## 4 Application of the SLaTCoW Method: Estimation of Ultrasonic Attenuation
The attenuation used to plot the theoretical curves shown in Figure 4 is 0 and 0.4 dB/cm/MHz for the cases without and with accounting for the attenuation, respectively, so that the reader can appreciate the necessity to account for the attenuation in order to retrieve the correct path of the complex branches. The value 0.4 dB/cm/MHz is demonstrated below to be the value allowing the best fit of the theoretical branch of the \(S_{2}\) mode to the experimental one. Note that the attenuation in the model is supposed to vary linearly with frequency. Although this is a strong assumption, the minimization process presented below is achieved on a small frequency range, i.e., from 40.5 to 41 MHz, where this assumption is acceptable. This small range of frequency is driven by the experimental frequency range in which the \(S_{1b}\) mode is observed (see Figure 3e). In Figure 4, the calculated dispersion curves and the results obtained from the measurements and analyzed by the SLaTCoW method are shown. Figure 4a,b shows the dispersion curves in the real \((k_{r},f)\) and imaginary \((k_{i},f)\) wavenumber-frequency planes for the aluminum plate, respectively. The experimental results for the \(S_{2}\) mode are depicted by red circles, and those for the \(S_{1b}\) mode by blue crosses. The dispersion curves calculated without attenuation are in black dash-dotted and doted lines for the \(S_{2}\) and \(S_{1b}\) modes, respectively. The dispersion curves calculated with attenuation are in red dashed and blue solid lines for the \(S_{2}\) and \(S_{1b}\) modes, respectively. Note that, for the sake of the presentation, the measured curve corresponding to the \(S_{1b}\) mode (\(k_{r}\) is negative) in Figure 4a is flipped at the position of the mode \(S_{-1b}\) (positive \(k_{r}\)). Note also that, when close to the position of the ZGV point, the signal-to-noise ratio (SNR) decreases due to the attenuation, which explains why there is no experimental points shown in Figure 4 close to the position of the ZGV point. In Figure 4b, it can be noticed that the recovered imaginary parts of the wavenumber for the \(S_{1b}\) mode are larger than those recovered for the \(S_{2}\) mode, which is consistent with the theory showing a slight asymmetry of the \(S_{1b}\) and \(S_{2}\) branches shifting along the \(k_{i}\)-axis. As the frequency decreases towards the ZGV frequency, the experimental \(S_{1b}\) branch deviates from the theoretical one, whereas both experimental and theoretical \(S_{2}\) branches remain coincident. The fact that a deviation from theory of experimental results is seen for the \(S_{1b}\) mode and not the \(S_{2}\) one could be associated to a poorer recovery of the complex wavenumber for the \(S_{1b}\) mode due to a lower amplitude of the measured \(S_{1b}\) mode, which can be seen in Figure 3e where the \(S_{2}\) mode has clearly more spectral energy than the \(S_{1b}\) mode. Note that the amplitude of the \(S_{1b}\) mode being lower than the amplitude of the \(S_{2}\) mode is actually expected because of the experimental monitoring of the gradients in normal displacement at the surface of the plate, which is theoretically smaller in amplitude for the \(S_{1b}\) mode than for the \(S_{2}\) mode, at least when thermo-elastically excited. We looked at each of the reported data points to check wether the \(S_{1b}\) mode complex wavenumber was picked within the noise floor or could be trusted, and we stopped the recovery where we were not confident. Therefore, although a biased recovery may explain the deviation, we are not excluding that other unknown factor(s) participate in it. Nevertheless, the experimental results show good agreement with the theoretical calculations when the attenuation is accounted for. In the real plane, no difference between the case with or without attenuation is noticed. However, the effect of the attenuation (repulsion of the branches) is clearly observed experimentally when the imaginary part of the wavenumber is recovered (see Figure 4b).