# Development and Characterization of a Flexible Soundproofing Metapanel for Noise Reduction
## 3 Results
### Structure of Flexible Soundproofing Metapanel
#### 3.1.1 Sound Transmission Loss of FSM  
We numerically calculated the STL of the FSM to determine the geometric parameters for achieving high STL in the target band. A two-step approach was used: (1) We found geometric parameters that position the frequency where the frequency-dependent effective mass density and bulk modulus of the FSM become zero between the target bands, allowing us to position the STL reduction outside the target band range. (2) We finally determined the mechanical properties satisfying the flexible FSM under the constraint that the STL in the target band should be higher than 20 dB (blocking more than 99% of incident wave energy). The unit cell of the FSM can be analyzed as a spring-mass system consisting of a thin membrane attached to a frame, with the thin membrane having multiple natural frequencies of various modes. The dynamic effective mass of the unit cell is expressed by the following equation shown in Figure 4a. \[M_{eff}=\frac{M+\sum_{i=1}^{n}\frac{m_{i}}{1-(\omega/\omega_{i})^{2}}}{1+\sum_ {i=1}^{n}\frac{\omega_{i}}{1-(\omega/\omega_{i})^{2}}} \tag{1}\]  
Figure 3: (**a**) Schematic illustration of the FSM, showing the box-shaped frame and vibrating membrane. (**b**) Detailed dimensions and parameters of the FSM unit cell. In this equation, \(M\) represents the mass of the frame structure responsible for radiating acoustic energy. The term \(\omega_{i}(=\sqrt{k_{i}/m_{i}})\) denotes the eigenfrequency of the \(i\)-th spring-mass system, where \(m_{i}\) and \(k_{i}\) are the equivalent mass and stiffness of the \(i\)-th eigenmode, respectively. The parameter \(\alpha_{i}(=f_{i}/f)\) is a coupling coefficient, indicating the ratio of the force acting on each equivalent mass to the force exerted on the frame. This formulation comprehensively represents the FSM's dynamic behavior, accounting for multiple vibrational modes and their interactions with the frame structure. Near the natural frequencies of the thin membrane, the system's mass approaches zero. Subsequently, as the force and acceleration responses of the FSM become opposite, an anti-resonance region with negative dynamic mass is formed. This anti-resonance region is expected to have high soundproofing performance due to its characteristically high dynamic mass. These characteristics can be confirmed through the FSM's STL. The equation for STL is as follows:  
\[STL=10\text{log}_{10}\left(\frac{\int_{s}|p_{i}|^{2}dA}{\int_{s}|p_{t}|^{2}dA}\right) \tag{2}\]  
The STL graph shows a sharp decrease (dip) at the resonance frequency. Compared to the mass law, a considerably larger acoustic reduction rate is observed in the anti-resonance region. Figure 4b illustrates the simulated STL of the FSM unit cell. For comparison, the STL of an equivalent unit cell calculated using the mass law is shown as a dashed line. The static area density of this equivalent unit cell is 1.9 kg/m\({}^{2}\), identical to that of the soundproofing flexible FSM unit cell. The STL curve of the FSM unit cell (blue solid line) exhibits three peaks at 2100, 3100, and 4400 Hz, indicating high STL bands at these frequencies, which suggests that near-total reflection occurs at these frequencies due to the soundproofing flexible FSM unit cell. Conversely, three dips observed at 1500, 4200, and 5300 Hz show lower STL than conventional mass law, implying that more acoustic energy can pass through the soundproofing flexible FSM at these frequencies than through the equivalent cell. However, although these dips show relatively lower soundproofing performance than the equivalent mass law, they maintain STL values above 20 dB in most cases. These results indicate that the soundproofing flexible FSM demonstrates excellent soundproofing performance in specific frequency bands while maintaining effective noise-blocking capabilities across a wide range of frequencies. Figure 4: (**a**) Concept diagram of the dynamic effective mass. A theoretical model of the FSM unit cell represented by equivalent spring–mass subsystems.