## 3 Numerical Studies
### Impact of Stochastic Tree Parameters on Attenuation Performance  
In contrast to mechanical metamaterials, the spatial distribution of trees in practice exhibits inherent randomness rather than strict determinism. Therefore, it is necessary to account for the variability in tree properties when evaluating mitigation efficiency. [PERSON] et al. [45] examined the attenuation performance of forest trees exhibiting disordered spatial arrangements, whereas [PERSON] et al. [44] studied tree rows with random geometric pa  
Figure 11: The vertical displacement fields for (**a**) a forward metawedge, (**b**) a reverse metawedge, and (**c**) the case without trees for two carriages at \(f\) = 22 Hz. The dashed lines outline the area of the tree grid. Figure 12: The ARS values at observation points A (**a**) and C (**b**) for the forward and reverse metawedges and the uniform height of trees in the frequency range of 1–80 Hz. Figure 13: The vertical displacement fields of \(x\) = 0 m for (**a**) a forward metawedge and (**b**) a reverse metawedge for two carriages at \(f\) = 22 Hz. rameters. Considering the discrete distribution characteristics of real forest trees in the longitudinal direction, this study further investigates the performance uncertainty of ground vibrations attenuation due to the variability in tree parameters for a two-dimensional tree grid. Specifically, the effects of stochastic fluctuations on tree height, radius, and density are investigated. Assuming that the parameters are uniformly distributed, each sample \(x_{n}\) (which could represent height, radius, or density) is determined by  
\[x_{n}=(1+\varepsilon_{n}\alpha)\overline{x} \tag{25}\]  
In this context, \(\overline{x}\) represents the mean value corresponding to a specific parameter of the tree, as documented in Tables 2 and 3. The term \(\varepsilon_{n}\) indicates a stochastic variable that follows a uniform distribution within the range \([-1,\,1]\), while \(\alpha\), which characterizes the extent of variation, is set to 0.4 in this study. A Monte Carlo simulation with 100 samples is conducted for each of the tree parameters--height, radius, and density--to evaluate the variations in attenuation efficiency. The train load consists of eight harmonic point loads (two carriages), and the other parameters for soil, the track, and forest trees are provided in Tables 1-3. The mean value and confidence interval spanning \([-2\sigma,\,2\sigma]\), where \(\sigma\) denotes the standard deviation, of the ARS evaluated at observation locations A-E (i.e., A (a), B (b), C (c), D (d), and E (e)) are illustrated in Figure 14. These results correspond to the scenario where the tree array is implemented over a uniform elastic half-space, with trunk length \(h_{\mathrm{ij}}\) varying within a range characterized by \(\alpha\) = 0.4. For comparison, results obtained under non-stochastic (fixed) parameter conditions are also included. The statistical ARS outcomes obtained from 100 realizations considering height variability in trees display a tendency that aligns well with the deterministic case. These findings indicate that the designed tree grid maintains substantial attenuation performance within the frequency ranges of 8-32 Hz and 63-80 Hz, even when considering the randomness in tree height. In addition, changes in tree height will lead to evident fluctuations in attenuation efficiency. For point A, the randomness of tree height leads to a large fluctuation in attenuation efficiency in the low-frequency range (8-16 Hz), reaching up to 1.7 dB, while the fluctuation in the high-frequency range (30-60 Hz) is less than 0.5 dB, indicating good stability. Points D and E, which are also close to the tree grid, exhibit a similar pattern. In contrast, the fluctuation amplitudes at observation points B and C, which are comparatively farther from the tree grid, are larger in the low-frequency range and smaller in the high-frequency range.