## 1 Introduction
### Experimental Results and Analysis
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline  & & \multicolumn{4}{c}{**Cross-Section Change from**} & \\ \multirow{2}{*}{**Fuel Volume/m3**} & \(\frac{S_{2}}{S_{1}}\) & \multicolumn{4}{c}{**Small to Large**} & & \multicolumn{4}{c}{**Cross-Section Change from Large to Small**} \\ \cline{3-10}  & & \(\Delta p_{3}\)/MPa & \(\Delta p_{4}\)/MPa & \(\mathbf{k}\) & \(\Delta p_{6}\)/MPa & \(\Delta p_{7}\)/MPa & \(\Delta p^{\prime}_{6}\)/MPa & \(\Omega\) & \(\overline{\mathbf{\gamma}}\) \\ \hline \multirow{4}{*}{0.18} & 1.23 & 0.7695 & 0.7155 & 1.075 & 0.6230 & 0.6422 & 0.6642 & 0.970 & 1.066 \\  & 1.78 & 0.7695 & 0.7011 & 1.098 & 0.5663 & 0.6446 & 0.8735 & 0.879 & 1.542 \\  & 2.42 & 0.7701 & 0.4622 & 1.666 & 0.5239 & 0.6192 & 0.9297 & 0.846 & 1.775 \\  & 3.16 & 0.7685 & 0.4635 & 1.658 & 0.4815 & 0.5741 & 0.9145 & 0.839 & 1.899 \\  & 4.00 & 0.7638 & 0.4646 & 1.644 & 0.4475 & 0.5299 & 0.9020 & 0.844 & 2.016 \\ \hline \multirow{4}{*}{0.23} & 1.23 & 0.9915 & 0.9013 & 1.100 & 0.7452 & 0.8710 & 0.7542 & 0.856 & 1.012 \\  & 1.78 & 0.9903 & 0.9263 & 1.069 & 0.7279 & 0.8734 & 1.1648 & 0.833 & 1.600 \\  & 2.42 & 0.9995 & 0.5995 & 1.667 & 0.6655 & 0.7745 & 1.1846 & 0.859 & 1.780 \\  & 3.16 & 0.9910 & 0.6014 & 1.648 & 0.6156 & 0.7451 & 1.1716 & 0.826 & 1.903 \\  & 4.00 & 0.9845 & 0.6078 & 1.620 & 0.5742 & 0.6978 & 1.1565 & 0.823 & 2.014 \\ \hline \multirow{4}{*}{0.28} & 1.23 & 1.2197 & 1.1328 & 1.077 & 0.9831 & 1.1473 & 0.9177 & 0.857 & 0.934 \\  & 1.78 & 1.2196 & 1.0067 & 1.211 & 0.9022 & 1.1859 & 1.3102 & 0.761 & 1.452 \\ \cline{1-1}  & 2.42 & 1.2194 & 0.7085 & 1.721 & 0.8361 & 1.1501 & 1.4748 & 0.727 & 1.764 \\ \cline{1-1}  & 3.16 & 1.2166 & 0.7156 & 1.700 & 0.7755 & 1.1123 & 1.4566 & 0.697 & 1.878 \\ \cline{1-1}  & 4.00 & 1.2140 & 0.7226 & 1.680 & 0.7197 & 1.0275 & 1.4125 & 0.700 & 1.963 \\ \hline \end{tabular}
\end{table} 
Table 2: Attenuation coefficient, increase coefficient, and reflection coefficient of shock wave peak overpressure in pipes with abrupt cross-section area changes. coefficient decreases as the initial peak overpressure rises before an abrupt cross-section decrease, whereas the reflection coefficient decreases as the initial peak overpressure rises. There is a specific relationship between the attenuation coefficient, initial peak overpressure, and the ratio of abrupt cross-section area change. Therefore, when a cross-section area changes abruptly from small to large, it is assumed that the relationship between the attenuation coefficient, initial peak overpressure, and the ratio of abrupt cross-section area change is as follows:  
\[k=A_{1}\cdot\left(\Delta p\right)^{B_{1}}\cdot\left(\frac{S_{2}}{S_{1}} \right)^{C_{1}} \tag{2}\]  
where \(A_{1}\), \(B_{1}\), and \(C_{1}\) are coefficients and exponents of nonlinear model. According to the data in Table 2, multivariate nonlinear regression of Equation (2) was carried out using Matlab 2018 Student Edition, which was is a comprehensive mathematical software launched by MathWorks in the United States. The Curve Fitting App tool in Curve Fitting Toolbox was used to fit functions. The following coupling relationships were computed:  
\[k=\left\{ \begin{array}{cc}1.063\cdot\left(\Delta p\right)^{0.112}\cdot\left( \frac{S_{2}}{S_{1}}\right)^{0.105}&1<\frac{S_{2}}{S_{1}}\leq 1.78\\ 1.752\cdot\left(\Delta p\right)^{0.055}\cdot\left(\frac{S_{2}}{S_{1}}\right)^{ -0.043}&1.78<\frac{S_{2}}{S_{1}}\leq 4\end{array} \right. \tag{3}\]  
Figure 5: Relationships between (**a**) attenuation coefficient, (**b**) increase coefficient, (**c**) reflection coefficient, and the ratio of abrupt cross-section area change. The fitting degree of the above relationships was found to be \(R^{2}\) = 0.747. Thus, the relationship given by Equation (3) between the attenuation coefficient, initial peak overpressure, and the ratio of abrupt cross-section area change is suitable for practical calculations.