## 4 Results  
### Comparison between Experimental Results and the Results Obtained by SAFE Based on [PERSON] et al. Comparison between Experimental Results and the Results Obtained by SAFE Based on [PERSON] et al. [25]. In the following section the comparison between experimental and finite element method results when a tone burst excitation is applied and the response captured in the scan line is presented and described in detail. Two tone burst excitations are applied in the sensor position with a central excitation frequency of 80 KHz in longitudinal and transversal direction. In Figures 10a and 11a, the displacements vs. the time measured over a scan line point are shown in the Figures 10b and 11b the displacements, over the scan line, at one specific time are presented. In the cited figures, the same measure in the time and space domain are indicated by using dots in different colors. Moreover, in Figures 10c and 11c the results captured in 150 points of the scan line during a time interval were presented after performing a double Fourier transform, superposed with the dispersion curves of the analyzed waveguide. Notice that the highest energy of the captured signal is concentrated on the excitation frequency in correspondence with the regions where the dispersion curves pass. Finally, Figures 10d-f and 11d-f show the displacement fields obtained by the finite element simulation where the time is integrated using an explicit scheme of integration performed using Ls-Dyna [42] applying the same excitations. Studies using this methodology applied to the investigation of wave propagation can be found in [43; 44]. This simulation illustrates the test that was performed where the geometry was discretized with cubic elements of a size of 0.5 [mm]. In the experimental tests and by using the PZT transducers polarized in two directions, it was possible to obtain the four fundamental modes in the dispersion curves within the range of frequencies of interest. Tests were performed from 10 to 100 [kHz] with a step of 10 [kHz]. The frequency and wavenumber parameters of each mode were extracted at every test from visual inspection of the graphs, similar to Figure 10a,b and Figure 11a,b. The data collected in the tests were organized in frequency versus wave number, thus, generating the dispersion curves experimentally presented and described with dots in Figure 12.  
The circle dots in Figure 12 indicate the pairs f and \(k=1/\lambda\) found in each test performed. Only one test of each configuration was conducted. However, despite this limitation in the experimental verification, a good agreement between the experimental and numerical dispersion curves performed with SAFE [28] method is presented in the same figure. The highest distortions are founded in the bending modes. Figure 10: A tone burst with the main frequency of 80 KHz in longitudinal direction. Experimental results: (**a**) displacements measured in a point over the scan line where T is the period of each wave propagation mode. (**b**) Displacements over the scan line at a specific time. Numerical results: (**c**) amplitude displacement in the frequency vs. wave number dominium (with a black line indicating the dispersion curves). (**d**) Bar analyzed with the map of displacement on it, at a specific time. (**e**) Transversal section with the spatial distribution of two wave modes. (**f**) Tridimensional mode views. Figure 11: A tone burst with the main frequency of 80 KHz in transversal direction. Experimental results: (**a**) Displacements measured in a point over the scan line where T is the period of each wave propagation mode. (**b**) displacement over the scan line at a specific time. Numerical results: (**c**) amplitude displacement in the frequency vs. wave number domininum (with a black line indicating the dispersion curves). (**d**) Bar analyzed with the map of displacement on it, in a specific time. (**e**) Transversal section with the spatial distribution of two wave modes. (**f**) Tridimensional mode views.