# Flexural Wave Propagation and Defect States of Periodic Slab Track Structure in High-Speed Railway
## 3 Bandgap Characteristics of Slab Track Structure
### Bandgaps  
For the periodic slab track structure, with fastening stiffness at 35 \(\times\) 10\({}^{6}\) N/m and isolation layer stiffness at 4 \(\times\) 10\({}^{8}\) N/m\({}^{3}\), the vertical bending wave dispersion curves of CRTS III slab track with slab length \(L\) = 5.6 m are shown in Figure 5.  
Characterized by its multiple periodicities, the dispersion curves of the CRTS III slab track structure exhibit complex bandgap characteristics compared to a single periodic track model [13]. Within a frequency below 1500 Hz, the real part of the dispersion curve shows multiple flat bands, while the imaginary part displays sharp peaks corresponding to these bands. These flat bands are attributed to the resonant modes of the slab, with the main flat bands within the 1500 Hz range being: 339 Hz, 485 Hz, 668 Hz, 883 Hz, 1130 Hz, and 1409 Hz. These flat bands correspond to the natural frequencies of the track slab as shown in Table 2. The elastic wave bandgaps resulting from these flat bands can be explained by the mechanism of local resonance. Discrete slabs act as scatterers, when the traveling waves in the rail resonate with the scatterers, the elastic waves become localized near the excitation point and cannot propagate forward. Additionally, due to the minor structural differences along the longitudinal direction of the track structure, the Bragg scattering effect is relatively insignificant, with the decay amplitude within the Bragg bandgap being less than that within the local resonance bandgap. For the propagation of the bending wave in the rail, according to the Bragg scattering mechanism, when the propagation wavelength of elastic wave in periodic structure is \(\lambda\) and the period is \(l\), the phase matching condition is \(2l\) = \(n\lambda\) (\(n\) = 1, 2,...) [39]. Therefore, the Bragg frequency of the rail is:  
\[f_{\mathrm{bragg}}=\frac{1}{2\pi}\Big{(}\frac{n\pi}{l}\Big{)}^{2}\sqrt{\frac {EI}{\rho A}} \tag{24}\]  
The Bragg frequencies of the CRTS III slab track are presented in Table 3.  
Figure 5: Flexural wave dispersion curves of CRTS III slab track (L = 5.6 m): (**a**) dispersion curves; (**b**) enlarged imaginary part. To further clarify the generation mechanism of the bandgaps in the CRTS III slab ballastless track structure, the wave modes at the band-edge were extracted. When \(\kappa=0\), the wave modes below 1500 Hz are shown in Figure 6. Mode (a) represents the starting frequency of the dispersion curve, and its wave mode is the in-phase resonance of the rail and the track slab, corresponding to the first-order resonance of the CRTS III slab ballastless track structure. Modes (b) and (c) form a Bragg bandgap, with the wavelength of the flexural wave equal to the length of the slab. Subfigure (d) shows the mode resulting from the bending resonance of the slab, and mode (e) presents the out-of-phase resonance of the rail and the slab. The bandgap composed of these two wave modes belongs to the local resonance bandgap. Modes (f) and (g) constitute a Bragg bandgap, in which the wavelength of the flexural wave is equal to the length of the slab while the rail and slab exhibit out-of-phase motion. Mode (h) indicates the local resonance of the track slab. Modes (i) and (j) constitute a Bragg bandgap, with the wavelength being half the length of the track slab. Modes (k) and (l) represent the local resonances of the track slab, corresponding to its 6 th and 7 th bending modes shown in Table 2. Mode (m) is a Bragg-type wave mode, with the wavelength being one-third of the length of the track slab. Modes (n) and (o) are the local resonances of the track slab, corresponding to its 8 th and 9 th bending modes. Mode (p) belongs to a Bragg-type wave mode, with the wavelength being one-fourth of the length of the track slab. Modes (q) and (r) represent the local resonances of the track slab, corresponding to its 10 th and 11 th bending modes.