# Parametric Study on the Effect of Rail Dampers on Track Decay Rate
## 3 Numerical Application
The bending wave is influenced by the evanescent wave only in the near field, and then it basically becomes a pure propagative wave (Figure 5c). Rail support does not change the nature of the two waves but modifies their mode of manifestation according to the main frequencies. For instance, Figure 6 shows the bending waves in the rail laid on soft rail pads and tampered ballast when the sleeper bay is 0.544 m, considering the following frequencies: 50 Hz, 110 Hz, 185 Hz, 600 Hz, 1135 Hz, and 1500 Hz; no damping is considered. Figure 4: Propagation constants (\(\eta=0\)). The frequencies are chosen so that each one is in one of the frequency ranges delimited by the main frequencies. The bending wave can take the form of attenuated standing wave (Figure 6a,c) or the form of standing wave (Figure 6e). On the other hand, the bending wave can be dominated by a propagative wave (Figure 6b,d,f). These features of the bending waves outlined above for a few frequencies do not change within a particular range of frequencies delimitated by the main frequencies, as can be seen in Figure 7, where the bending waves for four frequencies between the anti-resonance and the high resonance are displayed as examples. Figure 8 shows the features of the bending waves in rail depending on the frequency ranges delimited by the main frequencies of the track. The rail experiences standing waves within two ranges, namely, at frequencies lower than the low resonance and between the anti-resonance and the high resonance, and also around the pinned-pinned resonance. On the other hand, the bending waves in rail are dominated by the propagative wave when the frequency is within the range between the low resonance and the anti-resonance or within the range that stretches from the high resonance to near the pinned-pinned resonance, or its frequency is higher than the pinned-pinned resonance. Third, TDR for the track without rail dampers is analyzed. TDR is influenced by how the evanescent and propagate waves manifest in the different frequency ranges. Figure 9 presents TDR calculated for one-third octave bands between 100 and 2000 Hz considering spatial partitions with the step of \(1/32\) sleeper bay, with the active section above the sleeper and between sleepers (at middle). In general, all diagrams of TDR have a peak beyond the anti-resonance, within the second frequency range of the standing waves, between the anti-resonance and the high resonance. This peak corresponds to the one-third octave of 160 Hz/200 Hz for soft rail pad and tampered/settled ballast and to the one-third octave of 315 Hz for stiff rail pad. The value of this peak is higher in the case of stiff rail pads because the propagation of the bending waves is obstructed by the stronger coupling between the rail and the sleepers. The second local peak is more visible when the excitation point is situated between sleepers and it appears around the pinned-pinned resonance where the bending waves in rail are also standing waves, corresponding to the one-third octave of 1250 Hz. This time, the TDR peak is much smaller due to the pinned-pinned resonance. When the propagation of the bending waves is dominated by the propagative wave, TDR decreases to low values, as it happens beyond the high frequency and then, beyond the pinned-pinned resonance, where the rail vibration tends to be independently from the sleepers' vibration. However, the rail-sleepers coupling leads to higher TDR in the case of the stiff rail pads. Figure 8: Features of the bending waves in rail. Figure 7: Bending wave in the rail on soft rail pads and tampered ballast within the range delimited by the anti-resonance and the high resonance: (**a**) at 160 Hz; (**b**) at 200 Hz; (**c**) at 220 Hz; and (**d**) at 240 Hz; --, \(t=0\); --, \(t=T/12\); --, \(t=2T/12\); --, \(t=3T/12\), --, \(t=4T/12\), --, \(t=5T/12\), --, \(t=6T/12\). When the ballast is settled, this results in an increased TDR at low frequency, where the ballast stiffness matters.