# Review on Rail Damage Detection Technologies for High-Speed Trains
## 3 Non-Destructive Testing Methods for High-Speed Train Rails
### Ultrasonic Waveguide Testing Methods
The study by scientist [PERSON] [37] on the propagation of elastic waves on solid surfaces is considered the beginning of guided wave research, and these elastic waves are named Rayleigh waves. [PERSON] [38] studied elastic waves inside a flat plate based on [PERSON]'s research, derived the wave equation of elastic waves in a free state plate structure, obtained Lamb waves by solving the wave equation, and expressed the dispersion characteristics of the elastic waves using this wave equation, making a historic contribution to the development of guided waves. In 1959, [PERSON] [39; 40] studied the propagation of guided waves on the surface of tubular structures. By using Helmholtz decomposition to obtain arbitrary values of the physical parameters involved, characteristic equations applicable to hollow cylindrical structures were obtained, and complete analytical expressions for axisymmetric and non-axisymmetric guided waves were derived. In the research and design of ultrasonic guided wave excitation and reception devices, the commonly used excitation and reception guided wave methods are divided into three types: piezoelectric, electromagnetic, and laser pulse. [PERSON] et al. [41] studied a guided wave excitation sensor that can be used for rail detection and verified the feasibility of using piezoelectric transducers in rail detection through simulation analysis and experi  
Figure 7: Ultrasonic wave acquisition schematic diagram at different angles of ultrasonic head. mental verification. [PERSON] et al. [32] used non-contact air coupling and electromagnetic ultrasonic transducers (EMATs) to excite and receive ultrasonic guided waves at the rail head and rail waist and effectively detect the structural state of the rail. [PERSON] et al. [42] designed a transducer applied to the rail head and collected and analyzed the sensing signals received by the rail head. [PERSON] [43] and [PERSON] [41] designed and optimized, respectively, electromagnetic ultrasonic guided wave transducers for damage detection in steel rail bottoms. The phase velocity of a guided wave is the propagation velocity of a point with a fixed phase in the direction of its propagation, while the group velocity of a guided wave is the propagation velocity of an envelope composed of waves with similar frequencies. The group velocity represents the propagation velocity of the wave group. The general definitions of group velocity and phase velocity are  
\[C_{p} =\frac{\omega}{k} \tag{1}\] \[C_{g} =\frac{dw}{dk}\]  
Among them, \(\omega\) is the angular frequency, \(k=\frac{2\pi}{\lambda}\) is the wavenumber, and \(\lambda\) is the wavelength. Further deformation results in the relationship between group velocity and phase velocity, expressed as  
\[C_{g}=\frac{C_{p}^{2}}{C_{p}-\omega\frac{dC_{p}}{d\omega}} \tag{2}\]  
In the process of rail damage detection, the commonly used method for solving the dispersion curve is the potential function method, which is implemented with the particle motion equation derived by the Lame-Navier equation [42; 44]:  
\[\mu\cdot u_{i,j}+(\lambda+\mu)\cdot u_{i,j}+\rho\cdot f_{i}=\rho\cdot\ddot{u}_ {i}(i,j=1,2,3\ldots) \tag{3}\]  
Among them, \(\lambda\) and \(\mu\) are Lame Changshu parameters, \(\rho\) is the density, \(u\) is a unique particle, and \(f_{i}\) is the force acting on the particle. Although ultrasound can achieve the identification and localization of rail damage, there are still certain limitations in the refinement of damage types and quantitative analysis. The detection speed reaches 15 km per hour, and the detection rate can reach 91%. There is still significant research to be conducted in the field of the intelligent identification and detection of rail damage.