## 5 Conclusions  
In this study, numerical solutions were obtained for the dispersion characteristics of a DLLFP in the following three different physical domains using software: the pressure acoustic domain, the solid mechanics domain, and the acoustic-solid coupling domain. By comparing the numerical solutions with the analytical solutions, the feasibility and correctness of applying the Timoshenko beam model to solve the dispersion characteristics of layered fluid-filled pipes were validated. The frequency-thickness product dispersion curves and wavenumber-thickness product dispersion curves of the DLLFP were presented, and the variation patterns of the dispersion curves of different propagating guided wave modes with increasing frequency-thickness product were analyzed. It was found that the number of propagating guided wave modes in the DLLFP increases with increasing  
Figure 8: Absolute sound pressure and stress cloud of a pipeline. (**a**) f = 5008 Hz k = 10.619. (**b**) f = 12,755 Hz k = 9.7833. (**c**) f = 20,752 Hz k = 45.47. (**d**) f = 20,002 Hz k = 41.412. (**e**) f = 20,002 Hz k = 60,411. (**f**) f = 13,505 Hz k = 20.353.  
frequency-thickness product, and the guided wave dispersion curves vary with different propagation modes. The T(0,1) mode in the phase velocity dispersion curve is found to remain essentially smooth from low to high frequencies  
The pressure distribution cloud map of the DLLFP at specific frequencies was obtained through numerical simulation, and the characteristic of the intersection between the straight-through mode and the dispersion curve of the guided wave propagation mode was verified. The analysis of the characteristics of the dispersion curve and the pressure distribution of the DLLFP can provide a reference for the selection of modes in guided wave detection, can accurately identify pipeline damage, can reduce the possibility of misjudgment, and can ensure the safe and stable operation of pipeline systems. This finding has good commercial prospects in the field of non-destructive testing.  
Nonetheless, our proposed model only considers water as the liquid medium filled in the pipeline, and the fluids transported by the pipeline in actual engineering may also be comprised of complex fluids such as non-Newtonian fluids [51]. In future work, it is also necessary to verify the model with other types of liquids, or to carry out relevant experimental verification to meet the requirements of actual production.  
Conceptualization, J.Y. and J.L.; methodology, J.Y.; software, L.Z.; validation, C.W. and Z.T.; formal analysis, L.Z.; investigation, Z.T.; resources, J.Y.; data curation, L.Z.; writing--original draft preparation, J.Y.; writing--review and editing, J.L.; visualization, C.W.; supervision, D.Z.; project administration, D.Z.; funding acquisition, J.Y. All authors have read and agreed to the published version of the manuscript.  
This research is supported by the Natural Science Foundation of Guangdong Province (Project grant No. 2022A1515011562) and the Special Fund for Promoting High-quality Economic Development of Guangdong Province (Project grant No. [2020]161).  
Not applicable.  
The datasets used and analyzed in this work are available from the corresponding author upon reasonable request.  
The authors declare that they do not have any commercial or associative interest that represent conflicts of interest in connection with this work.