PROCEEDINGS of the
23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Anisotropic sound fields in reverberation-room measurements ofsound absorption coefficients: Wavenumber spectrum theory
Mélanie NOLAN 1
Acoustic Technology, Department of Electrical Engineering, Technical University of Denmark,Ørsteds Plads, Building 352, 2800 Kgs. Lyngby, Denmark and Saint-Gobain Ecophon, 265 75 Hyllinge, Sweden

ABSTRACT
Measured values of absorption coefficients obtained from standardized reverberation-chamber measurements
often differ across laboratories, mainly due to non-isotropic sound incidence on the absorbing specimen, anddifferences in the chambers’ shapes and dimensions. The present study describes an experimental method forcharacterizing sound field isotropy in reverberation rooms. The methodology relies on a plane wave
decomposition (i.e., estimation of the wavenumber spectrum) to describe the directional properties of thesound field in the vicinity of the absorbing specimen. By separating the incident from the reflectedcomponents in the wavenumber spectrum, it is possible to characterize the distribution of sound incidence
on the sample, and to infer angle-dependent absorption properties. Measurements are conducted in areverberation room with absorbing material on the floor using a programmable robotic arm to scan the soundfield. The results confirm that the distribution of sound incidence on the sample is not isotropic in the steadystate. The measured angle-dependent absorption coefficients show good agreement with theoreticalpredictions.
Keywords: Sound absorption, Isotropy, Array processing

1. INTRODUCTION
In reverberation-room measurements of sound absorption coefficients, a large sample of thematerial under test is placed in a reverberation room, and the absorption coefficient is deduced fromthe resulting reduction of the reverberation time, once the absorber has been introduced in the room[1]. The procedure is based on the simple reverberation formulas [2,3] for which a completely diffuse
sound field must be established in the test chamber. An ideal diffuse field can be defined as thesuperposition of an infinite set of plane waves with random phases and equal magnitudes, which
directions of propagation are uniformly distributed over all angles of incidence. In such a sound field,
the energy density is uniformly distributed in space and the energy flow is isotropic; i.e., the soundfield is homogeneous and isotropic. The absorption properties of the sample are represented by a(random-incidence) absorption coefficient, which corresponds to the average fraction of power
absorbed by the sample when sound is arriving on it equally from all directions; i.e., under theassumption of isotropic sound incidence.
The shortcomings of the measurement procedure have become increasingly evident as shown innumerous investigations [4-7]: several systematic comparisons of the absorption coefficientsdetermined in a number of standardized laboratories resulted in the conclusion that the absorption
coefficient of a given area of the same material can take virtually any value. A vast amount of literature
has grown around the subject and shows that the major cause of difficulties with the reverberation chamber measurement of sound absorption can be attributed to a lack of diffuseness of the sound field,
resulting in non-isotropic sound incidence on the absorbing specimen. Considerable experimental
effort has been spent on the problem of evaluating diffusion in reverberant sound fields, and numerous
methods have been proposed. An account of various descriptors is presented by Schultz [8], and more
recently by Jeong et al. [9]. In particular, a variety of methods have been proposed, where diffusion
is described on the basis of the angular distribution of sound energy [10-17].
Over the past few decades, technical developments in sensing methods have facilitated the threedimensional analysis of sound fields. In particular, microphone arrays are especially well suited for1

melnola@elektro.dtu.dk2313

applications where the sound waves impinge on the array from multiple directions. The idea of using
an array of microphones to measure the diffusion of reverberation chambers goes back to 1959, when
Schroeder suggested creating a microphone array by repeated reflections of a planar array at thesidewalls of a reverberation chamber [18]. Rather than an array dependent on the room symmetry,
Ebeling sampled the sound field in a reverberation chamber at every point of a three-dimensional
cubical grid, with the goal of measuring cross-correlation functions in the spatial-frequency domain[13]. More recently, Gover et al. [14] investigated how spherical microphone arrays can be used tocharacterize directional properties of reverberant sound fields. They adapted the method described inRef. 11 to spherical microphone arrays and measured the directional distribution of acoustic energy
by steering directional beams in every direction. Extending this approach, Berzborn et al. [16] recentlyintroduced the concept of Directional Energy Decay Curves (DEDC) derived from measurements ofdirectional impulse responses using a spherical microphone array. In Ref. 15, Nolan et al. proposed
an experimental method for evaluating isotropy in reverberant spaces in the steady-state, based on ananalysis of the wavenumber spectrum in the spherical harmonics domain. The underlying hypothesisis that, in a perfectly isotropic sound field, the wavenumber spectrum is spherically symmetric. The
method does not require a specific array configuration, as the analysis in the spherical harmonics
domain is performed on the wavenumber spectrum, which is defined over a sphere, and not on therecorded sound pressure directly. Besides, such spectral representation of the wave field can be used
to examine the distribution of the incident acoustic energy on the measuring sample and to reconstruct
the incident power flows over a three-dimensional domain [17]. The aim of the present paper is tosummarize the main findings from Refs. 15 and 17 and to discuss the limitations of some aspects ofthe work.

2. WAVENUMBER SPECTRUM THEORYA detailed account of the theory is given in Refs. 15 and 17, and only the main results will besummarized here.
2.1 Wavenumber spectrum and sound field isotropy
The conception that a given pressure field can be expressed uniquely as a superposition ofelementary waves is central to the methods outlined in this paper. Such plane wave decomposition
consists in applying a three-dimensional Fourier transform to a measured pressure field to estimate
the wavenumber spectrum [19], which is a representation of the sound field in the spatial-frequency
domain, or k-space:+∞pሺrm ሻ= ම Pሺkሻe-∞

L-j൫kx xm +ky ym +kzzm ൯෩ ሺkl ሻe-jkl ∙rm ,dk ≈ ෍ P

(1)

l=1

where the quantity Pሺkሻ=ȁPሺkሻȁejϕሺkሻ is the wavenumber (or angular) spectrum, with ȁPሺkሻȁ andϕሺkሻ its magnitude and phase, respectively. Each plane wave is traveling in a direction specified by
the wavenumber vector k , which satisfies the condition k2 ≥k2x +k2y with k2x +k2y +k2z =k2=ԡkԡ2(indicating that they are propagating waves).
The pressure field sampled at a discrete number R of receiver positions (with an array ofmicrophones) can be expressed in matrix form as(2)p=Hx,where p‫א‬CR is the measured sound pressure vector, x ‫ א‬CL is a complex coefficient vector෩ ሺkl ሻ in Eq. (1), and H ‫ א‬CR×L is the transfer matrix containing
containing the wavenumber spectrum P
the plane wave functions. The problem is typically underdetermined (L > R) and the estimation of x
obtained via a regularized matrix pseudo-inverse [20].
In order to evaluate isotropy, the magnitude of the wavenumber spectrum at frequency f = kc/(2π)
is expanded into a series of spherical harmonics∞n෩ ሺθ, ϕሻห = ෍ ෍ Amn ሺkሻYmหPn ሺθ, ϕሻ ,
n = 0 m = -n(3)where the complex coefficients Amn ሺkሻ can be calculated from the orthonormality relation of thesphericalharmonics
functions.
Therelative
monopole
strengthι =2314

nȁA00 ሺkሻȁΤσ∞n = 0 σm = -nȁAmn ሺkሻȁdetermines the degree of isotropy [15].

2.2 Sound field separation and reconstruction
From the estimated solution xො , it is possible to reconstruct the sound field anywhere in the domain(pressure, velocity, and sound intensity) [17]pr = Hr xො,(4)where pr ‫א‬CK is the reconstructed sound pressure vector estimated at a set of K positions rr, andHr ‫ א‬CKxL is the reconstruction matrix containing the plane wave functions evaluated at thereconstruction points rr. The particle velocity vector ur can be calculated from Euler’s equation of1
motion, and the active intensity vector is Ir = 2 Re൛pr u*r ൟ, where the superscript * denotes the complex

conjugate. Considering that xො =ൣxො (i) ;xො (r) ൧ , the reconstruction can alternatively be based on the(i)KxNcontaining the wave
complex coefficient vector xො ‫א‬CN and the propagation matrix H(i)r ‫א‬Ccomponents that describe the incident sound field (where N is the number of waves conforming theincident sound field). In this case, the reconstruction provides a complete characterization of the(i), velocity ur , and intensity I(i)incident acoustic field (sound pressure p(i)r ).
r
2.3 Angle-dependent absorption coefficient
Based on the sound field separation defined in Sec. 2.2, the angle-dependent absorption coefficient
can be determined according to [17]2π2෩ ሺθ, ϕሻห dϕ‫ ׬‬หPPabs,θ
= 1 - 2π0,α(θ) =Pinc,θ෩ ሺπ - θ, ϕሻห2 dϕ‫ ׬‬หP0≤θ≤02π2(5)2෩ሺθ,ϕሻห corresponds to the reflected power, and หP෩ ሺπ-θ,ϕሻห to the incident power. As such,where หP
the estimation of absorption follows from the ratio of absorbed to incident power observed in themeasurement aperture and does not assume any specific waveform incidence. The individual
wavenumber spectra are integrated on the azimuth angle ϕ to prevent biases at angles with noincidence (a pure-tone sound field in a reverberation room is not diffuse, and not all frequencies
contain energy in every direction [17]). This averaging procedure is not critical since, in the case of
isotropic absorbers, the azimuth angle is of marginal significance [18].

3. EXPERIMENTAL RESULTS
This section describes measurements conducted in a large (215 m 3) reverberation room, using a
programmable robotic arm to scan the sound field. Two damping conditions in the room were
examined: the empty (undamped) room and the room with extra absorption (10.8 m 2 glass wool ofthickness 100 mm and flow resistivity 12.9 kPa.s/m 2) on the floor. The room complies with the ISO
354 requirements, and is essentially box-shaped, although there are 85 built-in concrete boundary
diffusers and 12 panel diffusers. A scanning robot UR5 (Universal Robot, Odense, Denmark) was
programmed to move a free-field microphone (Brüel & Kjær, Nærum, Denmark) within a rectangularvolume of dimension 0.6 x 0.8 x 0.25 m, creating a random array of 290 measurement positions in thevicinity of the absorbing sample (the closest microphone is located ~ 30 cm away from the absorber).
The room was excited by a built-in loudspeaker driven with a logarithmic sine sweep ranging from 20
Hz to 20 kHz with a duration of 10 s. The frequency response was measured at the 290 positions witha spectral resolution of 0.1 Hz. The complex coefficient vector x corresponding to the wavenumber
spectrum (i.e., the amplitudes of the waves) is estimated using Eq. (2). A plane-wave basis of 2000
plane waves of unknown amplitudes is considered, whose directions of propagation a re distributed
uniformly based on a Thomson problem. Tikhonov regularization (i.e., a l 2 least-squares solution) isused for the regularized inversion, along with the L-curve criterion as a parameter-choice method [21].
Figure 1 compares the magnitude of the wavenumber spectrum and corresponding sphericalharmonic expansion at 1 kHz, measured in the undamped and damped rooms in the vicinity of thefloor/absorber. In the undamped room, as seen in Fig. 1(a), a few dominant directions are detected(i.e., a few waves carrying considerably more energy than others, seemingly corresponding to a few
interfering modes and early reflections from the neighboring rigid floor), indicating that the field is
not perfectly isotropic. In the damped room [added absorption, Fig. 1(b)], the wavenumber spectrum2315

is less omnidirectional, as there are no waves propagating in the positive z-direction (higherhemisphere in the wavenumber spectrum) because no sound is being reflected away from the absorber( α≈1 at 1 kHz). Besides, the incident sound field (lower hemisphere) exhibits some dominant
incident directions, indicating that the sound field incident onto the absorber is anisotropic. The results
are supported by the corresponding spherical harmonics expansions: in the undamped case, thewavenumber spectrum is best described by the monopole moment of its spherical harmonic expansion
although higher-order moments are required to explain the few dominant directions (the isotropyindicator in this case is ι = 0.43, confirming that the sound field is not isotropic); in the damped case,the spherical harmonic expansion is no longer dominated by the monopole moment, resulting in asound field that is less isotropic than in the undamped case (ι = 0.28). It can be remarked that theisotropy indicator values are lower than in Ref. 15 for the same room/frequency, because the array iscloser to the floor/absorber in this case. Another observation that can be drawn from these data is that
there are clear differences between the two incident fields (with and without the sample - lower
hemispheres in the wavenumber spectra), which demonstrates the effect of the absorber on the incident
sound field.

Figure 1 – Magnitude of the wavenumber spectrum and corresponding spherical harmonics expansion in(a) the undamped room; (b) the damped room. Frequency: 1 kHz. Truncation order: N = 5.
Figure 2 shows a volumetric reconstruction of the acoustic intensity incident onto the sample forselected third-octave bands (250 Hz, 500 Hz, 1 kHz, and 2 kHz).The incident active intensity vectors
are computed over a parallelepiped lattice of 252 points in the vicinity of the absorbing plane (9 x 7x 4 grid with lattice spacing of 10 cm). The length of the vectors is proportional to the intensity (inW/m 2) at the base of the cone, and the cones are color-coded to also show the magnitude of theintensity vectors. At all frequencies, the incident energy is not constant and uniform throughout space(in clear disagreement with theoretical predictions that follow from assuming at diffuse sound field ).
It should be remarked that the direction of incident energy flow is in general not in the direction of asingle wave front, but rather results from the interaction of the many superposed waves; as such, itdoes not indicate the directional distribution of sound waves arriving onto a sample (contrarily to thewavenumber spectrum representation). Nevertheless, analysis of incident intensity can indicate thedeparture from the ideal state of diffusion.

2316

Figure 2 – Volumetric reconstruction of the active intensity incident onto the absorbing sample for selected
third-octave bands. [Adapted from Ref. 17].

Figure 3 shows the angle-dependent absorption coefficient determined from the wavenumber
spectrum measurement for the third-octave bands ranging from 200 Hz to 1 kHz. The absorption
coefficient is averaged over the azimuth angle and calculated as mean values per third -octave bands.
As mentioned in Sect. 2.3, the frequency averaging is performed on the wavenumber spectrum toavoid biased estimates. The asterisks * represent the experimental data estimated every 6° (π/30 rad).
The solid line is calculated from a transfer matrix model in the case of a single layer of porous material
with a rigid backing [22]. The characteristic impedance and the wavenumber for the material are
obtained from Miki’s empirical model [23]. The experimental values fit reasonably well with theprediction. The agreement is particularly good in the frequency range between 315 Hz and 1 kHz andfor angles between 0 and 60° (π/3 rad). At lower frequencies, the incident energy is not uniformly
distributed with angle, and the spatial resolution of the array is limited. At frequencies above 500 Hz,
where the sample is highly absorptive, the absorption coefficient at grazing incidence isunderestimated as a result of the limited wavenumber resolution of the measurement system: some of
the incident energy on the absorber at grazing incidence (in the lower hemisphere of the wavenumber
spectrum) leaks into the upper half of the wavenumber, which corresponds to directions of reflectedcomponents.

4. SUMMARY AND DISCUSSION
The directional properties of the reverberant sound field can be characterized by analyzing thewavenumber spectrum, which results from expanding the sound field into a plane-wave basis.
Measurements in a standardized reverberation chamber show that the sound field is not isotropic ( i.e.,
the wavenumber spectrum is not spherically/hemispherically symmetrical). The accuracy of theestimation depends on (i) the measurement system (the sampling of the pressure field should besufficient to estimate the wavenumber spectrum correctly; yet, no specific array configuration isrequired); (ii) the choice of the regularization scheme. It can also be remarked that in a reverberationchambers with diffusing elements (such as panel or boundary diffusers), exponentially attenuated2317

waves (evanescent waves) will appear at the observation point, if the latter is located in close
proximity of the diffusers. Besides, evanescent waves can be caused by diffraction evoked at thesample edges. This work is however concerned with analyzing the sound field away from any sourceand diffracting element (that is, evanescent waves are not present) [18].
The isotropy of a sound field can be characterized by expanding the wavenumber spectrum in thespherical harmonics domain, and the relative monopole strength determines the degree of isotropy.
Yet, such quantitative indicator alone is of no help in interpreting the results. For example, it cannot
distinguish between anisotropy due to low modal density and anisotropy caused by non-uniform
placement of absorbing material. In this connection, the inspection of the wavenumber spectrum [e.g.,Fig. (1)] is much more informative. The proposed measure may nevertheless be useful for comparative
investigations in a given room (the influence of source position, diffusers placement, etc. can beexamined). Note however that the metric is not a full measure of the degree of diffusion, since it
disregards the correlation between waves incoming from different directions.
Measurement of the wavenumber spectrum makes it possible to reconstruct and characterize theincident energy flows over a three-dimensional domain. Measurement results confirm that the incident
flow is not constant and uniform throughout space despite the presence of diffusing elements. Thiscan be attributed to the coupling between the measuring sample and the sound field in its vicinity. The
complex intensity vectors in a room can also be determined and visualized, which provides a valuable
tool for understanding the relation between flows of net energy (active intensity) and oscillatingenergy (reactive intensity) in an empty reverberation chamber or with absorption on the floor.
Finally, the angle-dependent absorption coefficient (averaged over the azimuth angle) can bedetermined from measurement of the wavenumber spectrum in front of an absor bing plane. There isfair agreement with theoretical predictions.
It is anticipated that the experimental methods outlined in this paper may find application inmeasurements in ordinary rooms (especially in rooms with absorbent ceilings, e.g., classrooms,offices, etc.). The ability to characterize the spatial properties of the sound field in theses rooms may
supplement reverberation time measurements in tasks related to room acoustical design.

Figure 3 – Angle-dependent absorption coefficient determined from wavenumber spectrum measurements.
The asterisks * represent experimental data; the solid line is obtained from a transfer matrix method alongwith Miki’s model. [Adapted from Ref. 17].

ACKNOWLEDGEMENTS
This work was supported by the Oticon Foundation.

REFERENCES
1. ISO 354:2003: Measurement of Sound Absorption in a Reverberation Room (International2318

Organization for Standardization, Geneva, Switzerland, 2003).
2. W. C. Sabine, Collected Papers on Acoustics (Cambridge, Harvard University Press, 1922).
3. P. M. Morse and K. U. Ingard, Theoretical Acoustics (McGraw-Hill, New York, 1968), Sec. 9.5.
4. C. W. Kosten, “International comparison measurements in the reverberation room,” Acustica 10,
400-411 (1960).
5. Y. Makita, M. Koyasu, M. Nagatu, and S. Kimura, “Investigation into the precision of measurementof sound absorption coefficients in a reverberation room,” J. Acoust. Soc. Jpn. 24, 381-392 (1968).
6. R. E. Halliwell, “Inter-laboratory variability of sound absorption measurement,” J. Acoust. Soc.
Am. 73(3), 880-886 (1983).
7. M. Nolan, M. Vercammen, C. -H. Jeong, and J. Brunskog, “The use of a reference absorber forabsorption measurements in a reverberation chamber,” in Proceedings of Forum Acusticum 2014,Krakow, Poland (2014).
8. T. J. Schultz, “Diffusion in reverberation rooms,” J. Sound Vib. 16, 17-28 (1971).
9. C. -H. Jeong, M. Nolan and J. Balint, “Difficulties in comparing diffuse sound field measures anddata/code sharing for future collaboration,” in Proceedings of Euronoise 2018, Heraklion, Crete(2018).
10. R. Thiele, “Richtungsverteilung und Zeitfolge der Schallrückwurfe in Räumen,” Acustica 3, 291302 (1953).
11. E. Meyer and R. Thiele, “Raumakustische Untersuchungen in zahlreichen Konzertsälen undRundfunkstudios unter Anwendung neuerer Messverfahren,” Acustica, 6-425 (1956).
12. Y. Yamasaki and T. Itow, “Measurement of spatial information in sound fields by closely locatedfour-point microphone method,” J. Acoust. Soc. Jpn. (E) 10, 101-110 (1989).
13. K. J. Ebeling, “Statistical properties of random wave fields,” in Physical Acoustics, Principlesand Methods, edited by W. P. Mason and R. N. Thurston (Academic, New York, 1984), Vol. XVII,
Chap. 4, pp. 233-310.
14. B. N. Gover, J. Ryan, and M. Stinson, “Measurements of directional properties of reverberantsound fields in rooms using a spherical microphone array,” J. Acoust. Soc. Am. 116(4), 2138-2148(2004).
15. M. Nolan, E. Fernandez-Grande, J. Brunskog, and C.-H. Jeong, “A wavenumber approach toquantifying the isotropy of the sound field in reverberant spaces,” J. Acoust. Soc. Am. 143(4), 25142526 (2018).
16. M. Berzborn and M. Vorländer, “Investigations on the directional energy decay curves inreverberation rooms,” in Proceedings of Euronoise 2018, Heraklion, Crete (2018).
17. M. Nolan, S. A. Verburg, J. Brunskog, and E. Fernandez-Grande, “Experimental characterizationof the sound field in a reverberation room,” J. Acoust. Soc. Am. 145(4), 2237-2246 (2019).
18. M. R. Schroeder, “Measurement of sound diffusion in reverberation chambers,” J. Acoust. Soc.
Am. 31(11), 1407-1414 (1959).
19. E. G. Williams, Fourier Acoustics: Sound Radiation and Near-Field Acoustical Holography(Academic Press, New York, 1999).
20. E. Fernandez-Grande, “Sound field reconstruction using a spherical microphone array,” J. Acoust.
Soc. Am. 139, 1168-1178 (2016).
21. P. C. Hansen, Discrete Inverse Problems: Insight and Algorithms (Fundamental of Algorithms,Vol. 7) (SIAM, Philadelphia, PA, 2010).
22. J. -F. Allard, Propagation of Sound in Porous Media: Modelling Sound Absorbing Materials(Elsevier Science Publishers, Amsterdam, the Netherlands, 1993), p. 38.
23. Y. Miki, “Acoustical properties of porous materials – Modifications of Delany-Bazley models,” J.
Acoust. Soc. Jpn. 11, 19-24 (1990).

2319