## 4 Theoretical Results
### The Effect of Axle Pulses on a Randomly Varying Track Support Stiffness  
Next, the train was run over an irregular track support and the ground vibration spectra are shown in Figure 6d-f. Once again, the passage of static loads generates the quasi-static response of the soil. The typical spectra can be observed below 8 Hz. For frequencies above 8 Hz, a big difference between the regular and irregular track support is obvious. Whereas the quasi-static component tends to zero, the varying track support stiffness generates a new ground vibration component with considerable amplitudes. The low-frequency quasi-static and the mid-frequency response are clearly separated by a minimum at around 8 Hz. There is a second minimum at 25 or 32 Hz, so that the mid-frequency ground vibration component can be identified by a typical maximum region between these minima. This characteristic is related to the axle-sequence spectrum \(X_{A}\)(f) of a bogie. The mid-frequency component also has a completely different attenuation with distance. In contrast to the strong attenuation of the quasi-static component, the mid-frequency part has a rather weak attenuation.  
The mid-frequency stochastic ground vibration component is generated as follows. The static axle load runs over different excitation points of the track that have different stiffnesses. If the track is stiffer, the impulse is shorter and has a wider bandwidth. If the track is softer, the impulse is longer and the bandwidth is narrower. Therefore, low frequencies of the impulse are similar for all excitation points, but the amplitudes around the bandwidth vary considerably. The impulses from neighbouring excitation points cannot perfectly cancel each other. The remainder of the superposition constitutes the new mid-frequency ground vibration component (which is called \"the scattered axle pulses\").  
Figure 7a-d shows the one-third octave band spectra for \(q\) = 3, 10, 30 and 50% variation of the track support stiffness. The mid-frequency range strongly increases with the percentage of the stiffness variation. At the end for 50% variation, the mid-frequency amplitudes are as high as for the nearest quasi-static response. The stochastic mid-frequency component is only moderately higher for a softer soil (Figure 6d). Another variation in Figure 7d-f shows the influence of the track stiffness on the mid-frequency component. The bending stiffness \(EI\) of the track is increased from the rail to 10 \(EI\) and 100 \(EI\), where 100 \(EI\) is approximately the stiffness of a 0.5 m thick concrete plate. The mid-frequency amplitudes are clearly reduced by a stiff plate, to less than one-tenth. The stiff plate results in a wider distribution of the static load along the track. This means a smoother impulse, which has  
lower frequencies and lower amplitudes. Thus, a new ground vibration component was found, but also a possible mitigation strategy for this component.  
Figure 6: Train passage with \(v_{T}=160\) km/h over a regular (**a,c,e**) and an irregular ballast track (**b,d,f**), \(q=50\%\)), different soils with \(v_{S}=\) (**a,b**) 150, (**c,d**) 200, (**e,f**) 300 m/s, one-third of the octave band spectrum at distances \(x=\square 3\), \(\circlearrowleft\) 5, \(\Delta\) 10 m.