## 1 Introduction
### Modal Analysis of Coupling Structure of the Lower Plate and the Water-Filled Chamber
\begin{cases}\rho_{0}\mathbf{M}_{\rm A}\big{[}\tilde{\mathbf{\alpha}}(t)+\mathbf{\Omega}_ {\rm A}^{2}\mathbf{\alpha}(t)\big{]}-L\tilde{\mathbf{B}}(t)=\mathbf{0}\\ \mathbf{M}_{\rm B}\Big{[}\tilde{\mathbf{\beta}}(t)+\mathbf{\Omega}_{\rm B}^{2}\mathbf{\beta}(t )\Big{]}+\rho_{0}\mathbf{L}^{\rm T}\mathbf{\alpha}(t)=\mathbf{Q}_{\rm B}(t)\end{cases}  \tag{18}\]  
In the above equation, \(\mathbf{Q}_{\rm B}(t)=0\) for modal analysis and let:  
\[ \begin{cases}\mathbf{\alpha}(t)=\frac{1}{\rho_{0}}\big{(}\sqrt{\mathbf{M}_{\rm A}} \big{)}^{-1}\mathbf{\chi}_{\rm A}{\rm e}^{{\rm i}\omega t}\\ \mathbf{\beta}(t)=\big{(}\mathbf{\Omega}_{\rm B}\sqrt{\mathbf{M}_{\rm B}}\big{)}^{-1}\mathbf{ \chi}_{\rm B}{\rm e}^{{\rm i}\omega t}\end{cases}  \tag{19}\]  
where \(\mathbf{\chi}_{\rm A}=[\mathbf{\chi}_{\rm A0}\mathbf{\cdot}\mathbf{\chi}_{\rm A1}\mathbf{\cdot} \mathbf{\chi}_{\rm A2}\mathbf{\cdot}]^{\rm T}\) and \(\mathbf{\chi}_{\rm B}=[\mathbf{\chi}_{\rm B0}\mathbf{\cdot}\mathbf{\chi}_{\rm B1}\mathbf{\cdot} \mathbf{\chi}_{\rm B2}\mathbf{\cdot}\mathbf{\cdot}]^{\rm T}\) are column vectors of the fluid-structure coupled modal shape coefficients \(\mathbf{\chi}_{\rm A\prime}\) (\(r=0,1,2,\ldots\)) and \(\mathbf{\chi}_{\rm Bj}\) (\(j=0,1,2,\ldots\)) associated with the cavity sound field and the flexible boundary structure, respectively. The eigenvalue problem can be obtained:  
\[\mathbf{\Gamma}_{\mathbf{\chi}}= \begin{bmatrix}\mathbf{\Gamma}_{11}&\mathbf{\Gamma}_{12}\\ \mathbf{\Gamma}_{21}&\mathbf{\Gamma}_{22}\end{bmatrix}\begin{bmatrix}\mathbf{\chi}_{\rm A }\\ \mathbf{\chi}_{\rm B}\end{bmatrix} =\omega^{2} \begin{bmatrix}\mathbf{\chi}_{\rm A}\\ \mathbf{\chi}_{\rm B}\end{bmatrix} =\omega^{2}\chi \tag{20}\]  
where \(\mathbf{\Gamma}\) represents a symmetric characteristic matrix which can be represented in blocks as \(\mathbf{\Gamma}_{11}=\mathbf{\Omega}_{A}^{2}+\big{(}\sqrt{\mathbf{M}_{\rm A}}\big{)}^{-1} \mathbf{L}\mathbf{M}_{\rm B}{}^{-1}\mathbf{L}^{T}\big{(}\sqrt{\mathbf{M}_{\rm A}}\big{)}^{-1}\), \(\mathbf{\Gamma}_{12}=\mathbf{\Gamma}_{21}^{T}=\big{(}\sqrt{\mathbf{M}_{\rm A}}\big{)}^{-1} \mathbf{L}\mathbf{\Omega}_{\rm B}\big{(}\sqrt{\mathbf{M}_{\rm B}}\big{)}^{-1}\), \(\mathbf{\Gamma}_{22}=\mathbf{\Omega}_{\rm B}^{2}\mathbf{\cdot}\mathbf{\times}\) is the vector of fluid-structure coupled modal shape coefficients. The eigenvalues \(\omega_{\rm Ck}^{2}\) (\(k=0,1,2,\ldots\)) of the matrix \(\mathbf{\Gamma}\), and its corresponding eigenvector \(\Big{[}\mathbf{\chi}^{(k)}\Big{]}^{\rm T}=\bigg{[}\{\mathbf{\chi}_{\rm A}^{(k)}\}^{ \rm T},\Big{\{}\mathbf{\chi}_{\rm B}^{(k)}\}^{\rm T}\Big{]}=\Big{[}\mathbf{\chi}_{\rm A 0}^{(k)}\mathbf{\cdot}\mathbf{\chi}_{\rm A1}^{(k)}\mathbf{\cdot}\mathbf{\chi}_{\rm A2}^{(k)} \ldots,\mathbf{\times}_{\rm B0}^{(k)}\mathbf{\cdot}\mathbf{\chi}_{\rm B1}^{(k)}\mathbf{\cdot} \mathbf{\chi}_{\rm B2}^{(k)}\ldots\Big{]}\) are obtained from Equation (20), where \(\mathbf{\omega}_{\rm Ck}\) is the \(k\)th order liquid-solid coupling mode frequency. The velocity potential \(\Phi_{\rm Ck}(\sigma,t)\) of the acoustic field in the cavity and the vibration velocity \(u_{\rm Ck}(\sigma,t)\) of the cavity wall have a single-frequency vibration mode:  
\[ \begin{cases}\Phi_{\rm Ck}(\sigma,t)=\frac{1}{\mu_{\rm Ck}\rho_{0}}\big{[}\mathbf{ \Phi}_{\rm A}(\sigma)\big{]}^{\rm T}\big{(}\sqrt{\mathbf{M}_{\rm A}}\big{)}^{-1} \mathbf{\chi}_{\rm A}{\rm e}^{{\rm i}\omega_{\rm Ck}t}\quad(\sigma\in V)\\ u_{\rm Ck}(\sigma,t)=|\omega_{\rm Ck}[\mathbf{W}_{\rm B}(\sigma)]^{\rm T}\big{(} \mathbf{\Omega}_{\rm B}\sqrt{\mathbf{M}_{\rm B}}\big{)}^{-1}\mathbf{\chi}{\rm e}^{{\rm i }\omega_{\rm Ck}t}\quad(\sigma\in D_{F})\end{cases}