Sound absorption modeling of thin woven fabrics backed by an air cavity

Abstract

A theoretical model for the oblique incidence sound absorption coefficient of thin woven fabrics backed by an air cavity is presented where the fabric is acoustically described by its specific airflow resistance and its surface mass density. The theoretical model is illustrated by an equivalent electrical circuit and validated in the case of normal sound incidence by experimental results obtained from impedance tube measurements on three fabric types. The influence of the surface mass density on the absorption coefficient is discussed and recommendations for practical applications are derived. Further, a simple formula to predict the specific airflow resistance of woven fabrics based on geometrical parameters is deduced. The normal incidence absorption coefficient and geometrical parameters of a set of 24 fabrics with a large range of interyarn porosities and specific airflow resistances were measured and used to validate the proposed geometry-based model to predict the absorption coefficient. Measured and estimated absorption coefficients show excellent agreement, with mean value and standard deviation of the differences of 0.03 ± 0.10. The model is therefore suitable for the design of new fabrics with an intended absorption coefficient.

Keywords

equivalent electrical circuit sound absorption coefficient specific airflow resistance woven fabric

In room acoustics, sound absorbers play a central role. In practical applications often textiles are employed as sound absorbers, for example, textile interior parts of automobiles or carpets in rooms that are used to absorb sound energy.¹ In these applications the absorbing material is mounted directly onto an acoustical hard surface. The textile can then be regarded as a porous medium and its sound absorption characteristic highly depends on its thickness.²^–⁴

However, when using curtains an air gap between the textile and the hard wall exists. Under these conditions also very thin textiles can have comparatively high sound absorption coefficients.^5,6 The application of curtains as sound absorbers for room acoustical purposes has several substantial advantages: curtains are relatively cost effective, lightweight, flexible, easy to manage and they enable variable room acoustics.

Already in 1970 it was reported that the sound absorption of curtains depends on the mounting distance to the wall, the airflow resistance and the surface mass density of the fabric, as well as the draping.⁷ Today, common models to predict the absorption coefficient of curtains typically include the size of the air cavity and the specific airflow resistance of the fabric.^8,9 They assume that no sound-induced vibrations of the fabric occur, that is, the surface mass density of the fabric is high. However, several authors^10,11 proposed how to consider sound-induced vibrations in a mathematical model, but it was neither validated by measurements nor profoundly discussed. In 1990 it was phenomenologically shown that the intrinsic parameters of a textile, that is, its microstructure, has a substantial influence on the sound absorption coefficient.¹² In 1999 this finding was emphasized by the use of an established mathematical model for micro-perforated panel (MPP) absorbers,^13,14 which was applied to thin woven textiles.¹⁰ This model based on geometrical parameters is quite complex and was validated with only two fabric types.

In this article always 0% fullness is assumed, that is, flat curtains. In the next section of this article a theoretical model for the prediction of the sound absorption coefficient of thin woven fabrics is presented. The theory is elucidated by an equivalent electrical circuit that provides insight into the acoustical and mechanical behavior of the structure. The model includes the effect of sound-induced vibrations and yields simple formulas for the oblique incidence absorption coefficient. Furthermore, in order to predict the absorption coefficient based on geometrical parameters, an approximation formula of the MPP is deduced. In the following section measured and calculated normal incidence absorption coefficients of a set of thin woven fabrics are compared and discussed. The influence of the surface mass density on the absorption coefficient is examined and recommendations for practical applications are derived. The geometry-based model to predict the absorption coefficient is validated by measurements on 24 woven fabrics. In the final section the conclusion about the proposed model is drawn.

Theoretical model

For decades much research has been carried out in the sound absorption modeling of porous materials, such as foams or fibrous materials. Today, many different calculation models to describe the acoustical behavior of porous media exist.² These models typically incorporate numerous and often quite abstract material parameters, such as flow resistivity, porosity, tortuosity, pore shape factor, permeability, thermal and viscous characteristic lengths, elasticity, etc.

In the case of a fabric backed by an air cavity, the interesting part of sound absorption occurs inside the fabric. Therefore, in the proposed model, dissipative effects inside the air cavity are neglected and only dissipative effects of the fabric are considered. The fabric is assumed to be acoustically thin, which means the thickness is much smaller than the wavelength of sound. This assumption significantly simplifies the absorption modeling as no bulk reaction inside the fabric has to be modeled and dissipative effects can integrally be characterized. For this purpose the fabric is modeled by taking into account viscous and mechanical effects.

Description of the absorbent structure

The absorbent structure consists of a thin sheet of fabric that is mounted at a defined distance D to a reflecting surface forming an air cavity, as shown in Figure 1. A one-dimensional arrangement of infinitely extended layers and a sheet, which is thin compared to the wavelength of sound, are assumed. In practical applications the condition about the acoustically hard backing will be fulfilled in cases where flat curtains are mounted in front of a wall or a closed window. In the case of a closed window this holds at least in the middle and upper frequency range in which curtains can be expected to absorb sound energy.

Figure 1.

Illustration of the absorbent structure.

Mathematical model: characterization by a surface impedance

It is assumed that an acoustic plane wave travelling in air impinges upon the described arrangement at an angle θ to the normal vector of the absorber surface. Due to Snell’s law, the propagation angle of the incident wave inside the air cavity will be unaltered. The sound pressure of the incident plane wave inside the air cavity can be written as

(1)

where A is the complex pressure amplitude, j is the imaginary unit, k₀ denotes the wave number in air with k₀ = ω/c, c is the speed of sound in air, ω = 2πf is the angular frequency with frequency f, and t denotes time. The sound particle velocity component of the incident wave perpendicular to the surface is thus given by

(2)

with the characteristic impedance of air Z₀ = ρc where ρ is the density of air. At the hard surface the incident wave will be totally reflected. The reflected wave inside the cavity can be written as

(3)

with the reflection coefficient r_L = 1 and the particle velocity

(4)

The surface impedance of the air cavity at x = −D is defined as the pressure/velocity ratio at the input of the air cavity and can be calculated by the use of Equations (1)–(4) as

(5)

By use of Euler’s and trigonometric identities, Equation (5) can be simplified to obtain the well-known formula of the rigidly backed air cavity with extended reaction:¹⁵

(6)

Since the sheet is assumed to be acoustically thin it can fully be described by a pressure/velocity ratio given by impedance Z_s. This impedance has to be in series with the surface impedance of the air cavity given by Equation (6).^8,15 Consequently, the surface impedance of the absorbent structure is given by

(7)

Absorption coefficients

From the surface impedance Z_in, the absorption coefficient for oblique sound incidence α_θ can be computed by^8,10,16

(8)

where r denotes the complex reflection coefficient of the absorbent structure and z = Z_in/(Z₀/cosθ) is the normalized surface impedance. Typical absorption coefficient curves of thin fabrics for the case of normal sound incidence, that is, θ = 0, are shown in Figure 6.

For the calculation of the random incidence absorption coefficient several models exist. However, it must be said that until now no satisfying model with good predictions of the absorption coefficient measured in the reverberation chamber has been presented and validated. The simplest and most common model is Paris’ formula,⁸ which is based on several assumptions that are never fulfilled in practice. In 1980, Thomasson revised Paris’ theory and introduced the so-called edge effect caused by the finite size of the specimen in the reverberation chamber.¹⁷ Only recently an extension of Thomasson’s formula, which accounts for the unequal distribution of the angles of incidence of the sound waves, was proposed by Jeong.¹⁸

Equivalent electrical circuit

The proposed mathematical model can be well visualized and analyzed by an equivalent electrical circuit.^15,19 In this analogy, sound pressure equals voltage and sound velocity equals current. On the left-hand side of Figure 2, the incoming sound wave coming from air is represented by a voltage source and a source resistance. The central block (broken lines) illustrates the fabric layer that is drawn as a parallel connection of two impedances R_s and jωm, which will be expanded in the following section. Illustrated by two bold lines, the air cavity can be interpreted as a transmission line. The three electrical parameters of the transmission line are given by the characteristic impedance Z₀/cosθ, the wavenumber k₀·cosθ and the length D. The load impedance Z_L on the right-hand side represents the hard backing of the absorber and is equal to infinity, that is, an open load. Therefore the transmission line acts as an open circuited stub. Equation (6) implies that this open-ended transmission line can be replaced by an impedance Z_C, which is illustrated by a second electrical circuit at the bottom of Figure 2.

Figure 2.

Equivalent electrical circuit of the proposed model for a thin fabric in front of a hard wall.

Impedance of a thin fabric

Woven fabrics can usually be considered thin compared to the wavelength in the frequency range up to 8 kHz and can therefore acoustically be characterized by a discrete impedance. The model described in this section will be referred to as porous membrane model.

The most relevant parameter to describe the acoustical characteristics of a fabric is its specific airflow resistance R_s (Pa s/m) which is defined as²⁰

(9)

where Δp is the pressure difference across the test specimen and u the linear airflow velocity according to ISO 9053.²⁰ The specific airflow resistance describes the loss of acoustic energy inside the fabric, which is mainly due to viscous effects, that is, friction. R_s can now directly be used for Z_s in Equation (7), which is discussed by Kuttruff.⁸ However, another decisive effect, especially when considering lightweight fabrics, is the sound-induced vibration of the fabric. If the fabric can move freely for normal sound incidence, that is, θ = 0, the vibration effect can be described by Newton’s law:

(10)

with the surface mass density m (kg/m²) and u_M being the velocity of the fabric. In the frequency domain Equation (10) yields the mass reactance:¹⁵

(11)

which describes the membrane-like character of the fabric. For oblique sound incidence, the bending waves of the fabric are excited. As the coincidence frequency of thin, limp materials typically lies high above 10 kHz, Equation (11) also approximately holds for oblique sound incidence. In the equivalent electrical circuit in Figure 2, Z_M is represented by an inductor with inductance m. The current through Z_M thus corresponds to the velocity of the fabric. For the combination of the processes described by Equations (9) and (11), the velocity of the sound particles and the velocity of the fabric have to be summed up, which corresponds to a parallel connection of impedances R_s and Z_M leading to^11,15,21

(12)

The asymptotic behavior of Equation (12) is discussed by Moholkar and Warmoeskerken.¹¹ Alternatively, Equation (12) can be represented by a transfer function-like expression:

(13)

with the cutoff frequency

(14)

In electrical engineering the cutoff frequency is a well-established concept to characterize electronic systems, such as filters. From Equation (13) one can see that at frequencies f >> f_c, the effect of the mass can be neglected and Z_s ≈ R_s. At the cutoff frequency f_c, the apparent powers consumed by the impedances R_s and Z_M are equal. This means that the same amount of acoustical power is dissipated by the airflow through the fabric as is used to move the fabric. This effect has a significant influence on the absorption coefficient. Generally, at frequency f_c the normal incidence absorption coefficient of fabrics with R_s < Z₀ is reduced due to sound-induced vibrations. For instance, at quarter wavelength distance to the wall, that is, D = λ/4, this reduction of α₀ amounts to up to 0.28. Even one octave above f_c, at 2·f_c, the absorption coefficient can be reduced by a value up to 0.1 compared to the case of infinite surface mass density, that is, no vibrations. As opposed to this, for fabrics with R_s > Z₀ the influence of the surface mass density can even lead to increased absorption coefficients at frequency f_c. So, especially for lightweight fabrics, the influence of the surface mass density is relevant, since, for example, a fabric with a surface mass density of 0.1 kg/m² and a specific airflow resistance of 400 Pa s/m has a cutoff frequency of 640 Hz.

The formulas presented so far may be used for the design of lightweight, sound-absorbing fabrics. For a given surface mass density one may want to find the optimal specific airflow resistance in order to maximize the normal incidence absorption coefficient α₀ at a specific frequency. Let us consider the interesting case where the fabric is placed at quarter wavelength distance to the wall, that is, D = λ/4. At this frequency, the impedance of the air cavity Z_c vanishes and therefore Z_in = Z_s. Thus, at the given frequency, by inserting Equation (12) into Equation (8), an expression for the normal incidence absorption coefficient depending on the specific airflow resistance R_s, the surface mass density m and the distance to the wall D was obtained. This expression was subsequently differentiated with respect to R_s and set to zero. After some calculation the following formula for the optimal specific airflow resistance could be found:

(15)

From Equation (15) one can see that for large area densities, small distances to the wall and thus high frequencies, the optimal specific airflow resistance is equal to the characteristic impedance of air Z₀. When considering the equivalent electrical circuit in Figure 2, one may see that this case is known as impedance matching, where the power transfer is maximized. However, for smaller area densities and lower frequencies the optimal specific airflow resistance is decreasing according to Equation (15).

Geometry-based model for the specific airflow resistance

The airflow through the fabric consists of two contributions: the airflow between the yarns, the interyarn flow, and the airflow through porous yarns, the intrayarn flow. Depending on the yarn types and the weaving pattern, only one or both of them contribute to the total airflow. In the equivalent electrical circuit the two airflows can be modeled as a parallel connection of two resistive elements. However in this article, it is assumed that only interyarn flow occurs. It is assumed that interstices between the yarns, that is, open pores, exist which usually means that most of the airflow will occur inside these pores.

The pores between the yarns can be interpreted as short capillaries. The analytical solution for the airflow in a small tube having a circular cross section was found by Rayleigh and contains Bessel functions. This solution has recently been applied to model sound absorption of knitted textiles.²² In 1975 Maa^13,14 presented an approximation formula and included end correction terms that yielded the theoretical basis for the MPP absorber. The MPP consists of a panel with many short narrow tubes that are separated by distances much larger than their diameter. The impedance of the MPP with tubes of diameter d and length t is given by^13,14

(16)

(17)

(18)

(19)

where η is the dynamic viscosity of air and σ is the perforation ratio. The formula of the MPP has already been applied to predict sound absorption coefficients of woven metallic and glass-fiber structures.^10,23

It is now assumed that x < 1, that is, only lower frequencies and narrow tubes are considered. It is further assumed that the diameter d of the tubes is in the same order of magnitude as the tube length t. Equation (16) can then be approximated by

(20)

and becomes real-valued and therefore delivers a very simple geometry-based model for the specific airflow resistance of the fabric:

(21)

For σ the interyarn porosity, that is, the ratio of the open area to the total area, has to be inserted. However, for the length t and the diameter d of the tubes it is not so straightforward to estimate appropriate values. As the interyarn pores typically have a rectangular cross section, as is depicted in Figure 3, the diameter of the tubes was chosen according to Kang and Fuchs¹⁰ as

(22)

with a and b the average length and width of an open pore. The diameter d in Equation (22) corresponds to a circle of equal surface area of one pore.

Figure 3.

Macro photograph of a synthetic fiber fabric with rectangular interyarn pores (black areas).

Comparison of calculated and measured values

For a set of different woven fabrics consisting of synthetic fibers, the normal incidence absorption coefficient for two air cavity sizes was experimentally measured. In order to validate the proposed theoretical model, measured and calculated absorption coefficients have been compared. The validation procedure was performed in two steps. As a first step absorption coefficient curves have been calculated by the porous membrane model where the specific airflow resistance of each fabric was determined by curve fitting of measured and calculated values. In a second step the geometry-based model for the prediction of the absorption coefficient was examined. For that purpose geometrical parameters of 24 fabrics have been used to predict a representative normal incidence sound absorption coefficient.

Measurement of the normal incidence sound absorption coefficient

Normal incidence absorption coefficients of fabrics were measured according to the standard ISO 10534-2,²⁴ using a Brüel & Kjaer two-microphone impedance tube of type 4206, as illustrated in Figure 4. The tube had a diameter of 10 cm allowing measurements in the range from 100 Hz to 1.75 kHz with a frequency resolution of 5 Hz. To minimize the influence of mounting and to reduce the basic absorption of the empty tube, a special mounting was used which did not alter the diameter inside the tube, even including the mounting. However, it has to be noted that the mounting prevents the fabric from freely moving and therefore mechanical stiffness and resistance, which are not covered by the presented model in the Impedance of a thin fabric section, are introduced to the membrane. Figure 5 shows a specimen glued onto an aluminum ring. For each fabric type two measurements with different air cavity sizes D of 0.10 and 0.15 m were performed.

Figure 4.

Measurement setup for the normal incidence sound absorption coefficient of thin fabrics (two-microphone impedance tube).

Figure 5.

Photograph of a specimen glued onto an aluminum mounting for impedance tube measurement.

Determination of the specific airflow resistance by curve fitting

The standard ISO 9053²⁰ describes two measurement methods for the airflow resistance of porous materials, where either at static or at slowly alternating airflow the pressure drop across a specimen is measured. However, using these methods it is difficult to measure specific airflow resistances lower than 50 Pa s/m due to low signal-to-noise ratios.²⁵ Therefore, in the context of this study, an indirect method for the determination of the specific airflow resistance of the fabrics was adopted: For each fabric type the two measured situations, that is, D = 0.10 and 0.15 m, were reproduced by calculations with the formulas given in the Mathematical model: characterization by a surface impedance, Absorption coefficients and Impedance of a thin fabric sections, whereas the specific airflow resistance of each fabric was adjusted to best match the two measured absorption curves. A similar model-based indirect method to estimate the airflow resistance of thin screens was recently published²⁵ and showed no significant differences compared to the direct measurement.

The normal incidence absorption coefficient curves were calculated by inserting Equation (7) into Equation (8) and by use of the porous membrane model given by Equation (12). The calculations were performed at the center frequencies of 1/24 octave frequency bands. A temperature of 20°C (68°F) was assumed at which the characteristic impedance of air amounts to 413 Pa s/m. The surface mass density, m, of each fabric was measured by a precision scale. As only one parameter per fabric – the specific airflow resistance – had to be varied, the curve fitting procedure was performed by hand.

Comparison of absorption coefficient curves

Figure 6 compares measured and calculated absorption coefficient curves of three fabric types A, B and C (rows) with two air cavity sizes (columns). The calculated curves were obtained from a curve fitting procedure described in the previous section. Generally, measured and calculated curves coincide very well. This suggests that the porous membrane model is suitable to describe the sound absorption characteristic of thin, lightweight, woven fabrics in the frequency range of interest. Slight deviations only occur at lower frequencies and at the local minima of the absorption curves, around 1.7 and 1.2 kHz, respectively. The former can be explained by measurement uncertainties and the influence of the mechanical resistance, that is, internal damping, of the fabric. The latter are due to the residual absorption of the measurement system including dissipation effects inside the air cavity.²⁵

Figure 6.

Measurement (black curves) and calculation (gray curves) of normal incidence sound absorption coefficients of three fabric types A, B and C (rows) with two air cavity sizes D = 0.10 m (left) and D = 0.15 m (right). The cutoff frequency f_c of each fabric type is indicated with a triangle on the frequency axis. For fabric B additionally a simulation with the identical specific airflow resistance but infinite mass density is drawn (dashed lines) to illustrate the effect of sound induced vibrations.

Table 1 contains measured surface mass densities and obtained specific airflow resistances, as well as thereof calculated cutoff frequencies of the three fabric types A, B and C, having ascending specific airflow resistances of 55, 155 and 260 Pa s/m, respectively. The specific airflow resistances all lie below the characteristic impedance of air and lead to ascending absorption coefficients for the fabrics A–C. The local minima of the curves in Figure 6 are located at frequencies where the impedance of the air cavity reaches infinity, that is, at frequencies where D = nλ/4 with n an even number. The local maxima are located at frequencies where the imaginary part of the surface impedance Z_in vanishes, which in a first approximation is given by frequencies where D = nλ/4 with n an odd number. Fabric B reveals a relatively high cutoff frequency, above 500 Hz, which is nearly one octave higher than the cutoff frequencies of fabrics A and C. Therefore, for fabric B, the most significant influence of the mass on the absorption coefficient is expected. When looking at the absorption curves in Figure 6 one can see that for fabric B there exist remarkable differences between the absorption values at the local maxima, whereas for fabrics A and C the absorption values at the local maxima are approximately constant. For D = 0.15 m the absorption coefficient of fabric B at the first local maximum (at ∼ 570 Hz) is reduced by 0.2 compared to the second local maximum (at ∼ 1.7 kHz), corresponding to a reduction of 27%. As fabric B has a cutoff frequency of 525 Hz it is thus concluded that the surface mass density has a significant influence even for frequencies above f_c_. For practical applications it is recommended that the influence of the surface mass density on the absorption coefficient can only be neglected for frequencies f > 2 f_c, since in this frequency range at quarter wavelength distance to the wall the influence on the absorption coefficient stays below 10%.

Table 1.

Characteristic acoustical data of three fabric types A, B and C

	Fabric type
	A	B	C
Surface mass density m [kg/m²]	0.042	0.047	0.143
Specific airflow resistance R_s [Pa s/m]	55	155	260
Cutoff frequency f_c [Hz]	208	525	289
Optimal specific airflow resistance R_s_,_opt [Pa s/m] for D = λ/4 = 0.10 m	199	216	363
Optimal specific airflow resistance R_s_,_opt [Pa s/m] for D = λ/4 = 0.15 m	142	156	322

Fabrics A and B have similar surface mass densities and therefore similar values for the optimal specific airflow resistance R_s_,_opt, according to Equation (15). However, fabrics A and B considerably differ in their actual specific airflow resistance R_s, which for fabric B is approximately three times larger than for fabric A. For D = 0.15 m the specific airflow resistance of fabric B is nearly optimal regarding absorption at quarter wavelength distance, that is, R_s ≈ R_s_,_opt. This means that in this case there is no potential for increasing the absorption coefficient by modifying the specific airflow resistance of the fabric. As opposed to this, fabric A with its specific airflow resistance substantially below the optimal value has a considerable potential of increasing the absorption coefficient at quarter wavelength distance by increasing the specific airflow resistance.

Determination of geometrical parameters from macro photographs

The geometry-based model for the airflow resistance, presented in the Geometry-based model for the specific airflow resistance section, is based on knowledge about the microstructure of the fabric. The geometrical parameters of the fabrics were extracted from macro photographs, as shown in Figure 5. A total of 24 fabric types have been investigated. Table 2 contains the geometrical data interyarn porosity σ and average pore diameter d of the tested fabrics. The interyarn porosity ranges from 1.4% to 38% and the pore diameter from 60 to 140 µm. One has to note that the evaluated mean values exhibit quite a high uncertainty of approximately ± 20%, which is due to local variations inside the fabric and the measurement technique employed. As no knowledge about the thicknesses of the fabrics, or to be more precise, the lengths of the pores, was available, for all calculations the parameter t was set to 45 µm, which approximately corresponds to the diameter of one yarn.

Table 2.

Geometrical data obtained from macro photographs and specific airflow resistances (see the Determination of the specific airflow resistance by curve fitting section) of 24 fabric types

Fabric type	Interyarn porosity σ [%]	Pore diameter d [µm]	Specific airflow resistance R_s [Pa s/m]	Fabric type	Interyarn porosity σ [%]	Pore diameter d [µm]	Specific airflow resistance R_s [Pa s/m]
1	38	137	10	13	7.5	94	52
2	31	121	11	14	6.8	84	50
3	25	90	18	15	5.9	62	85
4	22	87	24	16	4.1	85	105
5	13	122	19	17	4	78	105
6	13	86	27	18	3.3	75	135
7	13	118	25	19	2.8	118	122
8	11	112	29	20	2.5	75	105
9	11	76	41	21	2.1	61	165
10	11	76	32	22	1.8	87	268
11	9.1	135	36	23	1.5	100	208
12	8.5	72	55	24	1.4	108	304

Validation of the geometry-based model to predict the absorption coefficient

In order to validate the proposed geometry-based model for the absorption coefficient of thin woven fabrics, a representative absorption coefficient α_R was defined. Thus, standard statistical methods could be applied to this representative single value.

As Figure 6 suggests, the largest variations across fabric types occur at the local maxima of the absorption curves. In fact these values are especially appropriate for a model validation in the sense of a worst-case analysis. Furthermore, the local maxima are independent of the cavity size. However, as shown in the Comparison of absorption coefficient curves section, the local maxima of the absorption curve are affected by the finite surface mass density at low frequencies. Therefore, in order to validate the geometry-based model to predict the absorption coefficient, the high-frequency approximation is used and thus the representative absorption coefficient is defined as

(23)

which is single-valued and only dependent on the specific airflow resistance. The formula for α_R can be found by the use of Equation (8) and setting z = R_s/Z₀, that is, membrane motion is neglected.

Subsequently, representative absorption coefficients α_R of 24 fabrics (see Table 2) were determined by two methods: (1) α_R was calculated based on indirectly measured specific airflow resistance (see the Determination of the specific airflow resistance by curve fitting section) and (2) α_R was predicted based on measured geometrical parameters and by applying Equation (21). Figure 7 shows the comparison of measured and predicted absorption coefficients α_R. The differences of true and estimated values exhibit a mean value and standard deviation of 0.03 ± 0.10. The mean value does not significantly differ from zero with a probability of 0.81 (one-sample two-tailed t-test). The gray lines in Figure 7 show the linear regression line and the 95% prediction interval band for future estimates of the absorption coefficient by the proposed model and measurement techniques employed.

Figure 7.

Comparison of measured and predicted absorption coefficients α_R of 24 fabrics (see Table 2). The straight black line represents the identity line. The gray lines show the linear regression line (R² = 0.89) and the 95% prediction interval band for future estimates.

For increasing absorption coefficients an increase in the deviations between measured and calculated values can be observed. A Monte Carlo simulation revealed that this effect can be explained by the uncertainties of the measured geometrical parameters, which for high absorption coefficients lead to larger variances of predicted values. In addition, for large absorption coefficients the intrayarn airflow, that is, the airflow through the yarns, which is not covered by the applied model, becomes more important. A third explanation may be found in the assumption about the shape of the pores. The model assumes pores with a circular cross section, which may be justified in the case of a square cross section. However, some of the highly absorbing fabrics had elongated pores with a length-to-width ratio a/b > 5. In these cases a model for the airflow in a slit could be more appropriate.²⁶

For low absorption coefficients the model seems to systematically underestimate the true values. One has to note that the three fabric types with the lowest absorption coefficients are the ones having the largest porosities of 25, 31 and 38%. For these fabric types the basic assumption of the MPP that the pores are separated by distances much larger than their diameter, that is, σ >> 25%, is violated. Nevertheless, the model delivers only small absolute errors in these cases.

Conclusions

A theoretical model for the oblique incidence sound absorption coefficient of woven fabrics was presented where the fabric is acoustically described by its specific airflow resistance and its surface mass density. The influence of the surface mass density on the absorption coefficient was discussed and recommendations for practical purposes were derived. The theoretical model was illustrated by an equivalent electrical circuit and validated in the case of normal sound incidence by curve fitting of high-resolution impedance tube measurement data of three fabric types.

Further, a set of simple formulas to predict the absorption coefficient of thin woven fabrics based on geometrical parameters was deduced. The normal incidence absorption coefficient of a set of 24 fabrics with a large range of interyarn porosities and specific airflow resistances was measured and used to validate the proposed formulas. The differences of measured and estimated values exhibit a mean value and standard deviation of 0.03 ± 0.10. Consequently, the formulas are suitable for the design of new fabrics with an intended absorption coefficient.

Conceptually the proposed model is extendable for a series of additional aspects, for example, the equivalent electrical circuit and its mathematical description can easily be extended to multiple layers of fabrics as they occur in panel curtain systems. Further research is being carried out into the prediction of the random incidence absorption coefficient. It should be noted that the presented modeling approach can also be used in order to predict sound absorption coefficients, as well as sound transmission coefficients of fabrics without an air cavity.²¹

Footnotes

Funding

This work was supported by the Swiss Innovation Promotion Agency (CTI) [grant number 10675.1 PFIW-IW].

Conflict of interest statement

None declared.

Acknowledgments

Special thanks are due to Kurt Heutschi for many stimulating conversations and Annette Douglas for the excellent collaboration.

References

Dias

Monaragala

. Sound absorbtion in knitted structures for interior noise reduction in automobiles. Meas Sci Technol 2006; 17: 2499–2505.

Allard

Atalla

. Propagation of sound in porous media: Modelling sound absorbing materials. Wiley, 2009.

Shoshani

Yakubov

. A model for calculating the noise absorption capacity of nonwoven fiber webs. Text Res J 1999; 69: 519–526.

Liu

. Sound absorption behavior of knitted spacer fabrics. Text Res J 2010; 80: 1949–1957.

Meier

Müller

. Schallreflektierende und schallabsorbierende Stoffe für raumakustische Anwendungen. In: Proceedings of DAGA 2010, Berlin, 2010.

Lancaster

Casali

Cho

. Sound absorption coefficients of micro-fiber fabrics by reverberation room method. Text Res J 2007; 77: 330–335.

Peutz

VMA

. Sound absorption of curtains. In: Proceedings of the 79th Meeting of the Acoustical Society of America, Atlantic City, 1970.

Kuttruff

. Room acoustics. Elsevier Applied Science, 1991.

Marshall Day Acoustics. Zorba - sound absorption prediction software, Version 2.8. Auckland, 2007.

10.

Kang

Fuchs

. Predicting the absorption of open weave textiles and micro-perforated membranes backed by an air space. J Sound Vib 1999; 220: 905–920.

11.

Moholkar

Warmoeskerken

. Acoustical characteristics of textile materials. Text Res J 2003; 73: 827–837.

12.

Shoshani

Rosenhouse

. Noise absorption by woven fabrics. Appl Acoust 1990; 30: 321–333.

13.

Maa

. Microperforated-panel wideband absorbers. Noise Control Eng J 1987; 29: 77–84.

14.

Maa

. Potential of microperforated panel absorber. J Acoust Soc Am 1998; 104: 2861–2866.

15.

Mechel

. Schallabsorber - band III: Anwendungen. Stuttgart: S. Hirzel, 1998.

16.

Mechel

. Schallabsorber - band I: äussere schallfelder, wechselwirkungen. Stuttgart: S. Hirzel, 1989.

17.

Thomasson

S-I

. On the absorption coefficient. Acustica 1980; 44: 265–273.

18.

Jeong

C-H

. Non-uniform sound intensity distributions when measuring absorption coefficients in reverberation chambers using a phased beam tracing. J Acoust Soc Am 2010; 127: 3560–3568.

19.

Beranek

. Acoustics. New York: American Institute of Physics, 1986.

20.

International Organization of Standardization. ISO 9053: Acoustics - Materials for acoustical applications - Determination of airflow resistance. 1991.

21.

Sakagami

Kiyama

Morimoto

Takahashi

. Detailed analysis of the acoustic properties of a permeable membrane. Appl Acoust 1998; 54: 93–111.

22.

Honarvar

Jeddi

AAA

Tehran

. Noise absorption modeling of rib knitted fabrics. Text Res J 2010; 80: 1392–1404.

23.

Fuchs

Zha

. Einsatz mikro-perforierter Platten als Schallabsorber mit inhärenter Dämpfung. Acustica 1995; 81: 107–116.

24.

International Organization of Standardization. ISO 10534-2: Acoustics - Determination of sound absorption coefficient and impedance in impedance tubes, Part 2: Transfer-function method. 1998.

25.

Jaouen

Becot

. Acoustical characterization of perforated facings. J Acoust Soc Am 2011; 129: 1400–1406.

26.

Randeberg

. Adjustable slitted panel absorber. Acta Acustica Acustica 2002; 88: 507–512.

Fabric type	Interyarn porosity σ [%]	Pore diameter d [µm]	Specific airflow resistance R_s [Pa s/m]	Fabric type	Interyarn porosity σ [%]	Pore diameter d [µm]	Specific airflow resistance R_s [Pa s/m]
1	38	137	10	13	7.5	94	52
2	31	121	11	14	6.8	84	50
3	25	90	18	15	5.9	62	85
4	22	87	24	16	4.1	85	105
5	13	122	19	17	4	78	105
6	13	86	27	18	3.3	75	135
7	13	118	25	19	2.8	118	122
8	11	112	29	20	2.5	75	105
9	11	76	41	21	2.1	61	165
10	11	76	32	22	1.8	87	268
11	9.1	135	36	23	1.5	100	208
12	8.5	72	55	24	1.4	108	304

Fabric type	Interyarn porosity σ [%]	Pore diameter d [µm]	Specific airflow resistance R_s [Pa s/m]	Fabric type	Interyarn porosity σ [%]	Pore diameter d [µm]	Specific airflow resistance R_s [Pa s/m]
1	38	137	10	13	7.5	94	52
2	31	121	11	14	6.8	84	50
3	25	90	18	15	5.9	62	85
4	22	87	24	16	4.1	85	105
5	13	122	19	17	4	78	105
6	13	86	27	18	3.3	75	135
7	13	118	25	19	2.8	118	122
8	11	112	29	20	2.5	75	105
9	11	76	41	21	2.1	61	165
10	11	76	32	22	1.8	87	268
11	9.1	135	36	23	1.5	100	208
12	8.5	72	55	24	1.4	108	304

Fabric type	Interyarn porosity σ [%]	Pore diameter d [µm]	Specific airflow resistance R_s [Pa s/m]	Fabric type	Interyarn porosity σ [%]	Pore diameter d [µm]	Specific airflow resistance R_s [Pa s/m]
1	38	137	10	13	7.5	94	52
2	31	121	11	14	6.8	84	50
3	25	90	18	15	5.9	62	85
4	22	87	24	16	4.1	85	105
5	13	122	19	17	4	78	105
6	13	86	27	18	3.3	75	135
7	13	118	25	19	2.8	118	122
8	11	112	29	20	2.5	75	105
9	11	76	41	21	2.1	61	165
10	11	76	32	22	1.8	87	268
11	9.1	135	36	23	1.5	100	208
12	8.5	72	55	24	1.4	108	304