## 3 Results and Discussion
### Local Phase-Amplitude Joint Correction  
For comparative analysis, the velocity signal of the pressure bar 01 was first corrected for dispersion using the global phase-amplitude joint correction method in the first-order propagation mode. The global phase-amplitude joint correction method corrected the phase and amplitude of frequency components over the full duration of signal. The corrected result is shown in Figure 14. For comparison, the time was selected to be aligned with the original signal, and the upper limit of the phase correction frequency was set to be 0.5 MHz. In Figure 14a, the black line is the stress curve obtained by multiplying the velocity by the conversion coefficient given by the elementary theory (Table 3), and the green line is the stress curve obtained by the global phase-amplitude joint correction. It is noteworthy that the oscillation around 3.83 ms is moved forward, and this part of the oscillation is \begin{table}
\begin{tabular}{c c c} \hline \hline
**Order** & **Lower Frequency Limit/MHz** & **Upper Frequency Limit/MHz** \\ \hline First & 0 & 1.0 \\ Second & 0.6 & 1.5 \\ Third & 0.6 & 1.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The frequency range of the first to third order modes of bar 04. Figure 12: Time-frequency joint spectrum of the free surface velocity of bar 04. Figure 13: The first to third order modes included in the time-frequency joint spectrum of bar 04.  
not the frequency characteristic described by the first-order propagation mode of the axial stress wave, indicating that the global phase-amplitude joint correction method will introduce new errors. The local phase-amplitude joint correction method proposed in this paper was then used to correct the velocity signal to obtain the average stress signal of the compression surface of the pressure bar. The upper frequency limit was set to 0.5 MHz, signals beyond this frequency limit were considered to be noise and were filtered out. Signals in the frequency band were not phase-corrected, except for the first-order propagation mode. As can be seen from Figure 15, the diffuse signal between 3.67 ms and 3.82 ms is well corrected, while impacts on oscillations around 3.83 ms are minimal, therefore, the additional errors that were introduced can be neglected. Figure 16 compares the stress curve of the end face under pressure obtained by elementary theory, global phase-amplitude joint correction, and local phase-amplitude joint correction. It can be seen that the local phase-amplitude joint correction method led to lower baseline oscillations in the signal and showed a better correction result than those of the global method. The rising edge of the stress signal given by the elementary theory is 4.13 \(\upmu\)s, and the rising edge after local correction is 2.70 \(\upmu\)s (10-90%). Moreover, a negative oscillation occurs at 3.66 ms at the leading edge of the main peak, indicating that the algorithm should be further improved. Figure 14: The result of the first order mode global phase-amplitude joint correction of bar 01′s free surface velocity **(a)** Comparison of corrected stress curve with elementary theory stress curve; **(b)** Time-frequency joint spectrum of corrected stress curve. The local phase-amplitude joint correction method was further used to process the signal of the bar 04. The frequency range of each order propagation mode was set by parameters in Table 4. Figure 17a compares the corrected curve and the curve given by the elementary theory. It can be seen that the amplitude of the oscillation following the main peak is greatly reduced, while the frequency component of the non-axial stress wave around 3.785 ms is minimally influenced. Figure 17b shows the time-frequency joint time spectrum of the corrected stress signal. Components higher than 1.8 MHz  
Figure 16: Comparison of the stress curve of pressure bar 01 with different methods.