# Flexural Wave Propagation and Defect States of Periodic Slab Track Structure in High-Speed Railway
## 2 Flexural Wave Analysis Model for Unit Slab Track Structure
The coordinates of fastenings and isolation layer springs in the global system are:  
\[x_{j_{\rm r}}^{n_{\rm c}}=\widetilde{x}_{j_{\rm r}}+n_{\rm c}L_{\rm s};\ \ \ x_{j_{\rm s}}^{n_{\rm c}}= \widetilde{x}_{j_{\rm s}}+n_{\rm c}L_{\rm s} \tag{6}\]  
where, \(\widetilde{x}_{j_{\rm r}}\) and \(\widetilde{x}_{j_{\rm s}}\) represent the coordinates of the fastenings inside the supercell and the supporting springs under the composite slab. For infinite periodic structures, the Floquet transformation can be used to study the dynamic behaviors of infinite periodic structures within the unit cell in the wavenumberdomain. For any nonperiodic function defined in the \(\mathbf{x}\) domain \(f(x)=f(\widetilde{x}+nL_{\mathrm{s}})\), the \(\mathbf{x}\) domain satisfies the period \(L_{\mathrm{s}}\). The distance \(nL_{\mathrm{s}}\) between the \(n\)-th supercell and the reference supercell can be transformed into the wavenumber domain \(\kappa\in\left[-\frac{\pi}{L_{\mathrm{s}}},\frac{\pi}{L_{\mathrm{s}}}\right]\), and the Floquet transformation is defined as follows [37]:  
\[\widetilde{F}(\widetilde{x},\kappa)=\sum\limits_{n=-\infty}^{+\infty}f( \widetilde{x}+nL_{\mathrm{s}})e^{inL_{\mathrm{s}}\kappa} \tag{7}\]  
The corresponding inverse transformation is:  
\[f(\widetilde{x}+nL_{\mathrm{s}})=\frac{L_{\mathrm{s}}}{2\pi}\int_{-\pi/L_{ \mathrm{s}}}^{+\pi/L_{\mathrm{s}}}\widetilde{F}(\widetilde{x},\kappa)\exp(- \mathrm{i}nL_{\mathrm{s}}\kappa)d\kappa \tag{8}\]  
For the rail, the Fourier transformation and Floquet transformation are performed on Equation (4):  
\[E_{\mathrm{r}}I_{\mathrm{r}}\frac{\partial^{4}\widetilde{u}_{\mathrm{r}}}{ \partial x^{4}}-m_{\mathrm{r}}\omega^{2}\widetilde{u}_{\mathrm{r}}+k_{\mathrm{ r}}\sum\limits_{n=-\infty}^{+\infty}\sum\limits_{n_{\mathrm{c}}=-\infty}^{+ \infty}\sum\limits_{j_{\mathrm{r}}=1}^{N_{\mathrm{r}}}(\hat{u}_{\mathrm{r}}( \widetilde{x}+nL_{\mathrm{s}})-\hat{u}_{\mathrm{s}}(\widetilde{x}+nL_{ \mathrm{s}}))\delta\big{(}\widetilde{x}+nL_{\mathrm{s}}-\widetilde{x}_{j_{ \mathrm{r}}}-n_{\mathrm{c}}L_{\mathrm{s}}\big{)}e^{\mathrm{i}nL_{\mathrm{s}} \kappa}=0 \tag{9}\]  
Eliminating infinite terms is completed as follows:  
\[E_{\mathrm{r}}I_{\mathrm{r}}\frac{\partial^{4}\widetilde{u}_{\mathrm{r}}}{ \partial x^{4}}-m_{\mathrm{r}}\omega^{2}\widetilde{u}_{\mathrm{r}}+k_{\mathrm{ r}}\sum\limits_{j_{\mathrm{r}}=1}^{N_{\mathrm{r}}}(\hat{u}_{\mathrm{r}}-\hat{u}_{ \mathrm{s}})\delta\big{(}\widetilde{x}  
\[\frac{d^{j}\widetilde{\mu}_{\rm s}}{d\widetilde{x}^{j}}\bigg{|}_{x=L}=\frac{d^{j} \widetilde{u}_{s}}{d\widetilde{x}^{j}}\bigg{|}_{x=0}=0,(j=2,3) \tag{15}\]  
The displacement response solution of periodic rail can automatically satisfy the periodic boundary conditions at the cell boundary. Since the ends of the track slab are free, and when described using the modal superposition of free beams, its boundary conditions are automatically satisfied. Therefore, the responses of the track slabs are given by mode superposition:  
\[ \begin{split}&\tilde{u}_{\rm s}^{n_{x}}(\widetilde{x}_{0}, \omega)=\sum\limits_{p=1}^{N_{p}}\varphi_{p}^{n_{x}}(\widetilde{x}_{0})\alpha _{p}^{n_{x}}(\omega)\\ &(n_{x}=1,\ldots,N_{x};\widetilde{x}_{0}=\widetilde{x}-(n_{x}-1) L;\widetilde{x}_{0}\in[0,L])\end{split}  \tag{16}\]  
In the formula, \(x_{0}\) is the local coordinate of each slab, \(\alpha_{p}^{n_{x}}(\omega)\) is the modal coordinate, and \(\varphi_{p}^{n_{x}}(\widetilde{x}_{0})\) is the orthogonal modal function of the slab of the \(n_{x}-\)th cell. A free beam is used to simulate the slab, and its modal shape function \(\varphi_{p}(\widetilde{x}_{0})\) can be referred to the reference [38]. \(N_{p}\) is the number of considered modes of slab, and \(N_{p}\) should be sufficiently large and exceed the highest analysis frequency. When \(N_{p}\)= 1, only vertical rigid body mode is considered. Due to the periodicity of the structure, the shape function of each slab is the same, but the modal coordinates are related to the position of cell number within supercell.