## 5 Calculated and Measured Results at a Specific Site
### Ground Vibrations from Static or Dynamic Axle Loads and from Measurement  
Now the ground vibrations from train passages are predicted and compared with the measurements. The response at different distances from the track (for a train speed of 160 km/h) and the response for different train speeds (at a distance of 30 m) are shown in Figure 10. Theoretical responses are shown for the dynamic loads from irregularities in Figure 10a,b and for the static loads on a varying track support. In addition to the transfer functions \(H_{S}\) of the layered soil, the axle sequence \(X_{A}\) Equation (22) of the bogies is included in both cases. The wave field for the dynamic loads (Figure 10a) is very similar to the transfer functions in Figure 9c, as the dynamic loads are (almost) constant and at 1 kN for 160 km/h. In addition to the characteristics of the layered soil, the maximum at 12.5 Hz is increased slightly because of the axle sequence spectrum. The different train speeds (Figure 10b) have different axle sequence spectra, so that the regular pattern of Figure 9d is lost. The strict order of the train speeds can only be found at the highest frequencies. A clear maximum occurs only for 125 and 160 km/h. The measured train-induced wave field (Figure 10c) shows the small low-frequency amplitudes of the deeper stiff soil and the strong increase at about 10 Hz. The maximum at 12.5 Hz is more pronounced than for the transfer functions (Figure 9c) as the curves of all distances come close together. The high-frequency amplitudes are approximately those of the upper layer. The curves of different train speeds (Figure 10d) show clearly elevated responses for 125 and 160 km/h, whereas the train speeds below 100 km/h induce only smaller ground vibrations. The ground vibrations from dynamic loads in Figure 10a  
Figure 9: Transfer functions for the soil at site W, distances x = \(\square\) 3, \(\Circ\) 5, \(\Delta\) 10, + 20, \(\times\) 30 m, (**a**) measured from hammer impact, (**b**) calculated for a point load, (**c**) calculated for a standard train load; (**d**) response to constant train loads between 0.15 and 1 kN per one-third octave band representing train speeds of \(v_{T}\) = \(\square\) 160, \(\Circ\) 125, \(\Delta\) 100, + 80, and \(\times\) 63 km/h. and b are clearly smaller than (less than half) the ground vibration from measurements in Figure 10c,d. The ground vibrations from the static loads are presented in Figure 10e,f. In the near field, the quasi-static component can be found between 4 and 6 Hz. All other amplitudes are the result of the varying track support stiffness. The maxima at 12.5 Hz are increased by the pulses of the static axle loads, and they are much higher than for the dynamic loads in Figure 10a,b. The minima are almost at the same frequencies (8 Hz and 20/25 Hz) for the static loads, the dynamic loads and the measurements. The minima are strongest for the static pulse loads, which have an extra reduction in addition to the minima from the axle sequence and the low frequency reduction in the layering. The maximum, on the other  
Figure 10: Comparison of measured and calculated ground vibrations; wavefield at distances x = 3, 5, 10, + 20, \(\times\) 30 m for the maximum train speed of 160 km/h (**a,c,e**), and for different train speeds of \(v_{T}\) = \(\square\) 160, \(\circ\) 125, \(\Delta\) 100, + 80, \(\times\) 63 km/h at a distance of 30 m (**b,d,f**), calculated with geometric track irregularities (**a,b**), measured at site W (**c,d**), and calculated by a varying track support stiffness (**e,f**). hand, is increased by the static pulse loads in addition to the nearby (moderate) maxima of the axle sequence and the resonance of the layering. Because of these effects, the static loads yield the highest near-field amplitudes at 12.5 Hz, and at 30 m, they generate a maximum (Figure 10f), which is clearly higher than the maximum of the dynamic loads (Figure 10b) and nearly reaches the amplitude of the measurements (Figure 10d).