## 3 Analytical Method for Spatial-Temporal Prediction of the Extreme Pressure
### Spatial-Temporal Distribution of Pressure State Model
#### 3.2.3 General Equations of Pressure State Distribution in the Tunnel  
Equations (9)-(14) describe the situation where, at time \(t\), the PWS or PP reaches \(x_{w1}\) or \(x_{pi}\) m away from the tunnel entrance. \(p_{s_{w_{1}}}(x,t)\) or \(p_{s_{p_{p}}}(x,t)\) show the spatial-temporal distribution of the pressure state influenced by PWS or PP. The subscript of \(i=1,2,3,\dots\) is used to distinguish different PWSs and PPs. Specifically, in Equations (10)-(13), \(n_{w}\) calculates the reflected times of the wave as the pressure wave state (propagates as a compression wave or an expansion wave) and propagation direction changes with different \(n_{w}\). \(x_{w_{1}}(t)\) describes the propagation path of PWS, and \((-1)^{n_{w_{1}}(t)}\) describes the wave propagation direction varied with \(n_{w}\). As mentioned in Section 3.2.1, the first term in the equation of \(p_{s_{w_{1}}}(x,t)\) describes the pressure state distribution approximately; the second term modifies the case when \(x=x_{w}\); and the third term modifies the initial pressure state distribution when \(n_{w}=0\). Additionally, \(t_{0}\) and \(t_{1}\) are used to describe the PWSs, which are generated at different times. For the nose-entry-induced PWS, it generates when the train nose enters the tunnel (\(t=0\)), the tail-entry-induced PWS generates when the train tail enters the tunnel (\(t=L_{tt}/v_{tr}\)), the nose-exit-induced one generates when the nose leaves the tunnel (\(t=L_{tt}/v_{tr}\)), and the tail-exit-induced one appears when the tail exits from the tunnel (\(t=(L_{tt}+L_{tr})/v_{tr}\)). (1)   Nose-entry-induced PWS  
\[\left\{ \begin{array}{l}p_{s_{w_{1}}}(x,t)=sgn(x_{w_{1}}(t)-x)+\delta(x-x_{w_ {1}}(t))(-1)^{n_{w_{1}}(t)}+\delta(n_{w_{1}}(t))H(x-x_{w_{1}}(t))\\ x_{w_{1}}(t)=(-1)^{n_{w_{1}}(t)}\Big{(}ct-2\Big{[}\frac{n_{w_{1}}(t)+1}{2} \Big{]}Lu\Big{)}\\ n_{w_{1}}(t)=\Big{[}\frac{ct}{L_{tt}}\Big{]}\end{array} \right. \tag{10}\]  
(2)   Tail-entry-induced PWS  
\[\left\{ \begin{array}{l}p_{s_{w_{2}}}(x,t)=sgn(x-x_{w_{2}}(t))-\delta(x-x_{w _{2}}(t))(-1)^{n_{w_{2}}(t)}-\delta(n_{w_{2}}(t))H(x-x_{w_{2}}(t))\\ x_{w_{2}}(t)=(-1)^{n_{w_{2}}(t)}\Big{(}c(t-t_{0})-2\Big{[}\frac{n_{w_{2}}(t)+1 }{2}\Big{]}L_{tt}\Big{)}\\ n_{w_{2}}(t)=\Big{[}\frac{c(t-t_{0})}{L_{tt}}\Big{]}\\ t_{0}=\frac{t_{tr}}{v_{tr}}\end{array} \right. \tag{11}\]  
(3)   Nose-exit-induced PWS  
\[\left\{ \begin{array}{l}p_{s_{w_{2}}}(x,t)=sgn(x-x_{w_{2}}(t))+\delta(x-x_{w _{2}}(t))(-,,,1)^{(t)}_{n_{w_{2}}}+\delta(n_{w_{2}}(t))H(x_{w_{2}}(t)-x)\\ x_{w_{2}}(t)=(-1)^{n_{w_{2}}(t)}\Big{(}\Big{(}2\Big{[}\frac{n_{w_{2}}(t)}{2} \Big{]}+1\Big{)}L_{tt}-c(t-t_{1})\Big{)}\\ n_{w_{2}}(t)=\Big{[}\frac{c(t-t_{0})}{L_{tt}}\Big{]}\\ t_{1}=\frac{L_{tt}}{v_{tr}}\end{array} \right. \tag{12}\]  
(4)   Tail-exit-induced PWS  
\[\left\{ \begin{array}{l}p_{s_{w_{2}}}(x,t)=sgn(x_{w_{2}}(t)-x)-\delta(x-x_{w _{2}}(t))(-1)^{n_{w_{2}}(t)}-\delta(n_{w_{2}}(t))H(x_{w_{2}}(t)-x)\\ x_{w_{2}}(t)=(-1)^{n_{w_{2}}(t)}\Big{(}\Big{(}2\Big{[}\frac{n_{w_{2}}(t)}{2} \Big{]}+1\Big{)}L_{tt}-c(t-t_{0}-t_{1})\Big{)}\\ n_{w_{2}}(t)=\Big{[}\frac{c(t-t_{0}-t_{1})}{L_{tt}}\Big{]}\end{array} \right. \tag{13}\]  
In Equations (9) and (14), \(x_{p_{t}}(t)\) describes the moving path of the PP induced by the train nose/tail, and the subscript \(i=1,2\) is used to distinguish the PP induced by the train nose/tail. For the nose-induced PP, its influence starts from the moment when the train nose enters the tunnel (\(t=0\)) to the moment when the train nose exits the tunnel (\(t=L_{tt}/v_{tr}\)). For the tail-induced PP, it generates from the moment when the train tail enters the tunnel (\(t=L_{tr}/v_{tr}\)) to the moment when the tail exits from the tunnel (\(t=(L_{tt}+L_{tr})/v_{tr}\)). \(p_{s_{p_{t}}}(x,t)\) describes the pressure state distribution in the tunnel influenced by PP, and the first term on the right describes the pressure state distribution \(p_{s}(x)\) accurately, except for the case of\(x=x_{p}\), then the second term on the right supplements the condition of \(x=x_{p}\), as mentioned in Section 3.2.2.