## 3 Numerical Studies
### Attenuation Performance of Forest Trees for Different Number of Loads  
The ARS at the different observation points (A, B, and C) distributed on the ground surface along the direction perpendicular to the track (\(y\)-direction) for two carriages eight harmonic point loads, is illustrated in Figure 6a. The tree grid obtains a significant attenuation efficiency near \(f\) = 16 Hz in the low-frequency range (\(f\) \(<\) 20 Hz), and observation point A, which is closer to the tree grid, exhibits higher ARS values. As observation points B and C are located farther from the tree grid, the attenuation effect is weakened. At higher frequencies (\(f\) \(>\) 20 Hz), the differences in attenuation performance among observation points A, B, and C become negligible. Figure 6b presents the ARS values at the observation points (A, D, and E) distributed on the ground surface along the direction parallel to the track (\(x\)-direction). Observation points A and D are located within the effective attenuation range of the tree grid and exhibit higher ARS values. In contrast, observation point E is located near the edge of this range, where the attenuation efficiency is poor in the low-frequency band. Notably, the displacement at point E is even amplified at \(f\) = 14 Hz.  
Figure 7 presents the vertical displacement fields with (a) and without (b) the tree grid on the ground surface for two carriages (eight harmonic point loads) at \(f\) = 17 Hz. It can be observed that the surface wave scattering generated by the interaction load acting on the tree-soil interfaces is superimposed, resulting in a complex wave field. By comparing Figure 7a,b, it is evident that the surface waves propagating through the center of the tree grid along the \(y\)-direction are significantly attenuated. In contrast, those passing through the sides of the tree grid exhibit minimal attenuation, and even slight displacement amplification is observed.  
In contrast to the single-point excitation case, the surface waves generated by multiple loads interact with each other to form a more complex spatial displacement field, leading to more intricate attenuation behavior of the tree grid. Therefore, this section further investigates the attenuation performance under different numbers of vehicle loads. Figure 8 shows the ARS values of observation points A and C for different load numbers (corresponding to one, two, and four carriages, respectively) in the frequency range of 1-80 Hz. It can be observed that point A shows little difference in attenuation efficiency under different numbers of loads, whereas point C exhibits significant variation, especially in the frequency  
Figure 6: ARS values of the vertical displacement at observation points distributed along the direction perpendicular (**a**) or parallel (**b**) to the track for two carriages.  
range of 5-30 Hz. This is because, in the low-frequency range, the wavelength is long, and the superposition of multiple point sources generates more low-frequency surface waves, which can propagate around the tree grid and reach a farther observation point (point C).  
In summary, the tree grid demonstrates considerable attenuation performance under different train loads. As the number of loads (wheel-rail contact forces) increases, the mitigation efficiency is weakened to varying degrees across the entire frequency range (1-80 Hz). Furthermore, the mitigation efficiency at different observation points is also influenced by the number of loads. With an increasing number of carriages, more surface waves--particularly low-frequency components--can propagate around the tree grid to farther observation points, resulting in a reduction in the attenuation performance of the trees.  
Figure 8: ARS values at observation points A (**a**) and C (**b**) for different number of carriages in the frequency range of 1-80 Hz.  
Figure 7: The vertical displacement fields without (**a**) and with (**b**) the tree grid of the ground surface for two carriages at \(f\) = 17 Hz. The dashed lines outline the area of the tree grid.