# Flexural Wave Propagation and Defect States of Periodic Slab Track Structure in High-Speed Railway
## 1 Introduction  
The ballastless track in high-speed railway can be regarded as an infinite periodic structure, in which the ballastless track structure is composed of identical unit cells that are regularly arranged along the longitudinal direction of the line. Previous research has demonstrated that the propagation of elastic waves in periodic structures exhibits distinctive bandgap characteristics, wherein vibration transmission will be significantly attenuated [1, 2, 3]. However, in practical engineering, local defects are inevitably found in ballastless track structures, thereby disrupting periodicity. When local defects occur in periodic structures, the phenomenon of elastic wave localization emerges, which is referred to as defect states [4, 5, 6]. Clarifying the propagation of elastic waves in high-speed railwaytrack structures, especially in the presence of local defects, is of substantial theoretical significance for structural vibration and damage control. Due to the periodicity of track structures, the wave motion characteristics and dynamic analysis methodologies have been extensively investigated. [PERSON] et al. [PERSON] et al. (2016) employed the 2.5-D finite element approach to formulate the eigen equation of the periodic track structure, subsequently obtaining the vibration transmission coefficient of the track structure and analyzing the resonance modes corresponding to the bandgap boundaries frequencies. Moreover, a solution strategy for infinite periodic structures under moving loads was put forward by leveraging the periodicity of the track structure [PERSON] et al. (2016). [PERSON] et al. [PERSON] et al. (2016); [PERSON] et al. (2016) established a periodic ballastless track structure model and calculated its vertical vibration bandgaps by means of the plane wave expansion method and variational principles. Additionally, the influence of parameters such as rail temperature and fastening stiffness of the bandgap characteristics was discussed. [PERSON] et al. [PERSON] et al. (2016) described rail vibration via the 2.5-D finite element method and developed a dynamic response model for ballasted track structure by coupling with the displacement admittance of sleepers. [PERSON] et al. [PERSON] et al. (2016) utilized the wave finite element method to analyze the wave motion of periodically supported rails, exploring the propagation of different types of elastic waves in rail. [PERSON] et al. [PERSON] et al. (2016); [PERSON] et al. (2016) established wave analysis models for periodic ballasted and ballastless track structures and calculated the dispersion curves of the track structures using the transfer matrix method. Based on the phononic crystal theory, the mechanism of bandgap formation in track structures was preliminarily elucidated. [PERSON] et al. [PERSON] et al. (2016); [PERSON] et al. (2016) studied the effect of rail temperature force and fastening stiffness on the bandgaps and vibration transmission characteristics of ballasted track structures using a three-dimensional wave propagation analysis model. Furthermore, based on the bandgap characteristics of elastic waves in track structures, the introduction of supplementary bandgaps can also provide measures for vibration and noise control in track structures [PERSON] et al. (2016); [PERSON] et al. (2016); [PERSON] et al. (2016). When local defects occur in ballastless track structures, the dynamic properties change and directly influence the dynamic response of the structure and even the ride stability of high-speed trains. There are several types of ballastless track structures in China's high-speed railway lines, including CRTS (China Railway Track System) I prefabricated slab track, CRTS II prefabricated slab track, CRTS III prefabricated slab track, and double-block slab track [PERSON] et al. (2016). [PERSON] et al. [PERSON] et al. (2016) summarized the characteristics and causes of defects and damages in different ballastless track structures after long-term service.