Ultra-broadband acoustic ventilation barrier based on multi-cavity resonators

Yu-Wei Xu(許雨薇), Yi-Jun Guan(管義鈞),2,?, Cheng-Hao Wu(吳成昊), Yong Ge(葛勇),Qiao-Rui Si(司喬瑞), Shou-Qi Yuan(袁壽其),?, and Hong-Xiang Sun(孫宏祥),2,§

1Research Center of Fluid Machinery Engineering and Technology,School of Physics and Electronic Engineering,Jiangsu University,Zhenjiang 212013,China

2State Key Laboratory of Acoustics,Institute of Acoustics,Chinese Academy of Sciences,Beijing 100190,China

Keywords: acoustic metamaterials,ultra-broadband sound reduction,acoustic barrier,ventilation

1.Introduction

Sound reduction has always attracted a great deal of attention due to its practical applications in the fields of noise control, environmental protection and architectural acoustics.Recent development of acoustic metamaterials[1-9]and metasurfaces[10-17]provides alternative methods for designing various sound insulation and sound absorption systems based on different types of unit cells, such as Helmholtz resonators,[18-24]coiled Fabry-Perot resonators,[25-27]sound membranes,[28-31]metasurface-based structures,[32-40]and split-ring-resonators.[41,42]Generally, these systems have the advantages of high-performance sound insulation and absorption.However,most of them are designed as closed structures,and it is difficult to use them in the application scenarios requiring an additional feature of ventilation.

To overcome it, a variety of ventilation structures with sound insulation and absorption based on different mechanisms have been realized sequentially,[43-48]which mainly include sound absorber based on weak coupling of two identical oppositely oriented split tube resonators,[43]open sound silencer based on Fano-like interference,[45-47]and ultra-sparse open sound-insulation wall based on artificial Mie resonances.[48]Beyond that, ultrathin metasurfacebased structures[49]have been proposed to design ventilated sound insulation systems based on phased modulation, such as acoustic metacages,[50]open tunnels,[51]and windows.[52]These types of structures can realize sound insulation/absorption and ventilation simultaneously, however,the working bandwidths are narrow owing to resonance nature.In order to solve this problem, several researches[53-59]have been devoted to optimizing working bandwidths of sound insulation/sound absorption, such as ultra-wideband ventilation barrier[53]based on sound dissipation mechanism and interference mechanism,and ventilated sound metamaterial and sound absorption metamaterial based on the coupling modulation of resonance energy leakage and loss.[59]However,it still poses a great challenge to further increasing the bandwidth of ventilated sound-reduction structures.

In this work,we propose an ultra-broadband acoustic ventilation barrier composed of periodic unit cells.The designed single-layer ventilation barrier can realize broadband sound reduction with a bandwidth of 1560 Hz.Such a phenomenon arises from sound absorption caused by the eigenmode of the unit cell and sound reflection by the plate structure on upper surface of the unit cell.The measured and simulated results accord well with each other.Moreover, by designing two types of three-layer ventilation barriers composed of the unit cells with different values ofa(the length of the hollow square region) andw(the width of the channel between the adjacent cavities),we can increase the working bandwidths of both barriers to 3160 Hz and 3230 Hz,respectively,which are demonstrated by the experimental measurement.

2.Design of acoustic ventilation barrier

Fig.1.(a)Schematic diagram of acoustic ventilation barrier composed of periodically arranged unit cells.Photograph of (b) upper surface,(c)bottom surface,and(d)cross section of the unit cell,(e)schematic diagram of cross section of the unit cell.

3.Performance and mechanism of acoustic ventilation barrier

To demonstrate the performance of the designed ventilation barrier,we conducted an experiment to measure transmittance spectra of a unit cell in a straight waveguide made of acrylic plates with a size of 2 m×0.1 m×0.1 m [as shown in Fig.2(a)],in which the sound-absorbing foams are placed on both sides of the waveguide to eliminate reflected sound energy.The sample is placed in the middle of the waveguide,and the width of the waveguide is the same as the length of the unit cell.Beyond that, a loudspeaker driven by a power amplifier is placed on the left side to generate incident sound signals.Four 0.25-inch(1 inch=2.54 cm)microphones(Br¨uel&Kj?r type-4954,marked as Mic.1,2,3,and 4)are inserted into the waveguide from four holes with the same size to detect sound signals.The measured data are recorded by the module of Br¨uel & Kj?r 3160-A-042, and they are analyzed by the software of PULSE Labshop.

The acoustic pressure on the left of the sample in the waveguide is the superposition of the incident sound signal and reflected sound signal, and that on the right of the sample is the superposition of the transmitted sound signal and reflected sound signal.Thus,the sound pressures at the positions 1,2,3,and 4 can be expressed as

wherekis the wave number of air, andpiandptare the incident acoustic pressure and transmitted acoustic pressure,respectively, andprandptrare the reflected sound pressure in the left region and that in the right region of the waveguide,respectively, andd1andd2are the distance between Mic.1 and Mic.2,and that between Mic.3 and Mic.4,respectively,andl1andl2are the distance from Mic.2 to the left surface and that from Mic.3 to the right surface of the sample,respectively.Here,we selectd1=d2=10 cm,andl1=l2=13.5 cm.According to Eqs.(1)and(2),we can obtain

Thus,the transmitted coefficient can be expressed as

From Eqs.(7) and (9), we can calculate the transmittance,reflectance, and absorptance of the sample to bet=|T|2,r=|R|2,andα=1-|R|2-|T|2,respectively.

Figure 2(b) shows the measured transmittance spectra caused by the unit cell, in which the simulated results for the unit cell under the conditions of the hard and periodic boundary are also provided for comparisons.We can see that in the range of 834 Hz-2395 Hz (black shaded region), the transmittances are lower than 0.1,and thus the working bandwidth can reach about 1560 Hz, showing a typical characteristic of broadband sound reduction.Both types of simulated transmittance spectra are almost the same, and thus we can use the performance of a single unit cell to characterize that of the designed acoustic barrier.Beyond that,the measured and simulated results are basically in agreement with each other, in which the disagreement between both results may arise from the leakage of sound energy from the acrylic plates of the waveguide and the structural deviation of sample caused by the three-dimensional (3D) printing.Furthermore, according to the proportion of the size of the hollow region in the unit cell, we can obtain that the ventilation ratio of the designed barrier is 16%.

Fig.2.(a)Experimental set-up.(b)Measured(ME)and simulated(SI)transmittance spectra caused by the unit cell,with solid black line and dashed blue lines denoting simulated results for the unit cell under numerical condition of hard boundary and periodic boundary,respectively.

To explore the mechanism of the ventilation barrier, we simulate the absorptance and reflectance spectra caused by the unit cell.As shown in Fig.3(a), the broadband sound reduction is closely related to both sound absorption and reflection.To further explain it, we divide the working band into four regions (I: 834 Hz-1150 Hz, II: 1150 Hz-1450 Hz, III:1450 Hz-1700 Hz, and IV: 1700 Hz-2395 Hz) according to different characteristics.In the bands I and III,the absorptance values are both larger than 0.1, and the transmittance values are both lower than 0.1,however,both absorptance value and transmittance value are lower than 0.1 in the bands II and IV.We can see that the band I is determined by both sound absorption and sound reflection simultaneously,and the bands II and IV are mainly caused by sound reflection, which arises from the plate structure on upper surface of each unit cell.However,in the band III,the absorption coefficients are larger than 0.1, and the maximum value can reach a value exceeding 0.9 at 1580 Hz, so the band III is attributed to sound absorption caused by the unit cell.

Next, to provide an insight into the mechanism of sound absorption in the bands I and III,we simulate the pressure amplitude eigenfunctions of the unit cell around two absorption peaks in the bands I and III.As shown in Fig.3(b), there exist two types of eigenmodes at 920 Hz and 1578 Hz, in which most of the sound energy is absorbed into four cavities at 920 Hz, and it is mainly concentrated into the channel between the cavities and the outer frame at 1578 Hz.Figure 3(c)shows the simulated pressure amplitude distributions in the unit cell excited by the normal incidence of sound at 920 Hz and 1580 Hz.We can see that the field distributions in the unit cell are consistent with those of two eigenmodes in Fig.3(b),indicating that the sound absorption is closely related to the eigenmodes of the unit cell.Moreover, we simulate the distributions of thermoviscous energy loss density in the unit cell at 920 Hz and 1580 Hz.As shown in Fig.3(d),the thermoviscous loss is mainly distributed in the narrow channel between the cavity and the outer frame at 1580 Hz, which is much stronger than that in the channels between the adjacent cavities at 920 Hz.Such a phenomenon can be used to explain the difference between the two absorption peaks in the bands I and III.Therefore,the absorbed sound energy is mainly dissipated by the thermoviscous loss inside the channels of the unit cell in the bands I and III.

Fig.3.(a)Simulated sound transmittance,reflectance,and absorptance spectra caused by the unit cell.Different colors represent four working bands (I: 834 Hz-1150 Hz, II: 1150 Hz-1450 Hz, III: 1450 Hz-1700 Hz, and IV: 1700-2395 Hz).(b) Simulated pressure amplitude eigenfunctions of the unit cell at 920 Hz and 1578 Hz.Simulated distributions of(c)pressure amplitude and(d)thermoviscous energy loss density in the unit cell excited by normal incidence of sound(red solid arrows)at 920 Hz and 1580 Hz.

Fig.4.Simulated(a)effective bulk modulus,(b)effective impedance,(c)effective sound velocity,and(d)absorptance spectra of the unit cell around two absorption peaks.

To further analyze the above phenomena, we also simulate the effective bulk modulus, the effective impedance, the effective sound velocity of the unit cell,which are presented in Figs.4(a)-4(c),respectively,and the absorptance spectrum of the unit cell[as shown in Fig.4(d)]is also exhibited for comparisons.We can see that the real part of the effective bulk modulus is close to zero at 920 Hz and 1578 Hz[as shown in Fig.4(a)] owing to the excitation of the eigenmodes at both frequencies.Additionally, the imaginary part of the effective impedance and sound velocity at 1578 Hz are larger than those at 920 Hz[as shown in Figs.4(b)and 4(c)],indicating the high concentration and dissipation of sound energy in the unit cell at 1578 Hz.[39]The theoretical analysis of these effective parameters accords well with the characteristics of the unit cell in Fig.3.

Fig.5.Simulated transmittance spectra through unit cell under the excitation of acoustic waves at different incident angles.

Figure 5 shows the simulated transmittance spectra through the unit cell under the excitation of the acoustic waves at the incident angleθ=30?, 45?, and 60?.We can see that the simulated transmittance spectra are almost the same at different incident angles of sound, indicating high-performance broadband sound insulation in a wide range of incident angle for the designed ventilation barrier.

4.Bandwidth optimization of acoustic ventilation barrier

To optimize the bandwidth of the designed ventilation barrier,we also study and measure the transmission characteristics caused by the unit cell with different values ofaandw,which is shown in Fig.6.We can observe that by adjusting the parameteraorw,the ventilation barrier has high-performance sound reduction below 3000 Hz,and the measured results are in good agreement with the numerical results.

Fig.7.(a)Schematic diagram of the first type of three-layer ventilation barrier composed of unit cells with the same value of w(w=10 mm)but different values of a(a=5 cm for N =1; a=4 cm for N =2; a=3 cm for N =3).(b)Measured and simulated transmittance spectra caused by the three-layer ventilation barrier.(c)Schematic diagram of the second type of three-layer barrier composed of unit cells with the same value of a(a=4 cm)but different values of w(w=5 mm for N=1;w=10 mm for N=2;w=15 mm for N=3).(d)Measured and simulated transmittance spectra caused by three-layer ventilation barrier.

Next,we design two types of three-layer ventilation barriers with different values ofaandwto realize an ultrabroadband feature.Figure 7(a) schematically shows the first type of three-layer ventilation barrier with a distance(denoted asd)of 4.5 cm,in which the values of parameteraof the unit cells in layers 1,2,and 3 are selected as 5 cm,4 cm,and 3 cm,respectively,and the other parameters are the same as those in Fig.1(e).Here,it is necessary to point out that the ventilation ratio of the three-layer barrier is 9%owing to different values ofa, which decreases obviously in comparison with that of the single-layer barrier.Figure 6(b) shows the measured and simulated transmittance spectra of the three-layer ventilation barriers.We observed that the transmittance values are all less than 0.1 in the frequency range of 636 Hz-3798 Hz, and the bandwidth can reach about 3160 Hz,demonstrating the ultrabroadband characteristic.The measured and simulated results match well with each other.

Figure 7(c) presents the second type of three-layer ventilation barrier, in which the values ofwof the unit cells in each layer (N=1, 2, and 3) are selected as 5 mm, 10 mm,and 15 mm, respectively, and the distance between two adjacent layers is the same as that in Fig.7(a).Here, the ventilation ratio of this type of three-layer barrier is 16%, which is consistent with that of the single-layer barrier.The measured and simulated transmittance spectra of the three-layer ventilation barrier are shown in Fig.7(d).It is observed that the transmittance values are all below 0.1 in the frequency range of 660 Hz-3890 Hz (black shaded region), and the working bandwidth can increase to 3230 Hz,indicating the realization of an ultra-broadband characteristic of the ventilation barrier.The measured results show excellent agreement with the numerical results.Such ultra-broadband working bandwidths of both three-layer ventilation barriers are closely related to the bandgaps of the unit cells with different values ofaandw(as shown in Fig.6)and sound scattering and reflection between the adjacent barriers.Thus, we can effectively optimize the bandwidth of the designed ventilation barrier by using multilayer systems.

5.Conclusions

In this work,we have designed and demonstrated an ultrabroadband acoustic ventilation barrier.By designing a singlelayer ventilation barrier, we can observe a broadband sound reduction with the working bandwidth of 1560 Hz, which is attributed to the multiple mechanism, such as the sound absorption caused by the eigenmodes of the unit cell and the sound reflection by the plate structure on upper surface of the unit cell.Additionally, we study the sound-reduction performances of the unit cell by adjusting the parametersaandw.As a result,we design two types of three-layer ventilation barriers composed of the unit cells with different values ofaandw,which can achieve the ultra-broadband characteristics with the working bandwidths of 3160 Hz and 3230 Hz,respectively.The performances of these ventilation barriers are verified by experimental measurements.The proposed ventilation barriers with ultra-broadband sound reduction have a great potential in environmental protection and architectural acoustics.

Acknowledgements

Project supported by the National Natural Science Foundation of China (Grant Nos.12174159, 12274183, and 51976079),the National Key Research and Development Program of China (Grant No.2020YFC1512403), and the Research Project of State Key Laboratory of Mechanical System and Vibration(Grant No.MSV202201).