## 2 Spatial-Temporal Evolution of Pressure Waves in a Tunnel  
The generation, propagation, reflection, and superposition mechanism of complex pressure waves in tunnels are shown in Figure 1. It shows the pressure time histories at 140 m from the entrance and the associated wave diagram when a 201.4-m-long train passes through a 1005-m-long tunnel at a speed of 300 km/h (the local velocity of sound c is 336 m/s) [36]. The solid and dashed lines represent the compression and expansion waves, respectively, which are generated when the train enters/leaves the tunnel. The train nose entry/exit generates a compression wave propagating throughout the tunnel, resulting in a pressure increase (corresponding to the labels 1, 6-7, 10-12, 17-20, and 25-28 in Figure 1). At the same time, the train tail entry/exit generates an expansion wave causing a pressure drop (corresponding to the labels 3, 5, 8-9,13-16, 21-24, and 29 in Figure 1). Both the compression wave and the expansion wave propagate backward and forward in the tunnel at the local velocity of sound c. Each time the pressure wave reaches the tunnel portals, it reflects backward, and the reflected wave is out-of-phase with the incident one [18]. As pressure waves can be described by linear equations, the principle of superposition is applicable [37; 38]. That is, the pressure change caused by two or more pressure waves is the sum of the change caused by each wave individually.  
In this paper, the complex pressure waves are divided into four categories. As shown in Figure 2, according to the time sequence, the first is nose-entry-induced PWS, which includes the train nose-entry-induced compression wave and its reflected waves. The second one is the tail entry-induced PWS, including the tail-entry-induced expansion wave and its reflected waves. The third is the nose-exit-induced PWS, which contains the nose-exit-induced compression wave and its reflected waves. The last is the tail-exit-induced PWS, containing the tail-exit-induced expansion wave and its reflected waves [39; 40].  
In addition, the passage of the train nose and tail, represented by the green dashed lines and solid lines in Figure 1, also causes a pressure decrease and increase, respectively, due to the induced pressure pulse (PP) [41; 42]. The difference between PWS and PP is that PWS propagates to and fro with the velocity of sound c while PP does not reflect and moves with the train speed of \(v_{tr}\). It is reported that PPs also have effects on the location of extreme  
Figure 1: Pressure time histories at 140 m and associated wave diagram for a 201.4-m-long train passing through a 1005-m-long tunnel at 300 km/h (adapted from [PERSON] et al. [36]).  
pressure during train passage [18; 19]. As shown in Figure 2b, pressure fluctuates in two stages, namely the passing stage and the post-train stage. The pressure fluctuation during train passage is caused by the superposition of two entry-induced PWSs and two PPs induced by the end of the train while the significant pressure fluctuations in the post-train stage consider the superposition of four PWSs (two more exit-induced PWSs) and two PPs. Different superimposed times and locations of the PWSs lead to different spatial distributions of extreme pressure in the tunnel [43].