## 3 Numerical Studies  
This section conducts a systematic investigation into the attenuation effectiveness of forest trees in mitigating ground vibrations induced by railway traffic. Existing research has primarily addressed the attenuation effect of forest trees on wavefields induced by a single harmonic point load [44; 45]. However, the train is composed of multiple carriages, each of which contains multiple bogies and multiple wheelsets, thus producing a series of wheel-rail contact forces. The wave field generated by the superposition of multiple point sources will be different due to the number of wheel-rail contact forces, which could influence the vibration attenuation effectiveness of forest trees. Therefore, this section extends the investigation to examine the attenuation efficiency of forest trees under multiple point source excitations. Figure 4 shows the train loads distribution of two carriages. Each carriage consists of two bogies and four wheelsets; therefore, two carriages generate eight harmonic point loads, which are denoted as \(P_{1}\), \(P_{2}\), \(\ldots\), \(P_{8}\). The length of a single carriage is \(l_{c}=24.5\) m, and the distances between wheelsets are \(w_{a}=2.5\) m, \(w_{b}=14.875\) m.  
As illustrated in Figure 4, the train model consists of two carriages with a total of eight wheelsets generating harmonic vertical loads. The longitudinal span of these loads is 44.375 m, which exceeds the 40 m width of the surface tree array along the \(x\)-axis. This ensures that the entire array is subjected to the moving train load, and thus the vibration attenuation characteristics within the region of interest can be accurately assessed based on this loading scheme. In this section, a typical ballasted track is used, and the corresponding parameters can be found in Table 1. The material properties of the soil and forest trees can be found in Table 2. In conjunction with Figure 1, the geometric constants of the forest are summarized in Table 3. \begin{table}
\begin{tabular}{l l} \hline \hline
**Parameter** & **Value** \\ \hline Mass of the rail beam per unit length of the track, \(m_{\text{R}}\) & 120 kg/m \\ Mass of sleepers per unit length of the track, \(m_{\text{S}}\) & 490 kg/m \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters for the ballast track. Figure 4: Schematic diagram of the train load distribution. \begin{table}
\begin{tabular}{l l} \hline
**Parameter** & **Value** \\ \hline \hline \multicolumn{2}{l}{Lateral spacing of the tree grid (\(y\)-axis direction), \(a_{\mathrm{ly}}\)} & 2 m \\ \multicolumn{2}{l}{Longitudinal spacing of the tree grid (\(x\)-axis direction), \(a_{\mathrm{ly}}\)} & 2 m \\ \multicolumn{2}{l}{Distance from the track, \(D_{\mathrm{t}}\)} & 10 m \\ \multicolumn{2}{l}{Number of lateral trees, \(N_{\mathrm{y}}\)} & 20 \\ \multicolumn{2}{l}{Number of longitudinal trees, \(N_{\mathrm{x}}\)} & 20 \\ \multicolumn{2}{l}{Radius of the tree, \(r\)} & 0.25 m \\ \multicolumn{2}{l}{Height of the tree, \(h\)} & 14 m \\ \hline \end{tabular}
\end{table} 
Table 3: Geometric parameters of the soil. \begin{table}
\begin{tabular}{l l} \hline
**Parameter** & **Value** \\ \hline \hline \multicolumn{2}{l}{Mass of ballast per unit length of the track, \(m_{\mathrm{B}}\)} & 1200 kg/m \\ \multicolumn{2}{l}{Bending stiffness of the rail beam, \(EI\)} & 1.26 \(\times\) 10\({}^{7}\) N/m\({}^{2}\) \\ \multicolumn{2}{l}{Rail pad stiffness, \(k_{\mathrm{P}}\)} & 3.5 \(\times\) 10\({}^{8}\) N/m\({}^{2}\) \\ \multicolumn{2}{l}{Ballast stiffness per unit length of the track, \(k_{\mathrm{B}}\)} & 3.50 \(\times\) 10\({}^{8}\) N/m\({}^{2}\) \\ \multicolumn{2}{l}{Width of the ballast, \(2a\)} & 2.70 m \\ \multicolumn{2}{l}{Rail pad loss factor, \(\eta_{\mathrm{R}}\)} & 0.15 \\ \hline \end{tabular}
\end{table} 
Table 1: _Cont._  
Figure 5: Layout of the observation points (top-down view).