynamics can be compared to the phenomenon happening in the capillaries. This model describes the phenomenon happening in a tube divided into two different pressure zones (high-pressure zone and low-pressure zone), just after the sudden burst of a thin diaphragm. The burst creates a shock wave propagating further through the lower pressure zone of the duct. The removal of the diaphragm can be compared with the moment where the crack reaches the capillary. In his work, [PERSON] [17] presented the analysis of this shock wave propagation through small-scale ducts. The presence of shock waves would complicate the localization technique since their propagation speed theoretically depends on the amplitude, in contrast to the acoustic pressure waves where the propagation speed is constant.