Sound absorption of textile curtains – theoretical models and validations by experiments and simulations

Abstract

Textile curtains can be designed to be good sound absorbers. Their acoustical performance, as usually described by the sound absorption coefficient, not only depends on the textile itself but also on the drapery fullness and the backing condition, that is, the spacing between the fabric and a rigid backing wall, or the absence of a backing in the case of a freely hanging curtain. This article reviews existing models to predict the diffuse-field sound absorption coefficient, which to date can only predict the case of flat curtains. A set of existing models is extended to the case of curtains with drapery fullness using a semi-empirical approach. The models consider different backing conditions, including freely hanging curtains. The existing and new models are validated by comparing predicted sound absorption coefficients with data measured in a reverberation room. Hereby, curtains consisting of different fabrics and with different degrees of fullness are considered. Besides situations with rigid backing, also the measurement data of textiles hung freely in space are included in this study. Comparisons reveal a very good agreement between measured and predicted sound absorption coefficients. Compared to currently available commercial sound absorption prediction software that can only handle the situation of flat textiles with rigid backing, the results of the presented models not only show a better agreement with measured data, but also cover a broader range of situations. The presented models are thus well applicable in the design and development of new textiles as well as in the room acoustical planning process.

Keywords

sound absorption curtains folding measurements modeling

Noise interrupts communication, reduces productivity and may lead to fatigue or even adverse health effects. Therefore, it is necessary to optimize the acoustical quality of rooms taking into account their usage. Materials such as glass and concrete commonly used in interior design are acoustically “hard”. These materials reflect sound energy very well, creating a very reverberant sound field in the room. Improvements of the acoustical environment, for example, in rooms where people work, communicate or relax, may often be achieved by introducing sound absorbing materials. These, by decreasing reverberation, increase speech intelligibility and make rooms quieter. Textile curtains can be designed to be good sound absorbers. For a long time, however, only heavy curtains of material such as velvet have provided sufficient absorption to fulfill acoustical requirements. Recently, new lightweight and even translucent textile curtains that absorb sound very well have been developed,¹ which has put them in the focus of interest. Further, the use of textile curtains hung freely in space as sound absorbers has recently been investigated.² In this paper, calculation models to predict the acoustic performance of such lightweight curtains in rooms under various conditions are presented and validated by comparison with measured data.

To quantify the acoustical performance of materials or the resulting quality of rooms, various measures are available.^3,4 In statistical room acoustics,³ the most important parameter is reverberation time. Reverberation time RT may be predicted by the well-known formula by Sabine,⁵ with room volume V and total absorption area A of the room as input parameters, using

RT = 0.16 \frac{V}{A}

(1)

A is calculated by

A = \sum_{i} S i \cdot α S, i + \sum_{j} A Obj, j + A air

(2)

The first term in Equation (2) is the sum of the product of the sound absorption coefficients $α S$ of all room surfaces and their respective surface area S. The second term is the sum of the equivalent sound absorption areas $A Obj$ of all objects inside the room and $A air$ denotes the total absorption of air.

$α S$ , $A Obj$ and $A air$ are frequency dependent. The larger the values are, the more sound is absorbed by the surface material, by the object or by air, respectively. For $α S$ and $A Obj$ , standardized measurement methods exist. Tests are conducted in a reverberation room where the incident sound is assumed to be ideally diffuse.⁶ However, prediction of the sound absorption coefficients from the material properties of an absorber prove to be much more demanding.^7–9

For textile curtains, several prediction models to calculate the sound absorption quantities have been developed. The classical model only considers the size of the air cavity between the fabric and a rigid wall and the specific airflow resistance of the fabric as input parameters.³ It is based on the assumption that no sound-induced fabric vibrations occur, which is fulfilled if the fabric is much heavier than the surrounding air layer. However, for lightweight fabrics sound-induced vibrations affect the sound absorption characteristics. Several authors have proposed methods to take this effect mathematically into account.^10–12

Shoshani and Rosenhouse¹³ found that the microstructure of the fabric substantially influences the sound absorption coefficient. Since then, several models based on geometrical fabric parameters, such as porosity, thickness or pore size, have been proposed.^10,14–18 The effect of flow distortions due to the constriction of the oscillatory flow through the material was described mathematically by Atalla and Sgard.¹⁹ Recently, methods and theoretical models have been developed that have a high potential to provide accurate predictions for multilayer curtains either mounted in front of walls or freely hanging in the room.¹⁸ In the prediction, the effects that occur during testing in a reverberation room are considered, such as edge diffraction at the specimen²⁰ and non-uniform intensity distribution of the incident sound,²¹ as well as the effects mentioned above already. However, to date the models have only been validated with measurements of the absorption coefficient for normal incidence in the impedance tube,²² but not with measured data from a reverberation room. Further, the published models have been limited to flat curtains so far.

The objective of this study is therefore to extend the existing models of Pieren and Heutschi,¹⁸ which are the most recent available models, to the case of folded curtains. Hereby, the focus lies on the general case of a single fabric layer curtain. In the following, the existing models of Pieren and Heutschi¹⁸ to predict the sound absorption performance of curtains as measured in the reverberation room are briefly outlined. The models are then semi-empirically extended to the case of folded curtains. Finally, predicted results are compared to measured data from a reverberation room.

Theory and model development

In this section existing models for flat curtains are briefly summarized and subsequently extended to the case of curtains with drapery fullness. A fullness of 0% corresponds to a flat curtain; a draped curtain having twice the amount of fabric is referred to as 100% fullness. The existing and extended models allow for a description of the following four cases:

case I: flat curtain in front of a rigid wall;

case II: freely hanging flat curtain (without wall);

case III: folded curtain in front of a rigid wall;

case IV: freely hanging folded curtain (without wall).

Existing sound absorption models for flat curtains (cases I and II)

The models discussed here originate from the model presented by Pieren.¹⁷ The final model¹⁸ describes the general case of sound absorption coefficients of flat, lightweight, multilayer curtains, that is, the above cases I and II. Below, the equations to predict the (diffuse-field) sound absorption in reverberation rooms are derived and simplified for the case of a single layer curtain.

For case I, the considered set-up consists of a thin sheet of fabric that is mounted at a well-defined distance d in front of a rigid, that is, fully reflecting, surface forming an air cavity, as shown in Figure 1. The thickness of the sheet is assumed to be small compared to the wavelength of sound. It is further assumed that a plane airborne sound wave is incident on this structure. The incident sound wave is then partially reflected at, partially transmitted through, and partially absorbed by the fabric. At the right-hand side of the air cavity, the sound wave is fully reflected at the rigid wall and returns to the fabric from the opposite side, where the wave is again partially reflected, transmitted and absorbed. Hence, a complex sound field with interference of incoming and reflected waves is being built-up in the cavity. This again strongly affects sound absorption and, in particular, its frequency dependency. In contrast to case I, in case II (Figure 1), where the fabric is far away from any boundary, the transmitted wave is not reflected, but propagates away from the curtain. According to standard ISO 354,⁶ case I is characterized by $α S$ and case II by $A Obj$ , as in the latter case of the freely hanging curtain the actual surface area that is exposed to the incident sound is not well-defined.

Figure 1.

Sketch of case I with a fabric sheet at distance d to a rigid wall and of case II with a freely hanging fabric sheet without air cavity.

The physical behavior of the above structures can be described by physical models, such as the equivalent circuit (EC) method²³ that is well suited and broadly used for this purpose.^{7,11,12,17,18,24} In this analogy, mechanical and acoustical quantities or elements are represented by equivalent electrical quantities or elements. Accordingly, the ratio of a sound pressure and a sound particle velocity yields an acoustical impedance.

The fabric impedance Z_f is defined as the ratio of the sound pressure drop across the fabric and the sound particle velocity inside the fabric. For thin fabrics this impedance fully characterizes the acoustical behavior of the textile. It can be calculated from the specific airflow resistance R_s (in Pa s/m)²⁵ and the area density m (in kg/m²) as^7,11,12

Z f = \frac{j ω mR s}{j ω m + R s}

(3)

where

j = \sqrt{- 1}

is the imaginary unit,

ω = 2 π f

is the angular frequency and f is the frequency in Hz.

As the geometric arrangement of the textile strongly determines the acoustic field (see above), different equations for the absorption, based on the fabric material [Equation (3)] and the backing condition, need to be established. Further, for the backed or freely hanging textile they have to be distinguished between $α S$ (case I) and $A Obj$ (case II), respectively.

For a flat fabric placed at a distance d (in m) to a rigid wall, the statistical absorption coefficient is described by¹⁸

α S = \frac{8 \cdot ℜ {Z f}}{Z 0} \cdot \int 0 π / 2 \frac{w (θ) \sin θ}{| \frac{Z f}{Z 0} - j \frac{\cot (k 0 d \cos θ)}{\cos θ} + \bar{Z} F (θ) | 2} d θ

(4)

where ℜ denotes the real part and k₀ the wave number in air (

k 0 = ω / c 0

Z 0 = ρ 0 c 0

is the characteristic impedance of air with air density

ρ 0

and speed of sound in air c₀, w is a weighting function (see below) and

\bar{Z} F

is the averaged radiation impedance.²⁰ The cotangent term in Equation (4) introduces the frequency dependency due to the air cavity. The integration is performed over all angles of incidence θ under which a plane wave is incident on the fabric. Hereby, the normal incidence case as measured in the impedance tube corresponds to θ = 0 and the grazing incidence case with waves propagating parallel to the surface corresponds to θ = π/2. The non-uniform intensity distribution across the specimen in the reverberation room is taken into account by a weighting function w. The frequency dependent angle weighting function by Jeong²¹ was approximated by Pieren and Heutschi¹⁸ as

w (θ) = {1.64 - 0.3 \cdot θ - 0.47 \cdot θ 2, f \geq 350 Hz 1, f < 350 Hz

(5)

where θ is the angle in radians.

The averaged radiation impedance $\bar{Z} F$ accounts for the edge effect (typical overestimation of absorption due to diffraction at the edges) of the specimen. $\bar{Z} F$ depends on frequency, the angle of incidence θ and the size and shape of the specimen.²⁰ It has to be evaluated numerically. For the case of a rectangular shape of defined length and width, different numerical solutions have been presented.^18,26 For very large absorbers it can be approximated by $\bar{Z} F ≅ 1 / \cos θ$ .

For case II, the equivalent absorption area of a freely hanging fabric of surface area S (in m²) may be obtained by¹⁸

A Obj = \frac{16 S \cdot ℜ {Z f}}{Z 0} \cdot \int 0 π / 2 \frac{\sin θ}{| \frac{Z f}{Z 0} + \frac{2}{\cos θ} | 2} d θ

(6)

It is interesting to note the similarities of Equations (4) and (6) in the presented form, which is not obvious in the general form of the equations presented by Pieren and Heutschi.¹⁸

If $\bar{Z} F$ is calculated in advance [case I, Equation (4)], then only three parameters need to be known to determine $α S$ from the above model. These input data are the specific airflow resistance R_s and the area density m to estimate the fabric impedance Z_f [Equation (3)], as well as the distance d. Also, the prediction of $A Obj$ [case II, Equation (6)] only requires three parameters, namely R_s, m and the surface area S.

New sound absorption models for folded curtains (cases III and IV)

In practical applications, instead of flat curtains often folded curtains, that is, with fullness larger than 0%, are used, where a fullness of 100% is most common. Fullness influences the sound absorption of curtains.^27,28 Until now, no model for the prediction of sound absorption by folded curtains is available. Based on theoretical considerations, numerical simulations and conclusions on measured data, a semi-empirical model for $α S$ and $A Obj$ is derived in the following.

Commonly folded curtains are mounted in parallel to a flat wall or window (case III), as depicted in Figure 2. By introducing a certain degree of drapery fullness, the distance of the fabric to the backing wall varies locally. To characterize such a situation it is useful to use the average distance $\bar{d}$ corresponding to the distance of the curtain rail to the backing. The folding of the curtain is described by only two additional geometrical parameters, namely the folding depth b and the folding period p (Figure 2). For this geometric situation, the model was derived based on the following two assumptions.

Figure 2.

Sketch of cases III and IV of a folded thin curtain with an average distance $\bar{d}$ to a rigid wall and with folding depth b and folding period p.

Firstly, for a given fabric, the depth of the air cavity strongly affects the frequency dependency of the absorption coefficient (as in the above case I). For a flat fabric (case I), the air cavity depth d primarily determines the frequencies of the local maxima and minima of the absorption coefficient.¹⁸ A variation of the cavity depth thus smoothes the frequency dependency of the absorption coefficient. Therefore, intuitively the variation of air cavity depth is accounted for by averaging absorption coefficients calculated for several discrete distances.

Secondly, the folding implicates that a portion of the incoming and reflected sound waves traverses the fabric more than once while passing the curtain. This effect is expected to play a role only at grazing incidence and at high frequencies. The corresponding wavelengths thus have to be small compared to the folding. To model this effect, a frequency and geometry dependent function ξ is introduced. ξ describes the portion of sound energy that traverses the fabric double compared to the flat case. It may take values of 0 ≤ ξ ≤ 1.

With these two assumptions, the model for case III may be derived by adopting the model of case I. We propose a semi-empirical model for the statistical absorption coefficient of folded curtains as follows

\bar{α} S = \frac{α S (d = \bar{d} - b / 2) + α S (d = \bar{d} + b / 2)}{2}

(7)

In Equation (7), $\bar{α} S$ is the arithmetic mean absorption coefficient of the two extreme cases, that is, the fabric placed at $\bar{d} - b / 2$ to the rigid wall and at $\bar{d} + b / 2$ . Both $α S$ values in Equation (7) are calculated using Equation (4) with R_s and m of the fabric as input data. From an energetic point of view, the sound absorption coefficient of the folded curtain, $α S, fullness$ , can then be written as

α S, fullness = \bar{α} S + ξ \cdot (1 - \bar{α} S) \cdot \bar{α} S

(8)

In Equation (8), the expression in parentheses describes the sound energy that is not absorbed by a single fabric layer. A portion of this energy (ratio ξ) passes through the textile multiple times and is reduced by $\bar{α} S$ . The ratio ξ depends on the frequency f and the geometry, namely folding depth b (in m) and folding period p (in m), as defined in Figure 2. ξ is set to

ξ (f, b, p) = {0.5, if f > F ci = 70 / b and b > p / 2 0, else

(9)

Equation (9) was established based on the following considerations. For frequencies below a certain cut-in frequency, $F ci$ , ξ is set to zero, as folding does not affect sound absorption at low frequencies. For frequencies above $F ci$ , ξ is set to a constant value for the matter of simplicity. The non-zero value of ξ = 0.5 was empirically determined from the available dataset (see below) by systematic variation and visual inspection to find the optimal fit to measured data. This value implies that at high frequencies 50% of the sound energy is affected by the folding, in the sense that it traverses the fabric twice as often as in the flat case.

Further, $F ci$ depends on the folding geometry (cf. Figure 2). For $b \to 0$ (flat curtain) $F ci$ tends to infinity. If the whole geometry is scaled by a factor, $F ci$ has to be divided by this factor. These considerations define the properties of $F ci$ . The second free parameter of Equation (9) is a constant scaling of $F ci$ . The factor of 70 was also empirically determined from the available dataset (see below) by visual inspection. For the studied geometries, the resulting $F ci$ is about 1 kHz. From the expression for $F ci$ , the corresponding wavelength yields $λ ci \approx b / 5$ . This means that in the model, multiple traverses only occur for sound waves with wavelengths smaller than a fifth of the folding depth.

Finally, the folding depth has to be of the same order of magnitude as or larger than the folding period. How much larger is not critical, as fullness larger than 100% does not provide any substantial change in $α S$ .²⁹ This is accounted for by the mutual condition for p and b. The condition $b > p / 2$ allows b to adopt somewhat smaller values but also any larger value than p.

Note that the area density m used as input data in Equation (7) is independent of fullness, even though the total area density of the folded curtain increases because of the total mass increase, whereas its area remains constant. In fact, numerical simulations with a doubling of the area densities (100% fullness) resulted in distinct overestimations of $α S, fullness$ compared to measurements. It thus appears that the fabric vibration is a local phenomenon and, consequently, the area density of the fabric equals the effective area density of the curtain.

In this study we also consider the situation with a folded curtain mounted freely hanging in the diffuse field of a room (case IV), as depicted in Figure 2. We introduce a new equation for this case as an extension of the equation of case II

A Obj, fullness = A Obj + η \cdot A Obj

(10)

with the fullness efficiency factor η.

A Obj

is calculated by Equation (6) by applying the curtain covering area S* (in m²) instead of the surface area S of the fabric. For fullnesses of 0% and 100% with the same textile sample, this means that S* = S and S* = 0.5S, respectively. From the dataset that is presented in the following sections, a value of η = 0.7 was empirically determined to optimally represent measurements for 100% fullness.

Materials and methods

For model development and validation, the diffuse-field sound absorption of four textiles with varying acoustical properties was measured under different backing conditions and fullnesses, and also predicted using the presented models.

Textile characteristics

In this study four different woven fabrics made of synthetic materials (polyester) were used. Each fabric consists of three to five different types of yarns with linear mass densities of 24–1200 dtex. The fabrics vary in terms of their weaving pattern. The warp and weft densities are 50–65 yarns per cm. Table 1 presents the characteristics of fabrics 1–4, which are necessary as input parameters for the above models for cases I–IV. The area density m was measured from a small sample with a precision scale. The specific airflow resistance R_s was determined from a sample of 10 cm radius by an indirect measurement using an impedance tube.¹⁷ The cutoff frequency f_c, a parameter to characterize the effect of the vibration of the textile, was calculated by

f c = R s / (2 π m)

, as proposed by Pieren.¹⁷ In terms of acoustical properties, the four fabrics cover a broad range of lightweight textiles. Fabrics 2 and 3 have similar m and R_s values, but strongly differ from fabrics 1 and 4.

Table 1.

Non-acoustical parameters of fabrics 1–4

Fabric ID	Area density m [g/m²]	Specific airflow resistance R_s [Pa s/m]	Cutoff frequency f_c [Hz]
1	64	45	100
2	83	150	290
3	97	155	250
4	130	350	430

Measurements

Statistical absorption coefficients

α S

and equivalent sound absorption areas

A Obj

were measured according to the standard ISO 354⁶ for cases I–IV in the reverberation room of the Laboratory for Acoustics/Noise Control at the authors’ institution, Empa. Table 2 gives an overview on the configurations of the measured fabrics. According to the standard, the measurements were performed in 1/3 octave bands from 100 Hz to 8 kHz. Further, a frequency independent single number rating, the so-called weighted sound absorption coefficient,

α w

, was determined from frequency dependent measured

α S

according to the standard ISO 11654.³⁰

Table 2.

Measurement overview for the textiles 1–4 (according to Table 1) and their configurations to obtain the statistical absorption coefficient $α S$ and the equivalent sound absorption area $A Obj$

Case	Description	Distance to wall		No wall
Case	Description	d or $\bar{d}$ = 10 cm	d or $\bar{d}$ = 15 cm	No wall
I	In front of wall, 0% fullness	1–4	1–4	–
II	Freely hanging, 0% fullness	–	–	1, 2, 4
III	In front of wall, 100% fullness	1	1–4	–
IV	Freely hanging, 100% fullness	–	–	1, 2, 4

Empa’s reverberation room has a volume of 215 m³ and a peculiar shape with all walls skewed and not vertical to the floor. The room is equipped with hanging sound diffusers to further increase its diffusivity and a humidifier ensures a minimum relative humidity of 60% to reduce air absorption to minimum. Four fixed loudspeakers and six microphones are distributed in the room to measure the spatially averaged reverberation time with and without a test specimen. The reverberation time is evaluated using the so-called impulse response method, where level decays and reverberation times respectively are found by backward integration of the measured room impulse response functions. For the measurement of the impulse response function, the reverberation room is excited with a maximum length sequence (MLS) signal. Based on the difference in reverberation time with and without a specimen, the (diffuse-field) sound absorption of a test object or a test material is calculated. The size of the reverberation room requires test specimens with dimensions of typically 3 m × 4 m. In the used reverberation room, flat absorbers as well as specimens that require a well-defined distance to a rigid backing, as for cases I and III in this study, are placed in a fixed specimen zone on the floor, which was determined during the qualification procedure of the facility. To mount the textiles with a well-defined distance to the floor, wooden rectangular frames of 3 m × 4 m with different heights with thin tensioned wires attached at their top that span across the test area were used (cf. Figure 3). The fabrics were placed over these wires. For the flat case (case I), fabrics of 3 m × 4 m were used. The distance d to the rigid surface (cf. Figure 1) was realized by using frame heights of 10 or 15 cm. Measurements of folded curtains (case III) were made with 100% fullness, that is, fabrics of 3 m × 8 m were used. Figure 3 shows a photograph of a fabric mounted on the wooden frame with 100% fullness. With a wire spacing of 9 cm, a folding period p = 9 cm was achieved (see close-up view in Figure 3), which is a typical value for curtain systems. For the folded fabrics this resulted in a folding depth b = 7 cm. The mean distance $\bar{d}$ to the rigid surface (cf. Figure 2) was thus realized with frame heights of 13.5 and 18.5 cm. For cases II and IV the fabrics were simply hung up in the middle of the room on a wire that span across the room (Figure 3).

Figure 3.

Photographs showing textiles during the measurement in the reverberation room for the configuration of a folded curtain (case III) and a freely hanging curtain (case II).

Predictions

Statistical absorption coefficients $α S$ and equivalent sound absorption areas $A Obj$ of the tested specimens listed in Table 2 were predicted using the above-described calculation models, which were implemented in the commercial software package MATLAB, with the following input data and parameters.

The calculations were performed at the center band frequencies of 1/27 octave bands, and only the final results were averaged to 1/3 octave bands. As a first step, for each fabric its fabric impedance Z_f was determined by Equation (3) using the values given in Table 1. Z_f was then inserted either into Equation (4) or (6). To numerically solve the integrals in these equations, the angle θ was discretized in steps of 2°, corresponding to 0.035 radians. Further, the area S (in m²) and cavity depth d (in m) was the same as in the experiments. In Equation (4) the weighting function given by Equation (5) was applied, and the averaged radiation impedance $\bar{Z} F$ of a rectangle of 3 m × 4 m was once numerically calculated by the method given in the appendix of Pieren and Heutschi¹⁸ and used for all predictions. Statistical absorption coefficients of folded curtains were predicted using Equations (7)–(10) and the same input data as the in experiment, and adopting the geometric parameters b and p as determined during the measurements, that is, b =0.07 m and p = 0.09 m.

From the predicted absorption coefficients $α S$ in 1/3 octave bands, the weighted absorption coefficients $α w$ were determined according to the procedure described in the standard ISO 11654.³⁰

Comparison of measured and predicted data

In this section the measured and predicted results of different textile specimens and configurations, according to Table 2, are presented and compared for model validation.

Flat curtains in front of wall (case I)

Figure 4 shows measured and predicted $α S$ for flat curtains in front of a rigid wall (case I). Measured and predicted results both confirm that $α S$ increases with increasing specific airflow resistance within the studied range (Table 1). Measured and calculated results generally agree well over the whole frequency range from 100 Hz to 8 kHz for all fabrics. Similar $α S$ values of fabrics 2 and 3 agree well with the model. Also, distinct differences to fabrics 1 and 4 are clearly reproduced. Further, the calculations closely match the measured spectral shape, that is, local maxima and minima of $α S$ . This means that the interference between incoming and reflected sound waves is predicted well by the model. It can be observed that the local minima, corresponding to half and full wavelength resonance of the air cavity, are shifted by one third octave band to higher frequencies in the present diffuse field case as compared to the normal sound incidence. As an example, for d = 10 cm the local minima are at 2 and 4 kHz (cf. Figure 4), while the minima occur at 1.7 and 3.4 kHz for normal sound incidence.¹⁷ Finally, the absolute $α S$ values are sufficiently well predicted although certain deviations may be observed.

Figure 4.

Measured and predicted statistical sound absorption coefficients for flat curtains in front of a rigid wall (case I) of fabrics 1–4.

In particular, the values at the local minima are clearly more pronounced in the calculations than in the measurements. Apparently, the model overestimates the destructive interferences, which are less prominent in measurements where non-ideal conditions, such as residual absorption and scattering within the air cavity, lead to a coherence loss, which remains unaccounted for by the model. Further, at frequencies below 300 Hz, deviations between measured and calculated data may be observed, particularly for fabric 1. Predicted data slightly underestimates $α S$ compared to measurement. However, predicted as well as measured $α S$ values are relatively small and therefore prone to greater relative uncertainty. In the measurements, $α S$ is determined from the difference of reverberation time in the empty reverberation room and when the test specimen is installed. In the case of small measured $α S$ values also the difference of the measured reverberation times are very small. Thus, in this case small uncertainties in measured reverberation time may lead to relatively large uncertainties in $α S$ . Therefore, the difference between measured and predicted data in this frequency range is not significant in case of the flat curtains.

Finally, at high frequencies (above 2–4 kHz) the model predicts systematically lower values than measurements. The reason for this is not known. Three hypotheses are now shortly discussed. A first possible reason is the non-uniform intensity distribution during the measurement. Although the model accounts for this effect with Equation (5), this approximation may be inadequate for high frequencies. However, the fact that similar differences also occur for freely hanging curtains (see the next section) does not support this hypothesis. Secondly, generally also peculiarities of the laboratories might play a role. Round robin tests between laboratories have revealed large inter-lab differences.³¹ A third possible reason is the fabric impedance model [Equation (3)], which assumes static airflow through the fabric. This assumption may become inappropriate at high frequencies where effects such as inertia of the air moving inside the pores or flow distortions on both sides of the fabric occur. These effects are omitted in the applied model. To account also for these effects, the fabric impedance model would have to be extended. However, for the development and validation of an extended fabric impedance model, impedance tube measurement data well above 2 kHz would be required, but are currently only available up to 1.6 kHz.¹⁷ To investigate this possible deficiency of the model, preliminary impedance tube measurements were conducted on a sample of fabric 4 up to 6.4 kHz. They in fact revealed higher absorption coefficients at high frequencies than what the model predicts. These findings support the hypothesis that the fabric model has to be refined to more accurately predict the sound absorption coefficient at high frequencies.

Freely hanging flat curtain (cases II and IV)

Figure 5 shows measured and predicted $A Obj$ for freely hanging curtains. The values were adjusted by the curtain covering area and normalized to a reference area of 10 m² for better comparison. $A Obj$ steadily increases with frequency until it approaches a constant value at a frequency depending on the area density and specific airflow resistivity.¹⁸ The smooth shape of $A Obj$ is very different from $α S$ for flat curtains (interference pattern) in front of a rigid wall (case I, see above), but similar to folded curtains in front of a rigid wall (case III, see below).

Figure 5.

Measured and predicted equivalent absorption areas for flat and folded freely hanging curtains (cases II and IV) of fabrics 1, 2 and 4. Shown values are normalized to a reference area of 10 m².

The measurements show that for curtains with identical covering areas, $A Obj$ of folded curtains is significantly larger than that of flat curtains. The data shows that introducing folding and increasing the amount of textile by 100% leads to a 70% increase of sound absorption. This demonstrates that fullness is also beneficial for sound absorption of freely hanging curtains.

Measurements and predictions closely agree from 100 Hz to more than 1 kHz. Above this, the model increasingly underestimates $A Obj$ . Interestingly, the underestimation at high frequencies is of similar magnitude as for $α S$ (Figure 4), which suggests that it is caused by the same effect. If this is true, it can be concluded, as for case I, that the discrepancies are not due to the non-uniform intensity distribution.

Folded curtains in front of wall (case III)

Figure 6 shows measured and calculated $α S$ for folded curtains in front of a rigid wall (case III). As for case I, $α S$ of folded curtains again increases with increasing specific airflow resistance within the studied range (Table 1). In contrast to case I (Figure 4), $α S$ shows a much smoother curve shape over the frequency range. Apparently, because of varying air cavity depth, folding significantly reduces the interference patterns, so that local maxima and minima are much less pronounced. Further, at high frequencies folding results in somewhat higher $α S$ values compared to flat curtains (case I, Figure 4).

Figure 6.

Measured and calculated sound absorption coefficients for folded curtains in front of a rigid wall (case III) of fabrics 1–4 for average distance to the backing wall $\bar{d}$ = 15 cm. For fabric 1 also $\bar{d}$ = 10 cm is shown in gray.

The model predicts the absorption satisfyingly well in the measured frequency range from 100 Hz to 8 kHz for all fabrics. As for case I, the model underestimates measured $α S$ values at low as well as at high frequencies. This suggests that the same mechanisms are responsible for the observed discrepancies. However, compared to case I the deviations are more pronounced. This was expected because the model for case III was developed based on case I and extended with relatively simple semi-empirical corrections. The largest differences occur for fabric 1 at frequencies below 700 Hz. As already explained above, for the flat curtains, relatively small $α S$ are prone to greater measurement uncertainty. Furthermore, these differences may also be induced by non-ideal measurement conditions, such as the wooden frame potentially acting as a membrane absorber, which introduces additional sound absorption at low frequencies.

Weighted absorption coefficient (cases I and III)

Figure 7 contains a comparison of measured and predicted weighted absorption coefficients determined from the above data of cases I and III. The measured and calculated values agree very well. The maximal deviation of the predictions amounts to 0.05. In six out of 13 data pairs, the prediction underestimates the measurement by 0.05, and zero otherwise. The differences of measured and predicted values thus exhibit a mean value and standard deviation of 0.02 ± 0.03. The mean value does significantly differ from zero with a probability of 0.99 (one-sample two-tailed t-test).

Figure 7.

Comparison of measured and predicted weighted absorption coefficients $α w$ of flat (case I) and folded (case III) curtains. The asterisk denotes two overlaying points. The linear regression line (R²= 0.956) and the 95% prediction band for future estimates (PI) are shown in gray.

This integral comparison of predicted and measured $α w$ thus further confirms the applicability of the calculation models. The models generally yield conservative results, which means that the performance in real rooms is better.

Thus, the models, amongst other purposes, can be used to optimize $α w$ values by changing fabric parameters or curtain configuration. Exemplarily calculations reveal that increasing the average air cavity depth $\bar{d}$ from 15 to 19 cm of a curtain with 100% fullness and made from fabric 4 (Table 1) would increase $α w$ by 18% from 0.55 to 0.65 (cf. Figure 8).

Figure 8.

Simulation result showing the weighted absorption coefficient $α w$ as a function of the average air cavity depth $\bar{d}$ for a curtain consisting of fabric 4 and with 100% fullness.

Comparison of the presented model with commercial software

This section is dedicated to a comparison of calculation results by the presented model, and existing models that are implemented in commercial software. To our knowledge, no software specifically for the prediction of sound absorption of curtains exists. However, there are a few programs available that allow treating multilayer structures in front of a rigid wall. In these programs a flat curtain in front of a rigid wall (case I) may be entered as two layers, that is, an air cavity of depth d and a thin layer of porous material. In the following comparisons, the three commercial software packages SoundFlow, Winflag and Zorba (randomly named Software 1, 2, and 3 to maintain anonymity in the results) were used as follows.

AFMG SoundFlow, Demo Version 1.0.11 (Ahnert Feistel Media Group, Berlin, Germany) with the following specific program settings. Porous absorber calculation model: Bies/Hansen³² (only available model of the basic version of SoundFlow); dimension ISO 354 (3 m × 4 m); direction of incidence: diffuse field. No information has been found concerning the calculation of random incidence absorption.

WinFlag, Demo Version 2.4 (Morset Sound Development, Trondheim, Norway) with the following specific program settings. Reverb room, square with side length 3.163 m; selected porous absorber calculation model: Delany–Bazley.³³ The calculation of the random incidence absorption coefficient is based on Thomasson.²⁰

Zorba, Version 3.0.1 (Marshall Day Acoustics Ltd Pty, Adelaide, South Australia). Selected porous absorber calculation model: Allard–Champoux.³⁴ The calculation of the random incidence absorption coefficient is based on Thomasson.³⁵

Figure 9 exemplarily compares the predicted sound absorption coefficients $α s$ of two curtains with measured data. In most cases the commercial programs clearly overestimate sound absorption at frequencies below 500 Hz. These deviations strongly depend on the used program and the textile. The largest deviations occur for fabric 4. Within the dataset this fabric features the highest cutoff frequency of more than 400 Hz (see Table 1), which suggests that sound-induced fabric vibrations may play a role in that frequency range. In contrast to the presented model, these vibrations are not taken into account in the commercial programs, which partially explains the observed deviations. However, at frequencies above ∼1 kHz all calculation models yield predictions mostly close to the measured data. At these frequencies two commercial programs closely agree with the presented model, whereas one commercial program predicts very different 1/3 octave band values. This can be attributed to the assumption of local reaction within that software.³⁶

Figure 9.

Comparisons of the sound absorption coefficient $α s$ of flat curtains predicted with the presented model and commercial software packages. Fabric 3 with air cavity depth 10 cm (left) and fabric 4 with air cavity depth 15 cm (left) are shown.

As an integral comparison, Figure 10 shows the predicted and measured weighted absorption coefficients $α w$ of eight flat curtains (case I in Table 2). As expected from Figure 9, the commercial programs exhibit different prediction performances. From the data points and their linear regression lines it can be concluded that the presented model outperforms the tested commercial programs.

Figure 10.

Comparison of predictions of weighted absorption coefficients $α w$ by the presented model and commercial software packages. For each model, predicted values of eight flat curtains (case I in Table 2) are compared to measured values (with some overlaying points). The dashed lines show the linear regression lines.

Conclusions

In this article, existing and new calculation models for the diffuse-field sound absorption of textile curtains were presented. The models cover flat as well as folded curtains, and the configuration with a rigid backing and without backing. For these cases modeling results were compared to measured results from a reverberation room. Generally, a very good agreement between measured and predicted values was observed. Comparisons revealed only at high frequencies the potential for further model improvements. Nevertheless, compared to currently available commercial sound absorption prediction software, the presented models not only yield more accurate results, but also cover a broader range of configurations, while the commercial software can only handle the case of flat textiles with rigid backing. The proposed method performs considerably better than commercial software and can be a viable alternative to experimentally measured values, which are prone to uncertainties of similar magnitude. The models are therefore applicable to textile development as well as to room acoustical planning purposes. Although the studied parameter range was quite remarkable, further validations should be performed to extend the applicability of the models, in particular regarding higher specific airflow resistances. The results further indicate that future prediction model improvement could be achieved by enhancing the fabric model.

Footnotes

Acknowledgements

We would like to thank four anonymous reviewers for their constructive comments and Martin Wü rzer for his help in the measurements.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The authors received no financial support for the research, authorship and/or publication of this article.

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