## 2 Principle
### Proposed Local Phase-Amplitude Joint Correction Method  
According to reference [12], first of all, the phase correction is made for the different propagation phase velocities of the elastic stress wave in the bar. The phase when a certain frequency component of the stress wave transmitted to a position \(z\) in the pressure bar is  
\[\phi(f)=2\pi f\frac{z}{C} \tag{9}\]  
Equation (9) indicates that the phase is related to the frequency of the stress wave and the propagation velocity of the frequency, that is, related to the propagation order. Different orders of the same frequency have different phases due to different propagation velocities. Figure 5: The first to fourth normal frequency and propagation time curves of Poisson’s ratio of 0.287. The phase difference from elementary theory is  
\[\phi^{\prime}(f)=2\pi f\left(\frac{z}{C}-\frac{z}{C_{0}}\right) \tag{10}\]  
Adding this phase difference to each frequency component and performing inverse Fourier transform can correct the dispersion caused by different phase velocities. Taking into account predictions from the Pochhammer-Chree theory for both the variation of axial strain over the bar radius and the relationship between axial stress and strain, reference [18] corrected the amplitude of the measured strain signal on the bar surface to obtain the average stress on the pressure end of the pressure bar. In this paper, the velocity history curve of the center of the free surface of the pressure bar is measured. It is necessary to derive the relationship between the center velocity of the bar and the average stress on the end surface of the bar so as to correct the non-uniformity of the velocity along the radius of the bar. According to reference [5], the axial displacement of the bar end is  
\[u_{z}=i\gamma A\frac{J_{1}(ha)}{ha}f_{2}e^{i(\gamma z+\omega t)} \tag{11}\]  
The axial velocity is the first derivative of the axial displacement with respect to time  
\[u_{z}^{\prime}=-\gamma\omega A\frac{J_{1}(ha)}{ha}f_{2}e^{i(\gamma z+\omega t)} \tag{12}\]  
The axial velocity of the bar center is  
\[u_{z}^{\prime}(r=0)=-\gamma\omega A\frac{J_{1}(ha)}{ha}f_{2}(r=0)e^{i(\gamma z +\omega t)} \tag{13}\]  
The axial stress in the bar is  
\[\sigma_{zz}=-2\mu\gamma^{2}A\frac{J_{1}(ha)}{ha}f_{1}e^{i(\gamma z+\omega t)} \tag{14}\]  
where \(A\) is amplitude; \(f_{1}\) and \(f_{2}\) are defined as  
\[ \begin{array}{l}f_{1}=(1+x-\beta x)\frac{haJ_{0}(hr)}{J_{1}(ha)}+\frac{1- \beta x}{x-1}\frac{kaJ_{0}(kr)}{J_{1}(ka)}\\ f_{2}=\frac{haJ_{0}(hr)}{J_{1}(ha)}+\frac{1-\beta x}{x-1}\frac{kaJ_{0}(kr)}{J_{1 }(ka)}\end{array}  \tag{15}\]  
In order to obtain the average axial stress in the bar, it is necessary to integrate and average Equation (14)  
\[\overline{\sigma_{zz}}=-2\mu\gamma^{2}A\frac{J_{1}(ha)}{ha}\overline{f_{1}}e^ {i(\gamma z+\omega t)} \tag{16}\]  
where \(\overline{f_{1}}\) is defined as  
\[ \begin{array}{lcl}\overline{f_{1}}&=&\frac{2}{a^{2}}\int_{0}^{a}rf_{1}dr=2 \int_{0}^{a}\frac{r}{a}f_{1}d\left(\frac{r}{a}\right)\\ &=&2\int_{0}^{a}\left(1+x-\beta x\right)\frac{haJ_{0}(hr)}{J_{1}(ha)}\frac{r}{ a}+\frac{1-\beta x}{x-1}\frac{kaJ_{0}(kr)}{J_{1}(ka)}\frac{r}{a}d\left(\frac{r}{a}\right)\\ &=&2\left[1+x-\beta x+\frac{1-\beta x}{x-1}\right]\end{array}  \tag{17}\]The ratio of the axial velocity of the pressure bar center to the average axial stress is  
\[\frac{u_{z}^{\prime}(r=0)}{\overline{\sigma_{zz}}}=\frac{-\gamma\omega A\frac{ 1(ha)}{ha}f_{2}(r=0)e^{i(\gamma z+\omega t)}}{-2\mu\gamma^{2}A\frac{1(ha)}{ha} \overline{f_{1}}e^{i(\gamma z+\omega t)}}=\frac{C(1+\sigma)}{E}\frac{f_{2}(r=0 )}{\overline{f_{1}}} \tag{18}\]  
Take the reciprocal of the Equation (18) to define the amplitude correction coefficient  
\[M=\frac{\overline{\sigma_{zz}}}{u_{z}^{\prime}(r=0)}=\frac{E}{C(1+\sigma)} \frac{\overline{f_{1}}}{f_{2}(r=0)} \tag{19}\]  
Equation (19) indicates that the relationship between the axial mean stress and the axial velocity at the center of the bar is related to the propagation phase velocity, or to the propagation mode and frequency of the elastic stress wave.