## 2 Research on the Method of Solving the Shallow Water Seismic Wave Field Equation
### Partial Solution Method
\begin{cases}\mathbf{M}_{\text{f}}\tilde{\mathbf{p}}_{n}+\mathbf{K}_{\text{f}} \mathbf{p}_{n}-\mathbf{Q}^{\text{T}}\tilde{\mathbf{u}}_{n}=\mathbf{f}_{n}\ (\text{a})\\ \dot{\mathbf{p}}_{n+1}=\dot{\mathbf{p}}_{n}+\Delta t[(1-\alpha_{\text{f}}) \tilde{\mathbf{p}}_{n}+\alpha_{\text{f}}\tilde{\mathbf{p}}_{n+1}]\ (\text{b})\\ \mathbf{p}_{n+1}=\mathbf{p}_{n}+\Delta t\dot{\mathbf{p}}_{n}+\Delta t^{2}\left( \left(\frac{1}{2}-\beta_{\text{f}}\right)\tilde{\mathbf{p}}_{n}+\beta_{\text{f }}\tilde{\mathbf{p}}_{n+1}\right]\ (\text{c})\\ \mathbf{M}_{\text{s}}\tilde{\mathbf{u}}_{n}+\mathbf{K}_{\text{s}}\mathbf{u}_{n}+ \frac{1}{\rho_{\text{f}}}\mathbf{Q}\mathbf{p}_{n}=0\ (\text{d})\\ \dot{\mathbf{u}}_{n+1}=\dot{\mathbf{u}}_{n}+\Delta t[(1-\alpha_{\text{s}})\tilde {\mathbf{u}}_{n}+\alpha_{\text{s}}\tilde{\mathbf{u}}_{n+1}]\ (\text{e})\\ \mathbf{u}_{n+1}=\mathbf{u}_{n}+\Delta t\dot{\mathbf{u}}_{n}+\Delta t^{2}\left[ \left(\frac{1}{2}-\beta_{\text{s}}\right)\tilde{\mathbf{u}}_{n}+\beta_{\text{s }}\tilde{\mathbf{u}}_{n+1}\right]\ (\text{f})\end{cases}  \tag{11}\]  
From Equation (11c), we have  
\[\tilde{\mathbf{p}}_{n+1}=\frac{1}{\beta_{\text{f}}\Delta t^{2}}(\mathbf{p}_{n+1} -\mathbf{p}_{n})-\frac{1}{\beta_{\text{f}}\Delta t}\tilde{\mathbf{p}}_{n}- \left(\frac{1}{2\beta_{\text{f}}}-1\right)\tilde{\mathbf{p}}_{n} \tag{12}\]Substituting Equation (12) into Equation (11a), we obtain  
\[\left(\frac{1}{\beta_{\mathrm{f}}\Delta t^{2}}\mathbf{I}_{\mathrm{f}}+ \mathbf{M}_{\mathrm{f}}^{-1}\mathbf{K}_{\mathrm{f}}\right)\mathbf{p}_{n+1}=\frac {1}{\beta_{\mathrm{f}}\Delta t^{2}}\mathbf{p}_{n}+\frac{1}{\beta_{\mathrm{f}} \Delta t}\dot{\mathbf{p}}_{n}+\left(\frac{1}{2\beta_{\mathrm{f}}}-1\right) \ddot{\mathbf{p}}_{n} \tag{13}\] \[+\mathbf{M}_{\mathrm{f}}^{-1}\mathbf{Q}^{\mathrm{T}}\ddot{\mathbf{ u}}_{n+1}+\mathbf{M}_{\mathrm{f}}^{-1}\mathbf{f}_{n+1}\]  
Similarly, according to Equation (11d,f), we have  
\[\left(\frac{1}{\beta_{\mathrm{f}}\Delta t^{2}}\mathbf{I}_{\mathrm{ s}}+\mathbf{M}_{\mathrm{s}}^{-1}\mathbf{K}_{\mathrm{s}}\right)\mathbf{u}_{n+1}= \frac{1}{\beta_{\mathrm{s}}\Delta t^{2}}\mathbf{u}_{n}+\frac{1}{\beta_{\mathrm{ s}}\Delta t}\dot{\mathbf{u}}_{n}+\left(\frac{1}{2\beta_{\mathrm{s}}}-1\right) \ddot{\mathbf{u}}_{n} \tag{14}\] \[-\frac{1}{\rho_{\mathrm{f}}}\mathbf{M}_{\mathrm{s}}^{-1}\mathbf{ Q}\mathbf{p}_{n+1}\]  
Equations (13) and (14) can be abbreviated as  
\[\left\{ \begin{array}{l}\mathbf{A}_{1}\mathbf{p}_{n+1}=\mathbf{B}_{n}+ \mathbf{Q}^{\mathrm{T}}\ddot{\mathbf{u}}_{n+1}+\mathbf{f}_{n+1}\ (\mathrm{a})\\ \mathbf{A}_{2}\mathbf{u}_{n+1}=\mathbf{G}_{n}-\frac{1}{\rho_{\mathrm{f}}} \mathbf{Q}\mathbf{p}_{n+1}\ (\mathrm{b})\end{array} \right. \tag{15}\]  
Among them,  
\[\left\{ \begin{array}{l}\mathbf{A}_{1}=\frac{1}{\beta_{\mathrm{f}}\Delta t^{2 }}\mathbf{I}_{\mathrm{f}}+\mathbf{M}_{\mathrm{f}}^{-1}\mathbf{K}_{\mathrm{f}} \\ \mathbf{B}_{n}=\frac{1}{\beta_{\mathrm{f}}\Delta t^{2}}\mathbf{p}_{n}+\frac{1}{ \beta_{\mathrm{f}}\Delta t^{2}}\mathbf{p}_{n}+\left(\frac{1}{2\beta_{\mathrm{ f}}}-1\right)\ddot{\mathbf{p}}_{n}\\ \mathbf{A}_{2}=\frac{1}{\rho_{\mathrm{s}}\Delta t^{2}}\mathbf{I}_{\mathrm{s}}+ \mathbf{M}_{\mathrm{s}}^{-1}\mathbf{K}_{\mathrm{s}}\\ \mathbf{G}_{n}=\frac{1}{\rho_{\mathrm{s}}\Delta t^{2}}\mathbf{u}_{n}+\frac{1}{ \beta_{\mathrm{s}}\Delta t}\dot{\mathbf{u}}_{n}+\left(\frac{1}{2\beta_{\mathrm{ s}}}-1\right)\ddot{\mathbf{u}}_{n}\end{array} \right. \tag{16}\]  
When calculating Equation (15), firstly, we need to replace \(\mathbf{p}_{n+1}\) in Equation (15b) with that in Equation (15a) to obtain \(\mathbf{u}_{n+1}\) and then obtain \(\ddot{\mathbf{u}}_{n+1}\), and then we substitute it into Equation (15a) to obtain \(\mathbf{p}_{n+1}\). Finally, we use \(\dot{\mathbf{p}}_{n+1}\) and \(\dot{\mathbf{u}}_{n+1}\) in Equation (11) to achieve the iterative calculation step by step.