## 4 Example Calculation and Numerical Analysis  
The effective stiffness matrix between the reverse force q considering the coupling of the riverbed water and riverbed foundation and the viscoelastic Euler beam displacement \(w_{b}\) is obtained by Formulas (15a,b), (17)-(19). The equivalent stiffness matrix is then inserted into Formulas (15a,b) and (18) to obtain the expressions of riverbed soil displacement, pore pressure, and stress in the frequency field-wave field. According to the above derivation, the variables in the frequency field can be expressed as commonly as:  
\[\widetilde{\widetilde{\Omega}}(\xi,z,\omega)=\widetilde{\widetilde{\Omega}}^{* }(\xi,z,\omega)F_{z}\delta(\omega+\xi v_{c}) \tag{20}\]  
The solutions can be obtained in both the time and space domains by applying the double Fourier inverse transformation to the above equation. Considering the nature of the Dirac-\(\delta\) function, the double inverse Fourier transformation can be reduced to:  
\[\Omega(x,z,t)=(\frac{1}{2\pi})^{2}F_{z}\int\limits_{-\infty}^{+\infty} \widetilde{\widetilde{\Omega}}(\xi,z,-\xi v_{c})e^{i\xi(x-v_{c}t)}d_{\xi}^{x} \tag{21}\]  
Due to the complex expression of the integrand function, it is difficult to produce the closed-form solution of the Fourier inverse transformation. In this paper, the IFFT method is adopted to complete the Fourier inverse transformation. In addition, because the layered soil adopts the viscoelastic model, there is no branch point or singularity appearing in the integral path of the horizontal wave number \(\xi\) because of the soil viscosity force. The number of discrete points of the wave number \(\xi\) is \(N=4096\)[23], spatial interval \(\Delta x=0.1\), and the spatial calculation interval \(L_{x}=N\Delta x\). The vibration characteristics of the immersed tunnel based on the built-in infinite-length viscoelastic beam model of a fluid-saturated soil system under different water depths, load speeds, saturated soil, and viscoelastic beam parameters are investigated. As shown in Figure 1, the concentration degree of the moving point load is \(F_{z}\), with a constant velocity \(v_{c}\) along the positive direction of the \(X\) axis in the fluid-saturated soil system with the built-in infinite length viscoelastic beam. The viscoelastic beam adopts the standard solid model, in which the viscoelastic parameters are \(E_{p}^{(1)},E_{p}^{(2)}\) and the viscosity coefficient is \(\eta\). For the convenience of calculation \(v_{s}=\sqrt{\eta/\rho_{s}}\), displacement, stress, and pore pressure without dimension as: \(u*=2\pi\mu u_{R}/F_{z}\), \(M*=M_{x}/(F_{z}a_{R}^{2})\), \(Q*=Q_{x}/(F_{z}a_{R}^{2})\), \(p*=p(x,z,t)A/F_{z}\), in which the reference length \(a_{R}=1.0\) m.  
Example 1: Model validation. For comparison with the literature results, if the viscoelastic beam \(E_{p}^{(1)}=E_{p}^{(2)}=2.010^{9}\), \(\eta=0.0\) Pa, then the viscoelastic beam is fully elastic. In addition, when the saturated soil parameters \(a\), \(b_{p}\), \(M\), \(\rho_{f}\), \(\phi\), \(a_{\infty}\) are close to 0, the saturated soil can degenerate into elastic soil. A relatively simple two-dimensional model is considered [26], which consists of an elastic layer possessing a small viscosity and a beam located inside the layer. It is assumed that the layer is infinitely long in the horizontal direction, is fixed along the bottom, and has a traction-free surface. Figure 3 shows the depth of the riverbed as \(h_{w}=0.0\), the buried depth of the tunnel as \(H=7.0\) m, and from the riverbed bedrock, \(h_{s}=15.0\) m, moving point load as \(F_{z}=1.0\)\(\times\)\(10^{4}\) N\(\cdot\)m\({}^{-1}\), speed \(v_{c}=30.0\) ms\({}^{-1}\), the vertical and horizontal displacement of the riverbed surface observation point \(A\) (0.0 m, 0.0 m) changes with time \(t\), and the calculation results of homogeneous elastic soil-beam conditions are given in the literature [26]. From Figure 3, the solution in this paper is almost consistent with the results of the literature [26].