## 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
The first correction term is based on \(f(x)=\delta(x-a)\), as it returns 1 when \(x=a\); otherwise, it returns 0. When \(a=0\) especially, \(f(x)=\delta(x)\) returns 1 when \(x=0\); otherwise, it returns 0. \[\delta(x-a)= \begin{cases}1,&x=a\\ 0,&x\
eq a\end{cases}  \tag{2}\]  
In this situation, the correction term is \(f(x)=\delta(x-x_{w})\) in the cases when \(n_{w}\) is even, which only returns 1 when \(x=x_{w}\). It is notable that the difference between Figure 3(b),c is \(p_{s}(x_{w})\). In Figure 3(c), \(p_{s}(x_{w})=1\), represents point \(x_{w}\) being swept by a compression wave and encountering a pressure rise. While in Figure 3(b), \(n_{w}=1\) (represents the cases when \(n_{w}\) is odd), \(p_{s}(x_{w})=-1\). It illustrates that point \(x_{w}\) is being swept by an expansion wave and encounters a pressure drop. Then, the corresponding correction term is \(f(x)=-\delta(x-x_{w})\) for the cases where \(n_{w}\) is odd, which returns \(-1\) when \(x=x_{w}\). The difference between Figure 3(a),c is \(p_{s}(x)\) when \(x>x_{w}\). Then, the second correction term is based on \(f(x)=H(x)\), which returns 1 for \(x>0\) and 0 for \(x<0\). \[H(x)= \begin{cases}1,&x>0\\ 0,&x<0\end{cases}  \tag{3}\]  
In this situation, the correction term is \(f(x)=H(x-x_{w})\), which returns 1 when \(x>x_{w}\). In brief, the spatial distribution of the pressure state along the tunnel \(p_{s}(x)\) is as follows:  
\[p_{s}(x)=sgn(x_{w}-x)+(-1)^{n_{w}}\delta(x-x_{w})+\delta(n_{w})H(x-x_{w}) \tag{4}\]  
where \(f(x)=sgn(x_{w}-x)\) is the main body of the function, and the other two terms are the correction terms. \(f(x)=(-1)^{n_{w}}\delta(x-x_{w})\) modifies the value of \(p_{s}(x_{w})\) so that when \(x=x_{w}\), it returns \(-1\) when \(n_{w}\) is odd and returns 1 when \(n_{w}\) is even. \(f(x)=\delta(n_{w})H(x-x_{w})\) only modifies the case with, and returns 1 for \(x>x_{w}\), which is consistent with the actual situation. Additionally, \(n_{w}\) and \(x_{w}\) vary with time \(t\) and need to be described using a piecewise function. For the nose-entry-induced PWS, \(n_{w}(t)\) is as follows:  
\[n_{w}(t)= \begin{cases}0,&0\leq t<\frac{L_{tu}}{2}\\ 1,&\frac{L_{tu}}{c}\leq t<\frac{\frac{3}{c}L_{tu}}{c}\\ 2,&\frac{2\bar{L}_{tu}}{c}\leq t<\frac{3\bar{L}_{tu}}{c}\\ \ldots&\ldots\end{cases}  \tag{5}\]  
Thus, one of the piecewise functions, the floor function \(f(x)=[x]\), is introduced. It gives the largest integer less than or equal to \(x\). Then, \(n_{w}(t)\) is given by the following:  
\[n_{w}(t)=\left[\frac{ct}{L_{tu}}\right] \tag{6}\]  
For example, when \(\frac{L_{tu}}{c}\leq t<\frac{2L_{tu}}{c}\), then \(1\leq\frac{ct}{L_{tu}}<2\), the corresponding \(n_{w}\) is 1. For the nose-entry-induced PWS, the position of the pressure wave in the tunnel \(x_{w}\) (\(t\)) is given by the following:  
\[x_{w}(t)=\left\{ \begin{array}{ll}ct,&n_{w}(t)=0\\ 2L_{tu}-ct,&n_{w}(t)=1\\ ct-2L_{tu},&n_{w}(t)=2\\ 4L_{tu}-ct,&n_{w}(t)=3\\ \ldots&\ldots\end{array} \right. \tag{7}\]  
With the help of the floor function, it is rewritten as follows:  
\[x_{w}(t)=(-1)^{n_{w}(t)}\bigg{(}ct-2\bigg{[}\frac{n_{w}(t)+1}{2}\bigg{]}L_{tu }\bigg{)} \tag{8}\]Substitute \(n_{w}=0,\,1,\,2,\,3,\,\dots\) into Equation (8), then Equation (7) is obtained. This is the calculation of the nose-entry-induced PWS; those for other PWSs are deduced in the same way.