# Flexural Wave Propagation and Defect States of Periodic Slab Track Structure in High-Speed Railway
## 3 Bandgap Characteristics of Slab Track Structure
### Parameter Analysis  
When the stiffness of the isolation layer is maintained at 4 \(\times\) 10\({}^{8}\) N/m\({}^{3}\) and the vertical stiffness of the fastening increases from 25 \(\times\) 10\({}^{6}\) N/m to 65 \(\times\) 10\({}^{6}\) N/m, the variation laws of the starting and cut-off frequencies of the various bandgaps in response to the variation of fastening stiffness are shown in Figure 8. It can be observed that an increment in the fastening stiffness induces a shift of all bandgap frequencies towards higher frequencies.  
Figure 7: Vibration transmission in periodic slab track: (**a**) vibration transmission of the rail; (**b**) vibration transmission of the slab.  
The fastening stiffness has a relatively small impact on the width of the bandgaps in high frequencies, but mainly influences the frequencies of bandgaps below 300 Hz.  
With the increase in fastening stiffness, the first-order bandgap gradually broadens, and the corresponding cut-off frequency increases from 107 Hz to 112 Hz. For higher fastening stiffness, the first-order resonance of the ballastless track is predominantly governed by the support stiffness beneath the slab. The second and third bandgaps exhibit relatively narrow widths and are less affected by parameters. The fourth, fifth, and sixth bandgaps are mainly generated by the coupling effect of local resonance bandgaps and Bragg bandgaps, forming an approximately broad bandgap. Moreover, as the fastening stiffness increases, these bandgaps expand significantly. This expansion is attributed to the generation mechanism of these bandgaps, which is affected by the relative deformation of the rail and the slab. The seventh bandgap shifts towards higher frequency with the stiffness increases while the bandwidth gradually decreases. For the eighth and ninth bandgaps, an approximate coupling between the Bragg bandgap and the local resonance bandgap occurs when the fastening stiffness is within the range of 40-45 \(\times\) 10\({}^{6}\) N/m, increasing the width of the bandgaps.  
At a vertical stiffness of the fastening of 35 \(\times\) 10\({}^{6}\) N/m, the variation laws of bandgaps with the change of the stiffness of the isolation layer are shown in Figure 9. When the stiffness beneath the slab increases from 4 \(\times\) 10\({}^{8}\) N/m\({}^{3}\) to 1.2 \(\times\) 10\({}^{9}\) N/m\({}^{3}\), the frequencies of all bandgaps, with the exception of the Bragg bandgap at the pinned-pinned resonance, move towards higher frequencies. Compared with the fastening stiffness, the stiffness of the isolation layer has a more pronounced effect on the bandgaps. With the increased stiffness of the isolation layer, the frequency range of the first-order bandgap broadens gradually, and the cut-off frequency increases from 110.6 Hz to 147 Hz. The second and third bandgaps are relatively narrow and are minimally affected stiffness parameters. The fourth to eighth bandgaps are predominantly generated by the coupling effect of the local resonance and Bragg bandgaps, resulting in the formation of an approximately broad bandgap. Moreover, as the stiffness increases, the range of these bandgaps broadens significantly. When the stiffness of the isolation layer is within the range of 0.9-1.0 \(\times\) 10\({}^{9}\) N/m\({}^{3}\), five bandgaps converge, leading to a coupled bandgap in the range of 180-230 Hz.  
Figure 8: Influence of fastening stiffness on bandgaps.