# Low-Frequency Sound Absorption in Air Pipelines and Water-Filled Pipelines Based on Metamaterials
## 3 Results
### Low-Frequency and Broadband Composite Absorbing Metamaterial in Water-Filled Piplines  
We established a composite sound-absorbing metamaterial model composed of HR-1, HR-2, and rubber via COMSOL finite element software. Figure 9a shows a cutaway view of the composite metamaterial. The rubber thickness values are \(lr_{1}\) and \(lr_{2}\) for HR-1 and HR-2. Due to the abrupt change in the cross-section, the cylinder of water and visco-thermal effects provide acoustic mass and acoustic resistance for HR. Additionally, the coated rubber layer and cavity contribute to the acoustic resistance and partial capacitance. The simulations are conducted using the preset solid mechanics module of COMSOL to verify its performance. The density, Young's modulus, Poisson's ratio, and the loss factor of the viscoelastic rubber layer in this paper are selected as \(\rho_{\rm r}=1100\) kg\({}^{-3}\), \(E\)= 5 MPa, \(\
u=0.45\), and \(\eta_{r}=0.1\) for calculation, and the thickness of the rubber layer inner cavities is \(lr1=lr2=3\) mm. As shown in Figure 9b, the HR pairs, when combined with rubber layers, exhibit sound absorption coefficients of 0.6 at 500 Hz and 0.8 at 700 Hz. In contrast, when placed in a water-filled environment, the HR pairs without the rubber layer demonstrate near-zero sound absorption coefficients at low frequencies due to the lack of viscoelastic damping mechanisms. Introducing the rubber layer with viscoelastic losses makes it challenging for HR-2 to maintain resistance close to zero. As a result, HR-2 struggles to maintain the soft boundary characteristics and instead behaves as a low-frequency sound absorber. As shown in Figure 9c, we calculated the sound absorption and reflection coefficients of HR-1 and HR-2, respectively. The results show that both resonant frequencies of the cavities are 600 Hz. However, due to incomplete impedance matching, partially incident acoustic energy is reflected. Because the geometric parameters of HR-1 and HR-2 are very similar, and after regulating the impedance by introducing rubber layers, both cavities exhibit nearly identical sound absorption coefficient curves. So, only the sound absorption performance of HR-1 has been researched. We plotted the absolute pressure \(|p|\) and particle velocity \(v\) distributions of HR-1 at 600 Hz in Figure 9d. Under the excitation of the incident sound wave, sound pressure accumulates within the HR-1 cavity at 600 Hz, and a high vibration velocity distribution is generated at the neck due to the pressure difference between the interior and exterior of the cavity. The results suggest that the absorption characteristics are caused by quasi-Helmholtz resonance caused by the Helmholtz cavity with the inside rubber layer. However, the precise mechanisms underlying sound energy dissipation in these structures require further investigation to fully understand the role of the rubber layer and its interaction with sound waves. Figure 8: (**a**) Absorptance of metamaterials constructed by HR-3 and HR-4 (black); HR-1 and HR-2 (red); HR-5 and HR-6 (blue) from left to right on frequency domain. Solid lines represent the theoretical calculation results, and dashed lines represent the numerical simulation results. (**b**) Impedances of metamaterials constructed by HR-3 and HR-4; HR-5 and HR-6. We use the Nelder-Mead algorithm to optimize the absorption coefficient of Helmholtz resonance (HR) cavities with a rubber layer. The optimization considers neck length, diameter, cavity depth, and rubber layer thickness as variables to maximize the absorption coefficient over a constrained frequency bandwidth. Figure 10a illustrates that the optimized HR-7 achieved an absorption coefficient of 0.85 at 446 Hz, with the rubber layer thickness set to 3 mm.