## 4 Conclusions  
In this paper, based on the GN generalized thermoelastic model, a theoretical model of guided wave propagation in hollow cylindrical pipes containing a temperature field is established. The longitudinal thermoelastic guided waves and the curves of displacement and temperature are solved. The effects of diameter-to-thickness ratio and temperature on the propagation characteristics of guided waves in hollow cylinders are investigated. The following conclusions can be drawn:  
(1) Firstly, based on the GN generalized thermoelastic theoretical model, the governing equations of the thermoelastic guided wave are obtained by coupling the heat conduction equations. On the basis of satisfying the stress-free as well as isothermal boundary conditions, the acquisition of the dispersion curve of the thermoelastic guided wave is realized by using Legendre polynomials.  
(2) It is analytically obtained that the coupled heat conduction equation leads to a thermal mode. The mode is similar to the elastic mode, with a cutoff frequency as well as a dispersion phenomenon. Also, by solving the displacement and temperature variation curves, it is determined that the amplitude of the mass displacement of the thermal mode is much smaller than that of the other modes, but the amplitude of their temperature variation is close to the same value.  
(3) The effect of the diameter-to-thickness ratio on the propagation characteristics is obtained by solving the longitudinal guided wave dispersion curves at different diameter-to-thickness ratios. For L(0,1) and L(0,2), the cutoff frequency gradually increases with the decrease in the diameter-thickness ratio. For the thermal mode, the cutoff frequency of the mode decreases as the diameter-thickness ratio decreases.  
(4) The phase velocity dispersion curve from 293 K to 593 K is calculated with a temperature gradient of 50 K. By analyzing the first three modes, the results show that with the gradual increase in temperature, the phase velocity shows a decreasing trend.  
(5) In this paper, the propagation characteristics of longitudinal thermoelastic waves in hollow cylinders are studied by coupling GN generalized thermoelastic theory without energy dissipation. Although the linear and small deformation assumptions based on the model simplify the calculation model and make it more feasible in numerical calculation, these assumptions may be challenged under some high-frequency or large-deformation conditions, and there are some deviations. Therefore, in future studies, we plan to expand the existing model, consider the addition of a nonlinear strain-displacement relationship, and supplement the influence of other high-frequency effects to continue to improve the model.  
Methodology, J.Z.; Writing, J.Z. and X.Z.; Investigation, X.Z. and Y.L.; Data curation, J.Z., X.Z. and Y.Z.; Validation, Y.Z., X.S. and Y.L.; formal analysis, X.S. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Key Research and Development Program of China, grant number 2022 YFC3005002, and the National Natural Science Foundation of China, grant number 62071433.  
Not applicable.  
The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.  
All data that support the findings of this study are included within the article.  
The authors declare no conflicts of interest.