## 2 Zero-Group-Velocity Lamb Modes in an Aluminum Plate  
Generally, the propagation of Lamb waves is dispersive. Thus, many studies, both experimental and theoretical/numerical, focused on the dispersion curves of such guided waves [28; 29; 30; 31; 32; 33; 34; 35]. The dispersion curves of the symmetric (\(S\)) Lamb modes can be plotted after numerically solving the well-known Rayleigh-Lamb equation that reads [36; 37]  
\[\frac{\tan(ph)}{\tan(qh)}+\frac{(q^{2}-k^{2})^{2}}{4k^{2}pq}=0, \tag{1}\]  
for the \(S\) modes, where \(h\) stands for half of the thickness of the plate, \(p^{2}=\omega^{2}(1/V_{L}^{2}-1/V^{2})\), and \(q^{2}=\omega^{2}(1/V_{T}^{2}-1/V^{2})\). In the latter expressions, \(\omega\) denotes the angular frequency and \(V_{L}\) and \(V_{T}\) stand for the longitudinal and shear acoustic waves velocities of the material, respectively. Note that \(k=k_{r}+ik_{i}\) is the complex wavenumber with \(k_{r}\) and \(k_{i}\) the real and imaginary parts, respectively. The dispersion curves of the \(S\) Lamb modes are plotted in Figure 1, for a 71.5 \(\upmu\)m thick aluminum plate with elastic properties presented in Table 1 and neglecting the attenuation. The thickness is chosen to match that of the plate used in the experiments (see Section 3). In Figure 1a, the dispersion curves are plotted in the real plane \(k_{r}\)-\(f\), for \(k_{r}\geq 0\), where parts of the branches where \(k_{i}\
eq 0\) have been omitted for clarity. The complete view of the dispersion curves is plotted in three dimensions in the complex wavenumber-frequency space (\(k_{r}\), \(k_{i}\), \(f\)) in Figure 1b. The symmetrical modes propagating in both directions with positive (solid lines) and negative (dashed lines) group velocity, with subscript \"b\" for backward mode (this is the case where phase and group velocities are opposite in signs), are plotted for \((k_{r},k_{i})\in\mathbb{R}^{2}\): \(S_{0}\), \(S_{2}\), \(S_{1}\), and \(S_{4}\) modes are in solid lines, whereas \(S_{-0}\), \(S_{-2}\), \(S_{-1}\), and \(S_{-4}\) are in dashed lines. The modes are here numbered according to the number of nodes present in the thickness of the plate [37]. The arrows indicate the position of the first ZGV mode, i.e., the \(S_{1}S_{2}\)-ZGV mode, on the dispersion curves. Indeed, at particular positions on the dispersion curves of the modes, the group velocity \(V_{g}=\partial\omega/\partial k_{r}\) vanishes, while the phase velocity \(V_{ph}=\omega/k_{r}\) remains finite. In the case where the dissipation due to the material or to the waveguide boundaries is neglected, it is known that there is a rigorous identity between the group velocity and the velocity of energy transport under very general conditions [38]. Therefore, for the ZGV points where the group velocity vanishes while the phase velocity remains finite, the resonances remain locally at the position of the excitation. In Figure 1b, it is clear that a ZGV mode results from the interference of guided modes coexisting for a unique couple of frequency/wavenumber. In the specific case where the attenuation is neglected, it is seen that the four modes, \(S_{2}\) and \(S_{-1b}\) as well as \(S_{-2}\) and \(S_{1b}\), are coincident two by two at ZGV points (black points in Figure 1b). At these points, the four modes have exactly the same frequency \(f_{ZGV}\) and the same wavelength, i.e., same absolute value \(|k_{r}^{ZGV}|\) of the real part of the wavenumber: \(k_{r}^{ZGV}=k_{S_{2}}(f_{ZGV})=-k_{S_{1b}}(f_{ZGV})=-k_{S_{-2}}(f_{ZGV})=k_{S_ {-1b}}(f_{ZGV})\). In the case where the material presents some attenuation (always the case in reality), the four branches have non-zero positive (\(S_{2}\), \(S_{1b}\)) and negative (\(S_{-2}\), \(S_{-1b}\)) imaginary parts \(k_{i}\) for all \(k_{r}\) and \(f\). Therefore, the branches of the modes \(S_{2}\) and \(S_{-1b}\), on the one hand, and of the modes \(S_{-2}\) and \(S_{1b}\), on the other hand, no longer coincide at the ZGV points, which actually no longer exist.