## 2 The Simplified Calculation Model and Control Equation of an Underwater Immersed Tunnel
\begin{array}{l}E_{b}=E_{bR}(\frac{1+i\omega\tau_{bc}}{1+i\omega\tau_{bc}}), \ E_{bR}=\frac{E_{b}^{(1)}E_{b}^{(2)}}{E_{b}^{(1)}+E_{b}^{(2)}}\\ \tau_{bc}=\frac{\eta}{E_{b}^{(1)}+E_{b}^{(2)}},\ \tau_{bc}=\frac{\eta}{E_{b}^{(2)}} \end{array}  \tag{5}\]  
In the formula: \(E_{b}^{(1)}\) and \(E_{b}^{(2)}\) is the elastic parameter of the viscoelastic beam in the standard solid model; \(\eta\) is the viscosity coefficient. Considering the riverbed surface (\(z=0\)) impervious condition, there are the following boundary conditions:  
\[ \begin{array}{l}u_{wz}(x,z=0,t)=u_{z}(x,z=0,t)\\ pf_{fw}(x,z=0,t)=-\sigma_{zz}(x,z=0,t)\\ \sigma_{zx}(x,z=0,t)=0\\ w_{z}(x,z=0,t)=0\end{array}  \tag{6}\]  
Considering the riverbed surface (\(z=0\)) permeability, the following boundary conditions are provided:  
\[ \begin{array}{l}u_{wz}(x,z=0,t)=u_{z}(x,z=0,t)+w_{z}(x,z=0,t)\\ pf_{fw}(x,z=0,t)=p(x,z=0,t)\\ \sigma_{zx}(x,z=0,t)=0\\ \sigma_{zz}(x,z=0,t)=-p_{fw}(x,z=0,t)\end{array}  \tag{7}\]  
Figure 2: Schematic diagram of the immersed tunnel structure. At the riverbed water surface of (\(z=-h_{w}\)), there are:  
\[p_{fw}(x,z=-h_{w},t)=0\]  
If the displacement of the tunnel is zero in the horizontal direction and there is continuous displacement between the tunnel structure and the soil layer in the vertical direction, the following boundary conditions are provided:  
\[ \begin{array}{c}u_{x}(x,H^{-},t)=u_{x}(x,H^{+},t)=0,\\ w_{z}(x,H^{-},t)=w_{z}(x,H^{+},t)=W_{b}(x,t)\end{array}  \tag{8}\]  
For bottom bedrock, the following displacement boundary conditions are provided:  
\[u_{z}(x,z=H+Hs,t)=w_{z}(x,H+Hs,t)=0. \tag{9}\]  
**3. The Frequency Domain and Wave Number Domain Solutions and the Equivalent Stiffness**  
The decoupling method is used to solve the Biot equation by adopting a potential function to represent each displacement component in the soil, and the displacement vectors \(U\) and \(W\) can be expressed by four potential functions \(\Phi\), \(\Psi\), \(\chi\), and \(\varphi\). \[U=\
abla\phi+\
abla\times\psi;\quad\text{ W =}\
abla\chi+\
abla\times\varphi\] (10a,b)  
By substituting the Equation (10a,b) into the Biot control Equation (2a,b), two decoupled equations can be obtained. \[ \begin{array}{c}[K_{p}][L]\Big{\{}\ddot{\phi}\Big{\}}=[N]\{\phi\}+[R]\Big{\{} \dot{\phi}\Big{\}}\\ [K_{s}][L]\Big{\{}\ddot{\psi}\Big{\}}=[N]\{\psi\}+[R]\Big{\{}\ddot{\psi}\Big{\}} \end{array} \] (11a,b)  
In formula:  
\[ \begin{array}{c}[K_{p}]=\left[\begin{array}{cc}\lambda+&\alpha^{2}M&+2G& \alpha M\\ &\alpha M&\text{M}\end{array}\right]\\ [K_{s}]=\left[\begin{array}{cc}G&0\\ 0&0\end{array}\right][L]=\left[\begin{array}{cc}\
abla^{2}&0\\ 0&\
abla^{2}\end{array}\right]\\ [N]=\left[\begin{array}{cc}\rho&\rho_{f}\\ \rho_{f}&\text{m}\end{array}\right][R]=\left[\begin{array}{cc}0&0\\ 0&b\end{array}\right]\end{array} \]  
Two-dimensional Fourier transformation from space to wavenumber \(x\rightarrow\xi\), time to frequency, \(t\rightarrow\omega\)  
\[\left. \begin{array}{c}\widetilde{\widetilde{f}}(\xi,\omega)=\int\limits_{- \infty}^{\infty}\int\limits_{-\infty}^{\infty}f(x,t)e^{-i(\xi x+\omega t)}dxdt \\ f(x,t)=\frac{1}{(2\pi)^{2}}\int\limits_{-\infty}^{\infty}\int\limits_{-\infty }^{\infty}\widetilde{\widetilde{f}}(\xi,\omega)e^{i(\xi x+\omega t)}d\xi d \omega\end{array} \right\}\] (12a,b)  
In formula: Superscript \({}^{-\sim}\) represents \(x\rightarrow\xi\), \(t\rightarrow\omega\) the value of the two-dimensional Fourier transform domain. According to the [PERSON] principle [27], Equation (11a,b) can perform the Helmholtz decomposition:  
\[\
abla^{2}\widetilde{\phi}_{f}+r_{1}^{2}\widetilde{\phi}_{f}=0\ \
abla^{2} \widetilde{\phi}_{s}+r_{2}^{2}\widetilde{\phi}_{s}=0\ \
abla^{2}\widetilde{\psi}+r_{3}^{3}\widetilde{\psi}=0 \tag{13}\]  
In the formula, \(r_{1}\), \(r_{2}\), \(r_{3}\) are the complex wavenumbers of compressed fast, slow, and shear waves, respectively. In the rectangular coordinate system, the soil skeleton, fluid displacement, stress, and pore pressure can be expressed by the potential function:  
\[