## 3 Results
### Damage in the Aluminium Liner  
The signals obtained from the evaluation points, shown in Figure 3, were analysed by means of a 2D Fast Fourier Transform (FFT). The resulting dispersion map is in the frequency-wavenumber domain. It reveals different modes propagating in the plate and reflecting from the crack in the aluminium liner. The frequency-wavenumber spectra were superimposed with the dispersion curves to identify different modes, and the results are shown in Figure 5. The negative wavenumbers represent incident waves, whereas positive wavenumbers represent reflected waves. Two cases are compared in this figure, for the excitation of Mode 2 at 475 kHz (see Figure 5a,b) and of Mode 5 at 475 kHz  
Figure 4: Dispersion curves of: (**a**) phase and (**b**) group velocities, calculated for the infinite aluminium-CFRP hybrid plate using SBFEM. The first ten guided wave modes are numbered in numerical order. The circles mark modes, as chosen for numerical modelling. Recreated from original data from [13]. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Mode** & **Frequency** & **Group Velocity** & **Phase Velocity** & **Wavelength** & **Number of Cycles** \\  & **kHz** & **m/s** & **m/s** & **mm** & \\ \hline
1 & 100 & 2033 & 1562 & 15.6 & 10 \\
2 & 475 & 2565 & 2507 & 5.3 & 20 \\
3 & 950 & 2864 & 2761 & 2.9 & 30 \\
4 & 400 & 4981 & 5839 & 14.6 & 20 \\
5 & 475 & 3175 & 7132 & 15 & 20 \\
6 & 700 & 4238 & 5431 & 7.8 & 20 \\
7 & 860 & 5440 & 10,005 & 11.6 & 30 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Modes used for numerical modelling. (see Figure 5c,d), where in the cases in-plane and out-of-plane components have been determined, respectively. These examples are shown for the evaluation points positioned at the interface between aluminium and CFRP. Even though the excitation was performed by applying the corresponding mode shape at the central frequency of the desired mode to be excited, the excitation of other modes could not be avoided. However, these modes have much smaller displacements when compared to the mode having been explicitly excited. In the case of Mode 2 excitation, Mode 4 was excited too, whereas excitation of Mode 5 led to the excitation of Modes 1, 2 and 4, as shown in Figure 5. Because of the same excitation frequency and the number of cycles in the pulse, but different mode shapes, different modes were excited. The results show that, regardless of how many modes were excited in the plate, only Modes 2 and 4 reflect from the crack in the frequency range used. Considering the shapes of Modes 2 and 5, shown in Figure 6a,b, respectively, it is observed that Mode 5 has a very small displacement in the aluminium at this frequency, compared to the CFRP part and Mode 2. This indicates that the energy (displacement) of Mode 5 at the frequency of 475 kHz is concentrated mainly in the CFRP layers, and thus the damage in the aluminium layer cannot be found. The frequencies for all modes chosen for modelling, including the detectability of all damages defined, are summarised in Table 4. As regards the crack in the aluminium liner only Modes 5 and 7 do not interact. Considering the mode shapes, Mode 5 has a very small displacement in the aluminium layer compared to the CFRP plies (see Figure 6b). The same holds for Mode 7, not shown here for brevity. In contrast, other modes that were investigated have comparable amplitudes in both aluminium and CFRP parts (see Figure 6a,c,d). In Figure 7a,b, the dispersion curves of the coupled  
Figure 5: Frequency–wavenumber spectra of wave modes propagating in the aluminium-CFRP plate and reflecting from the 1 mm crack in the aluminium: **(a)** for in-plane and **(b)** out-of-plane components while exciting Mode 2 at 475 kHz, and **(c)** for in-plane and **(d)** out-of-plane components while exciting Mode 5 at 475 kHz. 6 mm aluminium-CFRP plate are compared to the dispersion curves of a single 2 mm aluminium plate and a single 4 mm CFRP plate, respectively.