## 2 Methods of Calculation
\begin{table}
\begin{tabular}{l l} \hline \(EI\) & \multicolumn{1}{c}{bending stiffness of the track beams} \\ \(f\) & \multicolumn{1}{c}{frequency} \\ \(F\) & \multicolumn{1}{c}{force} \\ \(F_{S}\) & \multicolumn{1}{c}{force on the soil} \\ \(F_{T}\) & \multicolumn{1}{c}{force on the track} \\ \(F_{V}\) & \multicolumn{1}{c}{force on the vehicle} \\ \(F_{S^{\prime}}\) & \multicolumn{1}{c}{force on the soil per track length} \\ \(h_{B}\) & \multicolumn{1}{c}{height of the ballast} \\ \(H_{TS}\) & \multicolumn{1}{c}{force transfer of the track–soil system} \\ \(H_{VT}\) & \multicolumn{1}{c}{force transfer of the vehicle–track interaction} \\ \(H_{VS}\) & \multicolumn{1}{c}{total force transfer of the vehicle–track–soil system} \\ \(k_{K}\) & \multicolumn{1}{c}{stiffness} \\ \(k_{B}\) & \multicolumn{1}{c}{static ballast stiffness} \\ \(k_{M}\)\({}^{\prime\prime}\) & \multicolumn{1}{c}{stiffness per area of a ballast mat} \\ \(k_{R}\) & \multicolumn{1}{c}{rail pad stiffness} \\ \(k_{U}\) & \multicolumn{1}{c}{stiffness of the sleeper pad} \\ \(k_{T}\) & \multicolumn{1}{c}{dynamic stiffness of the track} \\ \(K_{V}\) & \multicolumn{1}{c}{dynamic stiffness of the vehicle} \\ \(\mathbf{K_{TS}}\) & \multicolumn{1}{c}{stiffness matrix of the track–soil system} \\ \(m_{IV}\) & \multicolumn{1}{c}{mass of the wheelest} \\ \(\mathbf{N_{TS}}\) & \multicolumn{1}{c}{inverse stiffness matrix of the track–soil system} \\ \(\mathbf{T}\) & \multicolumn{1}{c}{transfer matrices} \\ \(u\) & \multicolumn{1}{c}{displacement} \\ \(u_{T}\) & \multicolumn{1}{c}{track displacement} \\ \(\mathbf{u}\) & \multicolumn{1}{c}{displacement vector} \\ \(\mathbf{u_{T}}\) & \multicolumn{1}{c}{displacement vector of the track} \\ \(v\) & \multicolumn{1}{c}{velocity, particle velocity of the soil} \\ \(v_{B}\) & \multicolumn{1}{c}{wave velocity of the ballast} \\ \(v_{I}\) & \multicolumn{1}{c}{ground vibration velocity for the isolated track} \\ \(v_{U}\) & \multicolumn{1}{c}{ground vibration velocity for the un-isolated track} \\ \(x\) & \multicolumn{1}{c}{coordinate across the track, distance from the track} \\ \(y\) & \multicolumn{1}{c}{coordinate along the track} \\ \(\mathbf{z}\) & \multicolumn{1}{c}{state vector (\(F\),\(u\))} \\ \(\omega\) & \multicolumn{1}{c}{circular frequency, \(2\pi f\)} \\ \(\xi\) & \multicolumn{1}{c}{wavenumber} \\ \hline \end{tabular}
\end{table} 
Table 1: Symbols. where the dynamic stiffness \(K_{S}(\omega,\xi)\) of the layered soil is calculated from an integration in wavenumber domain; see the details in [18]. The transfer function of a support section is achieved by multiplying the transfer functions of all support elements (for example, sieeper pad, ballast, and ballast mat; see Figure 1) as  
\[\mathbf{z_{1}}=\mathbf{T_{1}}\mathbf{T_{2}}\mathbf{T_{3}}\mathbf{z_{3}}= \mathbf{T}\mathbf{z_{3}}. \tag{8}\]  
The transfer matrix \(\mathbf{T}\) is transformed into the stiffness matrix \(\mathbf{K_{TS}}\) of the track-soil system as  
\[ \begin{bmatrix}F_{1}\\ -F_{2}\end{bmatrix} =\frac{1}{T_{21}} \begin{bmatrix}T_{11}&-1\\ -1&T_{22}\end{bmatrix}\begin{bmatrix}u_{1}\\ u_{2}\end{bmatrix} =\mathbf{K_{TS}}\mathbf{u} \tag{9}\]  
(note that the sign definition is different for \(F_{2}\), namely, \(F_{2}\) is in the same direction as \(u_{2}\) for the stiffness matrix). Finally, the behaviour of this track-soil system is described by the following equation in the frequency-wavenumber domain  
\[\mathbf{K_{TS}}(\xi,\omega)\mathbf{u_{T}}(\xi,\omega)=\mathbf{F_{T}}(\omega)= \begin{bmatrix}F_{\mathrm{T}}(\omega)\\ 0\end{bmatrix}  \tag{10}\]  
where \(\xi\) is the wavenumber along the track axis.