# Low-Frequency Sound Absorption in Air Pipelines and Water-Filled Pipelines Based on Metamaterials
## 3 Results
### Asymmetric Sound Absorption in Air Piplines  
The graphs displayed in Figure 5a,b provide a representation of the absorption coefficient A (red lines), reflection coefficient R (blue lines), and transmission coefficient T (green lines) of the plane waves originating from the left and right ports. Both the theoretical (solid lines) and simulation (dashed lines) outcomes display satisfactory coherence. It is noteworthy that the transmittance values for waves originating from both directions are nearly identical. As a result, the different reflectance performance primarily contributes to the asymmetric absorptions. The reflectance of the system shows a significant dip at 403 Hz when the plane wave propagates from the left port to the right port. The theoretical maximum absorption is 90.5%, while the simulation result is 95.6% at 408 Hz. However, when the plane wave propagates from the right port to the left port, the majority of the acoustic energy is reverberated. In Figure 5b, the reflection coefficient increases to 76.9%, and the absorption coefficient diminishes to 18.8% in the theoretical calculation (16.9% absorptance at 408 Hz in the simulation) at 403 Hz. The asymmetric absorption performance is approached in a low-frequency bandwidth. Figure 5: (**a**) Reflection coefficient, transmission coefficient, absorption coefficient, and impedance of the asymmetric sound-absorbing metamaterial when waves incident from the left port. (**b**) The acoustic performance of the metamaterial when waves incident from the right port. The solid lines and dashed lines represent the theoretical and simulation outcomes. Real(\(Z\)/\(Z\)0) is the acoustic resistance, while \(\text{Imag}(Z/Z\)0) is acoustic reactance. The mechanism of asymmetric absorption performance can be better understood by expounding it from the view of the equivalent boundary. At the resonance frequency of 403 Hz, HR-2 owns maximum reflection and minimum transmission and the reflection mode is inspired. The inspired resonator achieves a \"soft\" boundary on the cross-section of the pipeline. Consequently, the impedance at HR-2's bottleneck approaches near-zero. By adjusting the impedance of HR-1, the resistance becomes close to 1 and the reactance approaches 0, indicating that the impedance of HR-1 matches the impedance of air. The adjusted impedance of \(Z_{1}^{Left}\) can be calculated by Formula (10) and is shown in the lower panel of Figure 5a. Under the condition of impedance matching, the absorption mode is excited by the incident wave from left port. However, when the plane waves enter via the right port, HR-2 maintains the \"soft\" boundary characteristic, meaning that the impedance of HR-2 (\(Z_{\text{HR-2}}\)) is almost zero. The impedance at the HR-2 neck can be calculated by Formula (11) and is shown in the lower panel of Figure 5b. Under the mode coupling between Helmholtz resonators, the system has different impedance characteristics for waves that are incident from different ports, and then an asymmetric sound absorber is constructed. However, the mechanism of asymmetric sound absorption requires further study. To better understand the mechanism of asymmetric absorption, we utilized the Acoustic-Thermoacoustic Interaction Frequency domain module in COMSOL to analyze the distributions of absolute sound pressure \(|p|\) and particle velocity \(v\). When waves are emitted from the left port, both HRs are stimulated, resulting in a highly concentrated energy distribution, as shown in the upper panel of Figure 6a. In the lower panel of Figure 6a, a significant pressure reduction from inside to outside of the cavity due to resonance leads to a high concentration particle velocity distribution in the necks. This dissipation can be attributed to strong thermal viscosity under high acoustic velocity gradients. To quantify the energy dissipation intuitively, we measure the acoustic energy dissipation density, and the distribution map is displayed in the upper panel of Figure 7a.