Composite sound-absorbing metamaterials via multiple resonance coupling and quality factor modulation

Abstract

Broadband absorbers based on resonant acoustic metamaterials often require intricate designs, yet this complexity inherently restricts their bandwidth, robustness, and manufacturability. To overcome these constraints, we present a composite sound-absorbing metamaterial that combines multiple resonance coupling with quality factor modulation, leveraging micro-perforated plates and porous materials. This metamaterial exhibits near-perfect broadband sound absorption across a frequency range spanning from 340 to 3200 Hz. In addition, composite metamaterials exhibit greater robustness compared to resonant metamaterials, demonstrating better noise control capabilities in diffuse sound fields. This work uses a new mechanism to revitalize traditional sound-absorbing materials and bring them back to prominence in noise control. We anticipate that this innovative solution will address noise control challenges in demanding environments and provide a reference for further development of sound-absorbing metamaterials.

Graphical abstract

Keywords

multiple resonance coupling / quality factor modulation / composite absorber / noise control

1 Introduction

Absorption, as one of the core physical phenomena in classical wave systems, has widespread applications in many fields. When projected into the context of acoustic systems, this phenomenon transforms into the sound absorption issue in noise control. In the field of noise control, sound-absorbing metamaterials have been widely applied [1-5] to achieve effective noise attenuation. Porous sound-absorbing materials were once the preferred solution in noise control endeavors [6-9], yet their low-frequency sound absorption performance was limited by structural dimensions and could not be further enhanced. As another landmark technology following porous materials, perforated plate resonant absorbers sparked extensive research interest [10-13]. However, due to the constraints of their resonant characteristics, the frequency range over which perforated plates can effectively absorb sound is relatively narrow, making it difficult to achieve effective broadband sound absorption.

With the rapid advancement of sound-absorbing metamaterials [14-17], their ability to control large-wavelength sound waves with small sizes has emerged prominently in the fields of imaging [18-20], sound field modulation [21-24], topological acoustics [25-29], and acoustic communication [30, 31]. The advent of sound-absorbing metamaterials has also introduced fresh perspectives to the design of sound-absorbing structures. However, there are common issues in the design of these structures. To achieve broadband impedance matching for effective sound absorption [32-42], the design of resonance structures is often complicated, with embedded tubes, helical structures, labyrinthine structures, and other complex designs undoubtedly increasing the cost and design difficulty of the sound-absorbing metamaterials in practical applications. Additionally, a perfectly absorbing structure designed for normal incidence is not necessarily the optimal solution for addressing sound absorption in complex incidence scenarios. This significantly limits the practical application of sound-absorbing metamaterials.

In this work, we harness multiple resonance coupling and quality factor modulation as powerful theoretical tools to construct a composite sound-absorbing metamaterial utilizing perforated plates and porous materials. The synergistic effect of multiple resonance coupling and quality factor modulation enables the structure to avoid complex structural designs while achieving better noise attenuation in diffuse sound field. Simulation and experimental results validate the effectiveness of the proposed structure. This structure achieves efficient sound absorption within the range of 340 to 3200 Hz, with sound absorption higher than 0.9. Starting from the physical mechanisms, this work realizes efficient broadband sound absorption through simple designs, potentially offering a new approach for the practical application of broadband sound-absorbing metamaterials.

2 Results

A resonant sound-absorbing material’s performance hinges on impedance matching: sufficient sound wave entry and internal dissipation. Single-frequency impedance matching is easy, but broadband absorption is challenging. Here, we propose a method combining multiple resonance coupling and quality factor modulation to achieve broadband sound absorption. Multiple resonance coupling ensures that more sound waves in the operational frequency band enter the structure, while quality factor modulation ensures that sound waves entering the structure are absorbed.

The first factor that limits a resonant structure from achieving broadband performance is the number of resonances a single structure has in the operational frequency band. Here, we take the perforated plate absorber as an example, which usually supports only one resonance frequency in the operational frequency band. The single-layer perforated plate structure shown in Fig.1(a) exhibits only one absorption peak. However, by considering multiple resonances in a single structure, we can introduce multiple resonance frequencies in the operational frequency band, and the corresponding absorption spectral line will then exhibit multiple absorption peaks. It should be noted that the multi-layer series configurations can directly achieve broadband absorption, but this often leads to complex structural design and an increase in structural thickness.Therefore, in this work we choose the combination of series and parallel, which can effectively use the radiation coupling between the units to effectively reduce the thickness of the structure.

The second factor limiting the sound absorption capacity of resonant structures is the quality factor of the structure. For a resonant structure, the sound absorption capacity is actually controlled by three factors: resonance frequency \omega, radiation coupling \gamma, and intrinsic loss \mathrm{\Gamma }. What we pursue in a single structure is that the quality factor [Q=\omega / (2(\gamma +\mathrm{\Gamma }))] corresponding to each high absorption resonance frequency is as small as possible. For a resonant structure with a constant sound absorption peak, the smaller its quality factor (Q factor) at the corresponding target frequency, the larger its average sound absorption coefficient will be. In actual quality factor modulation, we usually choose the intrinsic loss \mathrm{\Gamma } of the structure as the modulation target. The Q factor of the absorption coefficient decreases gradually with the increase of the intrinsic loss. As shown in Fig.1(a), when porous sound absorption is introduced into the perforated plate sound absorption structure, the corresponding Q factor becomes smaller, and the average sound absorption coefficient increases due to the increase in intrinsic loss. Therefore, the combination of several resonators with multiple resonance modes and quality factor modulation can easily achieve broadband sound absorption in the operational frequency band.

Based on the above principle, we construct a broadband composite sound-absorbing metamaterial using microperforated plates and porous materials. Fig.1(b) and (c) show the schematic diagram of the proposed composite metamaterial. The structure is divided into twelve chambers by twelve perforated plates. The thickness and aperture of all perforated plates are 1\text{mm}. The width of each unit is w=15\text{mm}, and the cavity depth is h=20\text{mm}. Additionally, the opening diameter of the perforated plates is d=1\text{mm}, and the plate thickness is t_p=1 \text{mm}. The total height of the composite sound-absorbing metamaterial is H=85\text{mm}, and the total width is W=50\text{mm}. Considering practical manufacturing constraints, we restricted the sample parameters. Specifically, only the perforation rate of each plate was varied, while maintaining constant values for the cavity depth h, opening diameter d, and plate thickness t_p. Notably, incorporating additional degrees of freedom for these parameters in structural design could theoretically enable broadband sound absorption. However, this approach inevitably increases structural complexity and associated fabrication costs.

In Fig.2(a), we give the resonance positions and sound absorption coefficient curves of the three units. The sound absorption peaks (resonance modes) of the three units are orderly distributed in the operational frequency band, which provides the basis for combining them to achieve broadband sound absorption. Their reflection responses are shown in Fig.2(b), where we construct the complex frequency domain as f=f_r+\mathrm{j}{f}_i. Fig.2(b) plots {\log }_{10} | r{|}^2 as a color map. The reflection response on the real axis (f_i=0) is the reflection response of the structure in the actual situation. From this, we observe that each resonant unit supports multiple resonant sound absorption peaks. Furthermore, due to multiple resonance coupling, a single unit also exhibits a certain sound absorption capacity in the non-resonant frequency band.

Radiation coupling between units causes non-uniform sound absorption characteristics across the target frequency band in the assembled composite structure. To compensate for this coupling effect, precise adjustment of structural parameters is necessary to flatten the absorption curve. With the quality factor modulation (introduction of porous materials), we can eliminate the requirement for fine parameter tuning while maintaining uniform absorption performance within the operational bandwidth. This phenomenon can be clearly explained in the context of complex frequency domain. As shown in Fig.3(a), the zero point of the reflection of the combined structure is frequency shifted due to the radiation coupling, which is far away from the real axis in the complex frequency domain. With the introduction of porous materials, we can effectively manipulate the zero point position of the reflection in the complex frequency domain. In Fig.3, we present the complex frequency domain diagram of multiple resonance coupling metamaterials, and the complex frequency domain diagrams after adding one layer of porous materials and two layers of porous materials. As shown in Fig.3, with the introduction of porous materials, the reflection response zero point of the structure gradually shifts to the upper side of the complex frequency domain, indicating that the structure gradually tends towards an overdamped state. This means that the surface impedance of the structure gradually moves in a direction greater than the air impedance. Considering that the surface impedance of the oblique incident angle is the surface impedance of the normal incident angle multiplied by the cosine of the incident angle, the structure has a good sound absorption performance for the oblique incident case. As demonstrated by the absorption responses of the three configurations under different incidence angles given on the right side of Fig.3, it can be concluded that the introduction of porous materials effectively improves the structure’s robustness to deal with the incidence of sound waves at various angles. It should be emphasized that as the porous material continues to be introduced until the entire structure is completely filled, the system is always in an overdamped state, and the oblique incident sound wave is always effectively absorbed. Moreover, considering that there are both adiabatic and isothermal processes in the system, this structure may provide a new way to understand the causality law of sound absorption [42].

In order to support our conclusion, impedance tube experiments were carried out to verify the multiple resonance coupling metamaterials and the composite sound-absorbing metamaterial. Fig.4(a) shows the sound-absorbing metamaterial based on multiple resonance coupling and the composite sound-absorbing metamaterial. The experimental device is a standard impedance tube sound absorption test system [43]. The experimental results show good agreement with theoretical predictions, demonstrating that our design works as expected. This compact absorber absorbs approximately 90% of incident energy (average absorption coefficient of 0.9) over the range of 340 to 3200 \text{Hz}, while the structural thickness is only 85 \text{mm} (approximately \lambda /12). The introduction of porous material can effectively improve the sound absorption performance of metamaterials and fill the sound absorption valley where multiple resonance coupling is insufficient.

As mentioned above, the proposed metamaterial also has a good noise attenuation effect in the diffuse sound field. Considering the application of sound-absorbing metamaterials in the environment of multi-angle incidence (diffuse sound field), we construct a method to compare and evaluate the noise attenuation ability of sound-absorbing metamaterials through a simple simulation model. As shown in the Fig.5(a), the sound-absorbing metamaterials with a periodic arrangement is covered with a semicircular domain where the background pressure field is set as the diffuse sound field. The perfect matched layers (PMLs) are used to enclose the circular domain. The structure consists of twenty unit cells in total, with a overall length of 1000 \text{mm}. The radius of the semicircle is 600 \text{mm}, and the thickness of the perfect matched layer is 100 \text{mm}. An evaluation of the material’s noise attenuation ability can be obtained by comparing it with a rigid reflecting wall. The insertion loss method is adopted, and the measuring points are five equidistant points (30-degree angular separation) along a 450 \text{mm} radius circular path. The background pressure field is defined as the sum of N (N=500 in the simulation) uncorrelated plane waves:

(1)p_{\text{random}}={\displaystyle \frac{1}{\sqrt{N}}}{\displaystyle \sum _{n=1}^N{\mathrm{e}}^{-\mathrm{j}(k_{n,x}x+k_{n,y}y)}{\mathrm{e}}^{\mathrm{j}{\varphi }_n},}

where k_{n,x}=k_0\cos \left({\theta }_n\right) and k_{n,y}=k_0\sin \left({\theta }_n\right) are the wave vector components in each direction. The polar angle {\theta }_n and phase {\varphi }_n are independent random numbers, and k_0 is the wavenumber of air. Fig.5(b) shows the noise attenuation spectrum of the sound-absorbing metamaterial in different configurations. The blue line represents the simulation result with multiple resonance coupling metamaterial, and the red line represents the noise attenuation performance of the composite metamaterial. In order to better show the noise attenuation effect of the composite metamaterial, we selected a representative frequency 2990\text{Hz} and plotted the scattered sound field under three working conditions. Compared with multiple resonance coupling metamaterials, the sound field of the composite material has obvious attenuation. It can be concluded that composite metamaterials have a better noise attenuation effect than multiple resonance coupling metamaterials under the condition of the same diffuse sound field incident.

3 Methods

The acoustic performance of the composite sound-absorbing metamaterial can be characterized by its acoustic impedance. The impedance of a perforated plate in a cavity Z_P can be represented by Maa’s model [12] as

(2)Z_P=-\mathrm{j}{\displaystyle \frac{{\rho }_0\omega (t+0.85d)}{\sigma }}{\displaystyle \frac{J_0(k_{v}d/2)}{J_2(k_{v}d/2)}},

where \omega is the angular frequency, {\rho }_0 is the air density, k_v represents the viscous wave number [34], and J_n is the Bessel function of the first kind. The specific perforation rate \sigma of the perforated plates is shown in Tab.1.

The first two chambers are filled with porous sound-absorbing metamaterials. In the construction of porous materials, we used the Johnson−Champoux−Allard (JCA) model [44]. Its complex sound speed c_c, complex density {\rho }_c, and bulk modulus {\kappa }_c are

(3)c_c=\sqrt{{\displaystyle \frac{{\kappa }_c}{{\rho }_c}}},

(4){\rho }_c= {\displaystyle \frac{\tau {\rho }_0}{ϵ}}(1 +{\displaystyle \frac{ϵR_f}{\mathrm{j}\omega {\rho }_0\tau }}\sqrt{1+4\mathrm{j}\omega {\rho }_0\mu {\xi }^2}),

(5)\begin{array}{rl}{\kappa }_c=& \frac{\zeta p_0}{ϵ}[\zeta -(\zeta -1)(1\\ & +\frac{8K}{\mathrm{j}\omega {\rho }_0{L}_{th}^2{C}_p}\sqrt{1+\frac{\mathrm{j}\omega {\rho }_0{L}_{th}^2{C}_p}{16K}}{)}^{-1}{]}^{-1},\end{array}

where \zeta, K, \mu, and C_p are the specific heat capacity ratio, the thermal conductivity, the dynamic viscosity, and the constant pressure heat capacity of air, respectively. The p_0 is the standard atmospheric pressure. The flow resistance R_f=10616.5 \text{Pa}\cdot {\text{s/m}}^2, the porosity ϵ=0.9928, the thermal characteristic length L_{th} =64 μm, the viscous characteristic length L_v=207 μm, the tortuosity \tau =1.05, and \xi =\tau / (R_f{L}_vϵ) are related parameters of the porous materials.

The acoustic impedance of each perforated plate can be described by the transfer matrix P_n which can be obtained:

(6)P_n=\left(\begin{array}{cc}1& Z_P\\ 0& 1\end{array}\right),

where n is the serial number of the perforated plates. The depth of the cavity filled with porous material is assumed to be h constant, then the transfer matrix S_p of the cavity filled with porous material is

(7)S_p=\left(\begin{array}{cc}\cos (k_{c}h)& \mathrm{j}{\rho }_c{c}_c\sin (k_{c}h)\\ \mathrm{j}\sin (k_{c}h)/({\rho }_c{c}_c)& \cos (k_{c}h)\end{array}\right),

where k_c is the complex wave number. The height of the cavity unfilled with porous material is assumed to be constant h, then the transfer matrix S_0 of the cavity unfilled with porous material is

(8)S_0=\left(\begin{array}{cc}\cos (k_{0}h)& \mathrm{j}{\rho }_0{c}_0\sin (k_{0}h)\\ \mathrm{j}\sin (k_{0}h)/({\rho }_0{c}_0)& \cos (k_{0}h)\end{array}\right),

where k_0 is the wavenumber of air. For a unit with a four-layer structure, the transfer matrix is

(9)T_i=P_1{S}_p{P}_2{S}_p{P}_3{S}_0{P}_4{S}_0,

where i is the layer number of the nuit. Moreover, the acoustic impedance of a unit can be expressed as

(10)Z_i=T_{11}/T_{21},

where T_{11} and T_{21} are elements of the transfer matrix. For the composite absorber, three coupled units contribute to the absorption, so that the acoustic impedance Z_s of the coupled units is given by

(11)Z_s={\left({\displaystyle \frac{1}{S}}{\displaystyle \sum _{i=1}^M{\displaystyle \frac{S_i}{Z_i}}}\right)}^{-1},

where M is the number of units, S is the total area of the perforated plates, S_i is the area of a single perforated plate, and Z_i is the acoustic impedance of the parallel units. The absorption coefficient is

(12)\alpha =1-|(Z_s /{\rho }_0{c}_0-1)/(Z_s /{\rho }_0{c}_0+1) {|}^2.

The simulations encompassed Frequency Domain analyses. The medium is set as air, with static density {\rho }_0=1.21\text{kg}/{\text{m}}^3, sound speed c_0=343 \text{m}/\text{s}, dynamic viscosity \mu =1.81\times {10}^{-5} \text{N}\cdot \text{s}/{\text{m}}^2, standard atmospheric pressure p_0=101.325 \text{kPa}, specific heat capacity ratio \zeta =1.40, thermal conductivity K=0.026\mathrm{W} /(\mathrm{m}\cdot \text{K}), and specific heat rate C_p=1.013 \mathrm{k}\mathrm{J}/(\mathrm{k}\mathrm{g}\cdot \text{K}).

The experimental system is derived from the impedance tube absorption measurement system. The signals (amplitude and phase) are detected by the two microphones (GRAS, type 46BD) fixed on the wall of the waveguide. The measured signals are processed by the PXI Multifunction I/O Device (National Instruments, type PXIe-4497).

4 Conclusion

In this work, we have developed composite sound-absorbing metamaterials by harmoniously integrating two conventional sound-absorbing structures. The multiple resonance coupling and quality factor modulation endow the metamaterial with broadband sound-absorbing characteristics, and demonstrate robustness against various incident angles. This structure achieves efficient sound absorption within the range of 340 to 3200 Hz (average absorption coefficient of approximately 0.9), while maintaining a compact thickness of 1 /12 of the wavelength at the lowest frequency in the range. Notably, our design completely eliminates complex structures, leveraging the advantages of both resonant metamaterials and porous materials, and thereby offers a promising new avenue for the practical application of advanced sound-absorbing solutions.

References

[1] U. Ingard, On the theory and design of acoustic resonators, J. Acoust. Soc. Am. 25(6), 1037 (1953)

[2] M. R. Stinson, The propagation of plane sound waves in narrow and wide circular tubes, and generalization to uniform tubes of arbitrary crosssectional shape, J. Acoust. Soc. Am. 89(2), 550 (1991)

[3] M. E. Delany and E. N. Bazley, Acoustical properties of fibrous absorbent materials, Appl. Acoust. 3(2), 105 (1970)

[4] J. F. Allard and Y. Champoux, New empirical equations for sound propagation in rigid frame fibrous materials, J. Acoust. Soc. Am. 91(6), 3346 (1992)

[5] L. T. Cao, Q. X. Fu, Y. Si, B. Ding, J. Y. Yu, Porous materials for sound absorption, Compos. Commun. 10, 25 (2018)

[6] N. Kino and T. Ueno, Comparisons between characteristic lengths and fibre equivalent diameters in glass fibre and melamine foam materials of similar flow resistivity, Appl. Acoust. 69(4), 325 (2008)

[7] N. Kino, Further investigations of empirical improvements to the Johnson-Champoux-Allard model, Appl. Acoust. 96, 153 (2015)

[8] C. Q. Wang, L. Cheng, J. Pan, and G. H. Yu, Sound absorption of a micro-perforated panel backed by an irregular-shaped cavity, J. Acoust. Soc. Am. 127(1), 238 (2010)

[9] Z. Q. Liu, J. X. Zhan, M. Fard, and J. L. Davy, Acoustic properties of multilayer sound absorbers with a 3D printed micro-perforated panel, Appl. Acoust. 121, 25 (2017)

[10] M. Toyoda,K. Sakagami,D. Takahashi,M. Morimoto, Effect of a honeycomb on the sound absorption characteristics of panel-type absorbers, Appl. Acoust. 72(12), 943 (2011)

[11] M. Arunkumar, M. Jagadeesh, J. Pitchaimani, K. Gangadharan, and M. L. Babu, Sound radiation and transmission loss characteristics of a honeycomb sandwich panel with composite facings: Effect of inherent material damping, J. Sound Vib. 383, 221 (2016)

[12] D. Y. Maa, Potential of microperforated panel absorber, J. Acoust. Soc. Am. 104(5), 2861 (1998)

[13] C. Q. Wang, L. Cheng, J. Pan, and G. H. Yu, Sound absorption of a micro-perforated panel backed by an irregular-shaped cavity, J. Acoust. Soc. Am. 127(1), 238 (2010)

[14] X. Wang, R. Z. Dong, Y. Li, and Y. Jing, Non-local and non-Hermitian acoustic metasurfaces, Rep. Prog. Phys. 86(11), 116501 (2023)

[15] N. S. Gao, Z. C. Zhang, J. Deng, X. Y. Guo, B. Z. Cheng, and H. Hou, Acoustic metamaterials for noise reduction: A review, Adv. Mater. Technol. 7(6), 2100698 (2022)

[16] Y. Jin, Y. Pennec, B. Bonello, H. Honarvar, L. Dobrzynski, B. Djafari-Rouhani, and M. I. Hussein, Physics of surface vibrational resonances: Pillared phononic crystals, metamaterials, and metasurfaces, Rep. Prog. Phys. 84(8), 086502 (2021)

[17] B. Assouar, B. Liang, Y. Wu, Y. Li, J. C. Cheng, and Y. Jing, Acoustic metasurfaces, Nat. Rev. Mater. 3(12), 460 (2018)

[18] C. Shen, Y. B. Xie, N. Sui, W. Q. Wang, S. A. Cummer, and Y. Jing, Broad-band acoustic hyperbolic metamaterial, Phys. Rev. Lett. 115(25), 254301 (2015)

[19] Y. Xie, C. Shen, W. Wang, J. Li, D. Suo, B. I. Popa, Y. Jing, and S. A. Cummer, Acoustic holographic rendering with two-dimensional metamaterial-based passive phased array, Sci. Rep. 6(1), 35437 (2016)

[20] K. Melde, A. Mark, T. Qiu, and P. Fischer, Holograms for acoustics, Nature 537(7621), 518 (2016)

[21] K. Tang, C. Y. Qiu, M. Z. Ke, J. Y. Lu, Y. T. Ye, and Z. Y. Liu, Anomalous refraction of airborne sound through ultrathin metasurfaces, Sci. Rep. 4(1), 6517 (2014)

[22] Y. Li, X. Jiang, R. Q. Li, B. Liang, X. Y. Zou, L. L. Yin, and J. C. Cheng, Experimental realization of full control of reflected waves with sub-wavelength acoustic metasurfaces, Phys. Rev. Appl. 2(6), 064002 (2014)

[23] J. J. Zhao, B. W. Li, Z. N. Chen, and C. W. Qiu, Manipulating acoustic wavefront by inhomogeneous impedance and steerable extraordinary reflection, Sci. Rep. 3(1), 2537 (2013)

[24] Y. Hou, R. Dong, H. Ma, X. Wang, and Y. Li, Broadband tunable sound transparency via underwater metasurfaces, Phys. Rev. Appl. 23(3), 034054 (2025)

[25] K. Ding, C. Fang, and G. C. Ma, Non-Hermitian topology and exceptional-point geometries, Nat. Rev. Phys. 4(12), 745 (2022)

[26] R. Dong, Y. Zhu, D. Mao, X. Wang, and Y. Li, Topological non-reciprocal robust waveguide transport, Sci. China Phys. Mech. Astron. 67(5), 254311 (2024)

[27] X. Xiang, Y. G. Peng, F. Gao, X. X. Wu, P. Wu, Z. X. Chen, X. Ni, and X. F. Zhu, Demonstration of acoustic higher-order topological Stiefel−Whitney semimetal, Phys. Rev. Lett. 132(19), 197202 (2024)

[28] F. Gao, X. Xiang, Y. G. Peng, X. Ni, Q. L. Sun, S. Yves, X. F. Zhu, and A. Alù, Orbital topological edge states and phase transitions in one-dimensional acoustic resonator chains, Nat. Commun. 14(1), 8162 (2023)

[29] F. Gao, Y. G. Peng, X. Xiang, X. Ni, C. Zheng, S. Yves, X. F. Zhu, and A. Alù, Acoustic higher-order topological insulators induced by orbitalinteractions, Adv. Mater. 36(23), 2312421 (2024)

[30] C. X. Zhang, X. Jiang, J. J. He, Y. Li, and D. Ta, Spatiotemporal acoustic communication by a single sensor via rotational doppler effect, Adv. Sci. (Weinh.) 10(10), 2206619 (2023)

[31] K. Wu, J. J. Liu, Y. J. Ding, W. Wang, B. Liang, and J. C. Cheng, Metamaterial-based real-time communication with high information density by multipath twisting of acoustic wave, Nat. Commun. 13(1), 5171 (2022)

[32] Y. Li, B. Assouar, Acoustic metasurface-based perfect absorber with deep subwavelength thickness, Appl. Phys. Lett. 108(6), 063502 (2016)

[33] R. Z. Dong, D. X. Mao, X. Wang, and Y. Li, Ultrabroadband acoustic ventilation barriers via hybrid-functional metasurfaces, Phys. Rev. Appl. 15, 024044 (2021)

[34] S. B. Huang, X. S. Fang, X. Wang, B. Assouar, Q. Cheng, and Y. Li, Acoustic perfect absorbers via helmholtz resonators with embedded apertures, J. Acoust. Soc. Am. 145(1), 254 (2019)

[35] S. B. Huang, X. S. Fang, X. Wang, B. Assouar, Q. Cheng, and Y. Li, Acoustic perfect absorbers via spiral metasurfaces with embedded apertures, Appl. Phys. Lett. 113(23), 233501 (2018)

[36] J. W. Guo, X. Zhang, Y. Fang, and R. H. Qu, An extremely-thin acoustic metasurface for low-frequency sound attenuation with a tunable absorption bandwidth, Int. J. Mech. Sci. 213, 106872 (2022)

[37] J. W. Guo,X. Zhang,Y. Fang,Z. Y. Jiang, A compact low-frequency sound-absorbing metasurface constructed by resonator with embedded spiral neck, Appl. Phys. Lett. 117(22), 221902 (2020)

[38] Y. Wang,H. G. Zhao,H. Yang,J. Zhong,D. Zhao, Z. Lu,J. H. Wen, A tunable sound-absorbing metamaterial based on coiled-up space, J. Appl. Phys. 123(18), 185109 (2018)

[39] C. Song, S. B. Huang, Z. L. Zhou, J. Zhang, B. Jia, C. C. Zhou, Y. Li, and Y. D. Pan, Perfect acoustic absorption of helmholtz resonators via tapered necks, Appl. Phys. Express 15(8), 084006 (2022)

[40] S. B. Huang, Z. L. Zhou, D. T. Li, T. Liu, X. Wang, J. Zhu, and Y. Li, Compact broadband acoustic sink with coherently coupled weak resonances, Sci. Bull. (Beijing) 65(5), 373 (2020)

[41] Z. L. Zhou, S. B. Huang, D. T. Li, J. Zhu, and Y. Li, Broadband impedance modulation via non-local acoustic metamaterials, Natl. Sci. Rev. 9(8), nwab171 (2022)

[42] S. B. Huang, Y. Li, J. Zhu, and D. P. Tsai, Sound-absorbing materials, Phys. Rev. Appl. 20(1), 010501 (2023)

[43] Standard test method for impedance and absorption of acoustical materials by impedance tube method, doi: 10.1520/C0384-04R22 (2022)

[44] Y. Atalla and R. Panneton, Inverse acoustical characterization of open cell porous media using impedance tube measurements, Can. Acoust. 33, 11 (2005)

RIGHTS & PERMISSIONS

Higher Education Press