# Development and Characterization of a Flexible Soundproofing Metapanel for Noise Reduction
## 3 Results
### SPL Test for Enclosure-Type FSM  
Figure 8a illustrates the Sound Pressure Level (SPL) measured in the 0-8000 Hz range for both the cylindrical FSM and a control group. The control group in this context represents an equivalent plate following the mass law, as shown in Figure 2c (left image), with a mass identical to that of the FSM. The equivalent plate data are included to demonstrate the superior sound insulation performance of the FSM compared to conventional materials of the same mass. The FSM exhibited superior soundproofing performance compared to the control group across most frequency bands, with notable results in specific high-frequency ranges. While the mass law suggests that STL typically increases with higher material density, our FSM demonstrated higher sound insulation performance than the control group, indicating that the FSM achieved high efficiency and low weight compared to traditional materials.  
Figure 8b compares SPL between the FSM and the control group in the target frequency band. Our experimental SPL measurements confirm that the FSM achieves higher STL than predicted by the mass law, especially in the 2000-5000 Hz range. The FSM demonstrates excellent soundproofing performance, particularly in 2130-2990 Hz, 3210-4110 Hz, and 4130-4960 Hz.  
Figure 8c presents the STL simulation data (Figure 4b) with specific regions highlighted. Green areas indicate where the FSM's STL performance exceeds the calculated mass law, while red areas show where it falls below. This visual representation facilitates a direct comparison with the experimental SPL data in Figure 8b. It is important to note that STL and SPL have an inverse relationship: peaks in STL correspond to troughs in SPL, and vice versa. Comparing Figure 8b,c, we observe that the experimental results slightly shift towards higher frequencies relative to the simulation. However, both datasets consistently demonstrate high noise reduction effectiveness in the frequency ranges of approximately 2000-2800 Hz, 3100-3850 Hz, and 4150-4800 Hz. These ranges align with the green areas in Figure 8c and correspond to the low SPL regions in Figure 8b, validating our FEM  
Figure 7: Vertical displacement of the FSM unit cell at anti-resonance frequencies, showing three primary modes (\(\overline{\omega}_{1}\), \(\overline{\omega}_{2}\), \(\overline{\omega}_{3}\)).  
model and confirming the FSM's superior soundproofing capabilities, remarkably where it outperforms the mass law prediction.  
Furthermore, we confirmed that the cylindrical FSM maintains flexibility while delivering excellent sound insulation. This combination of flexibility and performance highlights the FSM's potential for various applications where effective soundproofing and adaptable form factors are required. These results suggest that our FSM design could be valuable in diverse noise-insulation scenarios, including but not limited to acoustic enclosures, automotive applications, and architectural acoustics. The flexibility and high performance demonstrated by the FSM open up new possibilities for noise control in various applications, particularly where adaptable form factors are required alongside effective soundproofing.