## 2 Experimental and Analysis Method
### Signal Analysis Method  
Wavelet transform (WT) [18] is a time-frequency translation of the time-series signal, which is an alternative to the Short Time Fourier Transform (STFT). It is known that a sufficiently large time window is appropriate for characterizing low-frequency components in the time-series signal. On the contrary, a narrow time window is adequate for characterizing high-frequency components. The theoretical justification for this lies in [PERSON]'s uncertainty principle, which dictates that a fixed time window cannot simultaneously maintain adequate resolution in time and frequency [18]. This presents a disadvantage for the STFT and supports our choice of WT. WT has been applied to various fields to analyze ultrasonic signals [19; 20; 21], as the wavelet coefficients in the time-frequency co-ordinate can be presented as a contour map, which illustrates characteristic features of the ultrasonic signal at a glance. In this work, the WT is written as follows:  
\[WT(u,s)=\int_{-\infty}^{+\infty}U(t)\hat{\psi}(\frac{t-u}{s})dt, \tag{1}\]  
where \(u\), \(s\), \(U(t)\), and \(\hat{\psi}(t)\) are the time shift parameter, scaling parameter, ultrasonic signal, and complex conjugate of the mother wavelet function, respectively. The mother wavelet function \(\psi(t)\) is defined as:  
\[\psi(x)=\frac{1}{2\sqrt{\pi}\sigma}exp(-\frac{x^{2}}{\sigma^{2}})exp(2\pi x). \tag{2}\]  
The Gabor mother wavelet was selected because it has the best balance between time and frequency resolution, based on the uncertainty principle, and as previous studies using the Gabor mother wavelet have successfully characterized ultrasonic signals [19; 20; 21].  
\begin{table}
\begin{tabular}{l l l l l} \hline \hline
**Laser Spot Diameter** & **Pulse Width** & **Wave Length** & **Laser Power** & **Beam Profile** \\ \hline
5 mm & 10 nsec & 1064 nm & 14 m\()/pulse & Gaussian \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters of the pulsed-laser source.  
Figure 4: Schematics and actual photos of the specimens used for the experiment.  
Figure 5: Dimensions of the specimens.  
The group velocity dispersion curve is the frequency-velocity curve of the ultrasonic guided wave, which is obtained by calculating the characteristic equation [15]. In this experiment, we assumed that the symmetric logitudinal mode is dominant, as pulse-laser ablation was used for exciting the ultrasonic wave [13]. Therefore, the Pochhammer frequency equation was used to obtain the dispersion curves, as follows:  
\[\frac{2\alpha}{\beta}(\beta^{2}+k^{2})J_{1}(\alpha a)J_{1}(\beta a)-(\beta^{2} -k^{2})J_{0}(\alpha a)J_{1}(\beta a)-4k^{2}\alpha\beta J_{1}(\alpha a)J_{0}( \beta a)=0, \tag{3}\]  
\[\alpha=\left(\frac{\omega^{2}}{C_{L}^{2}}-k^{2}\right)^{\frac{1}{2}},\beta= \left(\frac{\omega^{2}}{C_{T}^{2}}-k^{2}\right)^{\frac{1}{2}},C_{p}=\frac{ \omega}{k}, \tag{4}\]  
where \(a\), \(\omega\), \(k\), \(C_{L}\), \(C_{T}\), \(J\), and \(C_{p}\) are the rod radius, circular frequency, wavenumber, longitudinal and transverse bulk wave velocities, Bessel function of the first kind, and phase velocity, respectively. The phase velocity of each vibration mode can be obtained by substituting \(k\) and \(\omega\) satisfying Equation (3) into \(C_{p}\) of Equation (4). Meanwhile, the group velocity dispersion curve is obtained from the following equation by substituting \(C_{p}\) into Equation (4).  
\[C_{g}(f)=\frac{\partial\omega}{\partial k}=C_{p}^{2}(C_{p}-\omega\frac{ \partial C_{p}}{\partial\omega})^{-1}=C_{p}^{2}(C_{p}-f\frac{\partial C_{p}}{ \partial f})^{-1}. \tag{5}\]  
The result of the WT can be mapped onto frequency-velocity co-ordinates by dividing the length of the rod, \(L\), used in this experiment by time shift parameter, \(u\). Then, \(WT(L/u,s)\) of the ultrasonic signal obtained from the experiment and \(C_{g}(f)\) are compared, and the modes contained by the ultrasonic signal are investigated.