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Abstract

Acoustic metamaterials with both ventilation and broadband asymmetric absorption have demonstrated great scientific significance and promising applicability. In this work, we design an asymmetric absorbing cell (AAC) consisting of a pair of detuned Helmholtz resonators (HRs) connected by sound channels that allows airflow with a ventilation ratio (ventilation area divided by sound incidence area) of 40%, which can achieve near-perfect sound absorption in the operating frequency range when sound waves are incident from the left port. However, when incident on the right port, the acoustic absorption coefficient does not exceed 40% at most, so asymmetric absorption is achieved. In addition, we form parallel three-cell asymmetric absorber (PTAA) by paralleling three AACs, which have broadband asymmetric absorption compared to AAC. Furthermore, we design multi-asymmetric absorber (MAA), which can achieve broadband asymmetric absorption range from 1000 Hz to 1750 Hz, and also allow air circulation. Moreover, experimental validation is conducted to demonstrate the effectiveness of fabricated MAA by 3D printing technology. Our designs open potential possibilities for developing ventilated functional devices capable of absorbing sound asymmetrically.

Keywords

Helmholtz resonator asymmetric absorption broadband sound absorption

Introduction

Acoustic metamaterials, by virtue of their outstanding capabilities, promise an extensive application prospects, such as subwavelength broadband sound absorption and flexibility features.¹ To achieve high-efficiency subwavelength absorption, micro-perforated plates^2–4 and metamaterial sound absorber have been constructed in single-port systems,^5–11 which only need cancel the reflected sound waves without allowing air circulation and heat exchange. In particular, high-efficiency broadband sound absorption in two-port open systems is of practical significance and is highly desired in scenarios which require simultaneously silence and ventilation. Recently, a number of two-port asymmetric absorbers have been proposed to address structural ventilation and heat dissipation requirements. Long et al. realized two-port asymmetric absorption through a pair of HRs with detuned resonant frequencies, when the sound waves are incident from the left port, it can achieve nearly perfect sound absorption of more than 99%, while when the sound wave is incident from the other end, almost all the sound waves are reflected.¹² Yang et al. realized near-perfect absorption in two-port systems by constructing monopole and dipole resonances.¹³ Hong et al. constructed a two-port absorber with both ends open by a pair of weakly detuned helical HRs placed on an acoustic duct.¹⁴ Ma et al. realized perfect absorption for a broadband low-frequency sound in a two-port system by employing micro-perforated panel resonators.¹⁵ Gao et al. proposed a mechanism that used coupling modulation of resonance energy leakage and loss in resonators to realize optimal high-efficiency and broadband sound absorption in two-port open ducts.¹⁶ Zhu et al. proposed a paradigm that parallel hybridizing resonators and non-resonant sound reactance-dominated boundaries to realize a ventilated sound absorber, which minimizes the system efficiently.¹⁷

The above literature on sound absorption of dual port system mainly refers to the realization of narrow band sound absorption or wide band sound absorption in theory. For engineering applications, it is often required to achieve efficient sound absorption in a wide frequency range, and the size of the sound absorption structure is required to be as small and compact as possible. In order to achieve high-efficiency absorption and ventilation on the basis of compact structure, we demonstrate an asymmetric absorber cell (AAC) consisting of a pair of detuned HRs connected by sound channels, which achieves near-perfect sound absorption when the sound waves are incident from left ports, while the sound is mostly reflected when the sound waves are incident from the other side. On the basis of AAC, we investigate the asymmetric sound absorption bandwidth of three-unit asymmetric absorber by paralleling three AACs. Its wider operating frequency range paves the way for the design of multiple asymmetric absorber (MAA) composed of multiple detuned resonators. The designed ventilated absorber has the potential to provide solutions for noise control engineering.

Methodology

Design description of asymmetric sound absorber

Figure 1(a) shows the structure of an asymmetric sound absorbing cell (AAC) composed of two HRs, which are connected to the upper and lower main sound channels through branch channels. It should be emphasized that since the cavity depths of the two resonators are slightly different, their resonant frequencies are detuned. Ascribed to the inherent asymmetry of dissipation loss that arise from the resonators, the unit responses are for the sound from opposite ports. Concretely, sound waves from left side are almost totally consumed, while ones from the right side are reflected. Moreover, airflow is allowed to circulate through the channels, the ventilation ratio (ventilation area divided by sound incidence area) can reach to 40%. In Figure 1(b), geometric parameters annotated as L and W are the length and width of the cell, and b and t are the length and width of the cross section of the main channels, and the width of the cross section of the branch channels is a. The overall thickness of the cell structure is about D. For the two resonators, except for the different cavity depths (w₁, w₂), the rest of the parameters (neck length = l, neck radius = r, cavity height = h, cavity width = t) are exactly the same. The utilized geometric parameters are listed in Table 1.

Figure 1.

(a) 3D view of the asymmetric sound absorbing cell (AAC). (b) Details of a unit, its geometry, and dimensions. (c) 3D view of the parallel three-cell asymmetric absorber (PTAA). (d) 3D view of the multi-combination asymmetric absorber (MAA).

Table 1.

Geometric parameters of the AAC.

Name	L (mm)	W (mm)	D (mm)	a (mm)	b (mm)	Member	w (mm)	t (mm)	h (mm)	l (mm)	r (mm)
AAC	44	27	12	5	2.5	HR-1	15	10	13	2	1
AAC	44	27	12	5	2.5	HR-2	17	10	13	2	1

In order to make the sound absorber have a wider operating frequency range, a parallel three-cell asymmetric absorber (PTAA) is designed as shown in Figure 1(c). The PTAA is composed of three AACs in parallel, which effectively widens the sound absorption range on the premise of ensuring air circulation. It should be emphasized that only the neck radius (r) of the two internal resonators of the AACs are different, and the other parameters are completely the same. Moreover, according to this design idea, multi-combination asymmetric absorber (MAA) is designed to achieve wider-band asymmetric sound absorption, as shown in Figure 1(d). The utilized geometric parameters are listed in Table 2 and Table 3.

Table 2.

Geometric parameters of the PTAA.

Member	w (mm)	t (mm)	h (mm)	l (mm)	r (mm)
HR-1	15	10	13	2	1
HR-2	17	10	13	2	1
HR-3	15	10	13	2	0.9
HR-4	17	10	13	2	0.9
HR-5	15	10	13	2	1.1
HR-6	17	10	13	2	1.1

Table 3.

Geometric parameters of the MAA.

Member	r (mm)	Member	r (mm)	Member	r (mm)
Cell-1	1.6	Cell-7	1.7	Cell-13	1.8
Cell-2	1.5	Cell-8	1.6	Cell-14	1.7
Cell-3	1.7	Cell-9	1.8	Cell-15	1.9
Cell-4	1	Cell-10	1.2	Cell-16	1.4
Cell-5	0.9	Cell-11	1.1	Cell-17	1.3
Cell-6	1.1	Cell-12	1.3	Cell-18	1.5

Theoretical calculation

For analyzing the acoustical scattering performance of the unit, a transfer matrix method (TMM) and a finite element method (FEM) conducted in COMSOL multiphysics are employed. In TMM, a total transfer matrix for the sound wave radiating from the left port can be given as $T_{t} = T_{△ s c} T_{H R - i} T_{s c} T_{H R - i} T_{△ s c}$ (1)where $T_{H R - i} = (\begin{array}{c} 1 & 0 \\ 1 / Z_{H R - i} & 1 \end{array})$ denotes for the matrix of the i th resonator with $Z_{H R - i}$ being its impedance and $T_{s c} = (\begin{array}{c} \cos (k_{s} l_{s c}) & j z_{s} \sin (k_{s} l_{s c}) \\ j \sin (k_{s} l_{s c}) / z_{s} & \cos (k_{s} l_{s c}) \end{array})$ for the matrix of the sound channel. $T_{△ s c} = (\begin{array}{c} 1 & j z_{s} k_{s} △ l_{s c} \\ 0 & 1 \end{array})$ describes the end correction due to the discontinuity from port to sound channels with $z_{s}$ and $k_{s}$ standing for the effective impedance and wave number of the sound channels, respectively. Thus, the scattering coefficients of the entire system can be expressed as $t = \frac{2 e^{j k_{0} l}}{T_{t, 11} + T_{t, 12} / z_{0} + T_{t, 21} z_{0} + T_{t, 22}}$ (2) $r^{L} = \frac{T_{t, 11} + T_{t, 12} / z_{0} - T_{t, 21} z_{0} - T_{t, 22}}{T_{t, 11} + T_{t, 12} / z_{0} + T_{t, 21} z_{0} + T_{t, 22}}$ (3) $r^{R} = \frac{- T_{t, 11} + T_{t, 12} / z_{0} - T_{t, 21} z_{0} + T_{t, 22}}{T_{t, 11} + T_{t, 12} / z_{0} + T_{t, 21} z_{0} + T_{t, 22}}$ (4)where $r^{L}$ ( $r^{R}$ ) stands for the reflection coefficient with sound wave radiating from the left (right) port and $t$ for the identical transmission coefficient of sound pressure. Hence, the absorptance with sound wave incident from the left and right ports can be obtained as $A^{L} = 1 - {| r^{L} |}^{2} - {| t |}^{2}$ and $A^{R} = 1 - {| r^{R} |}^{2} - {| t |}^{2}$ , respectively.

Due to the parallel connection of asymmetric sound absorbing cells in PTAA and MAA, a parallel transfer matrix method is developed to characterize the acoustic reponse of the absorber.¹⁸ The total transfer matrix for the structure in Figure 1(c) and (d) is given as $T_{t} = (\begin{array}{c} T_{t, 11} & T_{t, 12} \\ T_{t, 21} & T_{t, 22} \end{array}) = \frac{1}{\sum r_{n} Y_{n, 21}} (\begin{array}{c} \sum r_{n} Y_{n, 21} & - 1 \\ \sum r_{n} Y_{n, 22} \sum r_{n} Y_{n, 11} - \sum r_{n} Y_{n, 12} \sum r_{n} Y_{n, 21} & - \sum r_{n} Y_{n, 11} \end{array})$ (5)where $Y_{n}$ and $r_{n}$ are the admittance matrix and area ratio for each component, respectively. $Y_{n}$ can be expressed by the elements of its transfer matrix. $Y_{n} = (\begin{array}{c} Y_{n, 11} & Y_{n, 12} \\ Y_{n, 21} & Y_{n, 22} \end{array}) = \frac{1}{T_{n, 12}} (\begin{array}{c} T_{n, 22} & T_{n, 12} T_{n, 21} - T_{n, 11} - T_{n, 22} \\ 1 & - T_{n, 11} \end{array})$ (6)

Simulation

Figure 2(a) and (b) present the spectra for incident sound waves from the left and right ports of the AAC, where blue, black, and red lines (hollow symbols) correspond to reflectance, transmittance, and absorptance derived by the FEM. Since the cutoff frequency of the waveguide is 2425.4 Hz, only plane waves are considered, as in Ref. [19]. The asymmetric absorbing cell exhibits a high absorption of 94% at f = 970 Hz - the sound waves almost completely absorbed, and the bandwidth with sound absorption over 50% reaches 181.5 Hz, when sound waves are incident from left. However, for the other case that when the sound waves are incident from right, the highest sound absorption of the cell can only reach 40% at f = 950 Hz—most of the sound waves are reflected, but due to the symmetry of the system, the acoustic transmittance is the same in both cases. Moreover, the normalized impedance $Z r$ (= $\frac{p_{A}}{u_{A}} \frac{1}{ρ_{0} c_{0}}$ ) of AAC is given in Figure 2(c) and (d), where $p_{A}$ and $u_{A}$ are the average surface pressure and average velocity, $ρ_{0}$ and $c_{0}$ are the density and sound velocity of air, respectively. When the sound wave is incident from the left ports, the real part of the normalized impedance of AAC is 1 and the imaginary part is 0 at 970 Hz, which is nearly perfectly matched with the air impedance, thus achieving near-perfect sound absorption. When the sound wave is incident from right, the normalized impedance of AAC is mismatched with the air impedance within the calculated frequency range, and the sound waves are mainly reflected back to the incident ports.

Figure 2.

(a) Distribution of sound pressure and particle velocities at (a) 970 Hz or (b) 950 Hz when sound waves are incident from (a) left or (b) right ports and reflectance, transmittance, and absorptance of the asymmetric absorbing cell. (c) and (d) The respective normalized impedances to (a) and (b).

The phenomenon of asymmetric absorption can be explained by the effective acoustic boundary theory. At 950 Hz, which is near the resonant frequency of HR-2, HR-2 can be equivalent to a “soft boundary” because its impedance is near 0 at resonance, so when the sound wave is incident from the right ports, most of the sound waves are reflected by this “soft boundary,” while at 970 Hz—between the resonant frequency of HR-1 and HR-2, the impedance is matched to the air which results in near-perfect sound absorption when the sound waves are incident from left ports.¹² Moreover, the corresponding sound pressure and particle velocities are also plotted, it can be seen that when sound waves are incident from left, both resonators are excited simultaneously, and the sound pressure values in the two resonator cavities are equal but opposite in phase, at this time, the sound energy is largely dissipated by friction loss in the necks of the two resonators by the high particle velocity. And when the sound waves are incident from right port, according to the sound pressure distribution, it can be seen that only one resonator is excited, and the particle velocity at the neck of the corresponding resonator is very high, so the sound absorbing cell has a relatively weak dissipation on the sound energy, and most of the sound waves are reflected. Therefore, the AAC realizes the function of asymmetric absorption.

In addition, the effect of rectangular resonators and cylindrical resonators on the sound absorption performance of the AAC is considered, and found that the geometric shape of the resonators has almost no change on its absorption performance, which is different from Ref.²⁰ For the convenience of making PTAA and MAA, the rectangular resonators are chosen.

By paralleling absorbing cells with different neck radius, the broadband asymmetric sound absorptions can be achieved. As depicted in Figure 3, PTAA is composed of three absorption cells with different neck radius in parallel, it can be seen that a significantly wider asymmetric absorption band compared to asymmetric absorption cell is achieved as shown in Figure 3(c) and (d). In Figure 3(c), the band range with sound absorption over 80% is 300 Hz (890–1090 Hz), the sound absorption coefficient at the peak reaches 96% when sound waves are incident from left ports. But in another case where the sound waves are incident from the right ports, the highest sound absorption of PTAA is only 50%, the loss of the structure to the sound wave mainly depends on the reflection. In addition, from the perspective of impedance, as shown in Figure 3(e) and (f), when the sound wave is incident from the left ports, the surface impedance of PTAA can match the air impedance in a wide frequency range, while when the sound wave is incident from the right, the difference between the surface impedance and the air impedance is large, which cannot have an obvious sound absorption effect on the sound waves.

Figure 3.

(a) Distribution of sound pressure at 890, 940, 1000, 1050 Hz when sound waves are incident from left ports. (b) Distribution of sound pressure at 970 and 1050 Hz when sound waves are incident from right ports. Reflectance, transmittance, and absorptance of sound waves from (c) left and (d) right ports of PTAA. (e) and (f) The respective normalized impedances to (c) and (d).

To intuitively understand the mechanism of broadband asymmetric sound absorption of PTAA, the distributions of sound pressure at certain frequencies are plotted, respectively. In Figure 3(a), when sound waves radiate from the left ports, the asymmetric absorption cell in the middle of the PTAA is significantly excited at 890 Hz, and the sound energy at this frequency is largely dissipated by friction loss of the necks of the resonators. Similarly, at 940 Hz and 1050 Hz, the top and bottom cells of the PTAA are excited, respectively, and the sound energy is dissipated by friction. It is emphasized that there is coupling between the three asymmetric absorption cells of the PTAA at the peak frequency (1000 Hz) of the sound absorption coefficient, that is to say, six resonators inside the structure are all excited, which can explain the better absorption effect and broadband of PTAA than the AAC. However, when sound waves radite from the right port, the absorption is less than 50% in the frequency band except the ones around 970 Hz. By observing the distribution of sound pressure at 970 Hz, it can be found that only one cell (bottom) has a coupled sound absorption mode, so the sound absorption of the PTAA is weak. In addition, by the sound pressure distribution at 1050 Hz, it can be intuitively understand the strong reflection of the PTAA when the sound waves incident from the right ports, the sound pressure in the resonator is in the same phase, that is to say, there is no coupled sound absorption mode between the resonator.

On the basis of PTAA, MAA is designed to achieve more wider-band asymmetric sound absorption. See Table 3 for the geometric parameters of MAA, which has a wider operating frequency. Similar to the asymmetric sound absorption mechanism of AAC and PTAA, when the sound waves are incident from the left ports, HRs in the MAA have multiple coupling sound absorption modes in the range of 1000–1750 Hz, thus generating efficient broadband near-perfect sound absorption. When the sound waves are incident from the right pors, the sound absorption effect of the MAA within the calculated frequency range is weak, as shown in Figure 4(a) and (b). The surface normalized impedance of MAA is shown in Figure 4(c) and (d).

Figure 4.

The absorption of sound waves from (a) left and (b) right ports of MAA. (c) and (d) The respective normalized impedances to (a) and (b).

Experimental validation

To validate the acoustical performance of the designed asymmetric absorber, the standard impedance tube system is used for performing experimental measurements of MAA. The experimental setup is consistent with that adopted in numerical modeling as shown in Figure 5(a). The sample of the MAA, made of photosensitive resin via three-dimensional (3D) printing and fixed by a cylindrical kit in the impedance tube, is shown in Figure 5(b). The size of the designed MAA is 68 mm × 68 mm × 44 mm.

Figure 5.

(a) Equipment setup for measurement of sound absorption coefficient. (b) Test sample for experimental verification. (c) Simulation versus experimental sound absorption coefficient of the test MAA.

In the measurements, the input signal is generated by the computer and amplified by the audio power amplifier, and then the sound wave is fed to the impedance tube through a loudspeaker. The local pressure fields are recorded by inserting 1/4 inch condensed microphones into the top side of the impedance tube at designated positions. The microphones are calibrated by a calibration box and a piston phone before use.²¹ The sound pressure is collected by the signal collector with Smart Office software. Finally, the acquired information is processed in the MATLAB program to derive the corresponding sound absorption coefficient.

The relevant results are shown in Figure 5(c). Although the 3D printed samples have small accuracy problems, the experimental measurements agree well with the simulation calculations. It should be noted that the impedance tube test range used in this experiment is 0–1600 Hz, and the test results beyond this range are no longer accurate. The sound absorption effect of MAA from 800 Hz to 1600 Hz is very good when the sound waves are incident from the left side, but when the sound waves are incident from the right side, the sound absorption performance of MAA is not obvious. The experimental measurements confirm that the MAA can be practically in noise control engineering.

Conclusions

In this paper, a ventilated two-port asymmetric absorbing cell (AAC) is designed to achieve near-perfect sound absorption when sound waves are incident from the left ports, and mostly reflected when sound waves are incident from the right ports. Subsequently, a wider-band sound absorption was achieved by paralleling three asymmetric absorption cells. Based on this idea, MAA was designed to achieve ultra-broadband asymmetric sound absorption. The experimental results demonstrated the excellent asymmetric absorption performance of MAA. Furthermore, if the asymmetric absorption structure is applied to automobile exhaust systems, for example, the effects of bias flow and grazing flow should be concerned, as shown in Ref.[22, 23]. The structure designed in this paper has good application potential in the field of vibration and noise control.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research,authorship,and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research,authorship,and/or publication of this article: This work was supported by Natural Science Foundation of Shandong Province (ZR2020ME123).

ORCID iD

Deshi Meng

References

Shen

Díaz-rubio

, et al. Systematic design and experimental demonstration of bianisotropic metasurfaces for scattering-free manipulation of acoustic wavefronts. Nat Comm 2018; 9: 1–9.

Maa

. Potential of microperforated panel absorber. J Acoust Soc Am 1998; 104: 2861–2866.

Bravo

Maury

. Sound attenuation and absorption by micro-perforated panels backed by anisotropic fibrous materials: theoretical and experimental study. J Sound Vib 2018; 425: 189–207.

Min

Guo

. Sound absorbers with a micro-perforated panel backed by an array of parallel-arranged sub-cavities at different depths. Appl Acoust 2019; 149: 123–128.

Mei

Yang

, et al. Dark acoustic metamaterials as super absorbers for low-frequency sound. Nat Commun 2012; 3: 3–756.

Wang

Luo

Zhao

, et al. Acoustic perfect absorption and broadband insulation achieved by double-zero metamaterials. Appl Phys Lett 2018; 112: 021901.

Wang

Xie

, et al. A sound absorbing metasurface with coupled resonators. Appl Phys Lett 2016; 109: 091908.

Fang

Zhang

Zhou

. Acoustic porous metasurface for excellent sound absorption based on wave manipulation. J Sound Vib 2018; 434: 273–283.

Long

Shao

Liu

, et al. Broadband near-perfect absorption of low-frequency sound by subwavelength metasurface. Appl Phys Lett 2019; 115: 103503.

10.

Huang

Zhou

, et al. Compact broadband acoustic sink with coherently coupled weak resonances. Sci Bull 2020; 65: 373–379.

11.

Guan

Zhao

Low

. Experimental evaluation on acoustic impedance and sound absorption performances of porous foams with additives with Helmholtz number. Aerosp Sci Technol 2021; 119: 107120.

12.

Long

Cheng

Liu

. Asymmetric absorber with multiband and broadband for low-frequency sound. Appl Phys Lett 2017; 111: 143502.

13.

Yang

Meng

, et al. Subwavelength total acoustic absorption with degenerate resonators. Appl Phys Lett 2015; 107: 104104.

14.

Hong

Bai

, et al. Two-port network spiral type asymmetric absorption system. Appl Acoust 2022; 4: 192.

15.

Kim

Lee

S H

, et al. Quasi-perfect absorption of broadband low-frequency sound in a two-port system based on a micro-perforated panel resonator. Appl Acoust 2022; 186: 108449.

16.

Gao

Liang

, et al. Improving sound absorption via coupling modulation of resonance energy leakage and loss in ventilated metamaterials. Appl Phys Lett 2022; 120: 261701.

17.

Zhu

Long

Liu

, et al. An ultra-thin ventilated metasurface with extreme asymmetric absorption. Appl Phys Lett 2022; 120: 141701.

18.

Verdière

Panneton

Elkoun

, et al. Transfer matrix method applied to the parallel assembly of sound absorbing materials. J Acoust Soc Am 2013; 134: 4648–4658.

19.

Zhao

. A real-time plane-wave decomposition algorithm for characterizing perforated liners damping at multiple mode frequencies. J Acoust Soc Am 2011; 129(3): 1184–1192.

20.

Zhao

Yin

. Experimental investigation of geometric shape effect of coupled Helmholtz resonators on aeroacoustics damping performances in presence of low grazing flow. Aerosp Sci Technol 2022; 128: 107799.

21.

Zhao

Ang

C Z

. Numerical and experimental investigation of the acoustic damping effect of single-layer perforated liners with joint bias-grazing flow. J Sound Vib 2015; 342: 152–167.

22.

Zhao

Sun

, et al. Experimental and theoretical studies of aeroacoustics damping performance of a bias-flow perforated orifice. Appl Acoust 2019; 145: 328–338.

23.

Guan

Zhao

Ren

. Aeroacoustic attenuation performance of a helmholtz resonator with a rigid baffle implemented in the presence of a grazing flow. Int J Aerospace Eng 2020; 2020: 1916239–1916316.

Helmholtz resonator-based acoustic metamaterials enabling broadband asymmetric sound absorption and ventilation

Abstract

Keywords

Introduction

Methodology

Design description of asymmetric sound absorber

Theoretical calculation

Simulation

Experimental validation

Conclusions

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References