## 4 Conclusions  
In this study, based on the IDDES SST \(k-\omega\) turbulence model, a three-dimensional numerical model was used to transiently simulate the initial compression wave induced by a 400 km/h high-speed train entering a tunnel. This work focused on analyzing the spatial and temporal distribution of the initial compression wave and its one-dimensional planar wave characteristics. The conclusions are as follows:  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Case** & **Offset Running** & **Center Running** \\ \hline Simulation (Pa) & 2154.11 & 2075.30 \\ Hara’s Formula (Pa) & 2142.71 & \\ Relative Difference (\%) & \(-0.53\) & 3.25 \\ \hline \hline \end{tabular}
\end{table}
Table 2: The comparison of the initial wavefront amplitudes between the formula prediction and the simulation results.  
Figure 23: Comparison of the initial compression waves at point 3 in the 7_Di_ cross-section: (a) pressure–time history; (b) pressure gradient–time history.  
(1) Before the train enters the tunnel, the pressure at the measurement points inside the tunnel begins to rise. This is because the pressure disturbances generated by the train's movement in the open air propagate into the tunnel. As the part of the streamlined nose with the maximum cross-sectional area change rate enters the tunnel, the resulting compression wave causes the largest pressure gradient at the measurement point;  
(2) Pressure iso-surfaces can illustrate the process of the compression wave transitioning from three-dimensional characteristics to a one-dimensional planar wave. The pressure distribution near the train's nose inside the tunnel exhibits significant three-dimensional features. When using lower pressure values, the pressure iso-surfaces can represent the compression wave in front of the train. The compression wave close to the train is oblique, while the compression wave farther from the train becomes a one-dimensional planar wave;  
(3) The initial compression wave captured at the measurement points at a distance of 5_Di_ (_Di_ represents the equivalent diameter of the tunnel) from the tunnel portal has been fully characterized by one-dimensional features, which can provide a boundary input for the propagation of the initial compression wave towards the tunnel exit;  
(4) Regarding the offset effect of the train, the high-pressure iso-surfaces show significant differences between the offset running and center running cases. Due to the constraints of the ground and tunnel walls, the high-pressure region exhibits an irregular spherical shape in offset running, while in the center running case, the tunnel walls impose no restriction on the development of the pressure iso-surface, the iso-surface gradually develops symmetrically from the center to both sides. Compared to the offset running case, the initial compression wave amplitude and pressure gradient amplitude induced by central running are reduced by 3.66% and 6.87%, respectively.  
Conceptualization, Y.M. and X.H.; methodology, Y.M. and X.H.; software, Q.S. and Z.W.; validation, Z.W.; formal analysis, X.H.; investigation, Y.M. and X.H.; writing--original draft preparation, X.H. and Z.W; funding acquisition, Y.M.; writing--review and editing, X.H. All authors have read and agreed to the published version of the manuscript.  
This work was financially supported by the National Key R&D Program of China (Grant No. 2022 YFB2603400), and the Postdoctoral Fellowship Program of CPSF (Grant Number GZC20240628).  
Not applicable.  
The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.  
The authors declare no conflicts of interest.