# 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
Therefore, the response of the slab can be written as:  
\[\tilde{u}_{\rm s}^{n_{x}}(\widetilde{x}_{0}+nL_{\rm s},\omega)=\sum\limits_{p =1}^{N_{p}}\varphi_{p}^{n_{x}}(\widetilde{x}_{0})\alpha_{p}^{n_{x}}(n,\omega) \tag{17}\]  
Upon applying a Floquet transform to Equation (17), the slab responses in the frequency-wavenumber domain are:  
\[ \begin{split}\widetilde{u}_{\rm s}^{n_{x}}(\widetilde{x}_{0}+nL _{\rm s},\kappa,\omega)=\sum\limits_{p=1}^{N_{p}}\varphi_{p}^{n_{x}}( \widetilde{x}_{0})\widetilde{\alpha}_{p}^{n_{x}}(\kappa,\omega)\\ (n_{x}=1,\ldots,N_{\rm s};\widetilde{x}_{0}\in[0,L])\end{split}  \tag{18}\]  
where, \(\widetilde{x}_{p}(\kappa,\omega)\) represents modal coordinates in the wavenumber domain. By substituting the wavenumber domain solutions (13) and (18) of the rail and slabs into the governing Equations (10) and (12), and set \(G_{m}=\frac{2m\pi}{L_{\rm s}}\), the following equations can be obtained:  
\[ \begin{split}& E_{\rm r}I_{\rm r}\sum\limits_{m=-M}^{M}A_{m}( \kappa,\omega)(G_{m}-\kappa)^{4}e^{i(G_{m}-\kappa)\widetilde{x}}-m_{\rm r} \omega^{2}\sum\limits_{m=-M}^{M}A_{m}(\kappa,\omega)e^{i(G_{m}-\kappa) \widetilde{x}}\\ +k_{\rm r}\sum\limits_{j=1}^{N_{\rm r}}\left(\sum\limits_{m=-M}^{M }A_{m}(\kappa,\omega)e^{i(G_{m}-\kappa)\widetilde{x}}-\sum\limits_{p=1}^{N_{p }}\varphi_{p}^{n_{x}}(\widetilde{x}_{0})\widetilde{\alpha}_{p}^{n_{x}}(\kappa, \omega)\right)\delta\big{(}\widetilde{x}-\widetilde{x}_{j_{\rm r}}\big{)}=0 \end{split}  \tag{19}\]  
\[ \begin{split}& E_{\rm s}I_{\rm s}\varphi_{p}^{n_{x}}\sum\limits_{p=1} ^{N_{p}}\varphi_{p}^{n_{x}}(\widetilde{x}_{0})\widetilde{\alpha}_{p}^{n_{x}}( \kappa,\omega)-m_{\rm s}\omega^{2}\sum\limits_{p=1}^{N_{p}}\varphi_{p}^{n_{x}} (\widetilde{x}_{0})\widetilde{\alpha}_{p}^{n_{x}}(\kappa,\omega)\\ +&\sum\limits_{j_{\rm s}=(n_{x}-1)N_{\rm s}+1}^{n_{ \rm s}N_{\rm s}}k_{\rm r}^{j_{\rm s}}\sum\limits_{p=1}^{N_{p}}\varphi_{p}^{n_{ x}}(\widetilde{x}_{0})\widetilde{\alpha}_{p}^{n_{x}}(\kappa,\omega)\delta \big{(}\widetilde{x}-\widetilde{x}_{j_{\rm s}}\big{)}\\ -k_{\rm r}\sum\limits_{j=1}^{N_{\rm r}}\left(\sum\limits_{m=-M}^{ N_{\rm r}}A_{m}(\kappa,\omega)e^{i(G_{m}-\kappa)\widetilde{x}}-\sum\limits_{p=1}^{N_{p }}\varphi_{p}^{n_{x}}(\widetilde{x}_{0})\widetilde{\alpha}_{p}^{n_{x}}(\kappa, \omega)\right)\delta\big{(}\widetilde{x}-\widetilde{x}_{j_{\rm r}}\big{)}=0 \end{split}  \tag{20}\]  
\[ \begin{split}&(n_{x}=1,\ldots,N_{\rm s};\widetilde{x}\in[(n_{x}-1 )L,n_{\rm r}L])\end{split} \]Multiplying both ends of the rail Equation (19) by \(e^{-\mathrm{i}(G_{m}-\kappa)\widetilde{x}}(m=-M,\ldots,M)\), and integrating \(\widetilde{x}\) within the interval \([0,L_{\mathrm{s}}]\). For each Equation (20) of the slabs, multiply by \(\varphi_{p}^{n_{x}}(\widetilde{x}_{0})\), and integrate \(\widetilde{x}_{0}\) within the interval \([0,L]\) to obtain the following independent equations:  
\[ \begin{split} E_{\mathrm{r}}I_{\mathrm{r}}L_{\mathrm{s}}(G_{m}- \kappa)^{4}A_{m}(\kappa,\omega)-m_{\mathrm{r}}\omega^{2}L_{\mathrm{s}}A_{m}( \kappa,\omega)+k_{\mathrm{r}}\sum\limits_{j=1}^{N_{\mathrm{r}}}\left( \widetilde{u}_{\mathrm{r}}\big{(}\widetilde{x}_{j_{\mathrm{r}}}\big{)}- \widetilde{u}_{\mathrm{s}}\big{(}\widetilde{x}_{j_{\mathrm{r}}}\big{)}\right) e^{-\mathrm{i}(G_{m}-\kappa)\widetilde{x}_{j_{\mathrm{r}}}}=0\\ (m=-M,\ldots,M)\end{split}  \tag{21}\]  
\[ \begin{split} E_{\mathrm{s}}I_{\mathrm{s}}L\beta_{p}^{4} \widetilde{\alpha}_{p}(\kappa,\omega)-m_{\mathrm{s}}\omega^{2}L\widetilde{x}_ {p}^{n_{x}}(\kappa,\omega)+\sum\limits_{j=(n_{x}-1)N_{\mathrm{s}}+1}^{n_{x}N _{\mathrm{s}}}k_{j_{\mathrm{s}}}^{n_{x}}\widetilde{u}_{\mathrm{s}}\big{(} \widetilde{x}_{j_{\mathrm{s}}}\big{)}\varphi_{p}\big{(}\widetilde{x}_{j_{ \mathrm{s}}}\big{)}-k_{\mathrm{r}}\sum\limits_{j=1}^{N_{\mathrm{r}}}\left( \widetilde{u}_{\mathrm{r}}\big{(}\widetilde{x}_{j_{\mathrm{r}}}\big{)}- \widetilde{u}_{\mathrm{s}}\big{(}\widetilde{x}_{j_{\mathrm{r}}}\big{)}\right) \varphi_{p}\big{(}\widetilde{x}_{j_{\mathrm{r}}}\big{)}=0\\ \big{(}n_{x}=1,\ldots,N_{\mathrm{s}};p=1,\ldots N_{p}\big{)}\end{split}