## 2 Research on the Method of Solving the Shallow Water Seismic Wave Field Equation
### Partial Solution Method
It is worth noting that \(\mathbf{p}_{n+1}\) in Equation (15b) cannot be used to replace Equation (15a) because it not only involves the generalized inverse solution of the non-square matrix \(\mathbf{Q}\) but also when the initial value is set to zero; since \(\mathbf{Q}\) is a large and sparse matrix, the result of iterative computation of the loading term \(\left(\rho_{\mathrm{f}}\mathbf{M}_{\mathrm{s}}\mathbf{A}_{1}\mathbf{Q}^{-1} \mathbf{M}_{\mathrm{s}}\mathbf{A}_{2}+\mathbf{Q}^{\mathrm{T}}\right)\mathbf{f} _{n+1}\) is prone to be all zeros. Since \(\mathbf{M}_{\mathrm{f}}\), \(\mathbf{M}_{\mathrm{s}}\) are usually positive definite, the inverse of \(\mathbf{A}_{1}\), \(\mathbf{A}_{2}\) does not encounter the singularity problem. The implicit Newmark method is unconditionally stable when \(\alpha_{\mathrm{f}}=\alpha_{\mathrm{s}}=1/2\), \(\beta_{\mathrm{f}}=\beta_{\mathrm{s}}=1/4\). In this case, the expression for the iterative calculation of the sound pressure and displacement field variables after the simplification of Equation (15) is  
\[\left\{ \begin{array}{l}\mathbf{u}_{n+1}=\mathbf{b}_{1}\mathbf{u}_{n}+\mathbf{ b}_{2}\dot{\mathbf{u}}_{n}+\mathbf{b}_{3}\ddot{\mathbf{u}}_{n}-\mathbf{b}_{4} \mathbf{p}_{n}-\mathbf{b}_{5}\dot{\mathbf{p}}_{n}-\mathbf{b}_{6}\ddot{\mathbf{p }}_{n}\\ \quad-\frac{1}{\rho_{\mathrm{f}}}\Big{(}\mathbf{I}_{\mathrm{s}}+\frac{1}{\rho_{ \mathrm{f}}}\frac{4}{\Delta t^{2}}\mathbf{Q}\mathbf{d}_{1}^{-1}\Big{)}^{-1} \mathbf{d}_{2}^{-1}\mathbf{Q}\mathbf{d}_{1}^{-1}\mathbf{f}_{n+1}\\ \mathbf{p}_{n+1}=\mathbf{a}_{1}\mathbf{p}_{n}+\mathbf{a}_{2}\dot{\mathbf{p}}_{n} +\mathbf{a}_{3}\ddot{\mathbf{p}}_{n}+\mathbf{a}_{4}(\mathbf{u}_{n+1}-\mathbf{u }_{n})\\ \quad-\mathbf{a}_{5}\dot{\mathbf{u}}_{n}-\mathbf{a}_{6}\ddot{\mathbf{u}}_{n}+ \mathbf{d}_{1}\mathbf{f}_{n+1}\end{array} \right. \tag{17}\]  
where  
\[\left\{\begin{array}{l}\mathbf{b}_{1}=a_{0}\mathbf{q}_{1}^{-1}\mathbf{d}_{2}^ {-1}\mathbf{M}_{\mathrm{s}}+a_{0}\mathbf{q}_{1}^{-1}\mathbf{q}_{2}\mathbf{Q}^{ \mathrm{T}}\ \ \ \mathbf{q}_{2}=\frac{1  
3. Explicit-Implicit Mixed Method (EIM)  
The explicit-implicit mixed method is an explicit method for the fluctuation equations in the fluid region and an implicit method for the fluctuation equations in the elastomer region. In problems related to wave propagation in fluid media, the explicit method for the fluid domain is a reasonable choice. For the elastomer region, the center difference method is usually less suitable, as the low-frequency component is usually dominant in its structural dynamic response, allowing for a larger time-invariant time from computational accuracy considerations. The time step size depends mainly on the accuracy requirements for unconditionally stable implicit methods. The expression for the explicit-implicit hybrid method is given by  
\[\left\{ \begin{array}{l}\mathbf{M}_{\mathrm{f}}\tilde{\mathbf{p}}_{n}+ \mathbf{K}_{\mathrm{f}}\mathbf{p}_{n}-\mathbf{Q}^{\mathrm{T}}\tilde{\mathbf{u }}_{n}=\mathbf{f}_{\mathrm{n}}\\ \tilde{\mathbf{p}}_{n}=\frac{1}{\Delta t^{2}}\left(\mathbf{p}_{n-1}-2\mathbf{p} _{n}+\mathbf{p}_{n+1}\right)\\ \tilde{\mathbf{p}}_{n}=\frac{1}{\Delta t}\left(\mathbf{p}_{n+1}-\mathbf{p}_{n- 1}\right)\\ \mathbf{p}_{n-1}=\mathbf{p}_{n}-\Delta t\tilde{\mathbf{p}}_{n}+\frac{\Delta t^ {2}}{2}\tilde{\mathbf{p}}_{n}\\ \mathbf{M}_{\mathrm{s}}\tilde{\mathbf{u}}_{n}+\mathbf{K}_{\mathrm{s}}\mathbf{u }_{n}+\frac{1}{\rho_{\mathrm{t}}}\mathbf{Q}\mathbf{p}_{n}=0\\ \tilde{\mathbf{u}}_{n+1}=\tilde{\mathbf{u}}_{n}+\Delta t[(1-\alpha)\tilde{ \mathbf{u}}_{n}+\alpha\tilde{\mathbf{u}}_{n+1}]\\ \mathbf{u}_{n+1}=\mathbf{u}_{n}+\Delta t\hat{\mathbf{u}}_{n}+\Delta t^{2}\Big{[} \left(\frac{1}{2}-\beta\right)\tilde{\mathbf{u}}_{n}+\beta\tilde{\mathbf{u}}_{ n+1}\Big{]}\end{array} \right. \tag{18}\]  
where \(\alpha\), \(\beta\) are parameters related to the integration accuracy and stability. Equation (18) is transformed to obtain the expression of sound pressure and displacement field as  
\[\left\{