A Comparison of Designs for Broadband Sound Absorption

1. Introduction

Ideally, sound absorbing treatments for use in aircraft and vehicle structures, white goods and prefabricated building constructions, should provide broadband absorption over the audio-frequency range, while adding as little volume and weight as possible. Typical sound absorption coefficient spectra provided by thin hard-backed layers of synthetic porous materials such as glass wool or polymer foams are poor at low frequencies unless the layer thickness is increased or the layer is mounted on an air gap thereby adding volume. Many designs suitable for Additive Manufacture offer useful subwavelength absorption from thin layers.

An example of such a design is shown in Figure 1(a) [1]. A 10 mm thick, melamine foam layer covers an array of eighteen rectangular cavities. These include eight single cavities (numbered 1 – 8 in Figure 1(a)) and eight double cavities i.e., two connected single cavities (numbered 9 – 16 in Figure 1(a)). Sound enters the cavities through small holes of various widths (1.7 mm, 2.2 mm and 4 mm), so they act as Helmholtz resonators. The double cavities are folded beneath the single cavities to reduce the total height (31 mm) of the structure. Cavities 17 and 18 in Figure 1(a), which are 25 mm long and 10 mm wide, contain 20 mm and 12 mm thick layers of a different porous foam, henceforth called ‘cavity’ foam [1].

Also shown in Figure 1(b) are the frequency ranges encompassing the main contributions to the overall absorption coefficient spectrum from the cavity resonators (HR), cavities with porous infill and the thin, porous covering. Impedance tube measurements were made between 600 Hz and 1600 Hz on a 3D printed replica of the Figure 1(a) structure and analytical and numerical predictions were found to agree with the data from these measurements [1].

A second metamaterial design for broadband sound absorption is shown in Figure 2(a) [2]. Unit cells are filled with a porous material containing left- and right-hand series of regularly spaced embedded horizontal plates with lengths that increase with depth. Figure 2(b) shows a measured normal incidence absorption coefficient spectrum together with a corresponding numerical prediction (labelled simulation).

The acoustical properties of the melamine foam layer, the porous material infill used in the Figure 1(a) design [1] and the foam ‘sponge’ used in the Figure 2(a) design [2] are calculated using the Johnson-Champoux-Allard (JCA) model for a rigid-framed porous material. Corresponding expressions for the bulk complex density ${\rho }_1\left(\omega \right)$ and compressibility $C_1\left(\omega \right)$ are [3]

$${\rho }_1\left(\omega \right)={\displaystyle \frac{{\alpha }_{\mathrm{\infty }}{\rho }_0}{\varphi }}\left[1+{\displaystyle \frac{i\sigma \varphi }{\omega {\rho }_0{\alpha }_{\infty }}}G(\Lambda )\right],$$

(1)

$$C_1\left(\omega \right)={\varphi \left(\gamma P_0\right)}^{-1}\left[\gamma -\left(\gamma -1\right){\left[1+{\displaystyle \frac{i\sigma \varphi }{\omega {\rho }_0{\alpha }_{\infty }{N}_{Pr}}}{\mathrm{G}}^{\text{'}}\left({\Lambda }^{\text{'}}\right)\right]}^{-1}\right]$$

(2)

$$G\left(\Lambda \right)=\sqrt{\left(1+{\displaystyle \frac{4\mathrm{i}{{\alpha }_{\mathrm{\infty }}}^2\eta \omega {\rho }_0}{{\left({\sigma }_{\rho }\Lambda \mathsf{\varphi }\right)}^2}}\right)}$$

(3)

$${\mathrm{G}}^{\text{'}}\left({\Lambda }^{\text{'}}\right)=\sqrt{\left(1+{\displaystyle \frac{4\mathrm{i}{{\alpha }_{\mathrm{\infty }}}^2\eta N_{pr}\omega {\rho }_0}{{\left({\sigma }_C{\Lambda }^{\text{'}}\mathsf{\varphi }\right)}^2}}\right)}$$

(4)

where the $e^{-i\omega t}$ time convention is understood, $\omega =2\pi f$ (rad s^-1) is the angular frequency, $f$ (Hz) is the frequency, $t$ (s) is time, ${\alpha }_{\mathrm{\infty }}$ is tortuosity, $\varphi$ is porosity, $\sigma$ is flow resistivity with units of Pa s m^-2, ${\rho }_0$ is the density of air (kg m^-3). In equations (1c) and (1d), ${\sigma }_{\rho }={\displaystyle \frac{8\mu {\alpha }_{\infty }}{\varphi {\Lambda }^2}}$ and ${\sigma }_C={\displaystyle \frac{8\mu {\alpha }_{\infty }}{\varphi {{\Lambda }^{\text{'}}}^2}}$, where $\Lambda$ is a characteristic viscous length and ${\Lambda }^{\text{'}}$ is a characteristic thermal length. $N_{pr}$ and $\eta$ are the Prandtl number and dynamic viscosity coefficient for air respectively.

The characteristic impedance, $Z_1$, and the propagation constant, $k_1$, are calculated, respectively, from ${\rho }_1\left(\omega \right)$ and $C_1\left(\omega \right)$ using

$$Z_1=\sqrt{{\rho }_1\left(\omega \right)/C_1\left(\omega \right)}$$

(5)

$$k_1=\omega \sqrt{{\rho }_b\left(\omega \right)C_b\left(\omega \right)}$$

(6)

The specific surface impedance, $Z\left(d\right)$, of a hard-backed porous layer of thickness d is given by equation (10)

$$Z\left(d\right)=Z_1\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{h}\left(-ikd\right)$$

(7)

The plane wave reflection coefficient, R(d), and normal incidence absorption coefficient, $\alpha$(d), for a hard-backed porous layer of thickness d are given by equations (8) and (9), respectively.

$$R\left(d\right)={\displaystyle \frac{1-Z\left(d\right)}{1+Z\left(d\right)}},$$

(8)

$$\alpha \left(d\right)=1-\left|{\left(R\left(d\right)\right)}^2\right|,$$

(9)

The JCA parameter values used to calculate the acoustical properties of the three porous materials used in the first two structures [1,2] are listed in Table 1.

Figure 3(a) shows a third example of a broadband sound absorbing design based on the sonic black hole concept. It consists of parallel microperforated plates inside a conical shell [4]. Figure 3(b) shows the numerically predicted absorption coefficient spectrum for an arrangement of five MPP equally spaced at 12 mm inside a conical tube with a radius that decreases linearly from 30 mm to 1 mm at the back wall.

This paper investigates the extent to which sound absorption coefficient spectra comparable with or superior to those obtained with the designs in Figure 1(a), 2(a) and 3(a) could be achieved in simpler ways either by embedding a few thin solid vertical and horizontal partitions in hard-backed layers of the porous materials used in the Figure 1(a) and 2(a) designs, or by choosing porous materials with higher tortuosity.

Section 2 outlines theory for the acoustical properties of partitioned structures. Section 3 compares corresponding predictions of normal incidence sound absorption coefficient with data and predictions for the structures shown in Figure 1(a), 2(a) and 3(a). Section 4 contains predictions based on the use of alternative (unpartitioned) porous materials with higher tortuosity and Section 5 offers concluding remarks.

2. Theory for Partitioned Porous Layers

Hard backed layers of porous material containing adjacent unit cells containing partitions of types 1 and 2 are shown in Figure 4.

A third type of partition and the dimensions associated with all three types of partition are shown in Figure 5.

Using an effective medium approach, the overall impedance of unit cell type 1 (Figure 5(a)) is calculated by considering the two sub-cells shown in Figure 6 [4].

The left-hand sub cell of width $\left(W/2-t\right)$, where W is the total width of the unit cell and t is the thickness of the partition, has the characteristic impedance $Z_A^c$ given by [4]

$$Z_A^c={\rho }_0{c}_0{\displaystyle \frac{Z_1}{{\eta }_1}},$$

(10)

where $Z_1$ is given by equation (8).

The approximate impedance, $Z_B$, of the right-hand sub cell of width $\left(W/2-t\right)$ and length $\left(d_1-a-t\right)$, is given by [4]

$$Z_B\approx {\displaystyle \frac{-i}{{\eta }_1}}\left[\omega m-Z_A^c\cot \left(k_1\left(d_1-t-a\right)\right)\right],$$

(11)

$$m=\left(t+2\delta a\right){\rho }_1{\displaystyle \frac{W}{2a}},$$

(12)

$${\eta }_1=\left(W/2-t\right)/W,$$

(13)

where $\delta$ = 1.9, m is the acoustic mass of the gap and $a$ is the width of the gap. The gap is considered to form the neck of a resonator filled with porous material [4]. The approximation (in equation (11)) relies on the gap being sufficiently small that differences from standard radiation conditions are not important.

The specific surface impedance, $Z_{AB}$, of a unit cell with a type 1 partition is given by [4]

$$Z_{AB}={\displaystyle \frac{Z_A^c}{{\rho }_0{c}_0}}\left[{\displaystyle \frac{Z_B\cot \left(k_1{d}_1\right)-i{Z}_A^c}{-i{Z}_B+Z_A^c\cot \left(k_1{d}_1\right)}}\right].$$

(14)

The reflection and absorption coefficients are obtained by replacing $Z\left(d\right)$ by $Z_{AB}$ in equations (8) and (9) respectively.

Figure 6 shows the sub cells for type 2 partitions [4,5]. The left-hand part of the sub-cell is connected to the resonant absorbing volume of length $\left(d_1-d_2-t\right)$ and width $\left(W-t\right)/2$ in the right-hand half of the unit cell by the gap of width $a$ at the bottom of the vertical partition.

Figure 7. Sub cells of unit cell type 2 [4,5].

Figure 7. Sub cells of unit cell type 2 [4,5].

If, as for the type 1 arrangement, the gap is treated as the neck of a Helmholtz resonator, the specific surface impedance, ${Z_{AB}}^{\text{'}}$, for this sub-cell is given by equation (15)

$${Z_{AB}}^{\text{'}}={\displaystyle \frac{Z_A^c}{{\rho }_0{c}_0}}\left[{\displaystyle \frac{{Z_B}^{\text{'}}\cot \left(k_1{d}_1\right)-i{Z}_A^c}{-i{Z_B}^{\text{'}}+Z_A^c\cot \left(k_1{d}_1\right)}}\right],$$

(15)

$${Z_B}^{\text{'}}\approx {\displaystyle \frac{-i}{{\eta }_1}}\left(\omega m-{\rho }_0{c}_0{Z}_1\cot \left(k_1\left(d_1-d_2-t\right)\right)\right),$$

(16)

The specific surface impedance, $Z_C$, of the right-hand sub-cell is given by

$$Z_C=\left(Z_1/{\eta }_1\right)\cot \left(k_1{d}_2\right),$$

(17)

Hence, the specific surface impedance of a unit cell with type 2 partitions is given by

$${\left(Z_{type2}\right)}^{-1}={\left({Z_{AB}}^{\text{'}}\right)}^{-1}+{\left(Z_c\right)}^{-1},$$

(18)

The specific surface impedance of cell type 3 is the sum of the admittances of a sub cell of type 2 and that of sub cell of a hard backed porous layer of length $d_1$ corresponding the left-hand sub cell in Figure 6 after allowing that each sub cell has a narrower width ($W/3-t$).

$${\left(Z_{type3}\right)}^{-1}={\left({Z_{type2}}^{\text{'}}\right)}^{-1}+{\left(Z_D\right)}^{-1},$$

(19)

$$Z_D=\left(Z_1/{\eta }_2\right)\cot \left(k_1{d}_1\right),$$

(20)

$${\eta }_2=\left(W/3-t\right)/W,$$

(21)

$$m^{\text{'}}=\left(t+2\delta a\right){\rho }_{1}W/\left(3a\right).$$

(22)

${Z_{type2}}^{\text{'}}$ in equation (19) is calculated from equations (15) to (18) after replacing ${\eta }_1$ by ${\eta }_2$ and $m$ by $m^{\text{'}}$.

3. Predictions and Comparisons with Data

3.1. Predictions for Three Partition Types

To represent the acoustical properties of a porous foam, Yang et al. [5] use the Johnson-Champoux-Allard model with porosity 0.95, tortuosity 1.42, viscous characte5istic length 180 μm, thermal characteristic length 360 μm and flow resistivity 8.9 kPa s m^-2. Figure 8 shows normal incidence absorption coefficient spectra predicted for this foam containing each of the three types of partitions with $d_1=30$ mm, $d_2=21$ mm, $a=1$mm and $t=0.5$ mm. Also shown is the prediction for the foam layer without partitions.

Type 1 partitions are predicted to result in the highest absorption below 1 kHz but otherwise the resulting absorption spectrum is highly oscillatory. Although predictions for type 2 and 3 partitions do not show as much absorption as type 1 partitions below 1 kHz, they indicate higher absorption between 500 Hz and 5 kHz than predicted for the foam layer without partitions.

The predicted normal incidence absorption coefficient spectra for 30 mm thick layers of foam [5] with type 2 and 3 partitions are sensitive to the value of $d_2$. This is shown for 10 mm $\le d_2\le$ 25 mm and type 2 and type 3 partitions respectively in Figure 9(a) and 9(b).

3.2. Figure 1(a) Design Comparisons

Figure 10(a) and 10(b) show data from measurements of normal incidence absorption spectra made by Liu et al. [1] for the structure in Figure 1(a) over the frequency range from 600 Hz to 1600 Hz. These data are compared with predictions for type 2 partitions in each of the foams modelled with the parameters in Table 1 and with predictions for 31 mm thick foam layers without partitions.

The absorption predicted for a 31 mm thick layer of the ‘cavity’ foam is lower than the same thickness of melamine foam since it has a lower flow resistivity. But, with type 2 partitions, the ‘cavity’ foam is predicted to yield an absorption spectrum comparable to that measured for the Figure 1(a) structure between 700 Hz and 1200 Hz. The insertion of partitions in the ‘cavity’ foam is predicted to improve absorption spectra more than predicted if used in the melamine foam.

Figure 11 compares predictions of absorption coefficient spectra up to 10 kHz for the three types of partitions in the ‘cavity’ foam ($d_1=31$ mm, $d_2=15$ mm, $a=1$mm and $t=0.5$ mm), the same thickness of ‘cavity’ foam without partitions and the numerical prediction for the structure in Figure 1(a) [1].

A type 1 partition in the ‘cavity’ foam [1] is predicted to give higher absorption than predicted for the Figure 1(a) structure below 500 Hz but less at higher frequencies. However, Type 2 partitions in the ‘cavity’ foam are predicted to match the numerical prediction of absorption for the Figure 1(a) structure below 700 Hz, give less absorption than the numerical predictions between 1 and 2 kHz, but indicate higher absorption above 3 kHz. The predicted absorption coefficient spectrum for Type 3 partitioning is less than the numerical prediction for the Figure 1(a) structure between 500 Hz and 1.5 kHz but consistently higher thereafter. Moreover, the prediction for type 3 partitions is higher than predicted for the ‘cavity’ foam without partitions between 3 and 6 kHz. Overall, the predicted broadband absorption spectra with type 2 and type 3 partitions in the ‘cavity’ foam are higher than predicted for the design in Figure 1(a).

3.2. Figure 2(a) Design Comparisons

The Figure 2(a) design [2] represents an extension of that proposed by Yang et al. [6] who consider only the ‘right-hand’ arrays of the horizontal partitions shown in Figure 2(a). Figure 12 compares the measured normal incidence absorption coefficient spectrum for the Figure 2(a) design [2] with predictions for 50 mm thick ‘sponge’ and melamine foam and for the ‘sponge’ material containing the three types of partition (Figure 4).

The porous foam ‘sponge’ used in the Figure 2(a) design [2] has relatively low flow resistivity (see Table 1) and a 50 mm hard backed layer of this material offers a relatively low absorption spectrum (solid red line in Figure 12). Nevertheless, insertion of type 3 partitions in the ‘sponge’ is predicted to give a comparable absorption spectrum to that achieved with the multiple horizontal embedded plate Figure 2(a) design. Moreover, if 50 mm thick hard backed (unpartitioned) melamine foam, with the JCA parameter values listed in Table 1, were to be used, the corresponding predicted normal incidence absorption spectrum (broken brown line in Figure 12) would more than match that measured for the design in Figure 2(a) with ‘sponge’ infill.

3.2. Figure 3(a) Design Comparisons

Figure 13 compares the absorption spectrum predicted for the 60 mm thick Figure 3(a) design with that predicted for the same thickness ‘cavity’ foam used in the Figure 1(a) design with type 3 partitions ($d_2=25$ mm). The use of type 3 partitions in the ‘cavity’ foam layer is predicted to offer a comparable absorption spectrum to that predicted for the microperforated plate array.

4. Predictions for Naturally Sourced Materials

The low frequency absorption of a thin hard backed layer of a rigid porous material is improved by increasing the tortuosity of the material [7,8,9]. Naturally sourced porous materials tend to have higher tortuosity values than conventional synthetic materials [10] as well as having less environmental impact with respect to manufacture and disposal. Impedance tube measurements have been made on various naturally sourced samples and values of the JCA model parameters, other than flow resistivity and porosity which were measured independently, have been obtained by fitting the impedance tube data. Parameters for 30 mm and 50 mm thick samples made from sugarcane bagasse and Typha fiber (with bulk density 200 kg m^-3) are listed in Table 2 [10,11].

The JCA fitted tortuosity values for these materials, between 1.5 and 2.8, are higher than those for the porous foams used in the Figure (1a) design for which the highest fitted tortuosity value is 1.07 (see Table 1). Figure 14 compares normal incidence absorption coefficient data [10] for 30 mm thick layers made from sugar cane bagasse and Typha fiber with the numerical prediction for the 31 mm thick structure shown in Figure 1(a). Below 1.5 kHz the Typha sample offers superior absorption to the Figure 1(a) design. Above 1.5 kHz, the spectra for both Typha fiber and sugarcane bagasse are comparable.

Figure 15 compares normal incidence absorption coefficient data [11] for 50 mm thick layers made from Typha fiber with the numerical prediction for the 50 mm thick structure shown in Figure 2(a). The measured absorption coefficient values of the Typha fiber samples above 1 kHz are between 0.8 and 0.85, slightly lower than the measured absorption spectrum of the Figure 2(a) design. However the measured absorption coefficient values for the 7.5% binder Typha fiber samples are higher than those measured for the Figure 2(a) design below 1 kHz.

Figure 16 compares the normal incidence absorption coefficient spectrum predicted for the 60 mm thick Figure 3(a) design with absorption coefficient data for a 60 mm thick sample of Typha fiber [12].

The absorption coefficient data for the 60 mm thick Typha fiber sample above 1 kHz, with an average absorption coefficient value near 0.8, does not match the values predicted for the Figure 3(a) design. However, the absorption of the Typha fiber sample is higher between 500 Hz and 1 kHz.

5. Conclusion

Similar or higher broadband sound absorption spectra than measured and predicted for three relatively complex designs, based on resonant cavities, synthetic polymer foams, embedded arrays of horizontal plates in a porous layer and parallel microperforated plates, can be achieved with relatively simple partition arrangements in a porous layer or by using porous materials made from natural sources with higher tortuosity values than found for typical synthetic foams. Naturally sourced materials have higher bulk densities than typical synthetic foams, which would mean significant additional structural loading if 60 mm thick layers were to be used. However, it should be possible to reduce such additional weight by using thinner layers with airgaps behind them.