## 2 The Simplified Calculation Model and Control Equation of an Underwater Immersed Tunnel  
A simplified two-dimensional physical model is adopted for the underwater immersed tunnel, as shown in Figure 1, and the water depth of the river bed is \(h_{w}\), the buried depth of the underwater tunnel is \(H\), \(h_{s}\) from the river bed bedrock, the load concentration is \(Fz\), the velocity of the vertical mobile load is \(v_{c}\), considering the coupling of the riverbed water and riverbed foundation and underwater immersed tunnel, riverbed water using ideal fluid media, riverbed foundation layer using saturated porous medium theory simulation, underwater immersed tunnel structure simplified to infinite long viscoelastic Euler beam, using the standard linear solid model theory, describing the structure damping influence on vibration. Riverbed water is a theoretical fluid, and the dynamic control equation uses [PERSON]'s acoustic equation [23]:  
\[\
abla^{2}p_{fw}-\frac{1}{v_{w}^{2}}\frac{\partial^{2}p_{fw}}{\partial t^{2}}=4 \pi\delta(z-z_{\rm s}) \tag{1}\]  
In the formula: \(v_{w}\) respectively represents the fluid water pressure and the acoustic wave speed in the fluid, \(z_{\rm s}\) indicates the position of the action point of the sound source. The Biot theory of saturated soil [24] is used for the riverbed foundation description, and the momentum conservation equations of saturated pore medium and pore fluid are, respectively,  
\[ \begin{array}{l}\mu u_{i,jj}+(\lambda+\alpha^{2}M+\mu)u_{j,ii}-\alpha Mw_{j,ii }=(\rho\dot{u}_{i}+\rho\dot{v}_{i})\\ \alpha Mu_{j,ii}+Mw_{j,ii}=\rho_{f}\ddot{u}_{i}+m\ddot{w}_{i}+bp\ddot{w}_{i}\end{array} \] (2a,b)  
In the formula: \(u_{i}\), \(w_{i}\) (\(i\) = 1,2) are the displacements of the soil skeleton and the permeability displacements of the fluid relative to the soil skeleton. The point above the displacement indicates the derivative of time; \(\lambda\), \(\mu\) is the Lame constant; \(\alpha\), \(M\) are Biot soil skeleton and water compression parameters, respectively; \(\rho\), \(\rho_{f}\) is saturated soil and pore water density; \(f\) is porosity; \(m=a_{\infty}\rho/f\), \(a_{\infty}\) is the bending coefficient of the pore medium; \(bp\) represents the amount related to the viscosity of the pore fluid and the permeability coefficient of the soil. The constitutive equations for saturated soil and pore water are, respectively. \[ \begin{array}{l}\sigma_{ij}=\lambda\varepsilon\delta_{ij}+\mu(u_{i,j}+u_{j,i })-\alpha\delta_{ij}p\\ p=-\alpha Me+M\theta\end{array} \] (3a,b)  
In the formula: \(\varepsilon=u_{i,j}\), \(\theta=-w_{ij}\) is the incremental quantity of volumetric strain of the soil skeleton and the fluid volume in the unit volume pore medium, respectively; \(\sigma_{\ddot{y}}\), \(p\) is the total stress component and pore water pressure, respectively. Figure 1: A schematic illustration of the water-soil-viscoelastic beam coupling system under a moving load. The rectangular underwater immersed tunnel with a cross section shown in Figure 2 is simplified to an infinitely long viscoelastic Euler beam according to the paper [25,26] method, and the vibration equation is as follows:  
\[ \begin{array}{l}\rho_{B}\ddot{W}_{b}+E_{b}I_{b}\frac{g^{4}W_{b}}{gx^{4}}=Fz \delta(x-v_{c}t)\\ +a(\sigma_{zz}(x,h^{-},t)-\sigma_{zz}(x,h^{+},t))\end{array}  \tag{4}\]  
In the formula: \(E_{b}\), \(\rho_{B}\), \(W_{b}I_{b}\) are the structure density, vertical displacement, and tube section stiffness of the immersed tunnel, respectively; \(Fz\) is the load amplitude; and \(a\) is the equivalent average diameter of the cross-section of the underwater immersed tunnel. For the infinitely long viscoelastic Euler beam, the standard linear solid model theory is used to describe its damping situation, and the compound elastic modulus of the viscoelastic beam is:  
\[