## 3 Numerical Model  
The numerical model PiP (\"Pipe in Pipe\") [16; 17; 18; 19] was used for the generation of transfer functions between the tunnel invert and the ground surface. In brief, the PiP model is a semi-analytical model that accounts for the generation and propagation of vibration from an underground railway tunnel. The model performs the calculations in two stages. In the first stage, vibration at the tunnel-soil interface is calculated based on a model of two concentric pipes (and therefore the name of Pipe in Pipe), with the inner pipe representing a tunnel wall and the outer pipe representing the surrounding infinite domain of soil (with an internal radius equal to the external radius of the tunnel and external radius tending to infinity). The equations of motions for a linear elastic isotropic media are solved in the wavenumber-frequency domain in a cylindrical coordinate system, producing an efficient calculation for the displacements at the tunnel-soil interface [16; 17; 18]. In the second stage, fictitious forces are generated to produce the same displacement field but in a full-space media that does not contain the tunnel. These forces are then introduced in a half-space or layered-half-space media to calculate the displacements in the far-field. The model assumes that the displacement field at the tunnel-soil interface is not influenced by details in the \begin{table}
\begin{tabular}{c c c} \hline \hline
**Condition** & \(a\) [mm2*(1/m)2] & \(b\) [1/m] \\ \hline Worst & \(9.39\times 10^{-1}\) & \(6.89\times 10^{-2}\) \\ \hline Average & \(1.31\times 10^{-2}\) & \(2.94\times 10^{-2}\) \\ \hline Best & \(1.90\times 10^{-4}\) & \(9.71\times 10^{-3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: [PERSON]’s proposal: parameters values for different track conditions. Figure 3: Power spectral density function of vertical unevenness profile considering different proposals. far-field such as layering or existence of free-surface. The model was verified against a FE-BE model and good matching was observed. Subsequently, the dynamics of both the train and the track are addressed by integrating the tunnel-ground model with the train and track models. The track is modeled using a semi-analytical approach, where the rails and the slab are represented as infinitely long Bernoulli-Euler beams, while the rail pads are simulated as continuously supported spring-dashpot systems (see Figure 4). Consequently, the response of the track structure in the frequency/wavenumber domain, due to a point load \(\widetilde{P}_{z}\) acting on the rails, is determined through the following system of equations:  
\[ \begin{bmatrix}E_{r}I_{r}K_{1}^{4}-\omega^{2}m_{r}+k_{rp}^{*}&-k_{rp}^{*}\\ -k_{rp}^{*}&k_{rp}^{*}+E_{s}I_{s}k_{1}^{4}-\omega^{2}m_{s}+k_{eq}\end{bmatrix} \begin{cases}\widetilde{u}_{r}(k_{1},\omega)\\ \widetilde{u}_{s}(k_{1},\omega)\end{cases} = \begin{cases}\widetilde{P}_{z}(k_{1 },\omega)\\ 0\end{cases}  \tag{15}\]  
where \(\widetilde{u}_{r}\) and \(\widetilde{u}_{s}\) represent the vertical displacement of the rails and slab, respectively; \(EI\) and \(m\) represent the bending stiffness and mass, respectively; with the subscript '\(r\)' and '\(s\)' denoting the rails and slab, respectively. The equivalent complex stiffness of the rail pads, \(k_{rp}^{*}\), is defined as \(k_{rp}^{*}=k_{p}+i\omega c_{p}\), where \(k_{p}\) and \(c_{p}\) represent the equivalent stiffness and damping of the rail pads, respectively. \(k_{eq}\) is the equivalent stiffness of the foundation, evaluated by considering the displacement field at the tunnel invert computed using the PiP model. The consideration of the train-track interaction is essential in problems involving the generation of vibrations due to railway traffic. As initially discussed, the load applied by the train on the track can be divided into two components: (i) the static load, (ii) the dynamic load.