## 3 Analytical Method for Spatial-Temporal Prediction of the Extreme Pressure
### Spatial-Temporal Distribution of Pressure State Model  
#### 3.2.1 Distribution of Pressure State Influenced by Press Wave Systems  
Take the nose-entry-induced PWS as an example; as shown in Figure 3, with the tunnel entrance as the origin, the coordinate system was established along the tunnel as the \(x\)-axis. \(x=0\) and \(x=L_{tu}\) correspond to the tunnel entry and exit, respectively (\(L_{tu}\) is the tunnel length). The initial time \(t=0\) was recorded as the arrival of the train nose at the tunnel entrance, \(x_{w}\) is the position where the pressure wave reaches the tunnel, and \(p_{s}\) represents the pressure state, which is recorded as 1 for a pressure rise and \(-1\) for a pressure drop. For example, when a compression/expansion wave sweeps a point, its pressure state would be \(+1/-1\), accordingly, and remains stable until the next disturbance (PWS or PP) reaches; \(n_{w}\) is the reflected times of the wave at the tunnel port after its generation. As shown in Figure 3a,c,e, when the train nose enters the tunnel (\(t=0\)), the nose-entry-induced compression wave generates and moves from the tunnel entry (\(x_{w}=0\)) to the tunnel exit (\(x_{w}=L_{tu}\)), and the corresponding \(n_{w}\) is 0. As shown in Figure 3c, after being swept by the compression wave, the pressure rises, \(p_{s}=1\); the pressure state remains unchanged before the wave, \(p_{s}=0\). When the compression wave reaches the tunnel exit (\(t=L_{tu}/c\)), it reflects and then propagates as an expansion wave, and \(n_{w}\) becomes 1, as shown in Figure 3b,d,f. After being  
Figure 3: Schematic of pressure wave propagation and the associated distribution of pressure state: (**a**,**c**,**e**) \(n_{w}=0\); (**b**,**d**,**f**) \(n_{w}=1\). (\(p_{s}\): pressure state; \(n_{w}\): the reflected times of the wave at the tunnel port; \(x_{w}\): the location of pressure wave; \(L_{tu}\): tunnel length; \(v_{tr}\): train speed; \(c\): the velocity of sound). swept by the expansion wave, the pressure decreases, and \(p_{s}=-1\). Before the wave, the pressure state remains and is considered to be influenced by the previous compression wave, and \(p_{s}=1\), as shown in Figure 3d. The expansion wave propagates to the tunnel entry at the velocity of sound \(c\); it reaches the entry (\(t=2L_{tu}/c\)) and reflects as a compression wave; \(n_{w}\) becomes 2. As shown in Figure 4c, the pressure rises after the compression wave, where \(p_{s}=1\); before the wave, the pressure state remains and is considered to be influenced by the previous expansion wave, where \(p_{s}=-1\). Based on the propagation and reflection characteristics of the PWS, the propagation cycle of PWS is \(2L_{tu}/c\). Due to the periodicity of PWS, the pressure state fluctuation in the tunnel is classified into three categories: \(n_{w}=0\), \(n_{w}\) is odd, and \(n_{w}\) is even. Figure 4 shows the spatial distribution of pressure state \(p_{s}(x)\) in the tunnel when \(n_{w}=0\), 1, and 2. As shown in Figure 4, the spatial distribution of the pressure state in the tunnel \(p_{s}(x)\) should be described using a piecewise function. It is notable that, as shown in Figure 4c,d, \(p_{s}(x)\) can be regarded as the sign function \(f(x)=sgn(x)\) with different propagation directions. \(f(x)=sgn(x)\) returns 1 for a positive number, 0 for the number zero, and \(-1\) for a negative number. \[sgn(x)= \begin{cases}1,&x>0\\ 0,&x=0\\ -1,&x<0\end{cases}  \tag{1}\]  
As shown in Figure 4c, when \(n_{w}=2\) (represents the cases when \(n_{w}\) is even), \(p_{s}(x)\) can approximately be described by \(f(x)=sgn(x_{w}-x)\), but modifications are needed in the case of \(x=x_{w}\). Figure 4: Schematic of pressure state distribution under different \(n_{w}\): (**a**) \(n_{w}=0\), (**b**) \(n_{w}=1\), (**c**) \(n_{w}=2\), (**d**) Schematic of sign function (\(n_{w}\): the reflected times of the wave at the tunnel port; \(p_{s}\): pressure state; \(x_{w}\): the location of pressure wave; \(L_{tu}\): tunnel length; \(c\): the velocity of sound).