## 2 Research on the Method of Solving the Shallow Water Seismic Wave Field Equation
### Global Solution Methods  
The global solution is a direct solution to Equation (1). According to Equation (1), it can be seen that the coefficient matrices of the field variables are asymmetric, and symmetrization is required to facilitate the solution. According to the symmetry of \(\mathbf{M}_{\mathrm{f}}\), \(\mathbf{K}_{\mathrm{f}}\), \(\mathbf{M}_{\mathrm{s}}\), \(\mathbf{K}_{\mathrm{s}}\), two approaches can be used to symmetrize Equation (1). Method 1: Substitute Equation (6b) into Equation (6a), and then multiply the left and right sides of Equation (6b) by the same \(\rho_{\mathrm{f}}\mathbf{K}_{\mathrm{s}}\mathbf{M}_{\mathrm{s}}^{-1}\), and there are  
\[\left[ \begin{array}{cc}\mathbf{M}_{\mathrm{f}}&\mathbf{0}\\ \mathbf{0}&\rho_{\mathrm{f}}\mathbf{K}_{\mathrm{s}}\end{array} \right]\left[ \begin{array}{cc}\tilde{\mathbf{p}}\\ \tilde{\mathbf{u}}\end{array} \right]+\left[ \begin{array}{cc}\mathbf{K}_{ \mathrm{f}}+\frac{1}{\rho_{\mathrm{t}}}\mathbf{Q}^{\mathrm{T}}\mathbf{M}_{ \mathrm{s}}^{-1}\mathbf{Q}&\mathbf{Q}^{\mathrm{T}}\mathbf{M}_{\mathrm{s}}^{-1} \mathbf{K}_{\mathrm{s}}\\ \mathbf{K}_{\mathrm{s}}\mathbf{M}_{\mathrm{s}}^{-1}\mathbf{Q}&\rho_{\mathrm{f}} \mathbf{K}_{\mathrm{s}}\mathbf{M}_{\mathrm{s}}^{-1}\mathbf{K}_{\mathrm{s}} \end{array} \right]\left[ \begin{array}{c}\mathbf{p}\\ \mathbf{u}\end{array} \right]=\left[ \begin{array}{c}\mathbf{f}\\ \mathbf{0}\end{array} \right] \tag{20}\]  
Method 2: Substitute Equation (6a) into Equation (6b), and then multiply the left and right sides of Equation (6a) by the same \(\frac{1}{\rho_{\mathrm{f}}}\mathbf{M}_{\mathrm{f}}\mathbf{K}_{\mathrm{f}}^{-1}\), and we have  
\[\left[ \begin{array}{cc}\frac{1}{\rho_{\mathrm{f}}}\mathbf{M}_{\mathrm{f}} \mathbf{K}_{\mathrm{f}}^{-1}\mathbf{M}_{\mathrm{f}}&-\frac{1}{\rho_{\mathrm{f}} }\mathbf{M}_{\mathrm{f}}\mathbf{K}_{\mathrm{f}}^{-1}\mathbf{Q}^{\mathrm{T}}\\ -\frac{1}{\rho_{\mathrm{f}}}\mathbf{Q}\mathbf{K}_{\mathrm{f}}^{-1}\mathbf{M}_{ \mathrm{f}}&\mathbf{M}_{\mathrm{s}}+\frac{1}{\rho_{\mathrm{f}}}\mathbf{Q} \mathbf{K}_{\mathrm{f}}^{-1}\mathbf{Q}^{\mathrm{T}}\end{array} \right]\left[ \begin{array}{c}\tilde{\mathbf{p}}\\ \tilde{\mathbf{u}}\end{array} \right]+\left[ \begin{array}{cc}\frac{1}{\rho_{ \mathrm{f}}}\mathbf{M}_{\mathrm{f}}\mathbf{K}_{\mathrm{f}}^{-1}\mathbf{f}\\ 0&\mathbf{K}_{\mathrm{s}}\end{array} \right]\left[ \begin{array}{c}\mathbf{p}\\ \mathbf{u}\end{array} \right]=\left[ \begin{array}{c}\frac{1}{\rho_{\mathrm{f}} }\mathbf{M}_{\mathrm{f}}\mathbf{K}_{\mathrm{f}}^{-1}\mathbf{f}\\ -\frac{1}{\rho_{\mathrm{f}}}\mathbf{Q}\mathbf{K}_{\mathrm{f}}^{-1}\mathbf{f} \end{array} \right] \tag{21}\]  
The difference between Method 1 and Method 2 is that the symmetry treatment of the replacement variables is different when transforming Equation (1), where Method 1 replaces \(\tilde{\mathbf{u}}\), and the calculation involves the inverse of the elastomer mass matrix, and Method 2 replaces \(\mathbf{p}\), and the calculation involves the inverse of the fluid stiffness matrix. Although Equation (21) is formally more complex than Equation (20), the calculation of Equation (21) is superior according to the later analysis. Since only the upper surface boundary of the considered fluid region is a free surface boundary condition, which usually does not have zero frequency, the inverse of \(\mathbf{K}_{\mathrm{f}}\) does not present a singularity problem. Using \(\mathbf{U}\) to denote the field variables of the fluid and elastomer, \(\mathbf{F}\) to denote the acoustic source loading term, and \(\mathbf{B}_{1}\), \(\mathbf{B}_{2}\) to denote the coefficient matrices of the field variables, respectively, Equations (20) and (21) can be simplified as  
\[\mathbf{B}_{1}\dot{\mathbf{U}}+\mathbf{B}_{2}\mathbf{U}=\mathbf{F} \tag{22}\]  
The global solution of Equation (1) can be achieved by solving Equation (22) using the central difference method and the implicit Newmark method.