## 1 Introduction  
High-speed railways, as crucial national infrastructure, play an irreplaceable role in economic development. As high-speed train speeds continue to increase, the aerodynamic problems of high-speed trains and tunnel aerodynamic effects become more significant [1]; hence, tunnel micro-pressure waves (MPWs) have become an urgent problem to be solved [2]. The generation of tunnel MPWs involves four processes: the formation of initial compression waves, the propagation and evolution of initial compression waves within the tunnel, the radiation of micro-pressure waves, and their transmission through the air to the receiving end outside the tunnel [3], viz., buildings or people. MPWs cause vibrations of doors and windows in buildings near the tunnel and produce sonic booms, creating a serious environmental noise problem that is hazardous to people's physical and mental health [4]. The magnitude of the micro-pressure waves is directly proportional to the pressure gradient of the compression wave at the tunnel exit [5,6]. Therefore, reducing the pressure gradient is the main way to mitigate micro-pressure waves. At present, installing tunnel entrance hoods at the tunnel port [7, 8, 9, 10, 11, 12, 13] and optimizing the train head shape [14,15] are the main measures to mitigate micro-pressure waves, the basic principle of which is to extend the rise time of the initial compression wave pressure generated by the train entering the tunnel, viz., reducing the pressure gradient of the initial compression wave. Full-scale experimental measured data indicated [16] that when initial compression waves propagate in tunnels, the pressure gradient at the tunnel exit increases, sometimes even to several times the pressure gradient of the initial compression wave. Therefore, the evolutionary characteristics of the initial compression wave in the tunnel are the most crucial factors determining the magnitude of MPWs. Japan's Shinkansen cross-section area is relatively small compared to other countries, making it prone to micro-pressure wave issues. Therefore, Japan was the first country to begin researching such problems. [PERSON] et al. [17] conducted field measurements to research the propagation characteristics of compression waves along Shinkansen slab track tunnels and ballast track tunnels, where the train speed was 215 km/h, slab track tunnels were 3409 m and 8488 m in length, and ballast track tunnels were 16,250 m in length. The study indicated that the pressure gradient increases during the propagation of compression waves in the slab track tunnels, while the pressure gradient of compression waves decreases in the ballast track tunnel. [PERSON] et al. [18,19] and [PERSON] et al. [20,21] conducted full-scale experimental research into the evolutionary characteristics of compression waves in tunnels, in which the train speed was 128-256 km/h and 271-342 km/h and the tunnels were 3.064 km, 26 km, and 9.7 km in length. The research shows that the compression wave pressure gradient and propagation process are reduced in the tunnels' side branches. The train's head shape, the train's speed, and the form of the tunnel entrance hood all affected the propagation evolution process of the compression wave. After propagating through the very long tunnel, initial compression waves of similar pressure gradients but different waveforms will result in significant differences in the magnitude of micro-pressure waves at the tunnel exit. The German high-speed train tunnel cross-section area of 92 m\({}^{2}\) is the largest in the world. Therefore, the problem of micro-pressure waves appeared later in Germany. During the first test of the German ICE on the Nuremberg-Ingolstadt high-speed railway line in 2005 [22], sonic boom phenomena occurred at the exits of the Euerwang (7700 m) tunnel and the Irahull (7260 m) tunnel. Upon analysis, the reason for this was the nonlinear steepening of the compression wave propagation process in slab track tunnels. [PERSON] et al.