## 1 Introduction
### Calculation of the Transfered Power Flow of the Coupling System
From Equations (29), (36) and (37), the following equations can be obtained:  
\[\mathbf{F}_{\rm Eb}=\Big{[}\mathbf{S}_{21}+\mathbf{S}_{22}\mathbf{C}-\mathbf{A}^{(2)}(\mathbf{S}_{11}+ \mathbf{S}_{12}\mathbf{C})\Big{]}^{-1}\mathbf{A}^{(1)}\mathbf{Q}_{\rm E} \tag{39}\]  
\[\mathbf{U}_{n}=\mathbf{C}\Big{[}\mathbf{S}_{21}+\mathbf{S}_{22}\mathbf{C}-\mathbf{A}^{(2)}(\mathbf{S}_{11}+ \mathbf{S}_{12}\mathbf{C})\Big{]}^{-1}\mathbf{A}^{(1)}\mathbf{Q}_{\rm E} \tag{40}\]  
\[\mathbf{F}_{\rm Et}=(\mathbf{S}_{11}+\mathbf{S}_{12}\mathbf{C})\Big{[}\mathbf{S}_{21}+\mathbf{S}_{22} \mathbf{C}-\mathbf{A}^{(2)}(\mathbf{S}_{11}+\mathbf{S}_{12}\mathbf{C})\Big{]}^{-1}\mathbf{A}^{(1)}\mathbf{Q }_{\rm E} \tag{41}\]  
\[\mathbf{U}^{\prime}_{n}=(\mathbf{S}_{21}+\mathbf{S}_{22}\mathbf{C})\Big{[}\mathbf{S}_{21}+\mathbf{S}_{22 }\mathbf{C}-\mathbf{A}^{(2)}(\mathbf{S}_{11}+\mathbf{S}_{12}\mathbf{C})\Big{]}^{-1}\mathbf{A}^{(1)}\mathbf{Q }_{\rm E} \tag{42}\]  
The total input power flow for external excitation:  
\[ \begin{array}{ll}P_{\rm in}&=\iint_{D_{\rm E}}\frac{1}{2}{\rm Re}\{u^{\prime} _{n}(\sigma)P_{\rm E}^{\rm e}(\sigma)\}{\rm d}S\\ &=-\frac{\omega}{2}{\rm Re}\{{\rm j}\big{[}\iint_{D_{\rm E}}P_{\rm E}^{\rm e}( \sigma)[\mathbf{W}_{\rm E}^{\rm T}(\sigma)H_{\rm E}\mathbf{Q}_{\rm E}+\mathbf{W}_{\rm E}^{ \rm T}(\sigma)H_{\rm E}\widetilde{\mathbf{W}}_{\rm E}F_{\rm Et}]{\rm d}S\}\\ &=\frac{\omega}{2}{\rm Re}\{{\rm j}Q_{\rm E}^{\rm H}H_{\rm E}(\mathbf{Q}_{\rm E}+ \widetilde{\mathbf{W}}_{\rm E}F_{\rm Et})\}\end{array}  \tag{43}\]  
where the superscript \" \(\ast\)\" indicates that the conjugate is taken. Power flow into the lower plate-water-filled acoustic cavity coupling system:  
\[P_{\rm c}=\frac{1}{2}\mathrm{Re}\Big{\{}F_{\rm Eb}^{\rm H}\mathbf{U}_{n}\Big{\}} \tag{44}\]  
Power flow into the water-filled acoustic cavity:  
\[ \begin{array}{rl}P_{\rm tr}&=\iint\frac{1}{2}\mathrm{Re}\big{\{}U_{n}(\sigma) P^{*}(\sigma)\big{|}_{\sigma\in D_{\rm F}}\big{\}}\mathrm{d}S\\ &=\frac{\rho_{0}c_{0}\omega}{2}\mathrm{Re}\Big{\{}\mathrm{j}P_{\rm B}^{\rm H} \mathbf{T}^{\rm H}\mathbf{H}^{*}\mathbf{X}_{\rm A}^{\rm H}\big{(}\sqrt{M_{\rm A}}\big{)}^{- 1}\mathbf{L}\big{(}\varOmega_{\rm B}\sqrt{M_{\rm B}}\big{)}^{-1}\mathbf{X}_{\rm B}HT\mathbf{ P}_{\rm B}\Big{\}}\end{array}  \tag{45}\]  
where \(\mathbf{P}_{\rm B}=-\iint_{D_{\rm F}}\mathbf{W}_{\rm B}(\sigma)\Bigg{[}\sum\limits_{l= 1}^{N_{\rm I}}F_{\rm El}\delta(\sigma-\sigma_{l})\Bigg{]}\mathrm{d}S=-\overset{ -}{W}_{\rm B}F_{\rm Eb}\). Therefore, Equation (45) is further expressed as:  
\[P_{\rm tr}=\frac{\rho_{0}c_{0}\omega}{2}\mathrm{Re}\Bigg{\{}\mathrm{j}F_{\rm Eb }\overset{-}{W}_{\rm B}\overset{-}{\rm T}\mathbf{H}^{*}\mathbf{X}_{\rm A}^{\rm H}\Big{(} \sqrt{M_{\rm A}}\Big{)}^{-1}\mathbf{L}\Big{(}\varOmega_{\rm B}\sqrt{M_{\rm B}} \Big{)}^{-1}\mathbf{X}_{\rm B}HT\overset{-}{W}_{\rm B}F_{\rm Eb}\Bigg{\}} \tag{46}\]  
The ratio of \(P_{\rm tr}\) to \(P_{\rm in}\) reflects the absorption coefficient of the cavity and provides an evaluation of the energy absorption level of the sound insulation system. In acoustics, the average power flow per unit area is called sound intensity, and the sound intensity is proportional to the amplitude of the sound pressure. The sound pressure measured on a logarithmic scale is defined as the sound pressure level [28]. Therefore, there is a certain proportional relationship between the average power flow and the average sound pressure level [29]. Therefore, the average power flow, the average sound pressure level and the absorption coefficient can all evaluate the sound insulation performance of the system. However, due to the limited time and space, only different active control strategies are discussed for the system in this paper. So only the average power flow into the water-filled acoustic cavity is calculated and analyzed in the numerical simulation below.