PROCEEDINGS of the23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Measurement of oblique-incidence absorption coefficients ofvarious types of absorbing materials in a thin chamber
Tetsuya SAKUMA1; Naohisa INOUE2; Yuta SAKAYOSHI3

31

The University of Tokyo, Japan

2

The University of Tokyo, Japan

Nippon Sheet Glass Environment Amenity Co., Ltd, Japan

ABSTRACT
A novel measurement system for oblique-incidence absorption coefficients has been developed using a thinrectangular chamber and two arrays of microphones. In this method, a series of incident and reflected planewaves at corresponding angles are decomposed based on a propagation mode expansion technique.
Accordingly, absorption coefficients are determined at discrete incidence angles, of which numberincreases with frequency. First, a test measurement for glass wool was done in comparison with theoreticalestimates. As a result, the measurement system was basically validated, but abnormal values wereoccasionally observed at singular combinations of frequency and incidence angle. The faults theoreticallyoccur under two conditions: the interval of microphone arrays coincides with a half wavelength inincidence angle, and some modes are not generated due to loudspeaker position mismatch. Therefore, twopost-processes were incorporated to exclude abnormal values systematically. Then, measurements weredone for various types of absorbing materials, such as urethane foam, rock wool board, perforated panel,and honeycomb resonators, and their incidence angle dependence of absorption coefficients was examined.
Keywords: Absorption coefficient, Oblique incidence, Mode expansion

1. INTRODUCTION
There exist several methods for measuring oblique-incidence sound absorption coefficients in thefree or semi-free field, which generally requires a specimen with a large surface area [1-3]. On theother hand, we have recently proposed a novel method where the measurement is done for a strip ofspecimen in a thin rectangular chamber, and applying a propagation mode expansion technique with
two arrays of microphones [4]. In the thin-chamber method, a series of incident and reflected planewaves at corresponding angles are decomposed in the two-dimensional field, which is similar to theimpedance tube method for measuring normal-incidence absorption coefficients. Accordingly,
normal- and oblique-incidence absorption coefficients can be determined at a series of incidence
angles corresponding to the propagation modes, of which number increases with frequency. However,some faults can theoretically occur under two singular conditions: the interval of microphone arrays
coincides with a half wavelength in incidence angle, and some modes are not excited due toloudspeaker position mismatch.
In the first part of this paper, a test measurement is done for glass wool to validate themeasurement system in comparison with theoretical estimates, where two post-processes areincorporated to exclude abnormal values systematically under the two unsuitable conditions.
Furthermore, using the measured values at all the combinations of frequency and incidence angle,
oblique-incidence absorption coefficients at 1/3-octave bands are estimated by the least squares
method. In the latter part, measurements are done for various types of absorbing materials, such as
urethane foam, rock wool board, perforated panel, and honeycomb resonators. In comparison withtheoretical estimates, their incidence angle dependence of absorption coefficient is examined.

1

sakuma@k.u-tokyo.ac.jp
inoue@env-acoust.k.u-tokyo.ac.jp
3
Yuta.Sakayoshi@nsg.com
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2. MEASUREMENT METHOD AND SYSTEM
2.1 Measurement Principle
Considering a two-dimensional sound field with rigid boundary conditions at x = 0 and x = W(Fig. 1), sound pressure is expressed as the superposition of propagation modes, that is,∞()()

p ( x, y ) = ∑ ⎡⎣ an exp − jkyn y + bn exp jkyn y ⎤⎦ cos ( jkxn x ) ,n=0(1)where k xn and k yn are nth wave numbers in x and y directions. The nth mode satisfies kxn = nπ W
and kyn = k 2 − kxn2 , thus its traveling angle is defined as⎛ kn ⎞⎛ nc ⎞
θ n = arcsin ⎜ x ⎟ = arcsin ⎜.
(2)⎝ 2Wf ⎟⎠⎝ k⎠
Supposing that the surface of a specimen is at y = 0 , an and bn are the amplitudes of the incident
and reflected plane waves, respectively. Accordingly, the oblique-incidence absorption coefficient at2
the incidence angle θ n is given by α n = 1− bn an .
Multiplying Eq. (1) by mth x-directional basis, cos jkxm x , and integrating it over x axis can
extract the average sound pressure for mth propagation mode as follows:

()

εm W
p ( x, y ) cos ( jkxm x ) dx = am exp − jkym y + bm exp jkym y ,(3)
W ∫0
where ε m is 1 for m = 0 , and 2 for m > 0 . In the measurement, sound pressures are measured in
the lines y1 and y2 , and then, pm are calculated for each mode at the two lines. Finally, the(pm ( y ) =)()amplitudes of the incident and reflected waves of each mode can be determined by⎡ exp jk m y− exp jkym y1 ⎤ ⎧⎪ pm ( y1 ) ⎫⎪⎧⎪ am ⎫⎪y 21⎢⎥⎨⎬=⎨
⎬,m
⎢
mmb2
jsink
Δy− exp − jky y2 exp − jky y1 ⎥ ⎪ pm ( y2 ) ⎪y⎪⎩ m ⎪⎭⎢⎣⎥⎦ ⎩⎭

()

(())

(())

(4)

provided that kym Δy ≠ nπ where Δy = y2 − y1 .

Figure 1 – Propagation modes in two-dimensional sound filed.
2.2 Measurement System
Figure 2 shows the geometry of a thin chamber with dimensions of W = 1.3 m, D = 0.6 m, and H =0.05 m, where a strip of specimen is installed along a long side. A loudspeaker can be installed at oneof 10 positions on the opposite side to a specimen, and the position near a corner (no.10) is chosen inorder to generate all propagation modes. The interval of two lines for microphone arrays is 8 cm, and64 receiving points are located at an interval of 2 cm in each line. In actual, changing two circularholders with 4 microphones, 8ch impulse response measurements using a 1.37-second log swept sine
signal are done in turn. Subsequently, complex sound pressures are calculated at all points by theFourier transform, and the average sound pressures of each propagation mode are calculated at thetwo receiving lines using the trapezoidal rule in Eq. (3).

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Figure 2 – Geometry of the thin chamber with two microphone arrays and a loudspeaker.
2.3 Measurable Frequency Range and Angles
The above thin-chamber system has limitations on frequency and incidence angle in themeasurement. First, because of the requirement for two-dimensional field, the height of the chamber
gives the upper limit frequency, fu = c 2H , which is 3.4 kHz for this chamber. Second, the incident
and reflected waves cannot be determined by Eq. (4) under the condition, k ym Δy = π , where theinterval of microphone arrays coincides with a half wavelength in incidence angle. The singularfrequency for each mode is expressed by2c ⎛ m⎞1(5)⎜⎝ ⎟⎠ + 2 .
2 WΔyTherefore, measurement faults possibly occur at specific incidence angles above the lowest singularfrequency, fs0 = c 2Δy , which is 2.1 kHz for this chamber. If avoiding the faults under fu , theinterval of microphone arrays should be set less than the height of chamber, in a similar manner tothe impedance tube method. Figure 3 shows the measurable frequency range and incidence angles inthis chamber, where the number of propagation modes increases with frequency. For instance, the1/3-octave band of 500 Hz contains 4 oblique-incidence modes and a normal-incidence mode, which
could be enough to estimate the incidence angle dependence of absorption coefficient.
fsm =Figure 3 – Measurable frequency range and incidence angles in the chamber.

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3. VALIDATION OF THE MEASUREMNT SYSTEM
A test measurement of oblique-incidence absorption coefficients is done for glass wool with abulk density of 32 kg/m 3 and a thickness of 25 mm. First, two post-processes are examined for
excluding outliers due to the interval of microphone arrays and loudspeaker position mismatch,respectively. Subsequently, using the measured values, oblique-incidence absorption coefficients at
1/3-octave bands are estimated by the least squared method.
3.1 Treatment for Singular Frequencies
Figure 4 (a) shows the oblique-incidence absorption coefficients of the glass wool, measured in
the 1/3-octave band of 2000 Hz. As is illustrated in Fig. 3, this band includes singular frequencies
due to the interval of microphone arrays, thus causing abnormal values between 0 and 20 degrees.
Therefore, a post-process for excluding the outliers is introduced, where measured values areremoved if kym Δy is within (1 ± Δ)π . Figures 4 (b), (c) and (d) show the results after thepost-process with Δ = 0.01, 0.05 and 0.1. It is seen that the outliers below 20 degrees steadily
disappear with increasing Δ, however, those around 40 degrees still remain. In the following, Δ is
fixed at 0.1 for this post-process.

Figure 4 – Oblique-incidence absorption coefficients measured in 1/3-octave band of 2000 Hz:(a) raw, and (b) processed with Δ = 0.01, (c) Δ = 0.05, (d) Δ = 0.1.
3.2 Treatment for Unexcited Modes
Figure 5 shows the relative sound pressure levels of incident and reflected plane waves
decomposed for calculating the absorption coefficients in Fig. 4 (a). It is seen that the levels of
incident waves remarkably drop around 40 degrees, which is considered to result in the remaining
outliers in Fig. 4 (d). Figure 6 illustrates the positional relation of the loudspeaker position (no.10)
with the three propagation modes (9th, 10th and 11th). In theory, the 10th mode cannot be excited,
and there is a possibility that the two adjacent modes will be not sufficiently excited. Therefore,
another post-process for excluding unexcited modes is additionally introduced, where measured
values are removed if the amplitude of incidence wave is less than Γ times the average amplitude
over all modes. Figure 7 shows the results before and after the second post-process with Γ = 1/4, 1/2
and 3/4. It is seen that the outliers after the first post-process are sufficiently excluded with Γ = 1/2(-6 dB relative to the average level), thus this criterion is used in the following.

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Figure 5 – Relative sound pressure level of incident and reflected waves in 1/3-octave band of 2000 Hz.

Figure 6 – Loudspeaker position mismatch for three propagation modes.

Figure 7 – Oblique-incidence absorption coefficients measured in 1/3-octave band of 2000 Hz:(a) before the second process, and (b) processed with Γ = 1/4, (c) Γ = 1/2, (d) Γ = 3/4.
3.3 Estimation of Incidence Angle Dependence
Figure 8 (a) shows the measured values at the center frequency of each 1/3-octave band, butwhich are insufficient and uncertain to estimate the incidence angle dependence of absorption
coefficient over all angles. On the other hands, Fig. 8 (b), (c) and (d) show the measured values at all
combinations of frequency and incidence angle in the three bands, which will be useful to estimate
the incidence angle dependence in each band. Accordingly, estimations using all the measured values
excluding outliers are tested with two different curve fittings using the least squares method. One is a
global fitting with a high-order polynomial, and another is a piecewise estimation where lower-order1362

polynomials are used for the three angle ranges divided at 60 and 85 degrees. In the two fittings, theabsorption coefficient at 90 degrees is assumed to be zero, and the continuity of slope is imposed at
the two dividing angles in the piecewise fitting.
Figure 9 shows the absorption coefficients of glass wool with thicknesses of 25 and 50 mm,estimated by the two curve fittings at 500, 1000 and 2000 Hz, where 5th-order polynomials are usedin the global fitting, while for the piecewise estimation, 2nd-, 4th- and 1st-order polynomials areused below 60 degrees, between 60 and 85 degrees, and above 85 degrees, respectively. In addition,
theoretical lines are indicated, which are calculated with some parameters that are estimated fromnormal-incidence absorption coefficients based on Kato model [5]. It is seen that the estimatedresults with the two fittings are generally in good agreement with the theoretical ones, although theestimates tend to be greater around 70 degrees, especially at 500 Hz. Moreover, small unnaturalfluctuations appear below 60 degrees in the results with the global estimation, so that, the piecewise
estimation may be more suitable for this purpose. Finally, the feasibility of the developed
measurement system is validated in the case with glass wool.

Figure 8 – Oblique-incidence absorption coefficients measured in 1/3-octave bands:(a) center frequencies, (b) 500 Hz band, (c) 1000 Hz band, (d) 2000 Hz band.

Figure 9 – Incidence angle dependence of glass wool (32 kg/m3) with a thickness of (a) 25 mm, (b) 50 mm,
estimated by two curve fittings and by theory at 500, 1000 and 2000 Hz:

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4. MESUREMENTS FOR VARIOUS ABSORBING MATERIALS
4.1 Porous Materials
Measurements of oblique-incidence absorption coefficients are done for three types of porous
materials with different bulk densities and thicknesses: a glass wool (ρ b = 32 kg/m 3 , t = 25 mm), anurethane foam (ρ b = 24 kg/m 3 , t = 25 mm), and a rock wool board (ρ b = 325 kg/m 3 , t = 9 mm). Figure
10 shows the measured values excluding outliers in the 1/3-octave bands of 500, 1000 and 2000 Hz,
and the theoretical lines at the center and upper/lower cutoff frequencies. Although somediscrepancies are seen in the 500 Hz band, the measured results well correspond with the theoreticalones regardless of the type of porous materials. It is noted that theoretical absorption coefficientsaround 70 degrees are considerably sensitive to the parameters estimated under the normal-incidence
condition, and furthermore, the effect of anisotropy will be included in the measured results.

Figure 10 – Oblique-incidence absorption coefficients measured for three porous materials in 1/3-octave
bands of 500, 1000 and 2000 Hz: (a) glass wool, (b) urethane foam, (c) rock wool board.
4.2 Perforated Panels
In order to examine the incidence angle dependence of absorption coefficient of resonator-typeabsorbers, measurements are done for two types of perforated panel structure. One is a 5 mm thickboard with holes (diameter: 5 mm; pitch: 25 mm) and a 20 mm thick air layer, of which resonancefrequency is 750 Hz. The other is a honeycomb resonators panel composed of a 1 mm thick front
plate with holes (diameter: 1 mm; pitch: 8 mm), a 18 mm thick core, and a 1 mm thick back plate, ofwhich resonance frequency is 1015 Hz. Figure 11 shows the measured values around 1000 Hz, withtheoretical lines estimated by Atalla’s model [6]. It is seen that the ordinary perforated panel has
strong incidence angle dependence, where a resonance peak shifts with frequency. On the other hand,
the honeycomb resonators panel has no incidence angle dependence below 60 degrees, corresponding
to the theoretical estimates.

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Figure 11 – Oblique-incidence absorption coefficients measured for three porous materials in 1/3-octave
bands of 794, 1000 and 1260 Hz: (a) perforated panel, (b) perforated honeycomb panel.

5. CONCLUSIONS
The developed thin-chamber system for measuring oblique-incidence absorption coefficients wastested, incorporating two post-processes that exclude the outliers caused by the interval ofmicrophone arrays and the loudspeaker position mismatch. The test measurement for glass wool
demonstrated the behavior of the outliers, and confirmed the validity of the two post-processes.
Furthermore, using the measured values at all the combinations of frequency and incidence angle, theincidence angle dependence of absorption coefficient was estimated by least squares method. Thismeasurement method was applied for two types of absorbing materials such as porous materials andperforated panels, and their incidence angle dependences were successfully demonstrated incomparison with the theoretical estimates.

ACKNOWLEDGEMENTS
This project has been funded by the Grant-in-Aid for Challenging Exploratory Research fromJapan Society for the Promotion of Science (No. 15K14072).

REFERENCES
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3. Kimura K, Yamamoto K. A method for measuring oblique incidence absorption coefficient ofabsorptive panels by stretched pulse technique. Appl Acoust 2001;62(6):617-32.
4. Inoue N, Sakuma T. Development of a measurement method for oblique-incidence absorption
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