## 3 The Reference System: A Surface Line on a Homogeneous Soil  
The reference system is a surface track on a homogeneous medium stiff soil with the material parameters shear modulus \(G=8\times 10^{7}\) N/m\({}^{2}\), Damping ratio \(D=2\%\), a Poisson ratio of \(\
u=0.33\), and a mass density of \(\rho=2000\) kg/m\({}^{3}\) (resulting in shear and compressional wave velocities of \(v_{S}=200\) m/s and \(v_{P}=400\) m/s). The track is a rigid strip foundation of width \(b=3\) m, see also Table 1 for the standard parameters and the variations. All lose to firm non-cohesive soils (sands, gravels, boulders) and cohesive soils (clays) are included by the chosen range of wave velocities.  
Figure 4a shows the vibration mode at 80 Hz where waves can be observed along the surface as well as down to the interior of the soil. The wavelength at the surface is \(\lambda\approx 2.5\) m which means a wave velocity of \(v\approx 200\) m/s = v\({}_{S}\). According to the deformation pattern, this is the Rayleigh wave of the soil. The downward waves have twice the wavelength and  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & _Symbol_ & _Value_ & _Variation_ \\ \hline
**Soil** & & & \\ Shear modulus & \(G\) & \(8\times 10^{7}\) N/m\({}^{2}\) & \\ Damping ratio & \(D\) & 2\% & \\ Poisson ratio & \(\
u\) & 0.33 & \\ Mass density & \(\rho\) & 2000 kg/m\({}^{3}\) & \\ Shear wave velocities & \(v_{S}\) & 200 m/s & 100, 150, 300, 500 m/s \\ Compressional wave velocities & \(v_{P}\) & 400 m/s & \\ \hline
**Concrete** & & & \\ Young’s Modulus & \(E\) & \(3\times 10^{10}\) N/m\({}^{2}\) & \\ Mass density & \(\rho\) & 2500 kg/m\({}^{3}\) & \\ \hline
**Railway line** & & & \\ Track width & \(b\) & 3 m & \\ Embankment height & & 5 m & \\ Embankment width & & 5 m & \\ Embankment inclination & & 1:1 & 1:2, 2:1 \\ Cut depth & & 5 m & \\ Cut width & & 5 m & \\ Cut inclination & & 1:1 & 1:2, 2:1, \(\infty\) \\ \hline
**Tunnel** & & & \\ Width & & 6 m & 9 m \\ Height & & 6 m & \\ Wall thickness & & 0.5 m & 0.3, 0.7, 1.0 m \\ \hline
**Bridge** & & & \\ Width & 16 m & 6, 11 m \\ Span & 15 m & 5, 10 m \\ Height & 6 m & & \\ Wall thickness & & 0.5 m & \\ Pier dimension & 5 m \(\times\) 5 m & \\ \hline
**Reduction measures** & & & \\ Trench depth & 9 m & & \\ Trench thickness & 0.5 m & & \\ Wall depth & 9 m & & \\ Wall thickness & 0.5 m & & \\ Plate length & 9 m & & \\ Plate thickness & 0.5 m & & \\ Box dimension & 5 m \(\times\) 5 m & & \\ \hline \end{tabular}
\end{table}
Table 1: Standard parameters and variations.  
are compression waves with \(v_{P}\) = 400 m/s. (No reflections occur at the lower boundary indicating that the boundary dampers work well.) The compressional waves are a special effect of the wider excitation by the track (the strip foundation). In case of a concentratred line load (Figure 4b), no compression waves are visible, but instead slower body waves with a shorter wavelength appear, obviously shear waves. The Rayleigh waves at the surface are stronger than for the track load. The reduction of the strip load compared the line load in Figure 5 becomes stronger with frequency, approximately as \(u_{T}/u_{L}=2v_{S}/b\omega\) at higher frequencies [43].  
The frequency-dependent reduction effect of different railway lines and mitigation measures is calculated as the ratio of the (averaged) amplitudes \(v\) of the specific situation (for example with a trench) to the (averaged) amplitudes \(v_{R}\) of the reference system. The reference system with a surface track, which has been described in this section, will be used throughout the text. Only in Section 7, the concentrated line load solution has been used as the reference.  
Figure 4: The vibration mode at 80 Hz for (**a**) a rigid strip foundation (representing the track) and (**b**) a concentrated line load.