## 4 Results and Discussion
### Modal Analysis of Numerical Results: Identification of the Excited Modes
#### 4.1.1 Influence of the Geometry on the Excited Waves  
In the three geometries (EPL3, EPL16 and EPL80), we simulate a chirp signal (Figure 3) on the surface of each sample as a point force. The numerical signals calculated from the close vicinity of the source to the end of the specimen allow us to identify the different propagating modes through the 2D Fourier Transform (2 DFT). The results are the dispersion curves, i.e., the wave number as a function of frequency. The simulated dispersion curves represented on the left hand side, e.g., Figure 9a,c,e, are obtained from the simulated signal when applying a point force source (chirp) on the surface of  
Figure 8: Experimental results vs. simulation results for particle velocity on the surface of each specimen, 21 mm away from the pencil lead break for EPL3 (**a**), EPL16 (**b**), and EPL80 (**c**).  
each sample. These are compared to the theoretical semi-analytical ones (on the right hand side, e.g., Figure 9b,d,f). The theoretical semi-analytical curves are calculated for infinite waveguides of arbitrary cross section, e.g., plate, beam and bar in the cases of the EPL80, EPL16 and EPL3 samples, respectively, with a dedicated resolution algorithm using the semi-analytical discontinuous Galerkin method [53]. All the semi-analytical curves displayed in Figure 9b,d,f consist of black dots. However, for Figure 9b, we superimposed on the black dots the pure modes of torsion, flexion and the longitudinal modes of a beam in three different colours, while for Figure 9f, we superimposed the Lamb modes corresponding to the analytical solutions of the Lamb modes of an infinite plate in red and blue.  
The results show that the sample geometry evidently affects the propagated signals. For the thin sample (EPL3), bar modes are excited (Figure 9a,b). A good agreement is observed between the simulated modes and the theoretical modes, with the exception of the high wavenumber domain, i.e., above 1500 rad/m for the EPL3 and EPL16 specimens, and even more pronounced above 1000 rad/m for EPL80. This is easy to explain, since the discretisation of the FEM simulation would never be sufficiently sharp for the wideband chirp used as a source. Such chirp excitation is chosen to generate a large number of modes. However, some modes are not excited (resp. not measurable) on particular frequency bands (for instance \(S_{0}\) at low frequencies for EPL80) or not excited at all (for instance, \(S_{1}\)). This is because the chirp applies a point force on the surface. This asymmetry of the applied force favours Antisymmetric/Flexural modes. In the case of EPL16 (Figure 9c), the 2 DFT is more complex, and the identification of modes is more complicated due to the intermediate nature of the EPL16 geometry, half way between plate and bar. In this case, it is not possible to superimpose the theoretical modes on Figure 9d.  
For the large one (EPL80), very numerous plate modes can theoretically be excited (Figure 9e,f), and are indeed observed. Nevertheless, one can observe that these waves are predominantly excited close to Lamb modes, which strengthens the hypothesis of the EPL80 specimen being large enough to mimic a quasi-infinite plate.  
The excited modes are different from a geometry to another even if the source is the same. This study shows that AE tests done on a classical sample, e.g., EPL16 (Figure 9c), in the laboratory cannot replace an industrial case from a perspective of geometry. This observation leads to the conclusion that AE classification results obtained during a laboratory test on a laboratory sample cannot be directly applied to a structure whose geometry is completely different. In the following part, we examine the effect of the characteristics of the source on the recorded signal.  
Figure 9: Cont.