# Flexural Wave Propagation and Defect States of Periodic Slab Track Structure in High-Speed Railway
## 3 Bandgap Characteristics of Slab Track Structure  
### Model Parameters  
The propagation characteristic of vertical bending waves within the ideal periodic CRTS III slab track structure was analyzed firstly, with the track structure parameters detailed in Table 1. The fastening system is designed with a static stiffness of \(25\times 10^{6}\) N/m. However, due to the frequency-dependent property of rubber material, existing experimental studies have shown that its dynamic stiffness ranges from \(35\times 10^{6}\) N/m to \(65\times 10^{6}\) N/m. As for the isolation layer, which can be either geotextile or EPDM rubber, existing experimental results indicate that the stiffness of the isolation layer lies ranges from \(4\times 10^{8}\) N/m\({}^{3}\) to \(1.2\times 10^{9}\) N/m\({}^{3}\)[31].  
Firstly, it is necessary to discuss the impact of the number of rail waves and the modal numbers of the slab on the calculation results. When \(N_{p}\)\(=1\), that is, considering only the vertical rigid degree of the slab, the dispersion curves obtained from different numbers of rail wave solutions are presented in Figure 3. As the number of rail waves increases, the dispersion curves gradually converge. It can be observed that for a periodic slab track, with a unit cell length of \(5.6\) m, employing \(NR=31\) can yield accurate results.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Elastic Modulus** & **Rail Density** & **Rail Sectional Area (m\({}^{2}\))** & **Inertial Moment of Rail (m\({}^{4}\))** & **Stiffness of Fastening (N/m)** & **Fastening Spacing (m)** \\ \hline \(2.1\times 10^{11}\) & 7850 & 7.745 \(\times\) 10\({}^{-3}\) & \(3217\times 10^{-8}\) & \(35\times 10^{6}\) & 0.625 \\ \hline
**Slab Density** & **Elastic Modulus of slab (Pa)** & **Slab Thickness** & **SCC Thickness (m)** & **Length of Slab (m)** & **Stiffness of Isolation Layer (N/m\({}^{3}\))** \\ \hline
2500 & \(3.5\times 10^{10}\) & 0.2 & 0.09 & 5.6 & \(4\times 10^{8}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters of the CRTS III slab track.  
When \(N_{p}\)\(=12\), for a slab of length 5.6 m, the analysis frequency can reach up to 1700 Hz, as shown in Table 2. Therefore, considering 12 modal numbers of the track slab is sufficient to cover the frequency range up to 1500 Hz.  
To verify the accuracy of the theoretical model, the dispersion curves of the track structure were calculated using the transfer matrix method (TMM). The comparative results of the two computational methods are displayed in Figure 4. The Floquet method offers higher computational efficiency compared to the TMM. On a computer equipped with CPU i7-10510U @ 1.80 GHz and 16 GB of RAM, the computation time for the Floquet method is 4.8 s, whereas the TMM requires 14.6 s. When the phase velocity of the dispersion curve is low, the TMM has sparse data points, particularly at the band-edges, which complicates the precise determination of the bandgap frequency. When dealing with the supercell, the TMM encounters ill-conditioned matrix problems, leading to non-converging elastic wave dispersion curves.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline
**Modal Order** & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\ \hline
**Frequency (Hz)** & Rigid & 35.4 & 98.0 & 192.1 & 317.6 & 474.5 & 662.8 & 882.4 & 1133.4 & 1415.9 & 1729.6 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Modal frequencies for the free slab.  
Figure 4: Comparison of the two methods.  
Figure 3: Dispersion curves of slab track (\(N_{p}\)\(=1\)): (**a**) 0–500 Hz; (**b**) partially enlarged.  
Furthermore, for the slab track with defects, it is necessary to reevaluate the size parameters of the supercell and the number of rail waves. The stability of the computational results is attained when the size parameter is \(N_{x}\)= 21 and the number of waves is \(NR\) = 201 for the supercell.