## 4 Results and Discussion
### Notch Detection  
The second studied case is the detection of a notch machined with depth e/2 (see Figure 6) in a plate that has the same physical properties and thickness as previously. The notch is located at x = 52 cm. The emitted signal is the same as previously. One can observe two wave packets on the spatial time representation on the signals plotted in Figure 11a. The selected wave packet for the time reversal is chosen again as the most energetic one. It corresponds to the reflection on the notch.  
Figure 10: Horizontal displacements of the direct and adjoint problems at (a) x = 56.12 cm; y = 0 cm; (b) x = 70.2 cm; y = 0 cm; and (c) x = 80 cm; y = 0 cm.  
Figure 9: (a) Topological image in the zone of interest of the simulated plate; (b) Normalized sum of the topological energy at each position along the plate.  
The topological energy is computed using the horizontal displacements \(\mathrm{u}\left(\overset{\rightarrow}{\mathbf{r}},\mathbf{t}\right)\) in the region of interest, and the results are plotted in Figure 12. The maximum of the topological energy occurs at \(\mathrm{x}=52.86\) cm. The error of the location is higher than the uncertainty due to the regular grid (\(\Delta\mathrm{x}_{\mathrm{grid}}=0.61\) cm), but the difference remains lower than the wavelength of the mode A1 at 320 kHz, \(\lambda=2.13\) cm. So, the result is considered acceptable.  
The second wave packet shown in Figure 11 can also be time reversed and studied. In this case, the maximum of the topological energy plotted in Figure 13 occurs at \(\mathrm{x}=70.71\) cm and corresponds to the edge of the plate. A local maximum is also present at \(\mathrm{x}=50.51\) cm and is linked to the notch.  
Figure 11: (**a**) Envelope of the signals recorded by each element of the linear array—(x,t) representation; (**b**) 2D-FFT of the first wave packet—(k,f) representation.  
Figure 12: (**a**) Topological image in the region of interest of the simulated plate. (**b**) Normalized sum of the topological energy at each position along the plate.  
One can thus assume that the second wave packet contains information about both defects (notch and edge). Further investigations have shown that two modes coexist: the S0 and A1 modes. The A1 mode comes from the initial A1 mode that reflects on the edge. The S0 mode comes from the mode conversion that takes place at the reflection at the notch.  
The results obtained with the A1 mode on the edge and on the notch are reliable. The A1 mode has a low attenuation along the propagation at the studied frequency \(\times\) thickness \(\mathrm{fe}=2.496\) MHz-mm (Figure 2). Another mode, S0 or A0, would have leaked in the surrounded fluid before any reflected signal returned on the transducers. Lowly attenuated modes are preferred for a long-range inspection in a single plate. Nevertheless, considering the final aim of the work, that being the research of a defect in the second plate in a set of two parallel and immersed plates, the highly attenuated modes may be considered in order to maximize the energy incident to the second plate. This assumption will be studied in a future work.  
The results presented in this paper are in good agreement with the results of [PERSON] et al., which is the most similar work found in the literature in a free isotropic plate [PERSON] _et al._ (2018) and a free anisotropic plate [PERSON] _et al._ (2018). The locations of the defects are found to be good and accurate. Nevertheless, it has to be pointed out that those studies are not properly comparable. The emitter's positioning differs, and since the plates in [PERSON]'s studies are free, the Lamb waves do not leak into the surrounding medium.