# Achieving Enhanced Sound Insulation through Micromembranes-Type Acoustic Metamaterials
## 3 Response of the Supporting Frame  
As shown in Figure 1, the whole membrane can be regarded as being composed of a large number of independent A\(\mu\)Ms within the frame. When the vibration of the frame is small, it does not affect the A\(\mu\)Ms much; hence, the response of the whole assembly is given by Equations (2) and (3). This means that the TL of the whole assembly will be no different from that of a single A\(\mu\)M, which holds for the case of a small assembly. However, when the assembly is larger, the vibration of the lattice frame can no longer be ignored and, in this case, the boundary of any two nearby A\(\mu\)Ms cannot be regarded as rigid. So, now it is the case of a membrane mounted on a compliant frame, in which the effective stiffness \(K_{\rm e}\) and effective mass \(M_{\rm e}\) need to be calculated. First, the displacement of the cross-points of the supporting frame can be obtained according to Section 2. Then, the averaged displacement of each A\(\mu\)M cell can be approximately evaluated as the sum of the displacement of an A\(\mu\)M given by Equation (1) (local response) and the averaged displacement of the four cross-points of the surrounding frame. With the number of A\(\mu\)Ms assigned by their corner coordinates (top left corner) as shown in Figure 2, the displacement of the A\(\mu\)M, numbered \((\Psi,\xi)\), can be obtained as:  
\[\overline{x}^{\prime}_{(\Psi)(\xi)}=\overline{x}+\frac{\eta_{\Psi\xi}+\eta_{( \Psi+1)\xi}+\eta_{(\Psi+1)}+\eta_{(\Psi+1)(\xi+1)}}{4}=\overline{x}+\overline {\eta}_{(\Psi)(\xi)}\;, \tag{9}\]  
where \(\overline{x}\) is the average displacement of the A\(\mu\)M with rigid boundaries, \(\eta_{ij}\) is the displacement of the frame at cross-point \((i,j)\), and \(\overline{\eta}_{(\Psi)(\xi)}\) is the average displacement surrounding the A\(\mu\)M numbered \((\Psi,\xi)\). In the calculation, the square of the force should be used, so that \(\overline{x}\) should be based on \(Pr^{2}\) and \(\eta_{ij}\) should be based on \(P(r+a)^{2}\), where \(P\) is the unit force acting on the component. In general, the effective stiffness \(K_{\rm e}\) can be written as the force divided by the average displacement as  
\[K_{\rm e}=\frac{F_{(\Psi)(\xi)}}{\overline{x}^{\prime}_{(\Psi)(\xi)}}=\frac{P \cdot r^{2}}{\overline{x}^{\prime}_{(\Psi)(\xi)}}\;. \tag{10}\]  
Then, the effective mass \(M_{\rm e}\) can be obtained by calculating its kinetic energy as  
\[M_{\rm e}=t\rho r^{2}\frac{9\overline{x}^{2}/4+2\overline{x}\overline{\eta}_{ (\Psi)(\xi)}+\overline{\eta}^{2}_{(\Psi)(\xi)}}{\Big{(}\overline{x}+\overline {\eta}_{(\Psi)(\xi)}\Big{)}^{2}}\;. \tag{11}\]Up to this point, the equivalent stiffness and mass of each A\(\upmu\)M mounted in a real (compliant) frame are derived. The entire assembly consists of all A\(\upmu\)Ms and the supporting frame. The transmission coefficient of sound energy (in STL) can be calculated by considering both the transmission path through the membrane cells (A\(\upmu\)Ms) and the supporting frame. Therefore, the sound pressure over the two transmission paths can be written as \(p_{f}\) for the frame and \(p_{m}\) for the A\(\upmu\)Ms, which are calculated, respectively, as follows:  
\[p_{f}=\frac{2p_{i}R_{a}}{R_{fs}+1/\left(jwC_{fs}\right)+jwM_{fs}+2R_{a}}\, \tag{12}\]  
\[p_{m}=\frac{2p_{i}R_{a}}{R_{es}+1/(jwC_{es})+jwM_{es}+2R_{a}}\, \tag{13}\]  
where \(p_{i}\) is the incident sound pressure, \(R_{fs}=R_{f}/S_{f}\), \(M_{fs}=M_{f}/S_{f}\), \(C_{fs}=S_{f}/K_{f}\), \(R_{es}=R_{e}/S_{m}\), \(M_{es}=M_{e}/S_{m}\), and \(C_{es}=S_{m}/K_{e}\) is the specific acoustic impedance of the frame and A\(\upmu\)Ms. \(S_{f}\) and \(S_{m}\) denote the area of the frame and of each A\(\upmu\)M, respectively, so that \(S_{f}=L^{2}-(n+1)^{2}r^{2}\) and \(S_{m}=r^{2}\). \(R_{a}\) is the characteristic impedance of air, so \(R_{a}=\rho_{0}c_{0}\), where \(\rho_{0}\) and \(c_{0}\) are the density and sound velocity of air, respectively.