## 4 Example Calculation and Numerical Analysis
Compared with the vertical and horizontal displacement when the moving load is about to reach the observation point \(t<0\) and when the load leaves the observation point \(t>0\), it shows that when the load is about to reach the observation point \(t<0\), the vibration wave frequency at the surface observation point of the soil is higher. But with a smaller amplitude, Presented as the Doppler effect, that is, when the load approaches the observation point, higher-frequency vibrations will occur. The further the load is away from the observation point, the lower the vibration frequency. But the amplitude of the vibration wave is just opposite to the law of the vibration frequency. From Figure 9, the hole pressure at the observation point increases with the increase in load velocity. In addition, at low speed (\(v_{c}=0.2\;v_{s}\)), the pore pressure symmetry appears to point \(t=0.0\;\mathrm{s}\). However, at the high speed (\(v_{c}=1.2\;v_{s}\)), the symmetry disappears (Figure 7c). And there is negative pressure on \(t=0.0\;\mathrm{s}\). Example 3: Influence of the Permeability Coefficient of a Saturated Foundation load speed \(v_{c}=0.2\;v_{s}\), \(h_{w}=2.0\;\mathrm{m}\); the calculation parameters of riverbed saturated soil and underwater immersed tunnel beam are the same as in Example 2, investigating the influence of the permeability coefficient of saturated foundation \(b_{p}=1.0\times 10^{5}\)-\(1.0\times 10^{11}\;\mathrm{kg/m^{3}\cdot s}\) to the displacement, shear, and bending moment of the tunnel structure beam and the displacement and pore pressure of the internal observation point in river bed soil (A (0, 2.0 m)), as shown in Figures 10 and 11. From Figures 10 and 11, with the increase in the bp value of the saturated foundation (enhanced soil permeability), the vertical displacement, bending moment, shear force of the tunnel structure beam, the vertical and horizontal displacement, and the hole pressure of the observation point increase. However, when the bp value of a saturated foundation increases to a certain value \(h_{w}=4.0\;\mathrm{m}\), it has little influence on the vertical displacement, bending moment, shear force of the beam, vertical and horizontal displacement, and pore pressure of the observation point. The soil with poor permeability produces large pore pressure under the action of a load. Example 4: Influence of the viscosity coefficient of different viscoelastic beams. In the case of the viscosity coefficient of different viscoelastic beams: \(\eta=3.0\times 10^{5}\;\mathrm{Pa}\), \(3.0\times 10^{7}\;\mathrm{Pa}\), \(3.0\times 10^{9}\;\mathrm{Pa}\), \(3.0\times 10^{11}\;\mathrm{Pa}\), \(E_{p}^{(1)}=E_{p}^{(2)}=2.0\times 10^{9}\;\mathrm{Pa}\), the load speed \(v_{c}=0.2\;v_{s}\), \(h_{w}=2.0\;\mathrm{m}\), the calculation parameters of other saturated riverbed soil are the same as in Example 2. The elastic modulus and viscoelastic coefficient of the viscoelastic beam affect the displacement, shear, and bending moments of the tunnel structure beam and the  
Figure 6: Change of bending moment of the underwater immersed tunnel during different riverbed water depths. (**a**) v\({}_{\rm c}\) = 0.2 \(v_{s}\), (**b**) v\({}_{\rm c}\) = 0.5 \(v_{s}\), (**c**) v\({}_{\rm c}\) = 1.2 \(v_{s}\). Figure 8: Horizontal displacement change of riverbed foundation observation point A (0.0 m, 2.0 m) at different riverbed water depths. (**a**) \(v_{\rm c}=0.2\)\(v_{s}\), (**b**) \(v_{\rm c}=0.5\)\(v_{s}\), (**c**) \(v_{\rm c}=1.2\)\(v_{s}\). Figure 9: Change of hole pressure at the observation point of the riverbed foundation during different riverbed water depths. (**a**) \(\rm v_{c}=0.2\)\(\rm v_{s}\), (**b**) \(\rm v_{c}=0.5\)\(\rm v_{s}\), (**c**) \(\rm v_{c}=1.2\)\(\rm v_{s}\). Figure 10: Influence of different riverbed foundation permeability characteristics on the vibration of an underwater immersed tunnel structure beam. (**a**) vertical displacement, (**b**) shear force, (**c**) bending moment.