## Abbreviations \begin{tabular}{l l}
**a** & amplitude vector \\ \(b\) & width of the track \\
**C** & damping matrix of the multi-beam track model \\ d\(f\) & frequency step \\ d\(f_{T}\) & one-third octave frequency band \\ d\(t\) & time step \\ d\(y\) & load step \\
**e\({}_{1}\)** & base vector (top of the track) \\
**e\({}_{n}\)** & base vector (bottom of the track) \\
**E\(I_{j}\)** & bending stiffness of track beam \(j\) \\
**E\(I\)** & matrix of bending stiffnesses \\ \(f\) & frequency \\ \(f_{T}\) & one-third octave frequency \\ \(f_{0}\) & cut-off frequency of the track \\
**F** & force, axle load \\ F\({}^{\prime}\) & force per track length \\ \(F_{0}\) & static axle load \\ \(F_{S}\) & force on the soil \\ \(F_{T}\) & force on the track \\
**F\({}^{\prime}\)T** & force vector (forces on different track beams) \\ \(h\) & height of the soil layer \\ \(H_{zz}\) & transfer function of the homogeneous half-space in wavenumber domain \\ \(H_{H}\) & transfer function of the homogeneous half-space in space domain \\ \(H_{F}\) & filter function of the track \\ \(H_{kT}\) & transfer function for the stiffness irregularity \\ \(H_{sT}\) & transfer function for the stiffness variation of the track support \\ \(H_{S}\) & transfer function of the soil for a point load \\ \(H_{SS}\) & transfer function of the soil for a strip load \\ \(H_{T}\) & force transfer of the track \\ \(H_{V}\) & vehicle-track transfer function \\ \(H_{V}\)* & vehicle-track transfer function for stiffness-induced irregularities \\ \(k_{P}\) & stiffness of the sleeper pad \\ \(k_{R}\) & stiffness of the rail pad \\ \(k_{S}\) & dynamic stiffness of the track support \\ \(k_{S}\)* & dynamic stiffness of the Winkler support \\ \(k_{0}\) & average static stiffness of the track support \\ \(k_{I}\) & stiffness variation of the track support \\ \(K_{T}\) & dynamic stiffness of the track at the wheel-rail contact point \\ \(K_{V}\) & dynamic stiffness of the vehicle at the wheel-rail contact point \\ \(\bf{K_{0}}\) & average static stiffness of the track \\ \(\bf{K_{1}}\) & stiffness variation of the track \\ \(\bf{K}\) & stiffness matrix of the multi-beam track model \\ \(\bf{K_{S}}\) & dynamic stiffness matrix of the soil \\ \(\bf{K_{T}}\) & stiffness matrix of the track \\ \(\bf{K_{TS}}\) & stiffness matrix of the track-soil system \\ \(I_{B}\) & half the distance between two axles in a bogie \\ \(m_{W}\) & mass of the wheelset \\ \(m_{j}\) & mass per length of track beam \(j\) \\ \(\bf{M}\) & mass matrix \\ \(n\) & number of loading points \\ \(n_{B}\) & number of bogies in a train \\ \(p\) & stress distribution across the track \\ \(q\) & percentage of the variation in the support stiffness \\ \(s\) & (geometric) irregularity \\ \end{tabular}   
Appendix A Frequency-Wavenumber Domain Methods and Their Approximation for Dynamic Track and Soil Behaviour  
The dynamic behaviour of a layered soil is algebraically calculated in the frequency-wavenumber domain. The resulting vertical compliance \(H_{zz}\) of the soil surface is used twofold. The compliance for a point-load (the wave field as a function of distance \(r\) and \(\omega\)) is established by the integral [35]  
\[H_{S}(r,\omega)=\frac{u(r,\omega)}{F(\omega)}=\frac{1}{2\pi}\int_{0}^{\infty}H_{ zz}(\xi,\omega)J_{0}(\xi r)\xi d\xi \tag{1}\]  
where \(J_{0}\) is the Bessel function. The transfer function \(H_{S}(\omega\),\(r\)) of the wave propagation through a layered soil can be approximated by the transfer function \(H_{H}(\omega\),\(r\),\(\wp_{S})\) of the homogeneous soil and the frequency-dependent wave velocity \(\mathrm{v_{S}}(\omega)\) (the dispersion of the soil) [40]. Figure 11 shows the good agreement between the exact and approximate transfer functions for the wave propagation in a soft layer on a stiffer half-space.