# Research on a Rail Defect Location Method Based on a Single Mode Extraction Algorithm
## 1 Introduction  
Ultrasound guided waves are widely used in the non-destructive testing of continuous welded rails because of their wide coverage and rapid propagation over large distances [1; 2]. Compared with simple waveguide structures, such as plates and pipelines, rails have much more complex cross-sectional structures; thus, they require far more complicated ultrasonic wave modes for defect detection. With the increase in the frequency, the number of modes increases, making the research more difficult. However, the analytical method is no longer suitable for the analysis of the propagation characteristics of ultrasonic guided waves in rails. Therefore, numerical analysis methods have been broadly introduced, including the Boundary Element Method [3], the Finite Difference Method [4], the Finite Element Method [5], and the Semi-Analytical Finite Element Method [6]. Among them, the semi-analytical finite element method reduces the analysis dimensions and has a higher computational efficiency, thus is commonly adopted in this field of researches. The time difference between the main wave packet of the reflected wave and the excitation wave packet in the acquisition signal must be calculated; the position of the defect can be inferred by multiplying the group velocity of the main modes. Its accuracy is related to the selected detection mode. Therefore, it is necessary to have a good understanding of the propagation characteristics of guided waves in rails to select the appropriate modal type, frequency, and excitation mode for defect detection. [PERSON] and [PERSON] [7] used two-dimensional Fourier transform to separate the guided wave modes and calculate the corresponding phase velocities.T[PERSON] et al. [8] utilized the semi-analytical finite element (SAFE) method to calculate the group velocity and phase velocity dispersion curves of guided waves in rails at frequencies of 0-100 kHz and carried out experimental verification. [PERSON] et al. [9] used the vibration mode analysis method to analyze the guided wave modes at frequencies of 0-50 kHz; the dispersion characteristics and wave structures of five typical modes were analyzed, and the mode types and frequency ranges suitable for inspecting railhead and rail bottom defects were obtained. [PERSON] and [PERSON] [10; 11; 12; 13; 14] analyzed the guided wave mode data in rails at frequencies of 25 kHz and 35 kHz using the SAFE method and measured along the rails using a laser vibrometer. A large number of vibration data were obtained, the amplitudes of each mode were calculated, and the vibration waveforms of distant rails were successfully predicted. In 2017, [PERSON] et al. [15] studied the problem of the mode repulsion and crossing behavior of the approaching wave number versus frequency curves by analyzing the second derivative of the eigenvalue with respect to the wave number. These methods analyze the modal characteristics of guided waves propagating in rails, giving a modal analysis. However, it has never been possible to separate the reflected guided wave modes of defects to use them for defect location directly. Meanwhile, a large number of research teams have applied ultrasonic guided waves for defect detection in waveguide structures. Most of the research results focus on simple cross-sectional structures such as plates and pipes, while studies on complex structures such as rails remain relatively rare. [PERSON] et al. [2] adopted the finite element method to analyze the energy distribution characteristics of guided waves with different frequencies in rails. The simulation results showed that a high-frequency guided wave of 200 kHz is concentrated on the upper surface of the railhead, and a low-frequency guided wave of 30 kHz is distributed through the entire railhead. The guided waves at low frequencies are more sensitive to the transverse cracks of the railhead, while the guided waves at high frequencies are good at detecting defects in the shelling. These results have been experimentally verified. [PERSON] et al.