## 2 Zero-Group-Velocity Lamb Modes in an Aluminum Plate
The detected resonance would then correspond to the interference of the \(S_{2}\) and \(S_{1b}\) modes and/or the \(S_{-2}\) and \(S_{-1b}\) modes. The latter statement explains why the ZGV resonance is called \(S_{1}S_{2}\). Note that the term \"ZGV resonance\" is used to discuss the resonance effect associated with the existence of the ZGV Lamb mode but accounting also for the frequencies close to that of the ZGV mode where the Lamb waves have non-zero but really small group velocity. Although the branches of all the modes will be shifted in the positive or negative direction along the \(k_{i}\)-axis due to the attenuation and depending on their propagation direction, this shift will be easier to assess close to resonances than on propagative modes. Close to resonances, such as those at the ZGV points or else at \(k_{r}=0\) where there are the thickness mode longitudinal and shear resonances, the attenuation indeed induces a larger change on \(k_{i}\) than away from resonances where the group velocity is no longer negligible (note that, as discussed in [31], in the case of material absorption where the definition \(V_{g}=\partial\omega/\partial k_{r}\) of the group velocity leads to abnormal velocities, a possibility is \"to consider [instead] the velocity of energy transport which is defined as the ratio of energy flow to mechanical energy density\"). Therefore, although propagative modes do attenuate, good precision on the estimation of this attenuation will require a longer propagation distance where the group velocity is not closer to zero than in the vicinity of a resonance. This larger effect of the attenuation on \(k_{i}\) close to a resonance for a given mode is associated to the path of the Lamb mode branch that goes from a propagating branch in the plane \(k_{i}=0\) to a non-propagating branch, i.e., out of the plane \(k_{i}=0\), when the frequency is below the resonance (cut-off or ZGV) frequency (Figure 1b). For further details on the path of Lamb mode branches in absorbing plates, the readers are referred to the work of [PERSON] and [PERSON] [31]. We propose in the following to take advantage of this larger effect of attenuation on \(k_{i}\) in the vicinity of a ZGV resonance to estimate the attenuation. \begin{table}
\begin{tabular}{l c} \hline \hline
**Parameter** & **Value** \\ \hline Thickness (\(\mu\)m) \(2h\) & 71.5 \\ Density (kg/m\({}^{3}\)) \(\rho\) & 2700 \\ Longitudinal velocity (m/s) \(V_{L}\) & 6450 \\ Shear velocity (m/s) \(V_{T}\) & 3100 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Elastic properties for aluminum sample. Figure 1: Dispersion curves of the \(S\) Lamb modes in a 71.5 \(\mu\)m thick aluminum plate: (**a**) dispersion curves in the real plane \(k_{r}\)-\(f\) with a black arrow indicating the position of the \(S_{1}S_{2}\)-ZGV mode; (**b**) three-dimensional view (\(k_{r}\), \(k_{i}\), \(f\)) of the complex dispersion curves for the first symmetrical modes propagating in both directions with positive (solid lines) and negative (dashed lines) group velocity, with subscript “b” for backward mode: \(S_{0}\), \(S_{2}\), \(S_{1}\), and \(S_{4}\) modes are plotted in solid lines and \(S_{-0}\), \(S_{-2}\), \(S_{-1}\), and \(S_{-4}\) are plotted in dashed lines. The colors of \(S_{0}/S_{-0}\), \(S_{2}/S_{-2}\), \(S_{1}/S_{-1}\), and \(S_{4}/S_{-4}\) modes are blue, green, red, and black, respectively. Two black solid circles show the position of the ZGV point as indicated by the black arrows labeled “ZGV”. From the calculations of the dispersion curves in the case of the 71.5 um thick aluminum plate free of attenuation shown in Figure 1, it is expected to experimentally detect a ZGV resonance associated to the \(S_{1}S_{2}\)-ZGV mode with frequency \(f_{S_{1}S_{2}}=40.28\) MHz and wavenumber \(k_{S_{1}S_{2}}=21.93\) rad.mm\({}^{-1}\). The recovery of the complex wavenumber later on is therefore performed around those frequency and wavenumber.