## 2 Principle  
### Propagation Characteristics of Elastic Stress Wave in Cylindrical Bar  
#### 2.1.1 Phase Velocity Curve in Bar  
The material of the pressure bar is assumed to be isotropic, homogeneous, linear elastic, and undamped. The front end face of the pressure bar is the pressure bearing surface, which keeps a certain distance from the explosion center so that the pressure peak value is lower than the yield strength, which ensures that the elastic stress wave propagates in the bar. [PERSON] simplified the P-C frequency equation into:  
\[(x-1)^{2}\varphi(ha)-(\beta x-1)[x-\varphi(ka)]=0 \tag{1}\]  
where the definition of each parameter is shown in Table 1. The formula is a function of \(x\), \(\beta\), and \(\gamma a\), in which \(\beta\) is related to [PERSON]'s ratio of material. When the material is selected, its value has been determined. \(x\) is the relationship between the propagation phase velocity of the elastic longitudinal wave at a certain wavelength and the elementary theoretical wave velocity, and \(\gamma a\) is related to the ratio of bar radius to wavelength. Therefore, formula (1) actually gives the relationship between wavelength and wave velocity. Because of the properties of the Bessel function, Formula (1) has infinite sets of solutions, and it is difficult to obtain its analytical solution. Take \(C/C_{0}\) and \(D/L\) as values in a certain range, and then calculate the value of the function on the left of formula (1). The zero point of the function is the root of the formula. The function values show complex changes on the \(C/C_{0}\) and \(D/L\) planes. If the real part of the function value is greater than 0, the part is red, otherwise it is blue. Then, the boundary line of the zero point (Figure 1) is at the junction of the two colors, and the continuous change of function value is the root of each order of Formula (1). It should be pointed out that the positive and negative boundaries given by the program do not necessarily represent the continuity of function values. For example, the left adjacent point is positive infinity, while the right neighbor point is negative infinity, or vice versa; a program is developed to identify such discontinuous points and mark them in the graph with green dots. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Definition** \\ \hline \(C\) & phase velocity \\ \(C_{g}\) & group velocity \\ \(C_{0}\) & longitudinal wave velocity \\ \(L\) & wavelength \\ \(\gamma=2\pi/L\) & wave number \\ \(f\) & frequency \\ \(\omega=2\pi f\) & angular frequency \\ \(D\) & diameter of pressure bar \\ \(a=D/2\) & radius of pressure bar \\ \(\sigma\) & [PERSON]’s ratio \\ \(\beta=(1-2\sigma)/(1-\sigma)\) & intermediate parameters \\ \(x=(1+\sigma)(C/C_{0})^{2}\) & intermediate parameters \\ \(E\) & [PERSON]’s modulus \\ \(\mu=E/2(1+\sigma)\) & modulus of rigidity \\ \(h=\gamma(gx-1)^{0.5}\) & intermediate parameters \\ \(k=\gamma(2x-1)^{0.5}\) & intermediate parameters \\ \(\varphi(y)=yJ_{0}(y)/J_{1}(y)\) & composite function based on the Bessel function of the first kind \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Definition of elastic longitudinal stress wave parameters in the bar. Figure 1: The sign and discontinuous point distribution of the real part of the function value (\(\sigma=0.3\)). In Figure 1, the alternating positive and negative boundaries (except for green) are the phase velocity curves of each order of the elastic longitudinal wave described by P-C frequency equation. Under the specific Poisson's ratio, the real part symbol of the function and the distribution diagram of discontinuous points are drawn, and then several typical values corresponding to each order's curve are selected from the graph. Finally, the phase velocity curve of each order is obtained by solving the numerical equation by iterative method. Figure 2 shows the first-order phase velocity curves under different Poisson's ratios. It can be seen from the figure that the phase velocity of low-frequency stress wave is close to \(C_{0}\).