# Flexural Wave Propagation and Defect States of Periodic Slab Track Structure in High-Speed Railway
## 5 Conclusions  
To investigate the propagation characteristics of the flexural waves in the slab track structure, this study focuses on the CRTS III ballastless track. Considering the periodically distributed fastenings and slabs, a wave analysis model of the slab track structure considering local defects was established based on the Floquet transform and the supercell method. The following conclusions can be drawn:  
For the unit slab track structure, the Floquet transform method can be effectively applied to the elastic wave analysis of the track structure with and without defects, possessing the advantages of stable calculation and high efficiency. Within the frequency range of 1500 Hz, the CRTS III slab track structure exhibits 21 vertical flexural wave bandgaps. The formation mechanism of these bandgaps can be  
Figure 11: Variation of defect state frequency with defect size: (**a**) local defect at slab end; (**b**) local defect at the middle of the slab. elucidated by Bragg scattering and local resonance. The most prominent bandgap ranges are 0-110 Hz, 110-170 Hz, and 200-230 Hz. In the low-frequency range, the local resonance of the slab interacts with the waves in the rail, thereby giving rise to a broad bandgap through the coupling of the Bragg and the local resonance bandgaps. With the increase in fastening stiffness, the bandgaps shift towards higher frequencies. The range of the bandgap composed of the fourth to sixth bandgaps widens significantly due to the coupling effect of two types of bandgaps. With the increase in the isolation layer stiffness, the range of the fourth to eighth bandgaps widens markedly. Within the stiffness range of 0.9 to 1.0 \(\times\) 10\({}^{9}\) N/m\({}^{3}\) for the isolation layer, the five bandgaps interact with one another, resulting in the formation of a relatively broad bandgap in the range of 180-230 Hz. Local defects give rise to the emergence of defect states in slab tracks, manifested as the localization of elastic waves near the defect position. Local contact loss beneath the slab can generate six defect states below 500 Hz, and the formation mechanism of the defect states can also be explained by the local resonance of the structure at the defect position. The frequency of the first-order defect state is considerably influenced by the defect's size. The second-order defect state exhibits the characteristic of unidirectional propagation, and the third-order defect state demonstrates vibration localization. These defect states can serve as important indicators for defect identification. To further investigate the wave propagation and defect states within ballastless track structures, it is necessary to construct a three-dimensional wave model of the ballastless track structure in forthcoming studies, accounting for asymmetric motion and slab warping. A broader range of local defects can be discussed, including delamination between the slab and the SCC layer. Additionally, it is crucial to incorporate environmental factors and the influence of train loads, assessing the effects of temperature gradients and train loads on the wave propagation characteristics in ballastless track structures. Conceptualization, Q.Y. ; methodology, Q.Y. ; software, Q.Y. and Z.W. ; validation, Q.Y., Z.W., L.Z. and S.W. ; investigation, Z.L. ; resources, S.W. ; writing--original draft preparation, Q.Y. ; writing--review and editing, Q.Y. and L.Z. ; supervision, S.W. All authors have read and agreed to the published version of the manuscript. This research was funded by the National Natural Science Foundation of China (52208449) and the Research Fund Project of the China Academy of Railway Sciences (2024 YJ334). Not applicable. The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author. Authors [PERSON], [PERSON] and [PERSON] were employed by the China Academy of Railway Sciences Corporation Limited.