## 2 Research on the Method of Solving the Shallow Water Seismic Wave Field Equation
### Partial Solution Method  
Rewrite Equation (1) in the form of a system of equations  
\[ \begin{cases}\mathbf{M}_{\mathrm{f}}\mathbf{\ddot{p}}+\mathbf{K}_{\mathrm{f}} \mathbf{p}-\mathbf{Q}^{\mathrm{T}}\mathbf{\ddot{u}}=\mathbf{f}\ (\mathrm{a})\\ \mathbf{M}_{\mathrm{s}}\mathbf{\ddot{u}}+\mathbf{K}_{\mathrm{s}}\mathbf{u}+ \frac{1}{\rho_{\mathrm{f}}}\mathbf{Q}\mathbf{p}=0\ (\mathrm{b})\end{cases}  \tag{6}\]  
The partial solution is defined as solving Equations (6a) and (6b) using time-domain methods, respectively. Based on the explicit central difference method and implicit Newmark method, there are three kinds of partial solution methods: the explicit-explicit method, implicit-implicit method, and hybrid explicit-implicit method, and the specific solution process is as follows:  
1. Explicit-Explicit Method (EEM)  
The explicit-explicit method refers to the explicit integration method for both the fluctuation equations in the fluid and elastomer regions. The center difference method is applied to Equations (6a) and (6b), respectively, with the expressions\[ \begin{cases}\mathbf{M}_{\text{f}}\tilde{\mathbf{p}}_{n}+\mathbf{K}_{\text{f}} \mathbf{p}_{n}-\mathbf{Q}^{\text{T}}\tilde{\mathbf{u}}_{n}=\mathbf{f}_{n}\ (\text{a})\\ \ddot{\mathbf{p}}_{n}=\frac{1}{\Delta t^{2}}(\mathbf{p}_{n-1}-2\mathbf{p}_{n} +\mathbf{p}_{n+1})\ (\text{b})\\ \dot{\mathbf{p}}_{n}=\frac{1}{2\Delta t}(\mathbf{p}_{n+1}-\mathbf{p}_{n-1})\ ( \text{c})\\ \mathbf{M}_{\text{s}}\tilde{\mathbf{u}}_{n}+\mathbf{K}_{\text{s}}\mathbf{u}_ {n}+\frac{1}{\rho_{\text{f}}}\mathbf{Q}\mathbf{p}_{n}=0\ (\text{d})\\ \ddot{\mathbf{u}}_{n}=\frac{1}{\Delta t^{2}}(\mathbf{u}_{n-1}-2\mathbf{u}_{n} +\mathbf{u}_{n+1})\ (\text{e})\\ \dot{\mathbf{u}}_{n}=\frac{1}{2\Delta t}(\mathbf{u}_{n+1}-\mathbf{u}_{n-1})\ ( \text{f})\end{cases}  \tag{7}\]  
where \(\Delta t\) is the time step, the subscript \(n\) is the time, and \(n\)+1 is to perform a time step, e.g., \(\mathbf{p}_{n+1}=\mathbf{p}_{n+\Delta t}\). As an example, the fluctuation equations in the fluid region, the first three equations of Equation (7), give a specific procedure for calculating the sound pressure field. \(\mathbf{p}_{n+1}\) can be found from Equations (7b) and (7c). We have  
\[\mathbf{p}_{n-1}=\mathbf{p}_{n}-\Delta t\dot{\mathbf{p}}_{n}+\frac{\Delta t^{ 2}}{2}\tilde{\mathbf{p}}_{n} \tag{8}\]  
Substituting Equation (7b) into Equation (7a) gives  
\[\mathbf{p}_{n+1}=\left(2\mathbf{I}_{\text{f}}-\Delta t^{2}\mathbf{M}_{\text{ f}}^{-1}\mathbf{K}_{\text{f}}\right)\mathbf{p}_{n}-\mathbf{p}_{n-1}+\Delta t^{2} \mathbf{M}_{\text{f}}^{-1}\mathbf{Q}^{\text{T}}\tilde{\mathbf{u}}_{n}+\Delta t ^{2}\mathbf{M}_{\text{f}}^{-1}\mathbf{f}_{n} \tag{9}\]  
where \(\mathbf{I}_{\text{f}}\) is a unit matrix of the same size as the fluid mass matrix. Combining the given initial conditions, Equations (8) and (9), the sound pressure field \(\mathbf{p}_{n+1}\) in the fluid can be found. The fluctuation equations in the elastomer region are calculated by the same procedure as for the fluid. The final simplification of Equation (7) yields the expression for the sound pressure and displacement field as  
\[ \begin{cases}\mathbf{p}_{n+1}=\left(2\mathbf{I}_{\text{f}}-\Delta t^{2}\mathbf{ M}_{\text{f}}^{-1}\mathbf{K}_{\text{f}}\right)\mathbf{p}_{n}-\mathbf{p}_{n-1}+ \Delta t^{2}\mathbf{M}_{\text{f}}^{-1}\mathbf{Q}^{\text{T}}\tilde{\mathbf{u}} _{n}+\Delta t^{2}\mathbf{M}_{\text{f}}^{-1}\mathbf{f}_{n}\\ \mathbf{u}_{n+1}=\left(2\mathbf{I}_{\text{s}}-\Delta t^{2}\mathbf{M}_{\text{s} }^{-1}\mathbf{K}_{\text{s}}\right)\mathbf{u}_{n}-\mathbf{u}_{n-1}-\frac{1}{ \rho_{\text{f}}}\Delta t^{2}\mathbf{M}_{\text{s}}^{-1}\mathbf{Q}\mathbf{p}_{n} \end{cases}  \tag{10}\]  
where \(\mathbf{I}_{\text{s}}\) is a unit matrix of the same size as the elastomer mass matrices. 2. Implicit-Implicit Method (IIM)  
The implicit-implicit method is the implicit integration method for the fluctuation equations in the fluid and elastomer regions. The implicit Newmark method is used for Equations (6a) and (6b), respectively, with the expressions  
\[