## 5 Conclusions  
Based on the theory of ideal fluid and the dynamic behavior of saturated soil, the infinitely long viscoelastic beam model embedded in the fluid-saturated soil system is adopted to analyze the vibration response of the underwater tunnel. The analysis considers different factors such as water depths, load velocities, and viscoelastic beam parameters of the riverbed foundation. The results indicate that:  
1. The riverbed depth affects the dynamic response of the tunnel during operation. As the riverbed depth increases, the vertical amplitude of the tunnel structure beam increases. Moreover, when the riverbed depth is significant, even at low speeds, the vibration of the tunnel structure beam becomes fluctuating. This indicates that the riverbed water significantly reduces the Rayleigh wave velocity of the fluid-saturated soil system in the riverbed foundation. Therefore, it is necessary to control the driving speed during high water levels.
2. When the load speed reaches a very high level, the bending moment and shear force of the tunnel structure beam decrease as the load velocity increases. The distribution range is also relatively narrow. As the moving speed increases, more of the load is transferred to the riverbed foundation, thereby reducing the internal forces in the tunnel structure beam.
3. With the enhanced permeability of the saturated riverbed foundation, the vertical displacement, bending moment, and shear force of the tunnel structure beam increase. For riverbed-saturated foundation soils with poor permeability, the operating load of the subway will generate significant pore pressures in the foundation soil.
4. As the viscosity coefficient of the viscoelastic beam increases, the vertical vibration amplitude of the tunnel structure beam will decrease. However, further increasing the viscosity coefficient of the beam will have little effect on its vibration amplitude. Therefore, the standard solid model of the viscoelastic beam can effectively simulate the creep and relaxation phenomena of the material. objectively reflect the working condition of the concrete structure of the tunnel. It can objectively reflect the working conditions of concrete structures in tunnels.  
H.H.: Conceptualization, Software, Writing--original draft; Y.R.: Investigation; X.X.: Writing--review & editing; B.X.: Methodology, Validation. All authors have read and agreed to the published version of the manuscript.  
This research was funded by Key project of advantageous science and technology innovation team of Jiangxi province in 2017 (\"5511\"project), Grant number 20171 BCB19001.The funding program for major disciplines academic and technical leaders of Jiangxi Province in 2017, Grant number 20172 BCB2022. The key science and technology research project in Jiangxi province Department of Education, Grant number GJJ151096. The key projects of the natural science foundation of Jiangxi Province, Grant number 20202 ACBL204016. The Key Laboratory of Hydraulic and Waterway Engineering of the Ministry of Education, Chongqing Jiaotong University, Grant number SLK2023B04.  
Not applicable.  
Data Availability Statement: Data is contained within this article.  
The authors declare no conflict of interest.