# Elastic Slow Dynamics in Polycrystalline Metal Alloys
## 3 Materials and Methods
### Data Analysis  
For each waveform recorded at time \(t\) as mentioned above, the corresponding propagation velocity in the sample can be derived using a semianalytical formulation (MoDaNE) described in ref. [41; 42]. As already mentioned, each acquired waveform was post-processed by discrete-time Fourier transform with 2 Hz resolution to obtain the resonance curve; an example of results is shown in Figure 3 (red symbols).  
Figure 2: Schematic of the experimental set up. The sample is between the transmitter (T) and receiver (R) transducers.  
In the MoDaNE approach, the displacement solution in \(x=L\) (i.e., at the receiver position) was found for a sinusoidal forcing in \(x=0\). The amplitude of the solution as a function of frequency is expressed as  
\[u_{theor}=\frac{U_{0}}{\sqrt{\cosh^{2}(\alpha L)-\cos^{2}(kL)}} \tag{2}\]  
where \(L\) is the sample length, \(U_{0}\) is the source amplitude (after proper calibration), \(\alpha\) is the damping factor and \(k\) is the wavenumber, \(k=\omega/c\), being \(c\) the wave velocity. By fitting the experimental results (either \(u_{t,pre}\) or \(u_{t,t,tx}\)), the wave velocity can be derived (together with the other parameters \(U_{0}\) and \(\alpha\)).  
The fit result for the raw experimental data in Figure 3, shown as a red line, is not optimal, due to the distortions induced by two sources of error, which need to be corrected:  
* the transducers frequency response, which is not completely flat in the frequency range considered. A tilting correction to experimental data could then be obtained by dividing the experimental resonance curves with the transducer response \(\Psi_{t}(\omega)\), shown as a black line in Figure 3;
* the source function distortions, which is not a perfect square function in the frequency domain. The corresponding deconvolution \(\Psi_{s}(\omega)\) is given by the DTFT of the source signal (green line).  
The resulting tilted resonance curve can be obtained  
\[A_{tilt}(\omega)=A(\omega)/(\Psi_{t}(\omega)\Psi_{s}(\omega)) \tag{3}\]  
and it is shown as blue symbols. Here, the symbol \(A\) denotes the amplitude of the DTFT for each frequency \(\omega\). The fit of the tilted resonance curve (blue line) is significantly improved.  
The procedure discussed above is applied to signals acquired at different times during relaxation. It follows that curves of velocity vs. time could be derived as shown in Figure 4, where we plot resonance frequency (proportional to velocity) vs. time. Figure 4 allows also to appreciate the correctness and accuracy of the approach proposed. Here, the relaxation  
Figure 3: Discrete time Fourier transform of the output signal is shown as red symbols. The resulting fit using Equation (2) is shown as a red line. The tilting corrections introduced to compensate for the transducer and source frequency responses are shown as solid black and green lines, respectively. The corrected experimental data (blue symbols) and corresponding fitting curve (blue line) are perfectly superimposed. For the sake of readability, the number of data-points was reduced by a factor of 5. Vertical dashed lines show the fitting interval for MoDANE.  
curve of velocity for sample Alu-C20 is derived using the MoDaNE approach (red symbols) and compared with that obtained deriving velocity from the peak of the resonance curve (cyan symbols), obtained from a parabolic fit around the maximum. The two approaches provide the same result, but fluctuations are significantly smaller when MoDaNE is applied, indicating a higher accuracy in the determination of velocity.