## 1 Introduction
[23] took data from a high-speed train entering the 7700 m Euerwang tunnel at 330 km/h for field analysis. The research showed that due to the compression wave propagation in the tunnel, the compression wave pressure gradient increased, but installing acoustical absorbers can inhibit steepening in the tunnel while also reducing the compression wave pressure amplitude. [PERSON] et al. [24] conducted field tests to research the propagation characteristics of compression waves in Wuhan's Guangzhou high-speed railway tunnels. The research shows that when the train speed is 200 km/h, there is no phenomenon of nonlinear steepening observed during the propagation of compressionwaves in the tunnel. [PERSON] et al. [25] conducted experimental tests to investigate the influence of the unsteady boundary layer on the propagation characteristics of compression waves. The data obtained from full-scale experiments are the most accurate, but the cost is high; considering the limitations of completed lines and tunnels of high-speed trains, full-scale experiments could not achieve research on the propagation characteristics of initial compression waves under different tunnel entrance hoods for high-speed trains of 400 km and above. [PERSON] et al. [26] and [PERSON] et al. [27] used one-dimensional theory to investigate the evolutionary characteristics of the initial compression waves. The research findings indicate that the propagation characteristics of compression waves not only depend on the pressure amplitude and maximum pressure gradient of the initial compression waves but also rely on the waveform of the initial compression waves. [PERSON] et al. [28] determined the critical tunnel length for different train velocities and friction coefficients based on the steepening rate, which is important for the study of compression wave propagation. [PERSON] et al. [29] investigated the formation and propagation of weak shock waves. The results show that the formation distance of the weak shock wave decreases with the increase in the maximum pressure gradient of the initial compression wave. The propagation of compression waves in short tunnels can be addressed using three-dimensional simulation methods, but with high computational costs and demanding requirements [30]. In studies of the propagation of compression waves within very long tunnels, the one-dimensional flow model is commonly chosen, which can provide accurate numerical simulation results and high computational efficiency [31; 32]. The one-dimensional flow model requires reasonable initial compression wave waveforms provided by a full-scale experimental or three-dimensional flow model. We used the initial compression wave waveform as the entrance boundary condition for the one-dimensional flow model. The unsteady friction model is crucial for the numerical simulation of the propagation process of initial compression waves. [PERSON] [33] and [PERSON] et al. [34] have made outstanding contributions to the unsteady friction model. In 1968, [PERSON] proposed a weighted function model applicable to laminar transient flows. In 1975, [PERSON] [35] proposed an approximate expression for the original weighted function to reduce the computational time of [PERSON]'s transient friction model. In 1995, [PERSON] et al. extended [PERSON]'s laminar transient friction model to high Reynolds number transient flows. In 2002, [PERSON] et al. [36] researched the propagation characteristics of compression waves based on the transient friction model. In 2004, [PERSON] et al. [37] proposed a transient friction model for fully rough pipe flows. In 2002, [PERSON] et al. [38] discovered an accurate method to approximate the Vardy-Brown transient friction model. The research indicated that approximate methods provided higher computational efficiency and better accuracy. Therefore, this paper adopts [PERSON]'s transient friction model.