# Flexural Wave Propagation and Defect States of Periodic Slab Track Structure in High-Speed Railway
## 4 Defect State Analysis of Slab Track
### Characteristics of the Defective State  
The local damage of SCC is located at the slab edge, and the support stiffness in the defective area is set to 0 in the analytical model. When the defect size of the support area beneath the slab is 0.1 m, six defective states are generated in the slab track structure below 500 Hz. Flat bands representing defect states are formed within the original bandgaps as shown in Figure 10. When an external load is close to the frequency of a defect state, vibration localization at the defective position will be formed [35]. Due to the failure of the support beneath the track slab, most of the defect states are formed within the local resonance bandgaps associated with the deformation of the slab, except the second defect state, which is located in the Bragg bandgap showing rail-slab in-phase motion.  
Following the wave modes corresponding to the defect states illustrated in Figure 10, structural deformation is concentrated near the defect, with the maximum vibration response occurring in the defective cell. Mode A-Mode F represents the wave modes of the supercell at the defect state frequencies. For any wavenumber, there is only one corresponding wave mode for a defect state. Defect mode A shows the vibration localization of the defect state and the rail and the slab move in-phase, with the primary deformation concentrated around the local defect. For defect mode B, it is located in the Bragg bandgap and has a broader influence range. Since the defect is located on the left side of the slab end, the flexural wave exhibits rapid attenuation when propagating to the left side. Defect mode C is primarily characterized by the bending deformation mode of the slab, and its influence range encompasses an approximate length of two track slabs. For defect mode D, the rail and slab move out-of-phase, which can induce noticeable vibrations within a range of 10 m on either side of the defect. For defect states in the higher frequency range, such as defect mode E and mode F, the localization of elastic waves is less pronounced, and the influence range of local vibrations can exceed 100 m.  
Figure 9: Influence of isolation layer stiffness on bandgaps.  
Since the local defect directly affects the support stiffness of the slab, most of the defect states are located within local resonance bandgaps except defect mode B. The main reason for forming a defect state within the Bragg bandgap is that the maximum deformation is located at the slab end, as illustrated by mode C in Figure 6. The presence of local defects alters the dynamic properties of the original periodic structure, inducing structural resonance around the defect at specific frequencies within bandgaps, which does not propagate throughout the entire structure. Therefore, the formation mechanism of defect states in periodic slab tracks can be explained by local resonance.  
Furthermore, it is noteworthy that the defect states in defective structures and dispersion curves resulting from local resonance in non-detective structures are flat bands. The group velocity corresponding to the defect states is equal to 0, whereas the flat bands formed by local resonances are not strictly equal to 0. Consequently, the waves corresponding to the defect states are incapable of propagating within the structure and are localized near defects. In contrast, the flat bands are formed by the local resonance in a periodic structure, the standing wave modes represent the structure's resonances and will not be localized at a specific position.