# Achieving Enhanced Sound Insulation through Micromembranes-Type Acoustic Metamaterials
## 5 Results and Discussions  
Given the practical arrangement of a partition (or a barrier) for which insulation is required, the aim of this study was to prevent low-frequency sound transmission through a fixed-size hole by inserting the partition (which here refers to the A\(\upmu\)Ms assembly). An important factor is how the frame affects the sound insulation provided by the A\(\upmu\)Ms. Here, we theoretically and numerically calculated the TL of a fixed-size A\(\upmu\)M assembly with different number of cells and the dimensions of the lattice. Figure 5 and Table 1, show how a partition for a \(10\times 10\) cm square hole can be successively divided into smaller parts. Here, the frequency range of interest ranged from 10 to 1000 Hz for estimating the low-frequency acoustic behavior of the A\(\upmu\)Ms. In Figure 5a, the hole is composed of four A\(\upmu\)Ms (\(n=1\)), each having a comparatively larger size (\(r=47.5\) mm). Such a large A\(\upmu\)M cell is not stiff enough to provide sound insulation, and the TL curve quickly reverts to the mass law region. The hole was then divided into 16 parts, as shown in Figure 5b, i.e., \(n=3\), but the stiffness of the A\(\upmu\)M was still too small. Sound insulation occurs mostly in the mass-control region. With the further increase in the cells in the frame, each A\(\upmu\)M became stiff enough to bring the structure into the stiffness-controlled low-frequency region, as shown in Figure 5c for \(n=7\) and Figure 5d for \(n=15\). When the hole was divided into \(16\times 16\) parts (\(n=15\)), the TL of the structure could reach more than 30 dB at 100 Hz, which is in extreme contrast to the same hole divided into just a few parts shown in Figure 5a,b. We found that the theoretical predictions (black solid lines) agree well with the numerical ones (red dashed lines), especially at frequencies below the mass-control region. At higher frequencies, the deviation of theory from the numerical results comes from the A\(\upmu\)M assembly vibrating in more complex, higher modes, which is the limitation of the currently adopt theoretical model. Figure 5e,f shows by simulation and calculation, respectively, how TL increases when the number of A\(\upmu\)M cells is doubled both in columns and rows. It can be seen that TL increases by about 22 dB at 10 Hz, when \(n\) increases from three to seven (i.e., the number of A\(\upmu\)Ms in a column increases from four to eight), but the increases become smaller as the frequency increases. When \(n\) further increases from 7 to 15, it provides an additional TL of about 24 dB over a relatively broad band of low frequencies. This result suggests that the strategy is efficient for noise control at low frequencies. Figure 4: (**a**) The COMSOL simulation model with perfectly matched layers (PMLs) on each side of the A\(\upmu\)M assembly. A sound field was created on the lower side. (**b**) The model grid showing the fixed constraint set by the frame (red line). Figure 5: Sound transmission loss in a \(10\times 10\) cm hole as the number of AuM cells increases. Thickness of frame \(b=4\) mm, width \(a=1\) mm, edge length of the AuM \(r=8\) mm, and thickness of the AuM \(t=0.2\) mm. The number of A\(\upmu\)M cells is (**a**) \(2\times 2\), (**b**) \(4\times 4\), (**c**) \(8\times 8\), and (**d**) \(16\times 16\). (**e**) Incremental TL (simulation result) when the number of AuM cells in a column is doubled. (**f**) Incremental TL (calculation result) when the number of AuM cells in a column is doubled. As indicated in Figure 5, a structure can be successively divided into more and more parts so that, eventually, the hole could be made up of a plate made only of the frame material. What would its TL be then? At that point, the TL would be high, but the weight of the component should also be taken into consideration.