## 6 Thermoelastic Topography of GESA Coating: Thermal Dispersion Curve
For the T91 substrate, literature values of the thermoelastic parameters of [42] were assumed, while the obtained best fitting parameters (coating thickness, coating density and acoustic velocities) of the elastic experiment in Section 4 were used as fixed coating parameters. The heat capacity and thermal expansion coefficient of the coating layer were set equal to those of the substrate. These _ad hoc_ assumptions are feasible in view of the variability of values of the specific heat capacity and thermal expansion coefficient of similar solids being quite limited, compared to the thermal diffusivity, which, via the thermal conductivity, is strongly structure dependent. The values of the fixed parameters are listed in Table 4.  
Figure 15 shows the experimental data together with the resulting best fitting curves according the two-layered model. For the sake of easy comparison and validation, a curve following a complementary  
Figure 14: (**a**) Normalized \((x,t)\)-signal of the thermal decay at the coating surface of the GESA modified sample excited with a fringe of 35 μm. The diffraction grating was scanned (dx = 0.5 μm) in the wave propagation direction. (**b**) Normalized \((k,t)\)-signal of the logarithmically scaled thermal decay profile obtained after performing a one-dimensional Fourier transform along the \(x\)-direction of the signal in (**a**). error function with an effective diffusivity was added for each individual trace. The two-layered model fits the experimental data reasonably well, with only a little discrepancy between the two-layered model and the curves corresponding to an effective diffusivity. This demonstrates that for this case the effective diffusivity is a good approximation of the full model. The best fitting thermal diffusivity of the coating layer was \(\alpha_{1}=(5.0\pm 1.6)\times 10^{-6}\) m\({}^{2}\)/s, as is also depicted in Table 4. This value lies in the range of thermal diffusivities of low-carbon steels, i.e. between \(2\times 10^{-6}\) and \(13\times 10^{-6}\) m\({}^{2}\)/s [54; 55]. In Figure 16 the least-squares cost function was plotted as a function of the coating's thermal diffusivity, showing a unique minimum around the best fitting value. From the fitted parabolic function, a relative fitting uncertainty of about 32% could be extracted. The value of the extracted thermal diffusivity of the coating layer reveals a measurable contrast between the thermal properties of the coating and substrate material (that was fixed in the inverse problem), which should result in thermal dispersion. In Figure 17 it is shown that the effective thermal diffusivity starts from the coating's diffusivity at small wavelengths and then shifts to values intermediate between the coating and substrate diffusivity for increasing wavelengths before asymptotically reaching the substrate's diffusivity. At the maximum experimental wavelength over coating thickness ratio of about 5, the effective diffusivity should already be largely affected by the substrate. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \(d\) (\(\upmu\)m) & \(\rho\) (kg/m\({}^{3}\)) & \(c_{R}\) (m/s) & \(\alpha\) (\(10^{-6}\) m\({}^{2}\)/s) & \(\alpha_{\rm th}\) (\(10^{-6}\) K\({}^{-1}\)) & \(C_{p}\) (J/kg K) \\ \hline
**coating** & 22.3 & 5757 & 3314 & \(5.0\pm 1.6\) & 10.4 & 450 \\
**substrate** & & 7650 & 3024 & 7.5 & 10.4 & 450 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Assumed values for the fixed thermoelastic parameters in the fitting process aiming at the extraction of the thermal diffusivity of the coating in the GESA modified material. The best fitting thermal diffusivity of the coating layer is printed in bold. Figure 15: Set of time signals as extracted from the most dominant wavenumber components resulting from the thermoelastic excitation using grating spacings between \(\lambda=18\) and \(120\)\(\upmu\)m. As a result, of the long the thermal decay rate of the \(\lambda=120\)\(\upmu\)m line the signal was cut by the recording oscilloscope before it effectively became zero.