## 4 Application of the SLaTCoW Method: Estimation of Ultrasonic Attenuation
The error bars of the experimental measurements in the real wavenumber plane (red circle and blue cross in Figure 4a) stand for \(2\pi/L\), with \(L=4\) mm the scanning distance along the \(x\)-axis. The extra information given by \(k_{i}\) is useful for assessing the ultrasonic attenuation of the tested sample. To do so, a minimization process, with the attenuation as a parameter and allowing the best fit (least mean square) of the experimental dispersion curves (in the complex plane) to the theoretical dispersion curves accounting for attenuation, is performed. This process is illustrated in Figure 5. The experimental results shown in Figure 4b in the plane (\(k_{i},f\)) are reported in Figure 5a. The color map shown in Figure 5a stands for the change in the position of the \(S_{2}\) and \(S_{1b}\) branches with changing the ultrasonic attenuation \(\alpha\) from 0 (bright yellow) to 1 dB/cm/MHz (dark blue): the larger is the attenuation, the larger is the distance between the two branches in the plane (\(k_{i},f\)). The minimization process is illustrated in Figure 5b showing, for three cases, the root-mean-square deviation (RMSD) between the measured (\(k_{i}^{exp}\)) and calculated (\(k_{i}^{th}\)) imaginary parts of the wavenumbers as a function of the attenuation \(\alpha\), where RMSD(\(\alpha\)) is defined by:  
Figure 4: Dispersion curves in (**a**) the plane (\(k_{r},f\)) and (**b**) the plane (\(k_{i},f\)). The lines stand for the theoretical dispersion curves with (w/) or without (w/o) taking into account the attenuation. The symbols stands for the experimental data. In (**a**), the error bars of the experimental measurements in the recovery of the real part of the wavenumber is depicted by horizontal segments. \[\text{RMSD}(\alpha)=\sqrt{\frac{1}{N}\sum_{n=1}^{N}\left|k_{i}^{exp}(f_{n})-k_{i}^{ th}(f_{n},\alpha)\right|^{2}}\,. \tag{9}\]  
In Figure 5b, the red circles, blue crosses, and black triangles represent RMSD(\(\alpha\)) in the following three cases, respectively: (i) using only experimental and calculated results of the \(S_{2}\) mode, the attenuation is then named \(\alpha_{2}\) and is estimated to be \(0.40\pm 0.01\) dB/cm/MHz; (ii) using only experimental and calculated results of the \(S_{1b}\) mode, the attenuation is then named \(\alpha_{1}\) and is estimated to be \(0.54\pm 0.01\) dB/cm/MHz; (iii) using experimental and calculated results of both the \(S_{2}\) and \(S_{1b}\) modes, the attenuation is then named \(\alpha\) and is estimated to be \(0.47\pm 0.01\) dB/cm/MHz. The obtained ultrasonic attenuation values are shown in Table 2 together with those from the literature for aluminum [50]. Note that the proposed precision \(\pm 0.01\) comes from the step used in the calculation for the parameter \(\alpha\). It should be mentioned that the attenuation \(\alpha_{1}\) obtained by using only the results of the \(S_{1b}\) mode is larger than \(\alpha_{2}\) obtained with the \(S_{2}\) mode, which may be due to a biased complex wavenumber recovery, as discussed above. As shown in Table 2, the values of the ultrasonic attenuation \(\alpha\) estimated in this work are not similar to those measured in the literature [50]. However, the mechanisms for phonon attenuation are not the same. Note that the values of the attenuation \(\alpha_{lit}\) (see Table 2) are from [50] (Table III, p. 1053), in which the attenuation is measured along different propagation directions in aluminum. We selected each time the value in the table measured at the closest frequency to our case: 30.7 MHz in the [100] propagation direction, 35.4 MHz in the [110] propagation direction, and 31 MHz in the [111] direction.