## 4 Results and Discussion
### Edge Plate Detection  
Leaky Lamb waves reflect on the edge of the plate and propagate back toward the transducer. The experimental temporal signals are extracted from each element of the linear array, and each line in Figure 8a is the envelope of the signal acquired on an element. The signals are then processed with 2D Fast Fourier Transform (2D-FFT) to transform the space-time domain \((\mathrm{x},\mathrm{t})\) into a wavenumber-frequency \((\mathrm{k},\mathrm{f})\) representation as shown in Figure 8b. This transformation allows measuring the real part of the wavenumber, the leaky attenuation, and the phase velocity of the Lamb wave. Superimposing the theoretical dispersion curves allows verifying the nature of the Lamb wave mode involved in the propagation. The emitted wave packet propagates along the \(x\)-axis and is reflected by the right edge of the plate. The wavenumber in Figure 8b is negative because of the propagation of the reflected wave along the decreasing \(x\)-axis.  
Figure 7: Sketch of the simulation under COMSOL Multiphysics®.  
One can verify that the reflected mode corresponds to the initial propagative A1 mode: no conversion has occurred.  
The selected wave packet for following the time reversal process is chosen as the more energetic one in the negative wavenumbers that corresponds to a reflective wave. Considering the mode A1 at this product frequency \(\times\) thickness, the horizontal displacement has more amplitude than the vertical displacements. Thus, the topological energy is calculated using \(\mathrm{u}\left(\overset{-}{r},t\right)\) at each point of the plate.  
In this particular case, the computation of the source in the adjoint problem is different as described previously in Section 3. One can assume that in the reference medium in the direct problem, the plate is infinite, and there are no reflected waves (\(\mathrm{u}_{\mathrm{ref}}=0\)). Thus, the calculation of the source of the adjoint problem is straightforward: the wave packet delimited by the white lines in Figure 8a corresponds to the signal that will be time reversed and reinjected in the adjoint problem. The topological energy is then computed using the horizontal displacements \(\mathrm{u}\left(\overset{-}{r},t\right)\) in the zone of interest (represented in Figure 6), and the result is plotted in Figure 9. One can note that the distribution of the maximum of topological energy in the thickness of the plate follows the displacement profile of the considered mode.  
The maximum of the topological energy occurs at \(\mathrm{x}=70.2\) cm. The results are in good agreement with the real location of the edge (\(\mathrm{x}=70\) cm). Indeed, the difference with the real location of the edge is under the uncertainty based on the regular grid used in the postprocessing (\(\Delta\mathrm{x}_{\mathrm{grid}}=0.61\) cm). At this location, the fields in the direct and the adjoint problems cross and lead to the highest value of the topological energy (Figure 10b). On the contrary, the correlation between the fields before and after the edge is much lower (examples on Figure 10a,c).  
Figure 8: (**a**) Envelope of the experimental signals recorded by each element of the linear array—(x,t) representation; (**b**) 2D-FFT of the signals marked by the white lines—(k,f) representation.  
One can notice the decrease in the amplitude of the direct problem along the propagation and the increase in the amplitude of the adjoint problem due to the leaky attenuation. To improve the contrast of the topological image, an idea would be to compensate the attenuation knowing the attenuation coefficient of the mode involved. Indeed, such a compensation can only work in a purely monomodal propagation. The compensation cannot be done if at least two modes propagate.