## 5 Conclusions  
In this study, the aerodynamic pressure on sound barriers induced by high-speed trains was computed using CFD simulations. Through a comparison between the simulation results and field measurements extracted from the literature, it was observed that there was a notable concurrence in the peak pressure data and development with time, indicating the efficacy of the simulation method. Subsequently, the essential characteristics of pressure variation on sound barriers and pressure distribution along the vertical direction were explored. Additionally, the impact of distance to the train center and the train nose length on aerodynamic pressure on sound barriers was analyzed. The following conclusions were drawn:  
1. When a high-speed train passes through the sound barrier, it produces obvious \"head wave\" and \"tail wave\" effects, and the peak value of the \"head wave\" is obviously larger than the peak value of the \"tail wave\"; in the vertical direction of the sound barrier, the pressure reaches the peak value at the bottom, decreases gradually from the bottom to the top, and decreases rapidly when it is near the top of the sound barrier.
2. As the distance between the sound barrier and the center of the train gradually increases, the peak aerodynamic pressure on the surface of the sound barrier gradually decreases, showing a non-linear development trend.
3. The nose length has a great influence on the aerodynamic force of the sound barrier, the aerodynamic pressure generated by a train with a long nose length is smaller than that of a train with a short nose length, and the formula for calculating the aerodynamic pressure generated by a high-speed train on the sound barrier with respect to the nose length has been established.  
This study focuses solely on the distribution of aerodynamic pressure on vertical sound barriers when a train passes. A limitation is that it does not address other types of sound barriers, such as semi-enclosed or fully enclosed sound barriers.  
The contributions of all the authors to this manuscript are as follows: Conceptualization, [PERSON]; methodology, J.J.; software, J.J.; validation, J.J. and D.L.; formal analysis, J.J.; writing--original draft preparation, J.J.; writing--review and editing, J.J. and D.L.; supervision, Y.T.; project administration, Y.T. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the Natural Science Foundation of China, grant number 51378104, the \"One belt, one road\" innovation cooperation project under the policy guidance plan of Jiangsu Province, grant number BZ2021011, and the Fundamental Research Funds for the Central Universities, grant number 2242022k30030, 2242022k30031.  
Not applicable.  
The data presented in this study are available on request from the corresponding author.  
The authors declare no conflicts of interests.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Working Conditions** & **Simulated Value (Pa)** &
\begin{tabular}{c} **Calculated Value by** \\ **Equation (5) (Pa)** \\ \end{tabular} & **Relative Error** \\ \hline \(a_{\text{g}}=2.0\) m, \(x=6.0\) m & 1725 & 1706 & 1.1\% \\ \(a_{\text{g}}=3.0\) m, \(x=7.5\) m & 797 & 781 & 2.0\% \\ \(a_{\text{g}}=2.5\) m, \(x=9.0\) m & 983 & 971 & 1.2\% \\ \hline \hline \end{tabular}
\end{table}
Table 6: The comparison of simulation pressure values and the pressure calculated using Equation (5).