# 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 flexural wave propagation equations for the slab track structure can be formulated as follows:  
\[E_{\rm r}I_{\rm r}\frac{\partial^{4}u_{\rm r}}{\partial x^{4}}+m_{\rm r}\frac{ \partial^{2}u_{\rm r}}{\partial t^{2}}+k_{\rm r}\sum\limits_{n_{\rm c}=-\infty }^{+\infty}\sum\limits_{j=1}^{N_{\rm r}}(u_{\rm r}-u_{\rm s})\delta\Big{(}x-x _{j_{\rm r}}^{n_{\rm c}}\Big{)}=0 \tag{1}\]  
\[E_{\rm s}I_{\rm s}\frac{\partial^{4}u_{\rm s}}{\partial x^{4}}+m_{\rm s}\frac{ \partial^{2}u_{\rm s}}{\partial t^{2}}-k_{\rm r}\sum\limits_{j=1}^{N_{\rm r}} (u_{\rm r}-u_{\rm s})\delta\Big{(}x-x_{j_{\rm r}}^{n_{\rm c}}\Big{)}+k_{\rm s} \sum\limits_{j=1}^{N_{\rm s}}u_{\rm s}\delta\Big{(}x-x_{j_{\rm s}}^{n_{\rm c}} \Big{)}=0 \tag{2}\]  
where, \(E_{\rm r}\) and \(E_{\rm s}\) denote the elastic modulus of the rail and the composite slab, respectively, \(I_{\rm r}\) and \(I_{\rm s}\) are the moment of inertia of the rail and composite slab, and \(m_{\rm r}\) and \(m_{\rm s}\) are the masses per unit length of the rail and composite slab. In the ballastless track structure, both the fastenings and the isolation layer are described using discretely distributed springs, and the isolation layer spring spacing needs to be adjusted according to the defect size. \(k_{\rm r}\) and \(k_{\rm s}\) are the stiffness of the fastenings and the isolation layer, respectively. \(u_{\rm r}\) and \(u_{\rm s}\) represent the displacements of the rail and the slab. \(n_{\rm c}\) denotes the cell number, and \(n_{\rm d}\) represents the number of the defective cell. Within a unit cell of length \(L\), there are \(N_{\rm f}\) fastenings and \(N_{\rm s}\) isolation layer support springs, with corresponding coordinates \(x_{j_{\rm r}}^{n_{\rm c}}\) and \(x_{j_{\rm s}}^{n_{\rm c}}\). For the defective cell, considering the contact loss between composite slab layer and baseplate, the wave equation for the rail remains unchanged, and the wave equation for the composite slab is as follows:  
\[ \begin{split} E_{\rm s}I_{\rm s}\frac{\partial^{4}u_{\rm s}}{ \partial x^{4}}+m_{\rm s}\frac{\partial^{2}u_{\rm s}}{\partial t^{2}}-k_{\rm r }\sum\limits_{j=1}^{N_{\rm r}}(u_{\rm r}-u_{\rm s})\delta\Big{(}x-x_{j_{\rm r }}^{n_{\rm c}}\Big{)}+\sum\limits_{j=1}^{N_{\rm s}}\widehat{k}_{j_{\rm r}}u_{ \rm s}\delta\Big{(}x-x_{j_{\rm s}}^{n_{\rm c}}\Big{)}=0\\ (n_{\rm c}=n_{\rm d};x\in[n_{\rm c}L,(n_{\rm c}+1)L])\end{split}  \tag{3}\]  
where, \(\widehat{k}_{j_{\rm r}}\) represents the stiffness under the composite slab within a defective cell. To construct a supercell model, the size of the supercell is selected as \(L_{\rm s}=N_{x}L\), where \(N_{x}\) is the quantity of unit cells in a supercell and the coordinate range within the supercell is \(\widetilde{x}\in[0,L_{\rm s}]\). The wave equations for the slab track structure are rewritten as follows:  
\[E_{\rm r}I_{\rm r}\frac{\partial^{4}u_{\rm r}}{\partial x^{4}}+m_{\rm r}\frac {\partial^{2}u_{\rm r}}{\partial t^{2}}+k_{\rm r}\sum\limits_{n_{\rm c}=-\infty }^{+\infty}\sum\limits_{j=1}^{N_{\rm r}N_{\rm s}}(u_{\rm r}-u_{\rm s})\delta \Big{(}x-x_{j_{\rm r}}^{n_{\rm c}}\Big{)}=0 \tag{4}\]  
\[E_{\rm s}I_{\rm s}\frac{\partial^{4}u_{\rm s}}{\partial x^{4}}+m_{\rm s}\frac{ \partial^{2}u_{\rm s}}{\partial t^{2}}-k_{\rm r}\sum\limits_{j=1}^{N_{\rm r}} (u_{\rm r}-u_{\rm s})\delta\Big{(}x-x_{j_{\rm r}}^{n_{\rm x}}\Big{)}+\sum \limits_{j=(n_{\rm x}-1)N_{\rm s}+1}^{n_{\rm x}N_{\rm s}}k_{j_{\rm s}}^{n_{\rm x }}u_{\rm s}\delta\Big{(}x-x_{j_{\rm s}}^{n_{\rm x}}\Big{)}=0 \tag{5}\]  
where, \(n_{\rm x}=1,\ldots,N_{x}\) and \(x\in[(n_{x}-1)L,n_{x}L]\). Since the rail is a continuous beam, the governing equation does not change, only the length is changed, and the supercell contains \(N_{x}N_{\rm r}\) fastenings. For the composite slabs in the supercell, there are \(N_{x}\) independent equations and each cell contains \(N_{\rm s}\) isolation layer springs. The stiffness of the \(j_{\rm s}-\)th isolation layer support spring in the \(n_{\rm x}-\)th cell of the supercell is represented by \(k_{j_{\rm s}}^{n_{\rm x}}\).