# Low-Frequency Sound Absorption in Air Pipelines and Water-Filled Pipelines Based on Metamaterials
## 3 Results
### Asymmetric Sound Absorption in Air Piplines
The acoustic energy is efficiently dissipated within the neck of the Helmholtz cavity, particularly in the thermal boundary layer and viscous boundary layer along the inner surface of the neck, where the dissipation density is several orders of magnitude higher than in the surrounding regions. Based on the simulation results of the absolute pressure \(|p|\) and particle velocity \(v\) distributions, the mode coupling between the Helmholtz resonators (HRs) is further discussed. As shown in the lower panel of Figure 7a, \(|p|\) and \(v\) distributions along the radial direction of the pipe centerline are plotted. The field profile of \(|p|\) exhibits a node at HR-2, while the field profile of \(v\) shows an antinode, consistent with the behavior of acoustic waves radiating to a soft boundary. At this point, the interface of HR-2 perpendicular to the duct is considered to exhibit soft boundary characteristics, with HR-2 functioning in a reflection mode. In contrast, HR-1 operates in an absorption mode. The coupling of these two modes facilitates the efficient dissipation of the incident acoustic waves from the left side of the pipeline. In the upper panel of Figure 6b, when the plane waves are incident from the right port, the HRs are scarcely inspired, and only a small amount of sound energy is trapped within them. The distribution of the velocity field in the lower panel of Figure 6b is much weaker than in the case of left incidence, leading to minimal sound energy confinement. As shown in Figure 7b, when the wave is incident from the right port, the energy dissipation density is significantly lower compared to the incidence from the left port, indicating asymmetry in the absorption of incident acoustic energy. The pressure node of the field profile of \(|p|\) and the antinode of the field profile of \(v\) further confirm the soft boundary effect of HR-2. The classical Helmholtz cavity offers multiple degrees of freedom in adjusting the center frequency, and the asymmetric sound-absorbing metamaterials based on mode coupling are supposed to possess similar characteristics. We discuss the absorption performance tenability based on parameter regulation. After designing the depth of the \(i\)th HR cavity \(l_{\mathrm{ci}}\) and the length of the \(i\)th HR neck \(l_{\mathrm{mi}}\), metamaterials composed by coupling HR pairs of HR-3, HR-4, and HR-5, HR-6 achieve near-perfect absorption in the band from 379 Hz to 432 Hz. The relevant geometric parameters are listed in Table 1. As shown in Figure 8a, the absorption peaks (\(\alpha>0.9\)) are distributed at 376 Hz, 403 Hz, and 428 Hz (metamaterials are formed by HR-3 and HR-4; HR-1 and HR-2; and HR-5 and HR-6, respectively), while the simulation absorbing peaks are at 384 Hz, 408 Hz, and 439 Hz. The simulation results (represented by dashed lines) are in good agreement with the theoretical results. One reason for the frequency shift is that we simplified the acoustic pressure field in the cavity for the sake of calculation. The variation in geometric parameters is reflected in impedance modulation, which contributes to achieving tunable sound absorption. In fact, achieving ideal perfect absorption is challenging because reactance and resistance are not independent but are influenced by the same geometric parameters. Generally, quasi-perfect absorption is attained when reactance approaches 0 and resistance is close to 1. Figure 8b depicts the matched impedances of metamaterials formed by HR-3 and HR-4 and HR-5 and HR-6, respectively, revealing that the proposed metamaterial units have adaptable absorption band tunability at low frequencies. Figure 6: Absolute pressure \(|p|\) and the respective particle velocities at 408 Hz when plane waves are incident from (**a**) left or (**b**) right port.