## 4 Results and Discussion
### Spatial Distribution of the Extreme Pressure  
Based on the SDPS model, the distribution of the extreme pressure and typical pressure regions under different cases (described by \(M\) and \(\varepsilon\)) were analyzed. Since the flow issue with a high Mach number would involve shock waves, this paper only investigated the pressure wave propagation with \(M\leq 0.8\). As the train lengths in operation in China are approximately 200 m and 400 m, the three-dimensional effect should be considered for pressure fluctuations in tunnels within 50 m of the tunnel portals [52; 53]. Therefore, \(\varepsilon\) is no more than 0.8 in the SDPS model.  
Based on the dimensionless analysis, the dimensionless location and the time of the appearance of extreme pressure are associated with \(M\) and \(\varepsilon\). Figure 8a,b depict the dimensionless spatial distribution of the MPP and MNP with \(M\) and \(\varepsilon\) as the abscissa, respectively (\(\varepsilon\) = 0.05, 0.1, 0.15,......, 0.8 and \(M\) = 0.05, 0.1, 0.15,......, 0.8). \(x^{\star}\) = 0 corresponds to the tunnel entry, and \(x^{\star}\) = 1 corresponds to the tunnel exit. The results show that the MPP and MNP are distributed in the region \(x^{\star}\in[0.33,0.71]\), and 99.2% of them are concentrated in the region \(x^{\star}\in[0.33,0.67]\).  
Figure 9 shows the change in the location of the MPP and MNP with variations in \(M\) and \(\varepsilon\) (\(M\leq 0.8\), \(\varepsilon\leq 0.8\)). The locations of the MPP and MNP are indicated by the dark and light one of the same colors, respectively. As shown in Figure 9, most of the MPPs and MNPs are symmetrical about the midpoint of the tunnel axis and appear alternately around it. This is because the MPP and MNP are mostly determined by the superposition of four PWSs. While in several cases, the determination of the MNPs needs to consider the influence of the PPs caused by the train tail. It results in the asymmetry in the location of the MPP and MNP, as shown in Figure 9a. For example, when \(M\) = 0.15 and \(\varepsilon\) = 0.2, the MPP appears at the midpoint of the tunnel axis \(x^{\star}\) = 0.5 while the MNP appears at the position of \(x^{\star}\) = 0.67, showing asymmetry.  
In addition, in the post-train stage, the MPP is found to be determined by the superposition of PWSs, and MNP is related to the PWS and PP, which are similar to that during train passage [18; 19].  
Figure 7: Comparison between the proposed SDPS model and field tests in two tunnels (MPP: maximum positive pressure, MNP: maximum negative pressure).  
Figure 8: Distribution of the normalized location of the extreme pressure under different conditions (\(M\leq 0.8\), \(\varepsilon\leq 0.8\)): (**a**) from the point of the train/wave velocity ratio \(M\); and (**b**) from the point of the train/tunnel length ratio \(\varepsilon\).  
Figure 9: Change in the location of the maximum positive and negative pressures with different train/wave velocity ratios \(M\) and train/tunnel length ratios \(\varepsilon\): (**a**) \(\varepsilon=0.1\)–\(0.4\), \(M\leq 0.8\); (**b**) \(\varepsilon=0.5\)–\(0.8\), \(M\leq 0.8\); (**c**) \(M=0.1\)–\(0.4\), \(\varepsilon\leq 0.8\); and (**d**) \(M=0.5\)–\(0.8\), \(\varepsilon\leq 0.8\).