## 3 Construction of the Noise Prediction Model
### Equivalent Source Strength
A least-squares optimization framework was implemented with \(L_{w1}\)-\(L_{w5}\) as the optimization parameters, aiming to minimize the total mean-squared error between the measured and predicted instantaneous SPLs at \(R\). The optimization process was conducted with three distinct step sizes (10, 1, and 0.1 dB(A)) and corresponding variation ranges (\(\pm\)30, \(\pm\)10, and \(\pm\)5 dB(A)) for parameter adjustment. By systematically exploring all possible values within the 65-125 dB(A) range, the total mean squared error between the measured and predicted instantaneous SPLs at \(R\) was calculated for each combination of parameters. The \(L_{w1}\)-\(L_{w5}\) values that achieved the optimization objective were identified as the optimal sound power levels per unit length. This procedure was repeated for Magley train operating speeds of 235, 300, and 430 km/h to determine the optimal speed-dependent \(L_{w1}\)-\(L_{w5}\) values, as shown in Table 1.  
The optimal values of \(L_{w1}\)-\(L_{w5}\) at different train operating speeds can be used in Equation (11) to calculate the instantaneous SPL at \(R\). As can be seen from Table 2, further calculation results show that the equivalent continuous A-weighted sound level \(L_{Aeq}\) was obtained during the train passing time, and the absolute error between the measured \(L_{Aeq,E}\) and predicted \(L_{Aeq,P}\) values was <0.4 dB(A). These error metrics confirm the high accuracy of the \(L_{w1}\)-\(L_{w5}\) values derived through the least-squares optimization method, validating the effectiveness of the proposed line source model for Magley train noise prediction across multiple operating speeds. After \(L_{w1}\)-\(L_{w5}\) were solved, according to Equation (11), the noise source intensities \(L_{p1}\)-\(L_{p5}\) when the sound sources \(S_{1}\)-\(S_{5}\) acted alone were solved. Figure 13 shows the variation in the sound power levels \(L_{w1}\)-\(L_{w5}\) per unit length of the line sources with train operating speed, where the reference speed \(v_{r\!rf}\) was 235 km/h. It can be seen from Figure 13 that the sound power levels per unit length of the line sources have a good linear relationship with the normalized velocity logarithm. The prediction formulae for \(L_{w1}\)-\(L_{w5}\) can be expressed as  
\[ \begin{cases}L_{w1}=69.8\log_{10}(v/v_{w\!rf})+76.3\\ L_{w2}=27.6\log_{10}(v/v_{w\!rf})+108.1\\ L_{w3}=41.0\log_{10}(v/v_{w\!rf})+110.2\\ L_{w4}=35.4\log_{10}(v/v_{w\!rf})+111.5\\ L_{w5}=67.6\log_{10}(v/v_{w\!rf})+100.4\end{cases}  \tag{12}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Train Operating Speed (km/h)** & \(L_{w1}\) & \(L_{w2}\) & \(L_{w3}\) & \(L_{w4}\) & \(L_{w5}\) \\ \hline
235 & 76.3 & 108.2 & 110.1 & 111.5 & 100.5 \\
300 & 83.6 & 110.8 & 114.7 & 115.3 & 107.3 \\
430 & 94.6 & 115.4 & 120.9 & 120.8 & 118.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Optimal values of \(L_{w1}\)-\(L_{w5}\) at different train operating speeds (dB(A)). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Train** & **Measured** & **Predicted** & \(L_{Aeq}\) \\
**Operating Speed (km/h)** & \(L_{Aeq,E}\) & \(L_{Aeq,P}\) & **Error**\(\mid\)A\(\mid\) \\ \hline
235 & 87.6 & 87.9 & 0.2 \\
300 & 91.8 & 92.2 & 0.4 \\
430 & 98.1 & 98.6 & 0.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Measured versus predicted SPL errors at different train operating speeds (dB(A)).