Damped resonance for broadband acoustic absorption in one-port and two-port systems

We demonstrate broadband perfect acoustic absorption by damped resonances through inclusion of lossy porous media. By minimally placing the lossy materials around the necks of single-resonance Helmholtz resonators, where acoustic energy is concentrated, we show an increase in absorption bandwidths (>100% of the resonance frequency). Using the damped resonance, we demonstrate three types of broadband acoustic absorbers in one-port and two-port systems: broadband absorbers (one-port), broadband sparse absorbers (two-port), and broadband duct absorbers (two-port). Our approach for broadband absorption allows to minimize the number of resonances for compact absorbers, while it is beneficial for practical applications owing to the minimum use of porous materials.

Damped resonance for broadband acoustic absorption in one-port and two-port systems taehwa

Lee , tsuyoshi nomura & Hideo iizuka
We demonstrate broadband perfect acoustic absorption by damped resonances through inclusion of lossy porous media. By minimally placing the lossy materials around the necks of single-resonance Helmholtz resonators, where acoustic energy is concentrated, we show an increase in absorption bandwidths (>100% of the resonance frequency). Using the damped resonance, we demonstrate three types of broadband acoustic absorbers in one-port and two-port systems: broadband absorbers (oneport), broadband sparse absorbers (two-port), and broadband duct absorbers (two-port). Our approach for broadband absorption allows to minimize the number of resonances for compact absorbers, while it is beneficial for practical applications owing to the minimum use of porous materials.
Acoustic metamaterials allow unprecedented control of sound waves [1][2][3][4] , enabling intriguing phenomena such as acoustic cloak 5 , asymmetrical transmission 6,7 , topological insulators 8,9 , and perfect absorbers [10][11][12][13] . Especially, acoustic metamaterials for perfect absorption lead to lots of practical applications for noise and vibration control, since they have advantages over conventional acoustic absorbers, e.g., for low-frequency sound absorption 10,11 . Despite such an advantage, resonance-based acoustic metamaterials often suffer from a relatively narrow absorption bandwidth. There have been extensive efforts devoted to increasing absorption bandwidths 12,13 . The most common approach uses multiple resonances, whose frequencies are closely placed for overlap of their absorption spectra [12][13][14] . Recently, multi-resonance metamaterials overlaid with a thin lossy medium have shown an extremely broadband acoustic absorption 14 .
Some complexities are involved in designing such multi-resonance metamaterials. For example, each resonance in such a metamaterial should fulfill critical coupling conditions [12][13][14] . In addition, combination of multiple resonators each having a different resonance frequency leads to a bulky structure. Thus, minimizing the number of resonators in a unit cell is practically desired for constructing compact broadband absorbers. One approach implements multi-order resonances of one resonator in the unit cell, simultaneously satisfying critical coupling conditions at these multi-order resonances 15 . In this device, the spacing between the individual resonances is restricted by the fundamental frequency and its higher harmonics. Alternatively, ultra-broadband absorption can be realized with a single-resonance resonator in the unit cell, which has been theoretically demonstrated 16 . In addition, single-resonance broadband absorbers operating in water have been reported 17 . For airborne absorbers, broadband absorption has been demonstrated by combining a lossy medium with acoustic resonances [18][19][20][21][22][23][24] , proving the effectiveness of damped resonance.
In this work, we demonstrate broadband airborne acoustic absorption by maximizing the contribution of damped resonance. We first discuss conditions for perfect absorption in systems with and without transmission channels. For damped resonance, a lossy porous medium is minimally implemented in the regions of Helmholtz resonators where acoustic pressure field is highly concentrated. The effect of the lossy material on acoustic performance is systematically investigated. We demonstrate three types of broadband absorbers based on damped resonance.

Results
perfect acoustic absorption in one-port and two-port systems. As  ). The perfect absorption requires different approaches, depending on presence of transmission channels. For one-port systems, where transmission is prohibited by reflectors backing resonators (i.e. T = 0), the perfect acoustic absorption condition is equivalent to a zero-reflection condition (R = 0), requiring the impedance matching 11,25 . As illustrated in Fig. 1(a), one-port systems are composed of acoustic resonators and an acoustic reflector, and the distance (l) between the resonator and the reflector can be chosen as either λ  l or λ ∼ l /4. For two-port systems, where acoustic transmission channels exist, besides zero reflection, transmission needs to be zero in order to demonstrate perfect acoustic absorption. Transmission in two-port systems results from the radiation symmetry. For zero transmission, breaking the symmetry of systems has been implemented for a relatively small open channel 13 . For systems with relatively large open channels, resonance degeneracy (or dual resonance) is required to demonstrate perfect absorption [26][27][28] . As seen in Fig. 1(b), for perfect absorption, one can use two approaches based on dual resonances consisting of lossy/lossy 28 or lossy/lossless resonators 29 .
To understand the need of such resonance degeneracy, consider a single resonance in a two-port system, as illustrated in Fig. 1(c). For a lossy subwavelength resonator, the theoretical absorption limit is A = 0.5 because of the radiation symmetry (R = 0.25 and T = 0.25) 30 . In this case, perfect absorption is possible only when two acoustic waves are coherently incident from the opposite directions (i.e. coherent perfect absorption mechanism) 31,32 . In contrast, a lossless subwavelength resonator in a two-port system works as a perfect reflector 33 , because the radiated wave from the resonator is out-of-phase with the incident wave (i.e. destructive interference).
From an interference perspective, perfect acoustic absorption is explained for one-port and two-port systems, as shown in Fig. 1(a,b). For zero reflection in one-port systems, the backward radiated wave of a single-resonance resonator should be cancelled by reflected waves from the reflector that is placed with a quarter-wavelength distance ( λ = l /4). When a single-resonance resonator is placed on a reflector, the reflected wave from the reflector should be directly cancelled with the radiated wave from the resonator 30 . In this case, the area ratio between the Figure 1. Perfect acoustic absorbers in one-port and two-port systems. (a) Single-resonance resonators in oneport systems. For a resonator away from a reflector ( ≠ l 0), a lossy resonator has perfect absorption when the backward-radiated waves from the resonator are cancelled by the reflected waves of the forward-radiated waves by the reflector, thus requiring a relative phase shift of π and a quarter-wavelength distance λ ∼ l /4. For a resonator on the reflector (l = 0), perfect absorption occurs when the reflected waves from the reflector is directly cancelled by the radiated waves from the resonator. (b) Dual-resonance resonators in two-port systems. For lossy/lossless resonators, perfect absorption occurs when λ ∼ l /4, as the lossless resonator functions as a reflector. For lossy/lossy resonators, de-tuning of their resonance frequencies is needed to cancel the radiated waves out of each resonator. The distance between the resonators can be much smaller than the wavelength ( λ  l ). (c) Single-resonance resonators in two-port systems. A lossy single-resonance resonator has 0.5 absorption, 0.25 reflection, and 0.25 transmission, whereas a lossless resonator functions as a reflector with no transmission (i.e. the forward-radiated waves from the resonator are out-of-phase with the incident waves).
www.nature.com/scientificreports www.nature.com/scientificreports/ resonator and the surrounding reflector is critical. On the other hand, for two-port systems, the perfect absorption of lossy/lossless resonators is realized by the lossless subwavelength resonator that functions as a reflector (i.e. no transmission). When dual resonances result from lossy/lossy resonators, they need to be slightly de-tuned for perfect absorption 28 . Such de-tuning leads to a π phase difference between the radiated waves from the two resonators, resulting in cancellation of these radiated waves. These de-tuned resonators can be placed in close proximity at a distance of λ  l .
theoretical absorption bandwidth limit. For single-resonance resonators in one-port systems, the peak absorption (A) is expressed by where C leak is the radiation leakage (kg/s), and C loss is the intrinsic loss (kg/s). The detailed derivation of Eq. (1) can be found in Methods. From Eq. (1), the maximum absorption occurs when = C C leak loss , i.e. the critical coupling condition stating that the radiation leakage (C leak ) out of the resonator should be matched with the intrinsic losses (C loss ) 18,34 . In general, the leakage rate has a complex value 35 . Under the assumption ( λ  d ), the leakage rate can be expressed as a simple form. For two-dimensional (2D) resonators on a reflector with a period of λ  d , the radiation leakage (kg/s per unit depth) is given by 17 When resonators are isolated in free field 36 , or the period is comparable to the wavelength, the acoustic reactance (i.e. imaginary part of C leak ) is non-negligible 37 . Note that C leak is proportional to s 2 and inversely proportional to d, indicating that the critical coupling (C leak = C loss ) can be realized by adjusting these geometrical parameters. In addition, C leak is proportional to Z s ; the radiation leakage from resonators in water is much higher than that from the same resonators in air, i.e. > C C leak leak ,water ,air . By using the fact that the quality factor ( , the full-width-at-half-maximum (FWHM) absorption bandwidth (Δf 0.5 ) is given by

Bandwidth of perfect acoustic absorbers.
To analyze the absorption bandwidth, we consider single-resonance Helmholtz resonators in a one-port system ( Fig. 1(a)). Figure 2(a) shows the numerically-calculated absorption spectrum of Helmholtz resonators with the period (d = 25 mm), the neck size (w n = 0.5 mm, l n = 4 mm), and the cavity size (w c = 5.5 mm, l c = 15 mm). The absorber demonstrates the perfect absorption with a bandwidth of Δf 0.5 = 395 Hz (0.22f res ), satisfying the critical coupling condition (C loss = C leak = 0 .0041 kg/s m). By using Eqs (2) and (3) , which is rather insensitive to the physical dimensions. In this regard, porous media are effective in increasing the intrinsic losses (C loss ) without increasing the vibrating mass (m 0 ) due to their high porosity. Evidently, lossy porous materials (e.g. foams) have been used in combination of resonators, increasing the absorption bandwidth 13,19,20 .
To maximize the effect of lossy media on the absorption bandwidth, we investigate Helmholtz resonators with different inclusions of lossy media (Design I, II, and III), as illustrated in Fig. 2(b). Figure 2(b) also shows the absorption spectra of the three configurations. The neck width (w n ) and the cavity length (l c ) of the resonators are adjusted to ensure = C C loss leak and the same resonance frequency, while the other dimensions remain constant (d = 18 mm, l n = 2 mm, and w c = 11 mm). With increasing C loss by the lossy foams, C leak should be increased by using a larger neck width (w n ) or smaller period (d) (see Eq. (2)). By covering with lossy foams (Design I; w n = 0.5 mm, l c = 12 mm), the absorption bandwidth is increased by a factor of two (Δ = . . ), compared to the resonator without a lossy medium ( Fig. 2(a)). Such an increase results from both increased leakage and intrinsic losses (C loss = C leak = 0.0037 kg/s m; C loss /m 0 = 1801 s −1 = 1.49c 0 ). Notably, filling the neck with lossy foams (Design II; w n = 2 mm, l c = 28 mm) significantly increases the bandwidth to Δ = .
. f f 0 96 res 0 5 due to C loss /m 0 = 3.29c 0 . By partially filling the cavity (Design III; w n = 4 mm, l c = 34 mm), the bandwidth is further increased to Δ = .
. For Design III, we further study the effects of the period (d) and the neck length (l n ) on the absorption bandwidth. With varying only d, the leakage rate (C leak ) is changed while the loss (C loss ) remains the constant. There is no change in the cavity size and thus constant resonance frequency. Figure 3(a) shows peak absorption for different d. The numerical results (symbols) show the good agreement with the analytical results (solid line), which are calculated from Eq. (1) by using C leak from Eq. (2) and constant C loss . The analytically-calculated C leak is plotted in Fig. 3(b), while the constant C loss is obtained from C leak for A = 1, i.e. C loss = C leak,A=1 . As shown in Fig. 3(c), the absorption bandwidth analytically calculated from Eq. (3b) with end correction ( = + .  www.nature.com/scientificreports www.nature.com/scientificreports/ With varying the neck length (l n ) for constant neck width (w n ), the loss rate (C loss ) is changed while the leakage rate (C leak ) remains the constant. For different l n , the cavity size is adjusted to ensure the constant resonance frequency. As shown in Fig. 3(d), from the perfect absorption (A = 1; = C C loss leak ), the peak absorption decreases with l n since C loss becomes larger than C leak . The estimated peak absorption (dashed line) is obtained by assuming a linear increase of ≅ . C l l 0 4 / loss n n,0 with l n,0 the neck length for = . l w / 05 n n ,0 ( Fig. 3(e)). Despite the increase in C loss , there is only a slight change in absorption bandwidth in Fig. 3(f) because of the increase of the vibration mass, as we have confirmed it without lossy foams. The estimated bandwidth (dashed line) is obtained by the assumed vibrating mass (Fig. 3(e)).
To demonstrate an absorption-bandwidth increase through judicious inclusion of lossy media, we fabricate different types of absorbers based on Design III: broadband absorbers (one-port system), sparse acoustic absorbers (two-port system), and duct absorbers (two-port system). We observe a significant increase in the absorption bandwidth of these absorbers.

one-port broadband absorbers.
Using damped resonances (Design III), we minimize the number of resonances in a one-port system, while obtaining broadband absorption. Figure 4(a) illustrates a one-port acoustic absorber consisting of dual resonances (without the top plate). The low-frequency resonator has a larger cavity area (WL − wl), which surrounds the smaller cavity (wl) of the high-frequency resonator. The two resonators have the similar neck dimensions (w n, L or H = 5 or 4 mm, l n = 2 mm) and are partially filled with a lossy material with a combined thickness of t = 20 mm + l n (Design III), as the 3D-printed prototypes of the unit cell with and without the foam are shown in Fig. 4(b). Figure 4(c) shows the acoustic absorption (A) and reflection (R) spectra of the dual-resonance absorber (W = 40 mm, L = 43 mm, w = 18 mm, l = 20 mm). We observe that high acoustic absorption (>0.8) extends from  from Eq. (3a)) and the same thickness of the foam is more lossy in the high frequency (i.e. increase in C loss ). The thickness of the foam is much smaller than the resonance wavelengths of the both resonators.
The absorption bandwidth can be increased by decreasing the period (W) and thus increasing C leak , as shown in Fig. 5(a). Increasing C leak through smaller W is feasible as long as it does not compromise the peak absorption. The period (W) cannot be practically too small, because the constituent resonators need spaces occupied by their cavities; for smaller W, a larger L is needed to keep the resonance frequencies unchanged. Without the use of the foam, the absorption bandwidth is very small. Figure 5(b) shows the absorption spectrum for a dual-resonance resonator with the same period W = 40 mm, but without foam. We observe perfect absorption for very small neck widths (w n, L = w n, H = 0.7 mm). These small neck widths lead to small radiation leakage (C leak ) that can be matched with small intrinsic losses without foams (C leak = C loss ). In addition, each resonance has a very narrow bandwidth, compared to Fig. 4(c). In Fig. 5(b), the absorption spectrum (dashed line) is plotted for the same device of the  Broadband sparse absorbers. Sparse absorbers are composed of an array of resonator pairs, which are sparsely arranged, as illustrated in Fig. 6(a). Such a sparse arrangement falls into two-port systems, allowing acoustic transmission channels ( Fig. 1(a)). The sparsity of the resonator is defined as the width (W) of the resonator to the ratio of the period (d) (i.e. W/d), and a high sparsity is desired in minimizing fluid flow resistance 29 . Each resonator pair consists of two resonators of the same resonance frequency, but only the resonator, facing acoustic sources, is combined with the lossy foam (Design III), as shown in Fig. 6(b). The other resonator without a lossy medium is considered a lossless resonator. The pair of the lossy/lossless resonators leads to the perfect absorption, when the distance (l) between the two necks of the resonators is chosen to cancel the reflected waves from the lossless resonator by the reflected waves from the lossy resonator ( Fig. 1(b)). To meet the distance requirement ( λ ∼ l /4), the neck of the lossless resonator is positioned on the side wall. Figure 6(c) shows the absorption (A), transmission (T), and reflection (R) spectra of the sparse resonators (W = 20 mm, L = 66 mm, l = 42 mm, w n = 4 mm, and l n = 2 mm) with a period of d = 40 mm. The cavity sizes of the two resonators are identical (w c = 16 mm, l c = 30 mm) for the same resonance frequency. We observe that the absorption bandwidth (Δf 0.5 ) is approximately 900 Hz. Notably, despite the high sparsity (W/d), the acoustic transmission is near zero around the resonance frequency. The resonance frequency is 1600 Hz. The experimental results (symbols) show a good agreement with the numerical result (solid lines).
Our approach for increasing the absorption bandwidth with lossy media is effective in the design of sparse absorbers that need a high sparsity. By minimizing the number of resonances, the sparse absorber is enabled with a compact structure of a high sparsity. Similar to the broad absorber in a one-port system, the absorption bandwidth increases with decreasing d, as shown in Fig. 7(a). Alternatively, to increase the absorption bandwidth, one can consider adding another layer of sparse resonators of a different frequency, resulting in dual resonances, unless fluid flow resistance is significantly compromised. Figure 7(b) shows pressure fields for resonators of lossy/ lossless, lossy alone, and lossless alone. For the lossy/lossless case, all the acoustic energy (black arrow) goes to the front lossy resonator, whereas the lossy resonator alone permits considerable acoustic transmission. For the lossless resonator alone, acoustic wave is reflected back without transmission.
Broadband duct absorbers. Our approach for the broadband sparse absorbers can be extended to broadband duct absorbers, as these two types of absorbers operate in two-port systems. In addition, for λ  d , the periodic boundary of the sparse absorber can be replaced by the hard-reflecting boundary, and thus the same physics can be applied to both the sparse absorber and duct. Broadband duct absorbers enable lots of practical applications such as HVAC (Heating, Ventilation, and Air Conditioning) systems for vehicles and buildings. Our duct absorber implementing damped Helmholtz resonators is illustrated in Fig. 8(a). Based on the perfect absorption mechanism in two-port systems ( Fig. 1(b)), we use a lossy/lossless combination of resonators of the identical resonance frequency; the constituent resonators have the same size (w n = 10 mm, l n = 1.5 mm, w c = 30 mm, and www.nature.com/scientificreports www.nature.com/scientificreports/ l c = 50 mm). The upstream resonator only includes a lossy medium around its neck (a thickness of t = 21.5 mm), while the downstream resonator is considered lossless because it has a large radiation leakage and a small loss owing to its short length and large width of the neck (i.e.  Q Q leak loss ). The distance (l) between these two resonators is chosen to cancel the reflected waves from the lossy resonator with the reflected waves from the lossless resonator. As shown in Fig. 8(b), the prototype of the duct absorber is fabricated with transparent acrylic plates, allowing to see the black foam in the lossy resonator. Figure 8(c) shows the absorption (A), transmission (T), and reflection (R) spectra of the duct noise absorber (l = 49 mm, d = 40 mm). The numerical peak absorption is approximately 90% at the resonance frequency of f res = 1000 Hz, while the measurement shows the peak absorption is about 70%. We observe the absorption bandwidth of ~650 Hz (=0.65f res ) compared to ~0.2f res without a lossy inclusion. Although the numerical results (solid lines) captures well the spectral characteristics of the experimental results (symbols), such a discrepancy between the numerical and experimental results comes unexpectedly large reflection in the measurement, possibly because of a small size difference between the two cavities and thus unwanted resonance de-tuning.
The absorption bandwidth can be further increased by reducing the width (d) of the duct and thus increasing the radiation leakage (C leak ), as shown in Fig. 9(a). For example, the radiation leakage is increased to C leak = 1.5C loss (i.e. a small deviation from the critical coupling condition), which leads to a 25% bandwidth increase (From Eq. (4a)) without significantly compromising peak absorption, i.e. = = .
. . + . Figure 9(b) shows pressure fields for Helmholtz resonators of lossy/lossless, lossy alone, and lossless alone. The acoustic energy is concentrated on the lossy resonator with no transmission. For lossy alone, considerable transmission is observed. For lossless alone, no transmission is seen, since the lossless resonator works as a reflector. De-tuning approach (Fig. 2(b)) has been used for perfect duct absorbers 28 . We expect that the length (l) of our prototype device can be further optimized to a length of λ = . l 0 2 res .

conclusions
We have demonstrated broadband perfect acoustic absorption by using lossy porous media in combination with acoustic resonators. By placing the lossy materials around the regions of the resonators, where acoustic energy is concentrated, we have shown a significant increase in absorption bandwidths (>100% of the resonance frequency), while minimizing the use of the lossy materials. Using the damped resonances, we have demonstrated bandwidth increases for three types of acoustic absorbers. Our design approach can benefit acoustic absorbers based on Helmholtz resonators for broadening absorption bandwidths.    where x(t) is the oscillating amplitude, and F(t) is the force acting on the oscillator. In the regime of the period d that is much smaller than the wavelength λ, we have the forms of = F t p t S ( ) 2 ( ) osc and η = Z S S / r s osc inc 2 , respectively, where p(t) is the pressure of the incident acoustic wave, Z is the acoustic impedance of the surrounding fluid, S osc is the area of the oscillator, and S inc is the area of the unit cell. By solving Eq. (6), the oscillating amplitude in the frequency domain (ω) is given by The peak absorption at resonance (ω ω = 0 ) is given by

Methods
leak loss leak loss 2 Perfection absorption occurs when C leak = C loss , i.e. critical coupling condition. Rigorously, the critical coupling condition states that the radiated power (P leak ) is balanced with the dissipated power (P loss ): , where dx t dt ( ) is the root-mean-square average of vibration velocity. Thus, the equality of P leak = P loss implies C leak = C loss .