# Measurement of the diffuse field sound absorption using a sound field synthesis method

Samuel Dupont, Maryna Sanalatii, Manuel Melon, Olivier Robin, Alain Berry, Jean-Christophe Le Roux

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## To cite this version:

Samuel Dupont, Maryna Sanalatii, Manuel Melon, Olivier Robin, Alain Berry, et al., Measurement of the diffuse field sound absorption using a sound field synthesis method. Acta Acustica, 2023, 7, pp.26. 10.1051/aacus/2023021. hal-04124226

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Submitted on 9 Jun 2023

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# Measurement of the diffuse field sound absorption using a sound field synthesis method

Samuel Dupont<fnref n="1" />,*, Maryna Sanalatii<fnref n="2" />, Manuel Melon<fnref n="1" />,*, Olivier Robin<fnref n="3" />, Alain Berry<fnref n="3" />,
and Jean-Christophe Le Roux<fnref n="2" />

Received 16 December 2022, Accepted 3 May 2023

**Abstract** – A method for measuring the diffuse field sound absorption coefficient of a material using sound field synthesis is proposed. A planar loudspeaker array is first used to generate acoustic plane waves with variable incidence angle on the surface of a material under test. Using a two-microphone probe positioned closely to the sample's surface, the angle-dependent sound absorption coefficients are then estimated. Finally, the diffuse field absorption coefficient is computed following Paris formula. Numerical simulations are used to evaluate the respective effects of the maximum incidence angle value and the number of individual incidence angles that are required for a robust calculation of the diffuse sound field absorption. Measurements are conducted on three different materials and compared with either simulation results obtained using the Johnson-Champoux-Allard theory, or with measurement results obtained using the standard reverberation chamber method. For all considered materials and over a wide frequency range, the proposed method leads to results that are in better agreement with theoretical predictions than those obtained using standardized methods.

**Keywords:** Sound field synthesis, Measurement method, Absorption coefficient, Diffuse field

## 1 Introduction

Sound absorbing materials are widely used in industrial sectors such as construction and transportation. Their performance is mainly characterized by the sound absorption coefficient. The values of this coefficient usually depend on the considered frequency and on the incidence angle of sound waves at the material surface. In practice, sound absorption values are usually averaged over the incidence angle (diffuse field measurement in a reverberation room), and over frequency (octave or third octave bands). Single-value ratings have also been proposed, such as the Noise Reduction Coefficient – NRC [1], the weighted alpha value or the sound absorption class [2].

The measurement of the diffuse field sound absorption coefficient by the reverberation room method has been standardized by the International Organization for Standardization [3] and the American Society for Testing and Materials [1]. This method is based on the use of the Sabine formula that requires the establishment of a diffuse acoustic

field which is not always achieved in practice. Indeed, the results obtained using the reverberation room method suffer from well documented limitations such as a poor interlaboratory reproducibility due to a large sensitivity to the sample's position, to microphones and sound source frequency response and position, lack of diffusors in the chamber, the presence of the tested material which modifies the acoustic intensity angular distribution, and edge diffraction effects [4–10]. All these effects result in large uncertainties in the measurement of the diffuse field sound absorption coefficient, with values that can exceed unity while the theoretical value shall be between zero and one. Although standards attempt to reduce the differences between laboratories by imposing a minimum room volume, minimum sample size, etc., each reverberation chamber deviates from the diffuse field assumptions in its own way, so the results obtained are not inherently reproducible. A future version of the ISO 354 standard is currently being drafted (Committee Draft stage) and proposes various solutions to improve the accuracy of the measurement method, such as the damping of the reverberation room, the use of a reference absorber, etc. However, the work of Scrosati et al. [9]

<footnote n="1">¹Laboratoire d'Acoustique de l'Université du Mans (LAUM), UMR 6613, Institut d'Acoustique – Graduate School (IA-GS), CNRS, 72000 Le Mans Université, France</footnote>

<footnote n="2">²Centre de Transfert de Technologie du Mans (CTTM), 20 Rue de Thalès de Milet, 72000 Le Mans Cedex 09, France</footnote>

<footnote n="3">³Centre de Recherche Acoustique – Signal – Humain (CRASH), Université de Sherbrooke, 2500 Bd de l'Université, Sherbrooke J1K 2R1, QC, Canada</footnote>

<footnote>*Corresponding author: Manuel.Melon@univ-lemans.fr</footnote>
has shown that although some of the proposals were bene-
ficial, others could in some cases lead to increased variance
between laboratories. The shortcomings of ISO 354 and
ASTM C423-02 have these two important consequences:
1) the high variance between laboratory results does not
allow comparisons of absorptive materials with confidence,
2) the overestimation of the absorption coefficient prevents
the accurate prediction of reverberation times or sound
pressure levels as reported in the literature [11, 12]. Indeed,
the coefficients must be truncated or corrected to be used in
building acoustics prediction software. This can lead to
biased predictions, as well as additional iterations between
simulation and field results to adjust the sizing of required
soundproofing or sound control material (iterations that
could be directly avoided with accurate data). This is par-
ticularly concerning, given that acoustic comfort has
become increasingly important over the past decade, lead-
ing to the creation of several acoustical standards and
regulations that have to be followed.

To overcome these issues, several approaches were pro-
posed like *in situ* measurement methods under approximate
diffuse field conditions. Takahashi *et al.* [13] used the back-
ground noise and a two-microphone probe placed in the
vicinity of the material to measure the random incidence
absorption coefficient in ordinary rooms. This technique
was improved by using an ensemble averaging to calculate
the normal surface impedance of a material under random
excitation in a reverberation room to obtain its sound
absorption coefficient [14].

Another approach is to use a microphone array and take
advantage of this refined spatial information [15–19] to esti-
mate the absorption coefficient for each incidence angle.
Separation methods have also been proposed to split the
incident and reflected energy flows and then calculate an
angle-dependent absorption coefficient. For instance, dou-
ble layer microphone array and Statistically Optimized
Near-field Acoustical Holography (SONAH) [20, 21] have
been used for *in situ* measurements while plane wave
decomposition of pressure measurements performed in a
small volume close to the tested material has been applied
in a reverberation room [22] or in an ordinary room [23].

Further work has also been carried out to convert the
values of the Sabine absorption coefficient, ideally intended
to represent the absorption of an infinite slab of material
excited by a perfectly diffuse sound field, into a random
incidence absorption coefficient [24], accounting for non
ideal conditions during the measurements. For this purpose,
the effect of material size and the non-uniform distribution
of the excitation field intensity are taken into account [25].
Good results have been obtained, but they depend on
assumptions about the properties of the sample under test
and cannot be generalized to all types of materials.

Another solution is to impose a target incident sound
pressure field, typically using a loudspeaker array to radiate
a target acoustic field. In practice, physical loudspeakers
arrays are seldom used. A cost-effective and more efficient
solution in terms of measurement resources is to use a
virtual array in which a single loudspeaker is moved succes-
sively to different positions that loudspeakers would occupy

in a physical array. Using measurements of transfer func-
tions between each source and sensors placed in the vicinity
of the material under test, it is possible to a *posteriori* gen-
erate target acoustic fields such as a plane wave at normal
[26] or oblique incidence [27] or a diffuse field [28, 29] at a
post-processing step, and calculate the corresponding sound
absorption coefficients.

Using a Sound Field Synthesis (SFS) technique with a
source array close to the material surface offers several
benefits. Firstly, this ensures a large Signal-to-Noise Ratio.
Secondly, this reduces the effect of the room reflections
and edge diffraction by providing a strong direct signal.
Finally, traditional microphone array methods deal with
the problem as a consequence, while sound synthesis
methods prevent the issues from occurring in the first place,
providing a potential lever for removing room effects in the
future.

This work falls into the SFS technique category, and
more specifically by extending the results presented in
Ref. [27] to calculate the diffuse field sound absorption
coefficient of materials. This constitutes an alternative
approach to the reverberation chamber method that would
provide unbiased measurements between 100 Hz and 4 kHz
by solving the classical issues resulting from the use of stan-
dardized approaches (ISO 354 or ASTM C423 methods).
This paper also investigates the impact of angle step and
maximum angle on the Paris formula, reports on the influ-
ence of sample size on the absorption coefficient results and
compares the results obtained with the SFS method with
those calculated with the JCA model or measured with
the reverberation chamber method. The proposed approach
leverages some of the limitations observed in the work
published in Refs. [28, 29] for the estimation of the diffuse
field sound absorption coefficient (maximum incidence
angle limit, simplified spherical source-image model that
leads to erroneous results at low frequency, i.e. when the
product of the acoustic wavenumber and the source micro-
phone distance is small). Furthermore, the theoretical
framework proposed in this paper differs from that of Refs.
[28, 29]: a plane wave formalism versus a spherical wave
formalism, respectively.

It should be noted that one of the strengths of the
proposed method is its relative simplicity, especially consid-
ering the complexity of the problem it addresses. The exper-
imental setup requires only one sound source mounted on
motorized stages and two microphones. The required calcu-
lations are based on a single measurement sequence and an
integral calculation.

The paper is organized as follows: Section 2 recalls the
principle of the method for measuring the oblique incidence
absorption coefficient by sound field synthesis and extends
it to the calculation of the diffuse field absorption. Numer-
ical simulations using point sources are then performed in
Section 3 to validate the method and to identify key param-
eters that influences the precision of the estimated results.
Section 4 then presents experimental results that are
compared with theoretical predictions and reverberation
chamber measurements before summarizing the outcomes
in Section 5.
# 2 Theory

## 2.1 Plane wave synthesis

Consider a planar, square, regular array of L loudspeakers, with $\sqrt{L} \in \mathbb{N}^+$, depicted in Figure 1 by dark bullets. The array's side length is denoted $L_b$, and therefore the source spacing is $\Delta_l = L_l/(\sqrt{L} - 1)$. The array is placed parallel to and at a distance $h$ from the surface of a material to be tested. This array is used to generate acoustic plane waves of arbitrary angle of incidence. To this end, the input signals of the loudspeakers are filtered out to reproduce target sound pressure fields on a square surface of side length $L_m$ at the surface of the material ($z = 0$). The latter is sampled by $M$ microphones, with $\sqrt{M} \in \mathbb{N}^+$, separated by $\Delta_m = L_m/(\sqrt{M} - 1)$ (see white bullets in Fig. 1).

For each frequency $f$, the acoustic pressure $p_{lm}(f)$ generated by loudspeaker $l \in [1, L]$ at microphone $m \in [1, M]$ position is given by $p_{lm}(f) = g_{ml}(f)u_l(f)$, where $g_{ml}(f)$ is the source to microphone transfer function and $u_l$ is its input signal. Note that in the simulations, a point source model is used for $g_{ml}$ and $u_l(f)$ then corresponds to a source volume velocity whereas in the experiments $g_{ml}(f)$ is the measured transfer function and $u_l(f)$ is then the input voltage of loudspeaker $l$.

The vector of total sound pressure $\mathbf{p}$ at the $M$ microphone positions due to the $L$ loudspeakers can be calculated using a matrix formulation

$$ \mathbf{p}(f) = \mathbf{G}(f)\mathbf{u}(f), \quad (1) $$

with $\mathbf{p} = [p_1(f), \dots, p_m(f), \dots, p_M(f)]^T$, $\mathbf{u} = [u_1(f), \dots, u_m(f), \dots, u_M(f)]^T$, where the superscript $^T$ denotes the transposition, and

$$ \mathbf{G} = \begin{bmatrix} g_{11}(f) & \dots & g_{1l}(f) & \dots & g_{1L}(f) \\ \vdots & \ddots & \vdots & \ddots & \vdots \\ g_{m1}(f) & \dots & g_{ml}(f) & \dots & g_{mL}(f) \\ \vdots & \ddots & \vdots & \ddots & \vdots \\ g_{M1}(f) & \dots & g_{Ml}(f) & \dots & g_{ML}(f) \end{bmatrix}. \quad (2) $$

The target sound pressure at the microphone positions, an acoustic plane wave $p_l(f)$ with prescribed incidence angles $(\theta, \phi)$, is given by $\mathbf{p}_t = [p_{t1}, \dots, p_{tM}, \dots, p_{tL}]^T$, with

$$ p_{t_m} = e^{-jk(\sin \phi \cos \theta_{x_m} + \sin \phi \sin \theta_{y_m})}, \quad (3) $$

where $x_m$ and $y_m$ are the coordinates of the $m$th microphone in the plane $z = 0$, $k = 2\pi f/c$ is the acoustic wave number and $c$ the speed of sound in the air. The input signals of the loudspeakers are calculated by minimizing the error between the reproduced and target sound pressures as well as the squared inputs of the loudspeakers using the following cost function

$$ \begin{aligned} \mathcal{L} &= (\mathbf{p} - \mathbf{p}_t)^H (\mathbf{p} - \mathbf{p}_t) + \lambda \mathbf{u}^H \mathbf{u} \\ &= (\mathbf{G} \mathbf{u} - \mathbf{p}_t)^H (\mathbf{G} \mathbf{u} - \mathbf{p}_t) + \lambda \mathbf{u}^H \mathbf{u}, \end{aligned} \quad (4) $$

<figure><img src="image_3.png" /><figcaption>Figure 1. (Color online) Sketch of the measurement set-up. The dark blue bullets represent the locations of the loudspeaker while the white bullets represent the microphone positions for which the incident sound pressure field is constrained.</figcaption></figure>

where the superscript $^H$ denotes the Hermitian transpose and $\lambda$ is a Lagrange multiplier. By differentiating equation (4), with respect to $\mathbf{u}$ and equating it to 0, the input signals $u_l$ that minimize the above cost function at each frequency are given by

$$ \mathbf{u}_t = [\mathbf{G}^H \mathbf{G} + \lambda \mathbf{I}]^{-1} \mathbf{G}^H \mathbf{p}_t. \quad (5) $$

## 2.2 Absorption measurement

The set-up described in Section 2.1 is used to synthesize an acoustic plane wave with prescribed incidence angles $(\theta, \phi)$. The two-microphone technique [30] is then used to calculate the reflection coefficient under such excitation using the following expression

$$ R(\phi) = \frac{H - e^{jkz_a \cos \phi}}{e^{-jkz_a \cos \phi} - H} e^{2jkd \cos \phi}, \quad (6) $$

where $z_a$ is the elevation of the closest microphone with respect to the tested material, $d$ is the microphone separation and $H = p_{p1}/p_{p2}$ is the transfer function between the two microphones. Sound pressures $p_{p1}$ and $p_{p2}$ are respectively obtained at $(0, 0, z_a)$ and $(0, 0, z_2 = z_a + d)$. Note that the dependence on the plane wave azimuth angle $\theta$ does not appear in equation (6) as the tested sample is assumed to be homogeneous, isotropic and with large lateral dimensions. Thus, the estimated absorption coefficient only depends on $\phi$, and is obtained as $\alpha(\phi) = 1 - |R(\phi)|^2$.

The calculation of the diffuse field absorption coefficient $\alpha_d$ from the angle dependent coefficient $\alpha(\phi)$ was first discussed in 1928 [31] thus proposing what became the so-called Paris law

$$ \alpha_d = \int_0^{\pi/2} \alpha(\phi) \sin 2\phi \, d\phi. \quad (7) $$
Improvements to this formula have been proposed to take into account the transition from a free field condition to an actual diffuse field [32, 33].

In this work, the Paris formula is used to estimate the diffuse field sound absorption coefficient, but with an upper integration bound. This limit is the maximum angle $\phi_{lim}$ above which the contribution of the absorption coefficient has a lower influence, because of the sin $2\phi$ factor. Knowing the effect related to this maximum angle is interesting in practice as it is usually difficult to measure absorption coefficients for large incidence angles. The diffuse field sound absorption is then calculated as

$$ \alpha_d = \frac{\int_0^{\phi_{lim}} \alpha(\phi) \sin 2\phi \, d\phi}{\int_0^{\phi_{lim}} \sin 2\phi \, d\phi}. \qquad (8) $$

Note that the idea of using a higher integration bound has been used before, but rather to take into account the fact that ideally diffuse conditions cannot usually be obtained in real rooms, see for instance Ref. [34].

## 3 Numerical simulations

Numerical simulations are first reported to test the method under ideal conditions. A porous layer with a rigid frame is modelled using the Johnson-Champoux-Allard (JCA) model [35], see Appendix. Two different numerical evaluations of equation (8) are performed:

* The first one is used as a reference and labelled “Ref eval” in Figure 3. Equation (8) is computed with the analytical value of $\alpha(\phi)$ detailed in Appendix depending on the JCA parameters using the function quad\_vec from the Python Scipy package [36] until a threshold error of $10^{-9}$ is reached.

* The second one, labelled “Dscrt eval”, is performed by simulating the sound field reproduction process using equations (1)–(6) and performing the integration in equation (8) using the function trapz from the Python Numpy package [37]. It mimics the plane wave reproduction process and discrete measurements of $\alpha(\phi)$ over $N_{\phi}$ angles, as performed in experiments. The incidence angle is thus sampled over $[0, \phi_{lim}]$, with a discrete angular step $\Delta\phi_n = \phi_{lim}/(N_{\phi} - 1)$. An approximation of equation (8) is therefore obtained from this calculation.

Using these numerical evaluations, it is possible to study the influence of two key parameters on the estimation of $\alpha_d$: (1) The discrete angular step $\Delta\phi$, and (2) the maximum incidence angle $\phi_{lim}$.

The sound absorption coefficient of a 5 cm thick sample of polyurethane foam, PU1, is computed for various incidence angles using the “Ref eval” method. The JCA parameters for this material are given in column PU1 of Table 1. Results are shown in Figure 2 and highlight the angle dependence of the absorption coefficient of such absorbing materials.

In the proposed measurement procedure, the diffuse field absorption coefficient is calculated by averaging results obtained for discrete plane wave incidence angles (method “Dscrt eval”). Results obtained for equation (8) with $\phi_{lim} = 88^{\circ}$ and for different discretization numbers $N_{\phi} = 3, 6, 12$ and $24$ (and thus different angular steps $\Delta\phi = 44^{\circ}, 18^{\circ}, 8^{\circ}$ and $4^{\circ}$) are plotted in Figure 3. It can be noticed that the absorption coefficient reaches its analytical and reference value even for a limited number of discrete incidences $N_{\phi}$: with 6 incidence angles the evaluation based on discrete angles almost perfectly coincides with the reference curve. To highlight this result, an error criterion is calculated: $\epsilon_x = |\alpha_{ref} - \alpha_{Dscrt}|$, where $\alpha_{ref}$ corresponds to the diffuse field absorption coefficient calculated with the “Ref eval” method while $\alpha_{Dscrt}$ is the one obtained with the discrete evaluation. In Figure 3, the error is less than 1% for $N_{\phi} \ge 6$. For a material exhibiting this type of angular dependence of the absorption coefficient, only a limited number of plane wave incidence angles (typically around 10) can be used to obtain a correct estimation of the diffuse field absorption coefficient.

Another significant parameter is the maximum incidence angle $\phi_{lim}$ in the computation of $\alpha_d$. This parameter is particularly important because, as reported in the work of [27], the evaluation of $\alpha(\theta)$ by the proposed sound field synthesis becomes inaccurate for large incidence angles, typically above 60°. This is due to the difficulty in reproducing grazing plane waves with the source and microphone configurations depicted in Figure 1. Results computed with the “Ref eval” method are plotted in Figure 4 for various values of $\phi_{lim}$ between 45° and 88°. It is observed that for $\phi_{lim} \ge 78^{\circ}$, the results are very close to the true absorption coefficient. Again, the error curves $\epsilon_x$ in Figure 4 provide a useful insight into the effect of the maximum angle of integration. A 78° value is consistent with the limit angle values reported in the scientific literature [13, 38] when using the Paris formula.

## 4 Measurements

### 4.1 Experimental set-up and method

The method described in Section 2 was tested experimentally. Measurements were performed in a semi-anechoic chamber using a single, moving loudspeaker instead of a full loudspeaker array. A FaitalPRO 3FE22 3" loudspeaker, mounted in a closed box, with a bandwidth of 150–20 000 Hz at –3 dB, was moved over a regular square grid 50 cm above the tested porous layer using two Zaber linear motorized stages (Fig. 5). The value of 50 cm was chosen from the simulations carried out in Ref. [27], it allows the spherical pressure fields radiated by each of the sources to mix well enough to obtain a plane wave on the surface of the material under test. The source grid consists of 8 × 8 points forming a regular array of 1.05 m side length, resulting in a spacing between the sources of 15 cm and a Nyquist frequency of 1140 Hz. At the surface of the material, the incident acoustic field is constrained using equation (4) on a 20 cm square surface, regularly sampled
<table><caption>**Table 1.** JCA model parameters of the two polyurethane foams (PU1 and PU2) and glass wool (GW) used in numerical simulations and experiments.</caption><thead><tr><th>Parameter</th><th>PU1</th><th>GW</th><th>PU2</th></tr></thead><tbody><tr><td>Porous layer thickness h<sub>p</sub> [m]</td><td>0.05</td><td>0.05</td><td>0.05</td></tr><tr><td>Flow resistivity σ [N s m<sup>-4</sup>]</td><td>3320</td><td>13112</td><td>10605</td></tr><tr><td>Porosity φ [-]</td><td>0.97</td><td>0.954</td><td>0.956</td></tr><tr><td>Viscous characteristic length Λ [μm]</td><td>163</td><td>55.6</td><td>71.0</td></tr><tr><td>Thermal characteristic length Λ' [μm]</td><td>255</td><td>116</td><td>317</td></tr><tr><td>Tortuosity α<sub>∞</sub> [-]</td><td>1.06</td><td>1.0</td><td>1.67</td></tr></tbody></table>

<figure><img src="image_4.png" /><figcaption>**Figure 2.** Theoretical absorption coefficient $\alpha(\phi)$ of material PU1 for various incidence angles. Values are computed using the “Ref eval” method.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>**Figure 4.** Absorption coefficient $\alpha_d$ and error criterion $\varepsilon_x$ for material PU1 computed for various values of $\phi_{lim}$ using the “Ref eval” method.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>**Figure 3.** Absorption coefficient $\alpha_d$ and error criterion $\varepsilon_x$ for material PU1 computed for different discretization numbers $N_\phi$ with $\phi_{lim} = 88^\circ$.</figcaption></figure>

by 13 × 13 virtual microphones, giving a sensor spacing $\Delta_m$ of about 1.7 cm. These geometric parameters have been optimized in [27] to provide an accurate estimation of $\alpha(\phi)$. The matrix G of source to constraint microphones in equation (1) was calculated assuming free-field conditions

and omnidirectional point sources. This assumption is further discussed in this section, and in [27].

A two-microphone probe with a $d = 3$ cm spacing was composed of two 1/4" Brüel and Kjær microphones plugged to Nexus conditioning amplifiers. The lower microphone of the probe was set at $z_1 = 0.5$ cm from the material's surface. Generation of the loudspeaker signal for each source position and acquisition of the probe signals were performed via a custom Python code which controlled an RME Madiface XT audio interface. The excitation signal was a 12.1 s logarithmic sine swept signal ranging from 100 Hz to 7 kHz, amplified by a HPA D604 unit. Four averages per source position were performed to improve the signal-to-noise ratio and to obtain accurate transfer functions from each source position to each of the probe microphones. The measurement duration for 64 loudspeaker positions is approximately 2 h.

The measured transfer functions are then arranged in the following matrix $\mathbf{H}_p$:

$$ \mathbf{H}_p = \begin{bmatrix} h_{11}(f) & \dots & h_{1l}(f) & \dots & h_{1L}(f) \\ h_{21}(f) & \dots & h_{2l}(f) & \dots & h_{2L}(f) \end{bmatrix}, \quad (9) $$

where $h_{1l}$ and $h_{2l}$ denote the transfer functions from loudspeaker $l$ to microphone 1 and 2 respectively.

The following procedure is followed to calculate the absorption coefficient under any combination of angles ($\theta \in [0, 360^\circ]$, $\phi \in [0, 90^\circ]$):
<figure><img src="image_7.png" /><figcaption>**Figure 5.** Experimental set-up installed in the Laboratoire d'Acoustique de l'université du Mans semi-anechoic room. The material under test is PU2, see Table 1.</figcaption></figure>

1. Following a target sound pressure field, equation (3), the loudspeaker input signals $u_t$ are calculated using equation (5) and the terms of the **G** matrix are calculated using a point source model. This assumption is satisfied up to approximately $ka = 1$, where $a = 3.2$ cm is the loudspeaker's radius, yielding an upper frequency limit of approximately $1700$ Hz. A value of $\lambda = 10^{-7}$ is used unless otherwise stated.

2. The sound pressures at the microphone probe $\mathbf{p}_b = [p_{p1}(f), p_{p2}(f)]^T$, are calculated for each frequency using $\mathbf{p}_b = \mathbf{H}_p \mathbf{u}_t$.

3. $H(f) = p_{b1}(f)/p_{b2}(f)$ is finally calculated from $\mathbf{p}_b$ and used in equations (6)–(8) to obtain the sound absorption coefficients $\alpha(\phi)$ and $\alpha_d$.

## 4.2 Experimental results

Nine square sections of 50 cm side length of polyurethane foam (PU1 material, see Tab. 1) were assembled to form a square of 1.5 m side length. The measured sound absorption coefficients for five values of $\phi$ are plotted in Figure 6. Note that the absorption coefficient $\alpha(\phi)$ was computed from the reflection coefficient $R(\phi)$ which has been linearly averaged over 10 values of the azimuth

<figure><img src="image_8.png" /><figcaption>**Figure 6.** Theoretical (JCA) versus measured (SFS) plane wave absorption coefficient $\alpha$ of material PU1 for various plane wave incidence angles. The shaded zones indicate the standard deviation calculated from 10 linearly spaced values of azimuth angle $\theta$.</figcaption></figure>

angle $\theta$ equispaced in [0°, 180°]. Averaging the reflection coefficient $R(\phi)$ along the azimuth angle assumes that the material has a behavior that does not depend on the angle $\theta$. This assumption seems valid for the materials tested here, indeed polyurethane foams or glass wool generally exhibit an isotropic or transverse isotropic behavior. Moreover, the size of the tested materials combined with a near field excitation seems to be sufficient to limit the diffraction from the sample edges (this last point will be verified later on in Fig. 9). The standard deviation of the absorption coefficient (indicated by the shaded area of Fig. 6), is evaluated from the averages over azimuth angle.

The measured values and the absorption coefficient derived from the JCA model are in good agreement for incidence angles up to 60°. For $\phi = 60°$, the standard deviation calculated for 10 values of the azimuth angle $\theta$ is rather large. For $\phi = 88°$, the measured curve follows the theoretical prediction up to 3 kHz and then deviates from the expected results. In the low and medium frequency range, the oscillations observed in the absorption coefficient curves may be due to diffraction at the edges of the tested sample, as well as potential contributions from room modes. At high frequencies, the oscillations can be attributed to acoustic reflections from the motorized linear stages, and also to the mismatch between the hypothesis of an omnidirectional point source and the actual and more complex directivity of the used loudspeaker.

The values obtained for 15 $\phi$ angles ranging from 0° to 88° are averaged using the “Dscrt eval” method to evaluate the diffuse field absorption coefficient $\alpha_d(f)$ using equation (8). Results are shown in Figure 7 and compared to theoretical values calculated with “Ref eval” and the parameters of the JCA model given in Table 1. The two curves are in very good agreement from 100 Hz to 3 kHz. Theoretical and measured values in third octave bands are indicated with bullet points, showing also a very good agreement between
<figure><img src="image_9.png" /><figcaption>Figure 7. Theoretical (JCA) versus measured (SFS) diffuse field absorption coefficient $\alpha_d$ of material PU1 with $\phi_{lim} = 88^\circ$ and $\lambda = 10^{-7}$. Averaged values for third octave bands are indicated by the round markers. The angle-averaged value of $\varepsilon_{th}$ is also plotted.</figcaption></figure>

experiments and theory. Figure 7 also shows the mean value of the reproduction error criterion $\varepsilon_{th}$ given by

$$ \varepsilon_{th} = \frac{|\mathbf{p}_t - \mathbf{G}\mathbf{u}_t|_2}{|\mathbf{p}_t|_2}, \quad (10) $$

which estimates the deviation between the sound field $\mathbf{G}\mathbf{u}_t$ radiated by the loudspeaker array assuming a point source model for the loudspeakers and the target plane wave $\mathbf{p}_t$. The $\varepsilon_{th}$ curve in Figure 7 shows that the discrepancy between the experimental and theoretical absorption coefficients logically increases when the reproduction error $\varepsilon_{th}$ becomes especially large (above 4000 Hz). In order to obtain accurate results at higher frequencies (and thus reduce $\varepsilon_{th}$), it is possible to reduce the distance between the source and constraint microphone arrays [27]. Reducing the size of the source array would, however, decrease the quality of the results at low frequencies. Further study should be carried out to optimize the bandwidth of the method for a given number $L$ of sources. Another source of error at high frequencies is the directivity of the source used, that will not follow the omnidirectional assumption since the radius of the loudspeaker diaphragm becomes small compared to the acoustic wavelength (the loudspeaker enclosure also brings diffraction effects). This problem could be solved by measuring the actual directivity of the loudspeaker and incorporating it into the $g_{ml}$ transfer functions.

As the accuracy of the proposed method decreases for large incidence angles, the effect of the maximum angle $\phi_{lim}$ considered in equation (8) is now investigated. The results given in Figure 8 show that a maximum angle of $55^\circ$ is not sufficient to reconstruct a diffuse field absorption coefficient while results obtained with $\phi_{lim} = 78^\circ$ and $88^\circ$ are almost superimposed. This is expected since plane waves with large incidence angles provide larger absorption in low frequency (see Fig. 2). Nevertheless, the $\varepsilon_{th}$ reproduction error curves in Figure 8 show that increasing $\phi_{lim}$ also

<figure><img src="image_10.png" /><figcaption>Figure 8. Theoretical (JCA) versus measured (SFS) diffuse field absorption coefficient $\alpha_d$ of material PU1 for different $\phi_{lim}$ with $\lambda = 10^{-7}$. The third octave band averages are indicated by the round markers. The angle-averaged value of $\varepsilon_{th}$ is also plotted.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Figure 9. Theoretical (JCA) versus measured (SFS) diffuse field absorption coefficient $\alpha_d$ of material PU1 with $\phi_{lim} = 88^\circ$ and $\lambda = 10^{-6}$. Measurements have been performed for three different material sizes. The third octave band averages are indicated by the round markers. The angle-averaged value of $\varepsilon_{th}$ is also plotted.</figcaption></figure>

increases the reproduction error since grazing plane waves are more difficult to reproduce with the experimental setup. As a consequence, the results obtained with $\phi_{lim} = 55^\circ$ are closer to the theoretical diffuse field coefficient than the results obtained with larger values of $\phi_{lim}$ above 3 kHz.

Finally, the effect of the size of the tested material is studied using square samples with three side lengths: 50 cm, 100 cm and 150 cm. The matrix $\mathbf{H}_p$ is measured for each configuration, and used to compute corresponding $\alpha_d$. Results are presented in Figure 9. The largest sample size provides the best agreement between theory and experiment, and over a wide frequency range. Absorption values obtained for the smallest side length largely deviate from the theoretical predictions, but the values obtained with a $1\,m^2$ area are very close to those obtained with the largest size ($2.25\,m^2$ area) above 300 Hz.
<figure><img src="image_12.png" /><figcaption>**Figure 10.** Picture of the measurement set-up in the reverberation room showing the material under test, 5 of the 8 microphones and a group of two sources.</figcaption></figure>

## 4.3 Comparison with reverberation chamber measurements

Comparison measurements were conducted in a reverberation chamber following a standardized procedure [3], see the picture in Figure 10. The volume of the reverberation chamber is 335 m³, and reverberation times are measured using eight Bruël & Kjær 1/2" microphones and two independent white noise signals generated simultaneously by two groups of two sources. Two materials are tested: a polyurethane foam (PU2) and a glass wool (GW), see Table 1 for their JCA parameters (the available surface for PU1 could not meet the requirement of ISO354 standard). A surface of 12 m² was used for the reverberation chamber measurements while an area of 1.5 m by 1.5 m was used for the sound field synthesis method. For the latter method, the following parameters were used: $\lambda = 10^{-7}$, $N_{\phi} = 15$, $N_{\theta} = 20$ and $\phi_{\text{lim}} = 88^{\circ}$.

Results obtained for the PU2 material are plotted in Figure 11. Between 200 Hz and 2 kHz, the results obtained using the sound field synthesis method are in line with the theoretical predictions, while results obtained using the standardized method show an overestimation of the absorption coefficient, with a maximum discrepancy of 0.18 at 400 Hz with theoretical predictions (33% overestimation). Above 2 kHz, the reproduction error increases rapidly and the sound field synthesis method leads to slightly underestimated sound absorption coefficient compared with theoretical predictions.

Note that the overestimation of absorption coefficients in the reverberation chamber is a known result. However, it seemed interesting to compare the results of the SFS

<figure><img src="image_13.png" /><figcaption>**Figure 11.** Comparison of the diffuse field absorption coefficient $\alpha_d$ of the PU2 material between JCA model predictions, reverberation room measurement and sound field synthesis method (with $\lambda = 10^{-7}$ and $\phi_{\text{lim}} = 88^{\circ}$). The third octave band averages are indicated by the round markers. The angle-averaged value of $\varepsilon_{\text{th}}$ is also plotted.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>**Figure 12.** Comparison of the diffuse field absorption coefficient $\alpha_d$ of the GW material for three methods: JCA model, reverberation room and field synthesis method (with $\lambda = 10^{-7}$ and $\phi_{\text{lim}} = 88^{\circ}$). The third octave band averages are indicated by the round markers. The angle-averaged value of $\varepsilon_{\text{th}}$ is also plotted.</figcaption></figure>

method to both a theoretical model and to what is currently proposed by standards in order to illustrate the potential of the technique proposed here.

Results for the GW material are presented in Figure 12. The results obtained using the reverberation chamber method now largely overestimate the absorption coefficient values predicted using the JCA model. Values even exceed unity between 400 Hz and 2 kHz, with a maximum difference of 0.35 at 500 Hz (56% overestimation). The sound field synthesis results, like for the PU2 material, are in line with the theoretical prediction but with a slight overestimation of the absorption between 200 and 800 Hz (maximum difference of 0.08). This discrepancy is attributed to the
non-isotropic behavior of the glass wool, since an isotropic model was used to calculate theoretical values from parameters obtained under normal incidence.

## 5 Conclusion

In this paper, a sound field synthesis method was proposed and used to estimate the diffuse field sound absorption coefficient of materials. The method consists of generating plane waves with prescribed incidence angles and then measuring the corresponding absorption coefficient using a two-microphone probe. Instead of a physical loudspeaker array, a single and mobile loudspeaker is used and sound field synthesis calculations are performed at a post-processing step. The diffuse field absorption coefficient is then obtained by using a discrete version of the Paris formula. The method has been tested on three different materials and provided measured absorption coefficients close to values predicted with the JCA model between 150 Hz and 3 kHz. Compared with measurements conducted following the reverberation chamber method, the required sample area is reduced by a factor larger than 5, and the estimated sound absorption coefficients never exceed unity.

One direction for improvements is a bandwidth extension, in order to be able to characterize a material over octave bands from 63 Hz to 4 kHz. At low frequency, a loudspeaker with a lower cut-off frequency or a microphone probe with a larger spacing could improve the measurements. At high frequency, the use of coaxial loudspeakers and a properly tuned crossover filter could provide an omnidirectional source up to a higher frequency. Another solution is to measure the G matrix for all sources and constraint microphone positions as was done in [27], at the expense of a much longer measurement time. Another direction for improvements at large incidence angles could be to use a non-planar loudspeaker array, convex as an example. It is indeed more difficult for a planar array to radiate a plane wave in a direction with a large angle from the normal direction of the array.

## Acknowledgments

This work was performed within the framework of the "Centre Acoustique Jacques Cartier", an International Research Project labeled by the Centre National de la Recherche Scientifique (CNRS). The authors would like to thank Nicolas Poulain at CTTM for performing the measurements used to obtain the parameters of the JCA model, James Blondeau and Félix Lebeuf for their help with the measurements.

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# Appendix

## A.1 Johnson-Champoux-Allard Model

The absorbing materials used in this paper are modelled using the equivalent fluid theory of Johnson - Champoux - Allard (JCA). The JCA parameters used to calculate the theoretical absorption curves are listed in Table 1. They were measured by taking circular samples from the tested materials and measuring their normal acoustic impedance in a tube equipped with an impedance sensor. The flow resistivity and the porosity are deduced from the low frequency behaviour of the normal impedance [39] while the other parameters are obtained by inverting an analytical model governing the wave propagation in the material [40, 41].

The absorption coefficient $\alpha(\phi)$ for any incidence angle $\phi$ is calculated using

$$ \alpha(\phi) = 1 - \left| \frac{z_s(\phi) - \frac{z_0}{\cos(\phi)}}{z_s(\phi) - \frac{z_0}{\cos(\phi)}} \right|^2, \quad (11) $$

where $z_0$ is the characteristic impedance of air and $z_s$ is the surface acoustic impedance of the material which is given by

$$ z_s = -j \frac{z_p}{\cos \psi_p} \cot(h_p k_p \cos \psi_p), \quad (12) $$
where $z_p$ is the characteristic impedance of the material and $\psi_p$ is the angle of refraction in the material which is obtained from

$$ \cos \psi_p = \sqrt{1 - \frac{k^2}{k_p^2} \sin^2 \phi}. \quad (13) $$

The characteristic impedance of the porous medium is equal to

$$ z_p = \sqrt{\rho_{\text{eff}} K_{\text{eff}}}, \quad (14) $$

while the complex wave number in the materials is given by

$$ k_p = \omega \sqrt{\frac{\rho_{\text{eff}}}{K_{\text{eff}}}}. \quad (15) $$

The expressions of the effective density $\rho_{\text{eff}}$ and bulk modulus $K_{\text{eff}}$ of the porous medium depend on the material parameters listed in Table 1 and are given in [35].

**Cite this article as:** Dupont S. Sanalatii M. Melon M. Robin O. Berry A, et al. 2023. Measurement of the diffuse field sound absorption using a sound field synthesis method. Acta Acustica, 7, 26.