## 4 Results and Discussion  
### Theoretical Remarks  
In order to properly interpret the numerical results, a short introductory theoretical study is useful. Its objective is to have approximated values for the velocities of the different kinds of waves that can be observed in the system. On that purpose, previous works on \"water hammer\" phenomena are used. In the studied problem, pressure waves in fluid and elastic waves in the structure have to be considered as well as their mutual interaction. The excitation source we focus on is the local over-pressure induced by the failure. In the numerical model, there is no other source. The local over-pressure mainly propagates as pressure waves in the fluid. In addition to that, because pressurized fluid exerts loads on the structure walls, elastic waves are also created and propagate in the structure. With common materials (such as steel, zircaloy, and water), elastic waves in the structure travel faster than pressure waves in fluid, so the elastic wavefront goes ahead of the pressure wavefront. Then, elastic waves can partially radiate back to the fluid. As a consequence, small pressure fluctuations, induced by structural waves radiation, precede the main pressure wavefront. On the pressure history calculated or measured at a given point, small fluctuations appear therefore before the main peak corresponding to the pressure wavefront. Those fluctuations are called \"precursor waves\". We refer to the main pressure wavefront as the \"primary wave\" ([20]). For both types of waves, plane wave assumption can be considered for long wavelengths (i.e., low frequencies). This assumption simplifies not only analytical equations but also qualitative interpretation of numerical or experimental results, especially concerning elastic waves. In an elongated structure, such as a tube, longitudinal and shear waves combine together, depending on boundary conditions, and create guided waves. Different stress profiles over the cross-section can be associated with guided waves. Each possible stress profile is identified as a propagation mode. Guided waves are usually dispersive (i.e., propagation speed depends on the frequency), and each mode has a specific dispersion curve. With plane wave assumption, guided wave phenomena can be neglected, and only the fundamental wave types (longitudinal and transverse waves, which have constant velocities) can be considered. Moreover, according to [3], longitudinal waves effects are predominant in the low-frequency range. Therefore, \"precursor waves\" in the low-frequency range are assumed to be mainly related to axial waves (Structure axial waves induce fluid pressure fluctuations because of the Poisson effect, which creates transversal motion associated with tension-compression stresses. Without the Poisson effect, pressure fluctuations could be induced by friction effects only and therefore would be very low and almost invisible). Considering that pressure wave resulting from the failure is similar to the one resulting from water hammer, we can use some results of the four equations model of water hammer introduced in [21] and extended in [4] (among many others). This model is not presented here; we only introduce some of its results concerning wave speeds. Let us consider a pipe with an internal radius \(R\) and a wall thickness \(e\). The pipe is made of a material with density \(\rho_{s}\), Young's modulus \(E\), and Poisson's coefficient \(\
u\) and is filled with a fluid of density \(\rho_{f}\).