# Flexural Wave Propagation and Defect States of Periodic Slab Track Structure in High-Speed Railway
## 3 Bandgap Characteristics of Slab Track Structure
### Bandgaps
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Order** & **Frequency (Hz)** & **Bandgap Description** & **Formation** \\  & & & **Mechanism** \\ \hline
1 & 0–110.6 & In-phase resonance between rail and slab & Local resonance \\
2 & 111–111.6 & Rail and slab exhibit in-phase motion, and the wavelength of the & Bragg scattering \\  & & bending wave in rail is twice the length of the slab. \(\lambda=2L\) (\(n=1\)) & \\
3 & 116.7–118.8 & Rail and slab move in-phase, and the length of the bending wave & Bragg scattering \\  & & of rail is equivalent to the length of the slab. \(\lambda=L\) (\(n=2\)) & \\
4 & 125–151.1 & Slab bending resonance & Local resonance \\
5 & 152.3–163 & Slab bending resonance, and out of phase motion between rail & Local resonance \\  & & and slab & Local resonance \\
6 & 163.9–164 & Rail and slab move out of phase, and the wavelength of the & Bragg scattering \\  & & bending wave of rail is twice the length of slab. \(\lambda=2L\) (\(n=1\)) & \\
7 & 173.3–176.7 & Rail and slab move out of phase, and the bending wave length of & Bragg scattering \\  & & rail is equivalent to the slab length. \(\lambda=L\) (\(n=2\)) & \\
8 & 209.7–216.4 & The half-wavelength of the rail bending wave is equal to 1/3 the & Bragg scattering \\  & & length of the slab. \(\lambda=2L/3\) (\(n=3\)) & \\
9 & 225.9–231.8 & Bending resonance of slab & Local resonance \\
10 & 304.3–306.7 & The half-wavelength of the rail bending wave is equal to 1/4 the & Bragg scattering \\  & & length of the slab. \(\lambda=2L/4\) (\(n=4\)) & \\
11 & 337.4–339.3 & Bending resonance of slab & Local resonance \\
12 & 440.5–441.3 & The half-wavelength of the rail bending wave is equal to 1/5 the & Bragg scattering \\
13 & 485.8–486.5 & Bending resonance of slab & Local resonance \\
14 & 614.7–615.1 & The half-wavelength of the rail bending wave is equal to 1/6 the & Bragg scattering \\  & & length of the slab. \(\lambda=2L/6\) (\(n=6\)) & \\
15 & 668.1–668.4 & Bending resonance of slab & Local resonance \\
16 & 824.9–825.1 & The half-wavelength of the rail bending wave is equal to 1/7 the & Bragg scattering \\  & & length of the slab. \(\lambda=2L/7\) (\(n=7\)) & \\
17 & 883–883.2 & Bending resonance of slab & Local resonance \\
18 & 1070–1070.1 & The half-wavelength of the rail bending wave is equal to 1/8 the & Bragg scattering \\  & & length of the slab. \(\lambda=2L/8\) (\(n=8\)) & \\
19 & 1130–1130.1 & Bending resonance of slab & Local resonance \\
20 & 1340–1357 & The half-wavelength of the rail bending wave is equal to 1/9 the & \\  & & length of the slab. \(\lambda=2L/9\) (\(n=9\)) & \\
21 & 1409–1409.1 & Bending resonance of slab & Local resonance \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Vertical flexural wave bandgaps of CRTS III slab track and their formation mechanisms. bandgaps are coupled to produce a relatively wide bandgap range. For the fourth-order local resonance bandgap (225.9-231.8 Hz), a bending resonance between the rail and the track slab occurs. For higher-order local resonance bandgaps, the coupling effect between the rail and the track slab gradually decreases, and the bandgap widths gradually decrease, thus forming flat bands. In fact, the Bragg and local resonance bandgaps can be coupled to each other in the low-frequency range. There is a coupling between the Bragg and local resonance bandgaps near 110-170 Hz and 200-230 Hz to form approximately broad bandgaps as shown in Figure 5. The relatively wide bandgaps in the CRTS III slab track structure mainly include: the first-order low-frequency bandgap, broad bandgaps at 110-170 Hz and 200-230 Hz resulting from the coupling effect of the local resonance, and the Bragg bandgap.