## 2 Semi-Analytical Solution for the Coupled Track-Ground-Tree Grid System
### Coupling of the Track to the Tree Grid-Ground System  
In this study, the rail and sleepers are idealized as infinitely long Euler-Bernoulli beams, while the rail pads and ballast are represented as continuously distributed elastic springs, as depicted in Figure 3.  
Within the frequency-wavenumber domain, the motion equations of the track system subjected to a harmonic load \(\hat{P}(\omega)e^{i\omega t}\delta(x)\) (positioned at \(x=0\)) are expressed in the following matrix equation:  
\[ \begin{bmatrix}EIk_{x}^{4}-\omega^{2}m_{R}+k_{P}&-k_{P}&0\\ -k_{P}&k_{P}+k_{B}-\omega^{2}(m_{S}+m_{B}/3)&-(k_{B}+\omega^{2}m_{B}/6)\\ 0&-(k_{B}+\omega^{2}m_{B}/6)&k_{B}-\omega^{2}m_{B}/3\\ \end{bmatrix}\begin{bmatrix}\hat{u}_{r}\\ \hat{u}_{sb}\\ \hat{u}_{sav}\\ \hat{u}_{sav}\\ \end{bmatrix} = \begin{cases}2\pi\hat{P}(k_{x},\omega)\\ 0\\ -\hat{F}_{sav}(k_{x},\omega)\\ \end{cases}  \tag{16}\]  
Figure 3: Illustration of the ballast track system and the reference coordinate framework used in the model. where \(\omega\) signify the angular frequency of the load, respectively, while \(EI\) denote the bending stiffness of the rail per unit length. \(m_{R},m_{S},m_{B}\) denote the mass of the rail, sleepers, and ballasts per unit length. \(k_{P},k_{B}\) denote the spring stiffness of the rail pads and ballasts per unit length. \(\hat{\overline{u}}_{r},\hat{\overline{u}}_{sb}\) represent the vertical displacements of the rail and sleeper. The vertical displacement \(\hat{\overline{u}}_{sw}\) and the corresponding interaction force \(\hat{\overline{F}}_{sw}\) escribe the dynamic response at the interface between the substructure and the track; specifically, it is located along the central axis of the railway. At this interface, the load is assumed to act perpendicularly and be evenly distributed along the longitudinal direction \(x\), spanning the interval \(y=-a\) to \(y=a\). The transverse \(y\)-axis is defined as perpendicular to the track direction, as depicted in Figure 3. The interaction between the rail and the supporting ground is governed by enforcing displacement compatibility and ensuring force balance at the shared interface, as described below:  
\[u_{g}(x,y=0,z=0,t)=u_{sw}(x,t) \tag{17}\]  
\[p_{z}(x,y,z=0,t)=\left\{ \begin{array}{c}F_{sw}(x,t)/2a\,|x|\leq 2a\\ 0,|x|>2a\end{array} \right. \tag{18}\]  
where \(u_{g}\) represents the response of the soil and \(p_{z}\) corresponds to the normal stress. Applying the Fourier transform described in Equation (2) and substituting it into Equations (17) and (18) leads to  
\[ \begin{array}{c}\hat{\overline{u}}_{sw}(k_{x},\omega)=\frac{1}{2\pi}\int \limits_{-\infty}^{+\infty}\hat{\overline{u}}_{g}(k_{x},k_{y},\omega)dk_{y}\\ \hat{\overline{p}}_{z}(k_{x},k_{y},\omega)=\frac{e^{\hat{k}_{y}\omega}a-e^{- ik_{y}a}}{2 ik_{y}a}\hat{\overline{F}}_{sw}(k_{x},\omega)\end{array}  \tag{19}\]  
The vertical displacement of the ground, as derived from Equation (8), is given by  
\[\hat{\overline{u}}_{g}(k_{x},k_{y},\omega)=\sum\limits_{j}^{R-\text{modes}} \frac{\hat{\overline{u}}_{g}}{2\hat{\overline{u}}_{x}+k_{y}^{2}-k_{j}^{2}} \sigma_{zj}^{0}\sigma_{zj}^{0} \tag{20}\]  
Combining Equation (21) with Equation (16) leads to  
\[ \begin{array}{c}\hat{\overline{u}}_{sw}(k_{x},\omega)=\hat{\overline{H}}_{ Ig}(k_{x},\omega)\hat{\overline{F}}_{sw}(k_{x},\omega)\\ \hat{\overline{H}}_{lg}(k_{x},\omega)=\sum\limits_{j}^{R-\text{modes}}\frac{ e^{-ia\sqrt{k_{j}^{2}-k_{x}^{2}}}}{2a(k_{j}^{2}-k_{x}^{2})}\sigma_{zj}^{0}\sigma_{zj}^{0} \end{array}  \tag{21}\]  
Substitute Equation (21) into Equation (16) to obtain  
\[\left\{ \begin{array}{c}\hat{\overline{u}}_{r}(k_{x},\omega)\\ \hat{u}_{sb}(k_{x},\omega)\\ \hat{u}_{sw}(k_{x},\omega)\end{array} \right\}= \tag{22}\]  
\[\left\{ \begin{array}{ccc}EIk_{x}^{4}-\omega^{2}m_{R}+k_{P}&-k_{P}&0\\ -k_{P}&k_{P}+k_{B}-\omega^{2}(m_{S}+m_{B}/3)&-(k_{B}+\omega^{2}m_{B}/6)\\ 0&-(k_{B}+\omega^{2}m_{B}/6)&k_{B}-\omega^{2}m_{B}/3+1/\hat{\overline{H}}_{lg }(k_{x},\omega)\end{array} \right\}^{-1}\cdot\left\{ \begin{array}{c}2\pi \hat{\overline{P}}(k_{x},\omega)\\ 0\\ 0\end{array}