## 3 Numerical Validations of 3D Embedded Flaws Simulations
### Shear Waves
#### 3.2.2 Contact Pulse Echo Mode: S45\({}^{\circ}\) Waves--Vertical Flaw  
The next numerical comparison case involves the inspection of a vertical rectangular crack inspected using SV45 waves in pulse echo mode at 2.25 MHz (Figure 8a). The used probe emitting at 2.25 MHz is a contact circular planar probe of 6.35 mm diameter. The first numerical comparison consists in studying the effect of the flaw extension on maximal flaw echo amplitude for the 5 mm high crack (Figure 8b). For small extensions (\(<\)4 mm), some amplitude oscillations occur, probably due to interferences between waves scattered from both the edges and the corners. Important differences are observed between the FEM and PTD models in terms of maximal echo amplitudes for all extensions. It is unexpected that differences are obtained even for large extensions (25 mm \(\sim\) beam width) in the case of a 5 mm high crack corresponding to a large dimensionless factor ka \(\sim\) 11.25. In order to analyze the previous differences, the Ascans simulated by 3D FEM and 3D PTD are plotted in Figure 9 for the 40 mm extent (the largest extension in Figure 8b) and 5 mm high rectangular crack both at 2.25 MHz (corresponding to the case studied in previous Figure 8) and also at 5 MHz. At 5 MHz, the Ascan simulated by 2 D FEM CIVA ATHENA is also provided. The first echo chronologically observed in Figure 9a,b is obviously the top edge diffracted wave. Echoes generated by bulk wave diffraction from the bottom edge (after 40.5 \(\upmu\)s) and by later waves can be better distinguished at the higher frequency (5 MHz). A head wave arrives at almost the same time as a wave diffracted from the bottom edge: it corresponds to the diffraction at the top edge of the incident S wave path into a P creeping head wave, which is then rediffracted into a bulk S wave at the bottom tip (see Figure 9c), and its time of flight, which can be theoretically calculated by ray theory, is close to that of the bottom edge diffraction (as explained later using Table 1). Chronologically, the first wave arriving after the bottom edge diffracted wave is a bulk diffraction of a Rayleigh wave. When the beam hits the top tip, a Rayleigh wave (R\({}_{1}\)) is generated. It propagates along each crack face towards the opposite tip. Upon reaching the bottom tip, R\({}_{1}\) sheds the bulk S\({}_{5}\) wave. The path of this Rayleigh wave then diffracted in the bulk corresponds to that of the so-called S\({}_{5}\) wave, which is due to a secondary diffraction (see its path drawn in Figure 6 of Ref. [26]). Figure 8: (a) Inspection of rectangular crack using SV45 waves in Pulse Echo mode at 2.25 MHz (circular planar probe of 6.35 mm diameter), (b) of 5 mm height and varying extension and (c) of a 34 mm extension and varying height. When considering the time of flight of the top edge diffracted wave as the time origin, the time of flight of the different waves can be theoretically calculated using ray theory and compared to that obtained after 3D FEM simulation (see Table 1). The comparison of the times of flight shows that the second echo chronologically observed corresponds to the bottom edge diffracted wave interfering slightly with the head wave. Such a head wave is of lower amplitude since it decreases quickly during its propagation along the flaw surface and since the incidence is not critical (in Figure 9c, \(\theta_{\text{inc}}\) = 45\({}^{\circ}\) whereas \(\theta_{\text{c}}\) = 33\({}^{\circ}\) for steel). The third echo corresponds to the secondary Rayleigh wave diffraction \(\text{S}_{\text{S}}\) PTD predicts correctly the amplitudes of the primary bulk waves diffracted by the two edges but the PTD model does not simulate the secondary Rayleigh wave diffraction \(\text{S}_{\text{S}}\). It is this latter wave which leads to the more important amplitude in the 3D FEM Ascan.