## 1 Introduction  
As a high-speed train travels through a tunnel, pressure waves are generated due to the restricted airspace within the tunnel and the compressibility of the air. When the train nose and tail enter or leave a tunnel, the local flow and cross-section dramatically change, successively causing four pressure wave systems (PWSs) at the portals of the tunnel. With the reciprocating propagation, reflection, and superposition of pressure waves, long-duration and large-amplitude pressure fluctuations are generated inside tunnels and act on the tunnel and train surface, producing alternating loads. It affects the crack initiation and propagation in the structure [1] causes high-cycle fatigue damage to train structures [2, 3], tunnel structures [4, 5, 6], and pressure-bearing tunnel facilities (including fire doors, control boxes, signal lights, etc.) [3, 7, 8]; and leads to vibration discomfort for passengers [9], threatening the safety of transportation systems [10, 11]. For example, in 1999, influenced by long-term aerodynamic loads, a block of lining detached from the Rebunhama high-speed railway tunnel in Japan, which caused train derailment and the cancellation of 51 subsequent trains [1, 12]. More importantly, compared with the train passing stage, pressure fluctuation in the post-train stage (after a train exits from a tunnel) has a longer duration and slower attenuation as there is no interaction between the wave and the train [13], but the pressure amplitude is similar or even higher. For example, after a train left a 2812-m-long tunnel at a speed of 300 km/h, the maximum pressure fluctuation in the tunnel was up to 5 kPa. Additionally, it lasted over 60 s before decaying to 50% of the amplitude, which is almost twice the train passing time of 36 s [8; 14]. Similar results were found in field measurements [15], scaled-model experiments [13], and numerical simulations [4; 16]. Additionally, [PERSON] et al. [1] found that damage accumulation in the post-train stage is approximately equal to that during the train-passing stage, indicating that the aerodynamic effects in the post-train stage cannot be ignored. Currently, several researchers have discussed the spatial distribution of extreme pressure in tunnels during train passage [17; 18; 19]. However, research concerning pressure changes in the post-train stage is still limited, and the spatial-temporal distribution law of extreme pressure is not clear. Therefore, it is necessary to investigate the spatial-temporal distribution of the typical pressure (extreme pressure and maximum peak-to-peak pressure) after train passage, revealing the locations of potential accidents caused by fatigue failure in tunnels. The research methods for investigating pressure waves in tunnels include field measurements [8; 14; 18; 20], moving model tests [13; 21; 22], numerical simulations [4; 23], and theoretical analysis [24; 25; 26]. However, field measurements and moving model tests are usually expensive and limited by the existing test conditions, and numerical simulations with three-dimensional models consume a lot of computing resources and time [23; 27; 28; 29]. Additionally, the above methods are all difficult when trying to meet the needs of rapid decision-making with the lowest resource consumption, especially in engineering design. Theoretical analysis is useful for predicting pressure waves in tunnels. As the length of a tunnel is obviously longer than the tunnel's hydraulic diameter, and the propagation time of the pressure in the cross-section is much less than that along the tunnel, it exhibits typical one-dimensional geometric and kinematic characteristics. In the post-train stage especially, only the pressure wave continuously propagates, reflects, and superposes in the tunnel. Therefore, a one-dimensional theoretical model has been proposed to obtain the pressure characteristics in the tunnel.