# Low-Frequency Sound Absorption in Air Pipelines and Water-Filled Pipelines Based on Metamaterials
## 4 Conclusions  
In conclusion, we proposed a Helmholtz resonator (HR) design paradigm for low-frequency sound absorption in pipelines with different media. Based on this design paradigm, asymmetric absorption metamaterials for low-frequency air pipelines and composite metamaterials for low-frequency sound absorption in water-filled pipelines were designed. For the asymmetric absorption metamaterials, the sound absorption performance was analyzed using both theoretical and numerical methods. At 403 Hz, when the plane wave is emitted from one side of the waveguide, almost all the sound energy is absorbed, with the sound absorption coefficient reaching 0.905, while most of the sound energy is reflected from the opposite side. The sound pressure and velocity distribution maps calculated by the finite element method show that the asymmetry in absorption is attributed to the coupling between the absorption and reflection acoustic modes of the HRs. Furthermore, we demonstrated that an efficient absorption peak can be modulated within the 376 Hz to 428 Hz frequency range. For sound absorption in water-filled pipelines, we introduced rubber materials into the HR design paradigm and designed composite sound absorption metamaterials. The rubber layer provides additional acoustic resistance for matching the structure with water because the modulus of water is far larger than air. Simultaneously, under quasi-Helmholtz resonance, incident longitudinal waves are converted to transverse waves within the viscoelastic rubber layer and greatly dissipated. Notably, due to the additional acoustic resistance provided by the viscoelastic rubber, the HR, which typically exhibits soft boundary characteristics in air pipelines, struggles to  
Figure 11: (**a–d**) The composite metamaterial is formed by HR-8, HR-9, HR-10, and HR-11, with rubber layers attached to the bottom of each cavity. The distributions of absolute sound pressure |P| of composite metamaterial at 388 Hz, 425 Hz, 456 Hz, and 496 Hz. (**e**) The absorption spectra of four basic units and the composite metamaterial unit. maintain near-zero impedance. Under resonance conditions, the HR no longer exhibits a reflection mode but functions as a low-frequency sound absorber. We optimized the composite metamaterial unit by using the Nelder-Mead optimization algorithm, achieving an absorption coefficient of 0.85 at 446 Hz for the optimized unit. Finally, by constructing a coupled composite metamaterial with four parallel basic composite metamaterial units in the same pipeline section, we achieved broadband sound absorption from 380 Hz to 508 Hz, with the absorption coefficients exceeding 0.8. Under the HR design paradigm, the thicknesses of the asymmetric absorption metamaterial and the composite absorption metamaterial are 71 mm and 31.5 mm, respectively, both being similar and extremely thin. The two types of metamaterials designed based on the proposed Helmholtz cavity basic paradigm can be applied to low-frequency sound absorption in both air pipelines and water-filled pipelines. When applied to systems composed of pipelines that transport both air and liquids, there is no need to design different configurations of silencers for pipelines with different media. This reduces the complexity of designing silencers for the system and, due to the use of a single paradigm, improves reliability. By addressing the issue of excessive thickness differences in silencers caused by the large impedance mismatch between air and water, the low-frequency silencers we propose, due to their relatively thin thickness, can still be applied in pipeline systems located in narrow spaces where sound absorption design is required. The sound absorption bandwidth of the designed metamaterials is still not wide enough. In the future, the sound absorption bandwidth will be further broadened by arranging metamaterial units along the axial direction of the pipeline. Conceptualization, B.L. and Z.L. ; methodology, B.L. and S.L. ; software, B.Z. ; validation, B.L. and A.L. ; formal analysis, Z.L.