# Achieving Enhanced Sound Insulation through Micromembranes-Type Acoustic Metamaterials
## 2 Basic Calculation of a Square Membrane and Its Cross-Frame  
Figure 1a shows the geometry of the A\(\upmu\)M array and its frames, which are investigated in this paper. The structure is composed of a regular array of square holes in a solid material that is covered by a rubber membrane. The flexible membrane is clamped at the edges of the square holes, and the frame divides the whole membrane into a large number of A\(\upmu\)Ms. Each cell of the membrane, i.e., each A\(\upmu\)M, has side length \(r\) and thickness \(t\). Each cell of the lattice has width \(a\) and thickness \(b\). In this work, the membrane is assumed to be made of latex rubber with a Young's modulus \(E_{m}=7\) MPa, mass density \(\rho_{m}=1000\) kg/m\({}^{3}\), and Poisson's ratio \(\mu_{m}=0.49\); the frame is made of acrylic with Young's modulus \(E_{f}=3.2\times 10^{9}\) Pa, mass density \(\rho_{f}=1190\) kg/m\({}^{3}\), and [PERSON]'s ratio \(\mu_{f}=0.35\). As shown in Figure 1a, the investigated A\(\upmu\)M was a 2D acoustic metasurface aiming at blocking low-frequency waves propagating in the out-of-plane direction. Although it is similar to a 2D crystal with a lattice parameter of \((r+a)\), the presented work is not a study of photonic crystals. Studies on phononic crystals investigate in-plane waves that propagate along the crystal's surface, in which wave-matter interaction (for example, Bragg reflections) occurs when the operating wavelengths are comparable with the lattice constant. In contrast, the A\(\upmu\)Ms here are used as a barrier for blocking incoming waves, in which the interaction of the metasurface with the out-of-plane waves are considered, and the lattice constant is much smaller than the operating wavelength. The first step is to calculate the local response of each A\(\upmu\)M cell. In this work, the membrane is made of latex rubber, and the size of an A\(\upmu\)M is so small that it can be considered a thin plate when performing the calculation [22]. The reason for treating the A\(\upmu\)M as a thin plate is because the membranes in each cell are rather small compared to the wavelength of the operating frequency. Such an assumption of thin plate approximation was adopted and verified in our previous work [17]. Assuming a thin plate is vibrating at a low frequency, the effective stiffness \(K_{\text{mem}}\) and effective mass \(M_{\text{mem}}\) of a A\(\upmu\)M can be obtained. For each A\(\upmu\)M, the vibration amplitude can be written as [17; 23; 24]:  
\[\eta=\frac{3 Pr^{4}\big{(}1-\mu^{2}\big{)}}{8\pi^{4}Et^{3}}\bigg{(}1+\cos\frac {2\pi rx}{r}\bigg{)}\bigg{(}1+\cos\frac{2\pi y}{r}\bigg{)}\, \tag{1}\]  
Figure 1: (**a**) Structure of the A\(\upmu\)Ms. Each A\(\upmu\)M is a small square held within a larger lattice frame. Each A\(\upmu\)M has length \(r\) and thickness \(t\); each cell of the lattice has width \(a\) and thickness \(b\). (**b**) Coordinate of an A\(\upmu\)M cell used for calculating vibration. where \(\eta\) is the vibration amplitude of the A\(\upmu\)M at position \((x,y)\); \(E\) is [PERSON]'s modulus; \(\mu\) is [PERSON]'s ratio; \(\rho\) and \(t\) are the density and thickness of the A\(\upmu\)M, respectively; and \(P\) is the pressure on the A\(\upmu\)M (for incident sound pressure \(p_{i}\), then \(P=2p_{i}\)) [17]. The coordinate system is shown in Figure 1b. The average vibration amplitude can be calculated by integration. Equating the kinetic energy calculated by lumping parameters and the sum of the energy in each infinitesimal part of the A\(\upmu\)M, the equivalent stiffness and mass of a vibrating A\(\upmu\)M can be written, respectively, as [17]:  
\[K_{\text{mem}}=\frac{8\pi^{4}Et^{3}}{3\pi^{2}(1-\mu^{2})}, \tag{2}\]  
\[M_{\text{mem}}=2.25\rho r^{2}. \tag{3}\]  
Now that the local response of each A\(\upmu\)M with rigid boundaries has been obtained, the next step is to calculate the vibration of the frame.