Indirect Acoustic Characterisation of Membranes for the Control of Sound Absorption ^†

Abstract

Membranes can be used in different ways to achieve improvements in sound absorption. They can be used on top of base materials (sound-absorbing materials or other kinds) in such a way that they barely increase the thickness while increasing the sound absorption. They can be positioned inside a sandwich structure for the same purpose. They can also be used independently. There are thin membranes with different characteristics (perforated or otherwise, made up of different thin layers, fabrics made of different yarns, etc.) that are applied to surfaces with a plenum chamber. In all of these foregoing cases it can be difficult to characterise the membrane. Membranes with a smaller thickness lead to difficulties in the positioning of measuring devices thereby increasing the test error. The aim of this work is to obtain airflow resistance and the characteristic acoustic impedance of membranes in a more stable manner. Measurements of the airflow resistance and the characteristic acoustic impedance of sandwich structures with inserted membranes are considered for this purpose. The base material comprising the sandwich was characterised prior to these measurements. These measurements indirectly produce the airflow resistance and the characteristic acoustic impedance of the membrane. The results obtained show more stable measurements of the airflow resistance and the obtaining of the characteristic acoustic impedance of the membrane under the conditions in which it is set.

1. Introduction

The use of membranes in sound-conditioning applications is quite common in order to control the frequency range where more absorption is needed. In some cases, these membranes are attached to a well-known sound-absorbing material, and it is interesting to find out the contribution of the membrane to the sound-absorbing effect. In other cases, it may be a good idea to add them individually. In general, adding the membrane to a base absorbing material aims to increase the overall sound absorption without significantly increasing its thickness.

Adding a thin layer of material on top of a base substrate is not a new solution [1]. Work proposing the use of textile layers on top of absorbing materials can also be found [2]. There are commercial solutions that add a textile veil over an acoustic absorbing material in solutions with perforated panels or sheets. This is true for both acoustic conditioning solutions and outdoor noise barriers. However, the development of new materials and the application of new technologies means that they are still relevant today. Work with nanofibres that improve the sound absorption at low frequencies with a slight increase in the thickness can be found [3]. Nanofibre veils at roughly one millimetre thick result in considerable increases in the sound absorption coefficient over the entire range of interest. Another example of nanofibres with a spinning technique and similar conclusions can be found in [4]. This work also studies the insertion of nanofibres into sandwiches. The sound absorption coefficient is not always increased in sandwiches. This indicates that further information on the nanofibre membrane is required. In [5], the authors used a non-woven fabric coated with core–shell and hollow nanofibre membranes. Also of interest is the review of fibre-based sound-absorbing structures [6]. Other solutions with woven fabric on polyurethane foams can be found [7].

Membranes are being added to base elements using different materials that are not strictly porous. In [8], the authors used the integration of a honeycomb core with a membrane made from recycled materials, forming a recycled membrane honeycomb composite. Very recent work with electrospun polyacrylonitrile nanofibre membranes can also be found [9]. There are also solutions in which the thin layer is made up of perforated panels with tapered hole geometries coupled with polyurethane foam [10,11].

Other applications are only thin layers. In [12], the authors used thin band yarn woven fabric with kapok fibre. In [13], the authors used the sound absorption performance of a wedge-like knitted composite. This work studies the airflow resistivity and the absorption coefficient, among other variables of interest. In [14], the authors provide a study of a textile curtain as a membrane and its applications as an acoustic absorbing material are explored. Prior works on the use of curtains for acoustic conditioning can also be found [15]. In [16], different textiles with different yarn structures and their effect on sound absorption are studied. In [17], the airflow resistance, absorption coefficient and impedance of fabrics with tiny holes were studied. It is interesting to understand the acoustic behaviour of a membrane in all the cases presented.

An important parameter for evaluating the acoustic behaviour of the membrane is the airflow resistance (σ). The airflow resistance of a porous material is defined as the ratio between the pressure difference across a sample and the velocity flow of air through that sample, which represents the resistance encountered by the air as it passes through the materials [18,19]. Its value determines some acoustic properties of porous materials, such as sound absorption and sound transmission coefficients, since it is directly related to the characteristic impedance and propagation constant of the materials. Current approaches use standard-based direct methods [20,21,22] and/or accepted acoustic-based indirect methods [23,24,25] to determine it. It can also be determined from measurements of the surface impedance in impedance tubes. However, the results are not always satisfactory since each method has its limitations, especially when measuring thin and flexible materials, such as thin mats, fabrics, papers, and screens. A good review of the measurements, calculations and applications of the airflow resistance can be found in [26]. Currently, the parameter of airflow resistance or its equivalent airflow resistivity (airflow resistance per unit length) remains of great interest for characterising new acoustic absorbing materials based on the circular economy [27,28,29,30].

This research studies the airflow resistance of textiles, both woven and non-woven, with a wide range of airflow resistance values. In order to obtain these values, Ingard and Dear’s method [23] was used, which is briefly described in Section 2.1. In their study, Ingard and Dear suggest locking flexible materials between stiff transparent screens in order to avoid oscillatory motion of the material, which is directly related to the flow impedance. In [31], there is a discussion of the method and its limitations. In Section 2.1, it is proposed to perform the measurement by assembling a sandwich structure with previously known acoustic absorbing materials. A preliminary study of different acoustic absorbing materials with low error in the measurement of flow resistance, as presented in Section 3.1, is conducted. Once those with the lowest error are selected, various measurements are taken with a sandwich structure. From this sandwich structure, the flow resistance value of membranes, which can be seen in Section 3.1, is obtained. This section improves on the technique presented in [32], using a sandwich structure to reduce the test error.

Sound-absorbing materials with membranes can be modelled by using the transfer matrix method [18,33,34]. The theory behind this method is described in Section 2.2 of this document. Using the sandwich structure described in Section 2.1, the characterisation of the specific acoustic impedance of the membrane was performed.

Different combinations of sound-absorbing materials and membranes were studied. Membranes with varying airflow resistance, from lower to higher, were used. The specific acoustic impedance value of the membrane was obtained by measuring the surface impedance and determining the propagation constant (Γ) and the characteristic impedance (Z) of the base material of the sandwich [35]. The results are presented in Section 3.2.

Section 4 discusses the indirect procedure for measuring the airflow resistance (σ) of the membrane as well as the indirect procedure for measuring its characteristic impedance (Z_m).

2. Theory

This section summarises the two methods used. Section 2.1. summarises the measurement of membrane airflow resistivity indirectly. Section 2.2. summarises the measurement of the acoustic impedance of the membrane, also indirectly.

2.1. Airflow Resistance Measurement Method

The airflow resistance measurement method to be used is the Ingard and Dear method [23]. In this method, airflow resistance is obtained by placing the test sample near the centre of a closed cylindrical tube with one rigid end, a speaker at the other end and a pair of calibrated microphones, as shown in Figure 1.

One of the microphones is placed just at the front of the material sample (P₁) and the second microphone is placed at the front of the rigid termination of the tube (P₃), to measure the sound pressure level at these points.

The loudspeaker produces a white noise, and frequencies that generate an odd number of quarter-wavelengths along the d + L₂ distance, from the rigid termination to the end of the material sample, are selected. This phenomenon occurs at frequencies $f_n=\left(2n-1\right)c_o/4L_2$, where $n$ is a natural number and $c_0$ is the speed of sound inside the tube. The condition $\mathsf{\lambda }\gg 1.7\mathrm{D}$ must be met, where $\mathsf{\lambda }$ is the sound wavelength and $D$ is the tube diameter.

Considering that losses inside the tube are negligible and that the flow reactance is small at low frequencies, flow resistance of sample, σ, is obtained from sound pressure measurements P₁ and P₃, by using Equation (1):

$$\mathsf{\sigma }=\left|Im{\displaystyle \frac{P_1}{P_3}}\right|$$

(1)

In this case, the sample of thickness d will be composed of a sandwich of the same sound-absorbing material of thickness d₁ and airflow resistance σ₁ with a membrane of thickness d₂ and airflow resistance σ₂ in the centre, where d = 2d₁ + d₂ and σ = 2 σ₁ + σ₂. The value of the membrane’s airflow resistance can be obtained from this last expression. σ₂, by performing a subtraction.

2.2. Specific Acoustic Impedance Measurement Method

The same sandwich structure as described in Section 2.1 is proposed. It can be modelled by using the transfer-matrix method. In [36] a prior work using a membrane on a base absorbing material is presented. The membrane is assumed to have a specific acoustic impedance Z_m. Following the work of [33] it is assumed that the absorbing material can be modelled with the following transfer matrix:

$$T=\left(\begin{array}{cc}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h} \left(\mathsf{\Gamma }d\right)& Z\mathrm{s}\mathrm{e}\mathrm{n}\mathrm{h}\left(\mathsf{\Gamma }d\right)\\ {\displaystyle \frac{\mathrm{s}\mathrm{e}\mathrm{n}\mathrm{h}\left(\mathsf{\Gamma }d\right)}{Z}}& \mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h} \left(\mathsf{\Gamma }d\right)\end{array}\right)$$

(2)

where Γ and Z are the propagation constant and the characteristic impedance of the absorbing material, d is its thickness. For the membrane, assuming concentrated behaviour given its lesser thickness, its transfer matrix:

$$T=\left(\begin{array}{cc}1& Z_m\\ 0& 1\end{array}\right)$$

(3)

The proposed system is as follows:

$$\left(\begin{array}{c}{p}_0\\ U_0\end{array}\right)=\left(\begin{array}{cc}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h} \left(\mathsf{\Gamma }d\right)& Z\mathrm{s}\mathrm{e}\mathrm{n}\mathrm{h}\left(\mathsf{\Gamma }d\right)\\ {\displaystyle \frac{\mathrm{s}\mathrm{e}\mathrm{n}\mathrm{h}\left(\mathsf{\Gamma }d\right)}{Z}}& \mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h} \left(\mathsf{\Gamma }d\right)\end{array}\right)\left(\begin{array}{cc}1& Z_m\\ 0& 1\end{array}\right)\left(\begin{array}{cc}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h} \left(\mathsf{\Gamma }d\right)& Z\mathrm{s}\mathrm{e}\mathrm{n}\mathrm{h}\left(\mathsf{\Gamma }d\right)\\ {\displaystyle \frac{\mathrm{s}\mathrm{e}\mathrm{n}\mathrm{h}\left(\mathsf{\Gamma }d\right)}{Z}}& \mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h} \left(\mathsf{\Gamma }d\right)\end{array}\right)\left(\begin{array}{c}{p}_3\\ 0\end{array}\right)$$

(4)

where p₀ and U₀ are the pressure and volumetric velocity at point 0 (where the sound hits the sandwich structure directly), and p₃ is the pressure at point 3 (rigid closure), where U₃ = 0. This system models the behaviour of the Kundt tube test [35]. The system can be developed in such a way that:

$$\left(\begin{array}{c}{p}_0\\ U_0\end{array}\right)=p_3\left(\begin{array}{c}\cosh \left(\mathsf{\Gamma }d\right)\cosh \left(\mathsf{\Gamma }d\right)+Z_m{\displaystyle \frac{\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h} \left(\mathsf{\Gamma }d\right)\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}\left(\mathsf{\Gamma }d\right)}{Z}}+\sinh \left(\mathsf{\Gamma }d\right)\sinh \left(\mathsf{\Gamma }d\right)\\ 2\cosh \left(\mathsf{\Gamma }d\right){\displaystyle \frac{\mathrm{s}\mathrm{e}\mathrm{n}\mathrm{h}\left(\mathsf{\Gamma }d\right)}{Z}}+Z_m{\displaystyle \frac{\mathrm{s}\mathrm{e}\mathrm{n}\mathrm{h}\left(\mathsf{\Gamma }d\right)}{Z}}{\displaystyle \frac{\mathrm{s}\mathrm{e}\mathrm{n}\mathrm{h}\left(\mathsf{\Gamma }d\right)}{Z}}\end{array}\right)$$

(5)

The value of the surface impedance is:

$$Z_s={\displaystyle \frac{p_0}{U_0}}={\displaystyle \frac{Z(2Z\cosh \left(2\mathsf{\Gamma }d\right)+Z_m\mathrm{s}\mathrm{e}\mathrm{n}\mathrm{h}\left(2\mathsf{\Gamma }d\right))}{Z_m\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}\left(2\mathsf{\Gamma }d\right)-Z_m+2Z\mathrm{senh}\left(2\mathsf{\Gamma }d\right)}}$$

(6)

If the specific acoustic impedance of the material is subtracted from the above ratio, the following is obtained:

$$Z_m=-{\displaystyle \frac{2\cosh \left(2\mathsf{\Gamma }d\right)Z^2-{2ZZ}_s\mathrm{s}\mathrm{e}\mathrm{n}\mathrm{h}\left(2\mathsf{\Gamma }d\right)}{Z_s\left(1-\cosh \left(2\mathsf{\Gamma }d\right)\right)+Z\mathrm{s}\mathrm{e}\mathrm{n}\mathrm{h}\left(2\mathsf{\Gamma }d\right)}}$$

(7)

The value of Γ and Z can be obtained with the two-thickness method, in accordance with ISO 10534-2:2023 [35] in Annex F thereof. The Z value can be determined with the same standard by inserting the membrane between the two samples. Applying Equation (7), Zm is estimated.

3. Results

This section describes the materials used and the results obtained by applying the methods described in Section 2.

3.1. Airflow Resistivity Measurements

To start the procedure, airflow resistance measurements of different materials are carried out. The airflow resistance (σ) is measured in a device built in the laboratory. This device is described in [31]. The apparatus consists of a cylindrical, polymethylmethacrylate (PMMA) tube 40 mm in diameter, 5 mm wall thickness and 169 cm in length. One end of the tube is equipped with a Beyma CP800TI high-frequency compression driver with a 49 mm throat diameter, which permits emission without considerable distortion at 100 Hz. The other end is closed with a rigid, highly sound-reflective termination. The distance between the first microphone and the rigid termination was 84.5 cm. This value was chosen to be one-quarter wavelength at approximately 100 Hz. The two microphones used are 1/2 inch and mounted flush into the tube wall.

A range of different materials is selected and the error in their measurement is studied. Those materials with the lowest direct measurement errors are then chosen as base materials.

Table 1. Materials initially analysed and flow resistance results.

Material		Density (kg/m3)	Thickness (mm)	Airflow Resistance (Ns/m3)	Relative Error (%)
Textile recycling 1 cm		55.4	11.30	147.8	6.0
Recycled foam 2 cm		106.0	20.17	423.4	14.3
Polyester fiberfill 3 cm		19.3	29.23	53.2	8.3
Polyurethane foam 4 cm		9.4	39.07	717.7	2.7
Polyurethane foam 0.5 cm		0.1	4.93	71.6	4.0
Polyurethane foam 1.2 cm		6.7	11.60	145.3	2.2
Polyurethane foam 2.5 cm		26.4	25.03	95.3	0.7
M1: Non-woven textile		53.1	2.38	53.3	46.1
M2: Woven textile		373.2	1.10	357.4	20.2
M3: Woven textile		757.9	0.25	245.2	17.1
M4: Plastic		649.8	0.43	1139.2	51.0

shows the materials analysed and the results. Some membranes (M1–M4) were also included.

From Table 1, the materials with the lowest error were selected as the base materials. Polyurethane foam 4 cm, Polyurethane foam 1 cm and Polyurethane foam 2.5 cm were chosen, with associated measurement errors of 2.7%, 2.2% and 0.7%, respectively. These materials were tested for airflow resistivity with the membranes M1 to M4 inserted.

Table 2. Airflow resistivity results of sandwich structures.

Material	Thickness (mm)	Total Airflow Resistance (Ns/m3)	Relative Error (%)	Airflow Resistance Membrane (Ns/m3)
Polyurethane foam 4 cm (PF4)	39.1	717.7	2.7
Polyurethane foam 2.5 cm (PF2.5)	25.0	95.3	0.7
Polyurethane foam 1.2 cm (PF1.2)	11.6	145.3	2.2
PF4 × 2	78.2	1438.9	1.7
PF4 + M1 + PF4	78.2	1463.1	0.1	24.2
PF4 + M2 + PF4	77.5	1659.5	3.8	220.6
PF4 + M3 + PF4	79.4	1938.6	2.2	499.7
PF4 + M4 + PF4	78.2	4692.0	10.8	3253.1
PF2.5 × 2	49.7	185.8	0.9
PF2.5 + M1 + PF2.5	52.8	209.9	1.1	24.0
PF2.5 + M2 + PF2.5	52.4	440.5	2.4	254.7
PF2.5 + M3 + PF2.5	51.3	612.8	12.3	427.0
PF2.5 + M4 + PF2.5	51.3	3212.5	10.2	3026.7
PF1.2 × 2	23.6	281.6	1.4
PF1.2 + M1 + PF1.2	25.4	306.2	0.6	24.6
PF1.2 + M2 + PF1.2	24.7	522.4	7.9	240.8
PF1.2 + M3 + PF1.2	23.9	695.8	8.7	414.2
PF1.2 + M4 + PF1.2	24.6	3630.3	17.4	3348.7

shows the results for each absorbent material individually. The measurement is then displayed with two layers. Finally, the sandwich measurement is shown. The last column shows the result obtained for the airflow resistivity of each membrane in its corresponding sandwich.

It is observed that the flow resistance of two samples is consistent with the flow resistance of one sample (which should be half). For example, the value of Polyurethane foam 4 cm × 2 is 1438.9 Ns/m³. Half is 719.5 Ns/m^3, which corresponds to Polyurethane foam 4 cm (717.7 Ns/m³ which is within the measurement error). The same applies to the other two samples. Furthermore, in all three sandwich combinations, practically the same flow resistance results are obtained for M1 to M4. For samples with higher airflow resistivity, the error increases, but it is always lower than the individual measurement shown in Table 1.

3.2. Measurements in Accordance with ISO 10534-2:2023

In this case, measurements are carried out according to the two-thickness method of the ISO 10534-2:2023 standard [35]. The surface impedance is measured with a sample of thickness d, Z₁, and the surface impedance with a sample of thickness 2d, Z₂. By applying Annex F of the standard, the propagation constant value Γ and the characteristic impedance Z of the 1.2 cm, 2.5 cm and 4 cm Polyurethane foam base samples from the previous section are obtained. Once the material was characterised, the surface impedance of the sandwich with the membrane inside, Zs, is measured and by applying (1), the value of its specific acoustic impedance Zm is obtained.

Figure 2 compares results for the sound absorption coefficient of Polyurethane foam 2.5 cm alone, as a double layer and combined with M3. This example illustrates how the insertion of a 0.25 mm thick membrane is capable of producing significant increases in the sound absorption coefficient, with almost no change in thickness. Using 5 cm instead of 2.5 cm of Polyurethane foam leads to a large increase. The increased sound absorption achieved by inserting M3 would require much greater thickness if it were only Polyurethane foam.

It was found that the two-thickness method did not provide stable values for the 1.2 cm Polyurethane foam sample. It was also observed that the 2.5 cm measurement produced clear clipping at low frequencies around 350 Hz. The 4 cm measurement was stable above 200 Hz. This material was used as a reference.

Figure 3 shows the measurement of the surface impedances with d, Z₁ thickness and with 2d, Z₂ thickness, in accordance with the ISO 10534-2:2023 standard for 4 cm Polyurethane foam [35]. The real part is shown as a solid line and the imaginary part as a dashed line.

Figure 4 shows the result obtained from the propagation constant Γ and the Z characteristic impedance of the 4 cm Polyurethane foam based on Annex F of the ISO 10534-2:2023 standard [35].

Figure 5 below shows the results of the specific acoustic impedance Zm of the four membranes (M1 to M4) by measuring the surface impedance and applying Equation (2). It can be observed how the real part of the characteristic impedance increases in value from M1 to M4. This is consistent with the increase in the flow resistance that can be seen in Table 2. As for the imaginary part, there is no single behaviour. It is possible for it to be mainly associated with the type of material. From the graphs, it can be seen that the low frequency trend of the real part of the impedance tends towards the values obtained from airflow resistivity shown in Table 2: 24.2 Ns/m³ for M1, 220.6 Ns/m³ for M2, 499.7 Ns/m³ and 3253.1 Ns/m³ for M4. In this regard, the values are consistent.

4. Discussion

Point 2.1 outlines the proposed method to indirectly measure the airflow resistance. The results are shown in point 3.1. Table 1 shows the individual membrane tests. These tests have a high error rate. Moreover, there is no guarantee that the result obtained is correct. This is because it is very difficult to position the membranes. The membranes move and bend and when the loudspeaker is operating at a high volume, they become displaced. Furthermore, if the membrane is not very sound absorbing, Ingard and Dear’s condition that the material should be sound absorbing is not met and expression (1) has limitations [23]. As can be observed in Table 2, the airflow resistance results indirectly obtained with three different materials are very similar. The error associated with the test is much lower and, in addition, it is ensured that the material is sound absorbing in the test. The limitations remain those that were analysed in [31]. In the event that the Ingard and Dear method cannot be used, other methods could be tried [26].

The procedure for measuring the characteristic impedance is outlined in point 2.2. and measurements are made, as shown in point 2.3. The method depends on the measurement of the characteristic impedance and the propagation constant of the base material. Therefore, the initial limitation is the measurement of the foregoing parameters. The two-thickness method of the ISO 10524-2:2023 standard was used. This is because two thicknesses were used in the case of the airflow resistance measurement. In this case, there have been problems with thin-base samples. Once the problem of the base sample has been solved, the method can indirectly obtain the characteristic impedance of the membrane. However, this procedure can be improved with another type of method for measuring the characteristic impedance and the propagation constant of the base element. The two-thickness method is the primary limitation. An alternative method may be the modification of the two-thickness technique proposed in [37].

5. Conclusions

This paper presents two indirect procedures for characterising membranes. First, the airflow resistivity is obtained indirectly by using sandwich structures. It can be seen that the proposed method is highly stable once the appropriate base materials are chosen. Membranes of different thicknesses were used. The same values of membrane airflow resistivity were obtained for all measured sandwiches, taking into account the measurement error range. This is true even when the airflow resistivity increases. It was also observed that measuring more than one layer of the same material appears to be more stable than just taking a single measurement. Therefore, this method is highly suitable for measuring the airflow resistivity of membranes in view of the results. The limitations of the method are those of the Ingard and Dear method. A future line of research involves indirect characterisation using other airflow resistance measurement techniques, such as the method developed by Dragonetti et al. [24].

Secondly, an indirect method is proposed to obtain the specific acoustic impedance of membranes. This method requires the measurement of the propagation constant and the characteristic impedance of the base material. In this regard, it was observed that, although a priori the most logical method to obtain these data is to use the ISO 10524-2:2023 standard in its Annex F for two thicknesses, the results are only stable from certain thicknesses onward. The 4 cm sample did provide stable data. Once the base material had been characterised, the specific acoustic impedance values of the membranes were obtained, where consistency was observed in the behaviour of the real part of the membranes.

This second method can still be improved upon. The proposed improvement is the use of the two-cavity method or other methods to obtain more stable propagation constant and characteristic impedance data for the base material, which is what really needs to be improved. This will form the basis for future research. One measurement proposal is the use of the modification of the two-thickness technique proposed in Kim and Park [37].