Ultra-ventilated sound absorption metamaterial lamina

In this work, we will first conceptually propose an ultrathin sound absorption lamina, and then will experimentally verify such metamaterial absorbers based on the lamina units. The lamina unit, based on weak coupling of high-order Helmholtz resonances, is shown in figure 1(a). In our example, a periodic unit, enclosed by the red dashed line, has a length of 74 mm. Adjacent units are coupled to each other via a notch of length A (the outmost coupling corners are sealed for a finite structure). In figure 1(a), the periodic unit in the red-dotted box consists of the basic structure in the blue-dotted box through a fourfold rotation. The basic structure is constructed by two series-connected embedded Helmholtz resonators, whose sketch and internal structural parameters are shown in figure 1(b). Here, the wall thickness w = 2 mm, the width of the outer cavity l = 22 mm, and the width of the lamina Lx = 146 mm are chosen for the present study. A schematic diagram of a finite sample of 2 × 3 units is shown in figure 1(c) for a direct visualization. One suitable application scenario of this sound absorber is also illustrated in figure 1(d), where the lamina is located on the inner wall of the ventilation duct with a high ventilation area ratio. Benefitting from the ultra-thin thickness of this metamaterial lamina, the fluid in the ventilation duct could flow freely. As the sound wave near the resonant frequency travels through this metamaterial lamina, the majority of sound energy is drawn into the lamina and dissipated due to thermal viscosity loss. The embedded Helmholtz resonator can effectively extend the neck length of the primary Helmholtz resonator, leading to a decrease in resonance frequency. Then connecting a central Helmholtz resonator and two embedded Helmholtz resonators in series, the resonance frequency can be further reduced. Consequently, subwavelength sound absorption can be achieved. Our results demonstrate this sound absorption lamina can operate at a high ventilation area ratio condition (>80%), and achieve quasi-perfect low-frequency sound absorption in broadband and near-perfect ventilation performance.

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To verify the acoustic performance of the designed sound absorber, we chose three sets of geometrical parameters for both simulations and experiments, the parameters are given in table 1. We use the pressure acoustics module of the finite element software COMSOL Multiphysics to perform numerical analysis. The inner surface of the waveguide and the wall of the sample are fixed as hard boundaries of the acoustic field in the simulation. The parameters for the physical properties of air are set as follows for the normal temperature and pressure: the density of air ρ= 1.2 kg m⁻³, the speed of sound c= 343 m s⁻¹, the dynamic viscosity coefficient μ= 18.5 μPa·s, the thermal conductivity κ= 0.024 W (mK)⁻¹, the heat capacity at constant pressure C_p= 1000 J (kg·K)⁻¹. For comparison with experiments, as will be discussed later, the samples are placed in a hard-boundary rectangular waveguide with a cross-section of 146 mm × 146 mm. The simulation results for the three samples are shown as the solid lines in figure 2(a), where the resonance frequencies for samples 1, 2, and 3 are 605 Hz, 728 Hz, and 899 Hz, respectively. The corresponding maximum sound absorptions are 0.92, 0.98, and 0.94. For the three samples described above, the thickness Lz is 16.8 mm, so the ventilation area ratios of the three samples are 88.65%. Furthermore, the proportion of resonance wavelengths to thickness (Lz) for each of the three samples stands at 33.75, 28.04, and 22.71, in that order. Moreover, these sound absorbers occupy a volume of 0.29%, 0.51%, and 0.96% of the cube of the corresponding wavelength, respectively. To reveal the physics behind the absorption, we plot the acoustic pressure distributions at the resonant frequency in figures 2(b)–(d). Perfect absorption can be known to benefit from the fundamental unit's local resonant coupling and weak coupling between adjacent units.

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To verify the simulation results, the acoustic performance of the laminas fabricated with polylactic acid (PLA) by a 3D printer are measured in a square impedance tube [34]. The experimental data is shown in figure 2(a) as the dotted lines (red, green, and cyan), which agree well with the simulation results. Moreover, compared with sound absorption foam (Basotect G+, BASF) of the same profile, the metamaterial lamina presents excellent sound absorption performance around the resonant frequency. Furthermore, both experimental and simulated results point out this metamaterial lamina could realize selective sound absorption in a wide region at a low frequency (∼605 Hz–899 Hz). In addition, we performed a demonstration of noise reduction using sample 2 at 740 Hz, with a sound transmission loss (STL) of about 26 dB as shown in figure 2(e) (Multimedia available online).

The physical model of this lamina structure can be interpreted using the transfer matrix method (TMM). The transfer matrix of the ith layer is defined as:

$$\begin{align}\left[ {\begin{array}{*{20}{c}} {p_{i + 1}^ + } \\ {p_{i + 1}^ - } \end{array}} \right] = {T_i}\left[ {\begin{array}{*{20}{c}} {p_i^ + } \\ {p_i^ - } \end{array}} \right],\end{align} \tag{ 1 }$$

in which $p_i^ + (p_i^ - )$ represents the acoustic pressure forward (backward) at the front end of the ith layer, or the sound wave forward (backward) at the back end of the (i−1)th layer. The transfer matrix of the ith layer structure can be expressed as:

$$\begin{align}{T_i} = \left[ {\begin{array}{*{20}{c}} {{t_i} - \frac{{{r_i}{r_i^{\prime}}}}{{{t_i^{\prime}}}}}&{\frac{{{r_i^{\prime}}}}{{{t_i^{\prime}}}}} \\ { - \frac{{{r_i}}}{{{t_i^{\prime}}}}}&{\frac{1}{{{t_i^{\prime}}}}} \end{array}} \right],\end{align} \tag{ 2 }$$

t_i (r_i ) represents the transmission (reflection) of the i-layer structure when sound waves incident in the y-direction, while t_i' (r_i') represents the transmission (reflection) of sound waves incident in the opposite direction. The periodic units for sample 1, sample 2, and sample 3 in the first part exhibit a fourfold rotational symmetry, with t_i = t_i' and r_i = r_i'. The total transfer matrix for a lamina with n layers is as follows: T = T_nT_{n− 1} ... T₂ T₁, so the transmission matrix of the sound absorption lamina composed of identical 2 × 3 units can be expressed as

$$\begin{align}T& = {T_3}{T_2}{T_1} \approx {T_1}{T_1}{T_1}\nonumber\\ & = \left[ {\begin{array}{*{20}{c}} {\frac{{ - r_1^6 + t_1^6 + r_1^4\left(2 + 3t_1^2\right) - r_1^2\left(1 + 2t_1^2 + 3t_1^4\right)}}{{t_1^3}}}&{\frac{{{r_1}\left(1 + r_1^4 + t_1^2 + t_1^4 - 2r_1^2\left(1 + t_1^2\right)\right)}}{{t_1^3}}} \\ { - \frac{{{r_1}\left(1 + r_1^4 + t_1^2 + t_1^4 - 2r_1^2\left(1 + t_1^2\right)\right)}}{{t_1^3}}}&{\frac{{1 + r_1^4 - r_1^2\left(2 + t_1^2\right)}}{{t_1^3}}} \end{array}} \right],\end{align} \tag{ 3 }$$

which can obtain the transmission and reflection coefficients,

$$\begin{align}t& = \frac{{t_1^3}}{{1 + r_1^4 - r_1^2\left(2 + t_1^2\right)}},\end{align} \tag{ 4 }$$

$$\begin{align}r& = \frac{{{r_1}(1 + r_1^4 + t_1^2 + t_1^4 - 2r_1^2(1 + t_1^2)}}{{1 + r_1^4 - r_1^2(2 + t_1^2)}}.\end{align} \tag{ 5 }$$

Then, according to the law of energy conservation, the sound absorption coefficient can be obtained

$$\begin{align}A = 1 - |t{|^2} - |r{|^2}\end{align} \tag{ 6 }$$

as shown in figure 2(a) with cycle lines.

To obtain a better understanding of the selective sound absorption performance of this lamina structure, the relationship between structural parameters and absorption coefficients is also investigated based on controlled variables. The initial values of the geometric parameters are set as Lx = 146 mm, Ly = 218 mm, Lz = 16.8 mm, l = 22 mm, w = 2 mm, a = 3.6 mm, A = 8.1 mm, d = 4.2 mm, D = 12.3 mm, m = 4.5 mm, s = 1.7 mm, where Lx, Ly, l, and w are fixed values. We find that the parameters a, d, D, and L_z have a strong influence on the absorption. Figure 3(a) shows the absorption spectrum of the parameter a. As a is increased, the resonant frequency shifts significantly to the high frequency, and the absorption and the bandwidth of the absorption also increase. Figure 3(b) shows the relationship between the parameter d and the absorption. As d increases, the resonant frequency first shifts significantly to a higher frequency and then tends to a limit. In addition, the absorption coefficient first increases and then tends to approach a limit, while the absorption bandwidth becomes significantly broader. The relationship of the spectrum between the parameter D, frequency, and absorption coefficient is shown in figure 3(c). From this Figure, it can be seen that with increasing D, the resonant frequency, the sound absorption coefficient, and the absorption bandwidth all increase significantly. Figure 3(d) shows the influence of the sample height Lz on the absorption. As the Lz is increased, the absorption peak shifts to a lower frequency, while the peak absorption first rises and then falls. On the other hand, the geometric parameters A, m, and s only have little influence on the absorption spectrum, as can be seen in figure S2 of the supporting materials.

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To obtain broadband absorption, we further construct the lamina with various resonance units. Here two laminas comprising 2 × 3 units and 2 × 4 units are optimized for broadband sound absorption in the corresponding frequency range, in which Lx= 146 mm, l= 22 mm, and w = 2 mm are kept unchanged for the two laminas. Black solid lines in figure 4(a) display optimization simulated results of 2 × 3 units, where sound wave propagates along the +y direction. From 506 to 669 Hz, the absorption coefficient of this lamina is greater than 0.5, whose average sound absorption rate is 0.64. The black dotted line of figure 4(a) presents the corresponding experimental result, which is in good agreement with the simulation. We also independently simulated and analyzed the acoustic performance of units 1–6 separately, as shown in the group of colored curves in figure 4(a). The sound absorption coefficients of all six single units are consistently less than 0.5. Because of the coherently coupled weak resonances [35] between these units, the overall absorption coefficient and bandwidth increase significantly. In comparison to figure 2(a), the working bandwidth (absorption coefficient >0.5) is significantly increased by decoupling the resonances without altering the number of units. The optimization results for a sample consisting of 2 × 4 units are shown in figure 4(b), a solid black line shows the simulation result. In the band of 483–683 Hz, the simulated absorption coefficient is higher than 0.5 whose average absorption coefficient is 0.67. The black dotted line represents the experimental result of the corresponding sample, which is well in accord with the simulations. Numerical analysis of the independent unit labeled 1–8 is also carried out. When each unit acts alone, the absorption coefficient is substantially small. Both experimental and simulated data demonstrate the absorption value and bandwidth of the assembling lamina could be efficiently increased by the weak coupling. Due to the asymmetric of the structures, the scattering of sound waves is asymmetric when they enter from the +y or −y direction, this leads to the asymmetric absorption of sound waves. Experimental and simulation results of sound incidence along the −y direction are shown in figures S3 of supplementary material.

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By comparing the results of 2 × 3 and 2 × 4, it is found that adding decoupling units to the direction of acoustic propagation can effectively improve the average sound absorption coefficient and bandwidth. Enhancing sound absorption can also be achieved by compromising some ventilation efficiency by symmetrically placing two laminas on the waveguide's internal wall. We simulated this scenario and observed an enhancement in the sound absorption coefficient. For more information, consult figure S4 in the supplementary materials.