## 4 Validation of the Dominant Frequency Estimation Procedure  
The method of Section 3.2 was validated following a similar validation procedure to that presented in [29] and summarized in Section 3.1. Direct simulations were implemented in a COMSOL Multiphysics(r) environment to solve conduction heat transfer equations within the FPPHP solid domain. In particular, the same analytical heat flux distributions \(q_{exact}\) used for the validation procedure of [29] were given as a boundary condition at the wall-fluid interfaces. Specifically, the guessed distribution for the analytical heat flux \(q_{exact}\) assumed the following sinusoidal form [29]:  
\[q_{exact}=Amp\cdot\cos(2\pi t\cdot f-\varphi)+\frac{Amp}{2}-\frac{z}{H}\cdot Amp \tag{4}\]  
where \(Amp\) is the heat flux amplitude in W/m\({}^{2}\),\(f\) is the oscillation frequency in Hz, \(\varphi\) is the phase shift (different from channel to channel to reflect the oscillatory behavior in the real device), \(z\) is the axial coordinate and \(H\) is the length of the portion of interest within the adiabatic section. The reader is referred to [29] for deeper details on the procedure. The resulting synthetic temperature maps were therefore spoiled with noise, filtered and used as inputs for the inverse procedure to evaluate back the heat flux, \(q_{restoral}\). In other words, noisy temperature distributions were used as inputs for Equation (1), solved through the finite element method. However, while in [29] the validation procedure was focused on the percentage error estimation in terms of amplitude of the restored heat fluxes with respect to the exact ones (i.e., the guessed ones), the present validation aims at considering differences in oscillation frequency between \(q_{exact}\) and \(q_{restoral}\). Hence, \(q_{exact}\) and \(q_{restoral}\) were used as inputs for the wavelet method. In Figure 4, \(q_{exact}\) (amplitude \(Amp\) = 2000 W/m\({}^{2}\), frequency \(f\) = 1.2 Hz) is shown, together with the wavelet method outputs. Due to the fact that the analytical formulation of \(q_{exact}\), for which the time-dependent signal is presented in Figure 4a, is a sinusoid having a fixed frequency over time (please refer to [29]), the scalogram (Figure 4b) exhibits the same signal power (described by the color bar on the right) at each time instant of the observation window. It has to be stressed that the white dashed lines in scalograms delimit what is known as cone of influence (COI) [34]. Specifically, the COI highlights areas (grey-shaded) potentially affected by edge-effects, i.e., where the wavelet extends over the boundaries of the considered time interval, and they might not be representative of an accurate time-frequency description of the considered samples. For this reason, the power \begin{table}
\begin{tabular}{l l} \hline \hline \(A\) [1/Hz] & 71 \\ \(\omega_{0}\) [rad/s] & 2\(\pi\) \\ \(f_{min}\) [Hz] & 0.1 \\ \(f_{max}\) [Hz] & 18 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters used for the wavelet method. spectrum evaluation does not consider such areas when integrating the power related to each frequency over time. In Figure 4c, the power spectrum is reported: here, the power related to low (<0.8 Hz) and high (>2 Hz) frequencies \(P_{w}\) is null, while it increases as long as the natural frequency of the treated signal (1.2 Hz) is approached. The absolute maximum of the power spectrum is found at 1.23 Hz. Hence, the error for the dominant frequency estimation procedure, i.e., the frequency resolution of the wavelet method for the present analysis, was assessed to be equal to 0.03 Hz. Such an error is intrinsic in the wavelet approach due to the shape of the considered wavelet function, having coefficients listed in Table 2. The lower limit of 0.5 Hz for the frequency scales has been chosen due to the limited observation window (20 s), which prevents the detection of a sufficient number of oscillations at lower frequencies, undermining the reliability of the method.