## 1 Introduction  
With the continuous development of ultrasonic guided wave research, great progress has been made in the basic theory, numerical calculation, and simulation experiments. The application of ultrasonic guided wave technology is more and more extensive in the field of nondestructive testing. [PERSON] et al. [1, 2] and [PERSON] et al. [3, 4] led their research groups to develop a pipeline detection system for non-destructive detection of buried pipelines based on ultrasonic guided wave technology. [PERSON] et al. [5, 6] developed a rail detection system. Researchers have carried out non-destructive testing of waveguide mediums, such as flat plates [7, 8], composite materials, pipes, rods [9], and rails [10, 11], by using ultrasonic guided wave technology. The results obtained have greatly promoted the development and application of ultrasonic guided wave technology in petrochemical, aerospace, high-speed railway, and other industries. Ultrasonic guided waves can cover the cross-section of the waveguide medium during propagation and have a higher detection efficiency [12]. However, guided waves have the characteristics of multi-mode and dispersion, which also makes it complicated to use guided wave technology for non-destructive testing. The dispersion curves [13] describe the guided wave modes that can propagate at a certain frequency in a waveguide medium, and contain information, such as phase velocity, frequency, wavenumber, group velocity, and so on. It is important to develop a theoretical framework for analyzing and designing guided wave detecting equipment. There are many methods for solving the dispersion curves of waveguide media, and the semi-analytical finite elements (SAFE) method [14; 15] is often used to solve the dispersion curves of guided waves in an arbitrary cross-section waveguide medium. After the finite element discretization on the cross-section of the waveguide medium, the wave process is expressed by the simple harmonic motion along the direction of the wave propagation and the wave equation is established based on the Hamiltonian principle of the variational method. The relationship between the wavenumber and frequency is obtained by solving the eigenvalue problems, and then the dispersion curves can be plotted. When applying the SAFE method to solve the dispersion curves, the eigenvector is the displacement of each discrete node in the cross-section, and contains the shape of the guided mode. The eigenvector is used to analyze the mode shapes, which is important to study the excitation of the guided wave modes and develop a guided wave transducer. [PERSON] et al. [16] analyzed the process of Lamb wave propagation in metallic and resin plates loaded with water on a single surface. The structure of the Lamb wave was analyzed by the SAFE method. [PERSON] et al. [17] solved the dispersion curves of the guided waves in a rod, analyzed the mode shapes, and plotted the wave structures for a square bar. [PERSON] [18] solved the dispersion equation of guided waves in cylinders and plotted a mode diagram of the n-order modes of the guided waves in a pipe. [PERSON] et al. [19] plotted the mode shapes in a rail at low frequency, and analyzed the vibration characteristics of each mode based on the mode diagram. In recent years, high-speed railways have developed rapidly [20]. Rails are important infrastructure components of railways. In order to guarantee the safe operation of high-speed railways, it is necessary to detect rails' internal defects and prevent potential fracture in advance. The cross-section of the rail is complex. In the discrete cross-section, there are more nodes and elements, and the modes of vibration are complicated [21]. When analyzing the guided wave mode shapes in the rail, the method of plotting the mode displacement vector diagram is generally adopted [22]. The displacements of all nodes in the x, y, and z directions are listed in the mode vector matrix.