## 1 Introduction
### Modal Analysis of Coupling Structure of the Lower Plate and the Water-Filled Chamber
For elastic cavity wall, it is usually a shell structure:  
\[Sw_{n}(\sigma,t)+m_{\rm B}\frac{\partial^{2}w_{n}(\sigma,t)}{\partial t^{2}}= p_{\rm A}(\sigma,t)-p_{\rm B}(\sigma,t)(\sigma\in D_{\rm F}) \tag{9}\]  
where \(S\) is a linear differential operator characterizing the stiffness of the elastic cavity wall; \(m_{\rm B}\) is the surface density of the elastic cavity wall; \(w_{n}(\sigma,t)\) is the vibration displacement response of the elastic plate; \(p_{\rm A}(\sigma,t)\) and \(p_{\rm B}(\sigma,t)\) are the excitation (pressure) on the elastic cavity wall from the sound field inside the cavity and from the outside, respectively. Similar to the case of Equation (1), if \(p_{\rm A}(\sigma,t)=0\) and \(p_{\rm B}(\sigma,t)=0\), Equation (9) has the modal solution \(W_{\rm Bj}(\sigma)\cdot\exp\big{(}\mathrm{j}\omega_{\rm Bj}t\big{)}\)\((j=1,2,\ldots)\), where \(\omega_{\rm Bj}\) is the \(j\)th order modal frequency of the elastic cavity wall and \(W_{\rm Bj}(\sigma)\) is the form function corresponding to \(\omega_{\rm Bj}\) with the following orthogonal properties:  
\[\iint_{D_{\rm F}}m_{\rm Bj}W_{\rm Bj}(\sigma)W_{\rm Bj}(\sigma)\mathrm{d}S= \left\{ \begin{array}{cc}0&(i\
eq j)\\ M_{\rm Bj}&(i=j)\end{array} \right. \tag{10}\]  
where \(M_{\rm Bj}\)\((j=1,2,3,\ldots)\) is the modal mass of the thin-walled structure in vacuum. The solution of Equation (9) is expressed in the form of a superposition of vibrations:  
\[w_{n}(\sigma,t)=\sum_{j=1}^{\infty}W_{\rm Bj}(\sigma)\beta_{j}(t)=[\mathbf{W}_{ \rm B}(\sigma)]^{\rm T}\mathbf{\beta}(t)\quad(\sigma\in D_{\rm F}) \tag{11}\]  
where \(\mathbf{W}_{\rm B}(\sigma)=[W_{\rm B1}(\sigma),W_{\rm B2}(\sigma),W_{\rm B3}( \sigma),\ldots]^{\rm T}\) and \(\mathbf{\beta}(t)=[\beta_{1}(t),\beta_{2}(t),\beta_{3}(t),\ldots]^{\rm T}\) are the column vectors of \(W_{\rm Bj}(\sigma)\) and its corresponding modal coordinate \(\beta_{j}(t)\), respectively. Substituting the above equation into Equation (9) and using Equation (10):  
\[M_{\rm B}\Big{[}\tilde{\mathbf{\beta}}(t)+\Omega_{\rm B}^{2}\mathbf{\beta}(t)\Big{]}= \mathbf{Q}_{\rm A}(t)+\mathbf{Q}_{\rm B}(t) \tag{12}\]  
where, \(M_{\rm B}=\mathrm{diag}[M_{\rm B0},\ M_{\rm B1},\ M_{\rm B2},\ \ldots]\) and \(\mathbf{\Omega}_{\rm B}=\mathrm{diag}[\omega_{\rm B0},\ \omega_{\rm B1},\ \omega_{\rm B2},\ \ldots]\) are the diagonal matrices of the modal mass \(M_{\rm Bj}\) and the natural frequency \(\omega_{\rm Bj}\), respectively; \(\mathbf{Q}_{\rm A}(t)\) and \(\mathbf{Q}_{\rm B}(t)\) are the column vectors of the general forces loaded on the thin-walled structure in vacuum due to \(p_{\rm A}(\sigma,t)\) and \(p_{\rm B}(\sigma,t)\), respectively, and:  
\[\mathbf{Q}_{\rm A}(t)=\iint_{D_{\rm F}}W_{\rm B}(\sigma)p_{\rm A}(\sigma,t) \mathrm{d}S \tag{13}\]  
\[\mathbf{Q}_{\rm B}(t)=-\iint_{D_{\rm F}}W_{\rm B}(\sigma)p_{\rm B}(\sigma,t) \mathrm{d}S \tag{14}\]  
Using Equation (5), \(p_{\rm A}(\sigma,t)\) is expressed as:  
\[p_{A}(\sigma,t)=-\rho_{0}\left.\frac{\partial\Phi}{\partial t}\right|_{\sigma \in D_{\rm F}}=-\rho_{0}\left[\mathbf{\Phi}_{\rm A}(\sigma)\right]^{T}\Big{|}_{ \sigma\in D_{\rm F}}\dot{A}(t) \tag{15}\]  
Let the coupling matrix \(L\) be as follows:  
\[L=\iint_{D_{\rm F}}\mathbf{\Phi}_{\rm A}(\sigma)[W_{\rm B}(\sigma)]^{\rm T} \mathrm{d}S \tag{16}\]Combining Equations (8), (12), (13), and (16) and \(u_{n}(\sigma,t)=\partial w(\sigma,t)/\partial t\), the modal coordinate equation of the coupling system is obtained:  
\[ \begin{cases}\rho_{0}\mathbf{M}_{\rm A}\Big{[}\dot{\mathbf{A}}(t)+\mathbf{\Omega}_{\rm A}^{ 2}\mathbf{A}(t)\Big{]}-L\dot{\mathbf{B}}(t)=\mathbf{0}\\ \mathbf{M}_{\rm B}\Big{[}\tilde{\mathbf{B}}(t)+\mathbf{\Omega}_{\rm B}^{2}\mathbf{\beta}(t) \Big{]}+\rho_{0}\mathbf{L}^{\rm T}\dot{\mathbf{A}}(t)=\mathbf{Q}_{\rm B}(t)\end{cases}  \tag{17}\]  
where \(\mathbf{0}\) denotes a column vector with all \(0\) elements. Let \(\mathbf{\alpha}(t)=\dot{\mathbf{A}}(t)\) and convert the above equation to the following form for an easy solution:  
\[