## 4 Results and Discussions
### Effect of Train Nose Length on Aerodynamic Pressure  
The nose length of the train refers to the extent of the streamlined section at the head of the train. A streamlined design typically involves shaping the head car to minimize its cross-sectional area, resulting in a smoother profile that leverages aerodynamic principles to decrease air resistance. Previous studies indicate that trains with streamlined noses exhibit superior aerodynamic performance compared to those with square noses [14; 36]. Figure 14 illustrates the aerodynamic pressure peaks on surfaces of sound barriers when high-speed trains with varying nose lengths--3 m, 4.5 m, 6 m, 7.5 m, and 9 m--pass through at distances of 2 m, 2.5 m, and 3 m from the center of the railway track. The comparison reveals that, for a given distance from the sound barrier to the track center, the peak pressure on the sound barrier decreases with increasing train nose length. When the distances between the sound barrier and the train's center are considered to be 2.0 m, 2.5 m, and 3.0 m, respectively, increasing the nose length from 3 m to 9 m will result in peak aerodynamic pressure decreases of 31%, 25%, and 21% on the surface of the sound barrier. This indicates that nose length has a significant effect on the peak aerodynamic pressure on the surface of the sound barrier caused by the running trains. As shown in Figure 15, different lengths of train noses result in varying rates of change in the cross-sectional area along the surface of the train head. The maximum rate of change occurs in the head section with a 3 m nose length, while the minimum rate of change occurs with a 9 m nose length. This streamlined difference leads to reduced air resistance during the travel of the train with a 9 m nose length, resulting in weaker aerodynamic pressure effects on the surface of the sound barrier. This is why trains with longer noses generate smaller aerodynamic pressures when passing through sound barriers. Figure 14: Variation of aerodynamic pressure peak values for different nose lengths. Figure 15: Heads with different nose lengths. At present, some standards [37] utilize the following calculation model to determine the aerodynamic pressure on the vertical noise barrier caused by high-speed trains, as depicted in Equations (1) and (2). \[p_{\mathrm{k}}=\frac{\rho v_{\mathrm{tr}}^{2}}{2}\cdot k\cdot C_{P} \tag{1}\]  
\[C_{P}=\frac{2.5}{\left(a_{\mathrm{g}}+0.25\right)^{2}}+0.02 \tag{2}\]  
where \(p_{\mathrm{k}}\) is the characteristic value of distributed load (Pa), \(\rho\) is the air density taken as \(1.225\) kg/m\({}^{3}\), \(v_{\mathrm{tr}}\) is the train speed (m/s), \(C_{P}\) is the air pressure coefficient, \(k\) is the train shape coefficient, and \(a_{\mathrm{g}}\) is the distance between sound barrier and train center (m). To eliminate the influence of speed, the pressure values are normalized according to Equation (3). The disparity between \(C_{P}\) and \(C_{P}\)' lies in the fact that \(C_{P}\)' represents the actual aerodynamic pressure coefficient, accounting for the train's shape. The \(C_{P}\)' values induced by high-speed trains with various nose lengths (3 m, 4.5 m, 6 m and 7.5 m) at different distances are presented in Table 5.  
\[p_{\mathrm{k}}=\frac{\rho v_{\mathrm{tr}}^{2}}{2}\cdot C_{P}\text{'} \tag{3}\]  
Based on the \(C_{P}\)' values provided above and the correlation between the aerodynamic pressure coefficient and distance, as illustrated in Equation (2), the corresponding \(k\) values for high-speed trains with different nose lengths can be determined, as depicted in Figure 16. It can be observed that as the nose length increases, the train shape coefficient gradually decreases. Figure 17 exhibits the changing trend of the train shape coefficient in relation to the nose length of the train, demonstrating an excellent linear relationship between the two parameters with a coefficient of determination of 0.997.