# Achieving Enhanced Sound Insulation through Micromembranes-Type Acoustic Metamaterials
## 5 Results and Discussions
Notably, as aforementioned, our theoretical model considering the coupling of the membrane cells (A\(\upmu\)Ms) and supporting frame is valid below the first resonant frequency of the frame, which is the big dip in the STL curves in Figure 5b-d. So far, our theoretical model does not consider those higher modes of the supporting frame, which correspond to those quickly changing sharp peaks and dips of STL curves at higher frequencies (numerical results). Therefore, it is only meaningful by comparing the theoretical and numerical results below the fundamental resonant frequency of the frame. In this low-frequency range, the theoretical predictions agree well with the numerical results. Moreover, we observed that at higher frequencies, the theoretical results still show reasonable predictions of the general trend of the STL curves compared to numerical ones, except those sharp peaks and dips. Figure 6 shows another case comparison, which is the STL of a thin acrylic plate and of a membrane-acrylic structure (A\(\upmu\)Ms) of the same weight. For a more practical use, the solid plate is now 20 \(\times\) 20 cm in size and 2 mm in thickness. Then, the hole is divided into 400 parts (20 \(\times\) 20 cells): the thickness of the frame \(b\) is now 8.7 mm, the membrane thickness \(t\) is 0.4 mm, the edge of each square \(r\) is 8.95 mm, and the width of the frame \(a\) is 1 mm. The total weights of the two arrangements are kept the same. When using the membrane-frame structure (i.e., the A\(\upmu\)M assembly), the STL at a low frequency is around 8.6 dB higher, as suggested by the simulated result, while the calculated result is around 7.1 dB. The difference between the simulated and calculated results is caused by errors in the TL calculation for a single acrylic plate. In this work, the modeling of each membrane cell (A\(\upmu\)M) is based on the thin plate approximation (i.e., shell model). This model usually holds when the plate thickness is an order of magnitude smaller than its side length. However, in this case, the plate thickness is 2 mm while its side length is 8.95 mm. Therefore, errors mainly come from the thin plate treatment for this extreme case. However, both methods indicate considerable advantages in terms of low-frequency insulation by using A\(\upmu\)Ms. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Number of A\(\upmu\)M**} & \multicolumn{2}{c}{**Resonant Frequency (Hz)**} & \multicolumn{2}{c}{**Average Displacement by Static Force (m)**} & \multirow{2}{*}{**Maximum Displacement of Frame by Static Force (m)**} \\ \cline{2-2} \cline{4-6}  & **Single** & & & **Single** & **Frame by Static Force (m)** \\ \hline
2 \(\times\) 2 & 14 & 124 & 2.9 \(\times\) 10\({}^{-4}\) P & 3.8 \(\times\) 10\({}^{-8}\) P & 8.6 \(\times\) 10\({}^{-8}\) P \\
4 \(\times\) 4 & 58 & 202 & 1.7 \(\times\) 10\({}^{-5}\) P & 4.1 \(\times\) 10\({}^{-8}\) P & 1.4 \(\times\) 10\({}^{-7}\) P \\
8 \(\times\) 8 & 254 & 287 & 8.7 \(\times\) 10\({}^{-7}\) P & 2.5 \(\times\) 10\({}^{-8}\) P & 9.8 \(\times\) 10\({}^{-8}\) P \\
16 \(\times\) 16 & 1220 & 406 & 3.8 \(\times\) 10\({}^{-8}\) P & 1.4 \(\times\) 10\({}^{-8}\) P & 5.3 \(\times\) 10\({}^{-8}\) P \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Displacements and resonant frequencies of the assembly when a 10 \(\times\) 10 cm hole is successively divided into smaller parts.