## 4 Application of the SLaTCoW Method: Estimation of Ultrasonic Attenuation  
The aim of this section is to estimate the ultrasonic attenuation in the polycrystalline aluminum plate using the comparison between experimental and theoretical dispersion curves in the complex plane. To achieve this, the SLaTCoW method [27] is used. This method enables the extraction of complex wave-number information from guided elastic wave measurements. In the SLaTCoW method, unlike the case of the conventional two-dimensional (space and time) Fourier transform [45; 46], the time-domain Fourier transform  
\[\widetilde{u}(x,\omega)=\int_{0}^{T}u(x,t)e^{i\omega t}\mathrm{d}t, \tag{3}\]  
is associated with a spatial Laplace transform:  
\[\overline{u}(s,\omega)=\int_{0}^{L}\widetilde{u}(x,\omega)e^{-sx}\mathrm{d}x, \tag{4}\]  
where \(s=s_{r}+\imath s_{i}\) is the complex variable of Laplace. According to Equation (4), it is interesting to note that the quantity \(\overline{u}(s,\omega)\) corresponds exactly to the results obtained with a spatial Fourier transform when the real part of \(s\) (\(s_{r}\)) is set to zero:  
\[\overline{u}[s(s_{r}=0,s_{i}),\omega]=\widetilde{u}(k_{r},\omega), \tag{5}\]  
therefore implying that the dispersion curves in the real plane could be retrieved from \(\overline{u}[s(s_{r}=0,s_{i}],\omega)\). The idea of the SLaTCoW method is to compare the quantity \(\overline{u}(s,\omega)\) obtained from the experimental data to a model. Let us assume that the wave field could be written as the sum of the contributions of each of the modes in the frequency domain [27]:  
\[\widetilde{u}_{th}(x,\omega)=\sum_{n}\overline{A}^{n}e^{i\pi s}\Pi(x-L), \tag{6}\]  
Figure 3: (**a**) A typical measured signal in the aluminum plate and (**b**) its spectra. (**c**) Spatiotemporal measurements (B-scan) obtained in Configuration (2) of the experimental setup (see Figure 2c). (**d**) Wavenumber-frequency diagram obtained by a 2D Fourier Transform of the B-scan shown in (**c**). (**e**) Zoomed-in part of (**d**) around the \(S_{1}S_{2}\)-ZGV mode. where \(x\) denotes the spatial coordinate, \(\Pi(x-L)\) is the gate function equal to 1 when \(x\in[0;L]\) and equal to 0 elsewhere, and \(\overline{A}^{n}\) and \(k^{n}\) are the complex amplitude and complex wavenumber (depending on \(\omega\)) of the \(n\)th mode, respectively. Note that, in our case, the number of modes \(n\) is equal to 3, which denotes the three detected Lamb modes: \(S_{1b}\), \(S_{2}\), and \(A_{1}\). After applying the Laplace transform to \(\widetilde{u}_{th}(x,\omega)\), the theoretical expression \(\overline{u}_{th}(s,\omega)\) to which the experimental results is compared takes the following form:  
\[\overline{u}_{th}(s,\omega)=\sum_{n}\overline{A}^{n}\int_{0}^{L}e^{(-s+ik^{n} )x}dx=L\sum_{n}\Bigl{|}\overline{A}^{n}\Bigr{|}\frac{\sinh[(-s+ik^{n})\frac{L }{2}]}{(-s+ik^{n})\frac{L}{2}}e^{i\phi^{n}+(-s+ik^{n})\frac{L}{2}}, \tag{7}\]  
where \(\phi^{n}\) stands for the phase of the complex amplitude \(\overline{A}^{n}\). The recovery of the complex wavenumbers of the three modes is finally performed, frequency by frequency, by minimizing the following cost function:  
\[f_{opt}(\omega)=\sqrt{\sum_{s_{r}}\sum_{s_{i}}\|\overline{u}(s,\omega)- \overline{u}_{th}(s,\omega)\|^{2}}. \tag{8}\]  
Note that the minimization is performed under constraints with the Nelder-Mead simplex algorithm and the use of the function \(\texttt{fminsearchbnd}\) in MATLAB. For details and other applications of the SLaTCoW method which are beyond the scope of this manuscript, the readers are referred to the works in [27; 47; 48]. For the comparison with the experimental results, the complex Lamb wave spectrum is computed applying to elasticity equations a numerical spectral method, as proposed in [49].