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

Experimental characterization of the decaying sound field in areverberation room
Mélanie NOLAN1,2,3, Marco BERZBORN4 and Efren FERNANDEZ-GRANDE2
2

Acoustic Technology, Department of Electrical Engineering, Technical University of Denmark,2800 Kgs. Lyngby, Denmark
3Saint-Gobain Ecophon, 265 75 Hyllinge, Sweden

4

Institute of Technical Acoustics, RWTH Aachen University, 52074 Aachen, Germany

ABSTRACT

Reverberation-chamber measurements of sound absorption coefficients are based on theassumption that the sound field is diffuse (i.e., homogeneous and isotropic). Such sound fieldshould be established both before and during the sound decay. Yet, the test conditions forabsorption measurements render the establishment of a completely diffuse sound field difficult,if not impossible. It is therefore of interest to discuss the concept of acoustic diffusion in areverberation chamber. A recent investigation examined sound field isotropy in reverberation
rooms in the steady state, based on an analysis of the wavenumber spectrum in the sphericalharmonics domain. The purpose of the present study is to analyze the isotropy of the soundfield in a reverberation chamber across time. The evolution of the sound decay is examined byestimating the wavenumber spectrum in successive time instances, leading to a time-dependent
analysis of isotropy. Experimental results based on measurements in a reverberation room infour configurations (empty with and without diffusers, as well as with absorbing material on
the floor, with and without diffusers) are presented.
Keywords: Sound absorption, Sound decay, Isotropy, Array processing
1. INTRODUCTION
Reverberation-room measurements of the statistical absorption coefficient of acoustic
materials [1] depend critically on the diffusion of the sound field in the test chamber. The
procedure is based on the simple reverberation formulas, provided that complete
diffuseness of the sound field is achieved both before and during the sound decay. An
ideal diffuse field can be defined as the superposition 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
uniform in space and the energy flow is isotropic [2]. This definition can be extended toa set of exponentially decaying plane waves and thus, to the decaying sound field in areverberation room. Yet, the test conditions for absorption measurements render theestablishment of a completely diffuse sound field difficult.
The shortcomings of the measurement procedure have become increasingly evident asa result of numerous investigations [3,4], which show that the major cause of difficulties
with the reverberation-room measurement of sound absorption can be ascribed to a lack1

melnola@elektro.dtu.dk

4035

of diffuseness of the sound field. A variety of methods for the measurement of sound field
diffusion have been reported in the literature, mainly concerned with its evaluation in thesteady-state. The measurement of diffuseness during the decay phase has not received
similar attention, although a few papers have looked into the matter: Reverberation timeand decay irregularities have been used as a criterion for diffusion [5,6]; Balachandranand Robinson [7] extended correlation measurements to the decaying state; Several
studies have measured the acoustic intensity over time [8-10]. Besides, a promising lineof approach describes diffusion on the basis of the angular distribution of sound energy:
Venzke and Dämmig [11] extended the concept of directional diffusion introduced by
Meyer and Thiele [12] to the decaying state, by measuring the angular variation of soundenergy at various instants during the decay using highly directional transducers ; Morerecently, Gover et al. [13] adapted the method to measurements with a spherical array of
microphones; Extending this approach, Berzborn et al. [14,15] introduce the concept of
Directional Energy Decay Curves (DEDC) derived from measurements of directional
impulse responses using a spherical microphone array.
A recent investigation proposed an experimental method for evaluating isotropy inreverberant spaces in the steady-state, based on an analysis of the wavenumber spectrum
in the spherical harmonics domain [16]. The aim of the present study is to extend thewavenumber spectrum analysis method to the decaying state. The evolution of the sounddecay is examined by estimating the wavenumber spectrum over various time windows,
leading to a time-dependent analysis of isotropy. The paper is organized as follows: thetheoretical background is presented in Sec. 2. In Sec. 3, an experimental study is presented
based on measurements in a reverberation chamber using a programmable robotic arm.
2. THEORETICAL BACKGROUND
2.1

Wavenumber (or angular) spectrum and sound field isotropy

Let us consider the sound field produced by a pure-tone source with angular frequency
ω = kc in a reverberation chamber. In steady-state, the resulting sound field can be
represented as a superposition of plane waves [17], each traveling in a direction specified
by the wavenumber vector k = (kx, ky , k z)+∞pሺr, ωሻ = ම Pሺkሻejሺωt - k∙r ሻ dk ,
T

(1)

-∞

where r = (x, y, z), and the time dependency ejωt will be omitted throughout. In otherwords, the sound field is expressed as a superposition of plane waves. The quantityPሺkሻ=ȁPሺkሻȁejϕሺkሻ is the wavenumber (or angular) spectrum, with ȁPሺkሻȁ and ϕሺkሻ its
magnitude and phase, respectively. In the following, it is assumed that all plane waves
satisfy the condition ԡkԡ2 =k2 =k2x +k2y +k2z with k2 ≥k2x +k2y (indicating that they arepropagating waves). Such plane wave decomposition is valid away from any source ordiffracting element (where evanescent waves are not present).
In order to evaluate isotropy, the magnitude of the wavenumber spectrum is expanded
into a series of spherical harmonics [16]∞nหP෨ ሺθ, ϕሻห = ෍ ෍ Amn ሺkሻYmn ሺθ, ϕሻ ,(2)

n = 0 m = -nwhere P෨ ሺθ, ϕሻ denotes the two-dimensional wavenumber spectrum expressed in sphericalcoordinates, and the complex coefficients Amn ሺkሻ can be calculated from the4036

orthonormality relation of the spherical harmonics functions. The relative monopole
strength [16]ι = ȁA00 ሺkሻȁൗ෍

∞
n=0

෍

n
m = -nȁAmn ሺkሻȁ(3)determines the degree of isotropy, the underlying hypothesis being that in a perfectly
isotropic sound field, the wavenumber spectrum is rotationally symmetric. The measure
ranges from 0 to 1: 1 in the case where the flow of energy is equal in all directions; 0 if
the incident waves propagate in a single direction.
2.2

Implementation of the method

In practice, the wavenumber spectrum P෨ ሺθ, ϕሻ is obtained using a discrete plane wave
expansion, based on a discrete approximation of Eq. (1)Lpሺrr ሻ  ൌ ෍ P෨ ሺkl ሻe-jkl∙rr ,

(4)

l=1

where the directions of propagation of the L plane waves are uniformly distributed over aspherical domain. The pressure field is sampled at a discrete number R of receiver
positions, and can be expressed in matrix form asp=Hx,(5)
RL
where p‫א‬C is the measured sound pressure vector, x ‫ א‬C is a complex coefficient
vector containing the wavenumber spectrum P෨ ሺkl ሻ in Eq. (4), and H ‫ א‬CR×L is thetransfer matrix containing the plane wave functions. The problem is typicallyunderdetermined (L > R) and the estimation of x obtained via a regularized matrix pseudoinverse [18]. Subsequently, a spherical harmonic expansion of the magnitude of ea chcomponent of x can be obtained based on a discrete approximation of Eq. (2).
2.3Time analysis

In the present work, the isotropy of the sound field is analyzed as a function of time.
Such analysis is particularly useful for examining its temporal evolution during the decay
process and detecting asymmetries in energy distribution both in time and direction. It
may be of interest, for instance, to exclude the early reflections and examine the later partof the decay only. In the present study we focus on an analysis of isotropy for time
windows starting at a given time t0 and extending until the end of the measured impulse
response. Depending on the chosen value of t0, it becomes possible to analyze the steadystate (t 0 = 0s), or to exclude the direct sound, early reflections, or late reflections. For ananalysis window wt0 , the corresponding frequency response Ht0 reads+∞

Ht0 ሺr,ωሻ= ‫׬‬-∞ wt0 ሺtሻhሺr,tሻe-jωt dt ,(6)

where h is the measured impulse response at position r. In the case of a rectangular
1 ‫ ׊‬t ‫ א‬ሾt0 , +∞ሾ, Eq. (6) becomeswindow wt0 ሺtሻ = ቄ
0 ‫ ׊‬t ‫ א‬ሾ0, t0 ሾ+∞

Ht0 ሺr,ωሻ= ‫׬‬t0

hሺr,tሻe-jωt dtǤ(7)

The case t0 = 0s includes the energy arriving over the full impulse response and thus
corresponds to the steady-state response of the room. The subsequent analysis isperformed on the frequency response Ht0 so that pሺr,ωሻ = Ht0 ሺr,ωሻ [see Eq. (1)]. The
evaluation of isotropy follows as explained in section 2.1, based on the windowed

4037

responses of the room. Alternatively, a short sliding time window can be used to isolate
individual reflections (and, potentially, problematic reflections) in time and direction.
3. EXPERIMENTAL RESULTS
The proposed methodology is examined experimentally in a large (245 m3 )reverberation room at the Technical University of Denmark, using a programmablerobotic arm to scan the sound field (see Fig. 1). The room is essentially box-shaped,
although there are 19 removable panel diffusers. Four configurations of the room areconsidered: the empty (undamped) room with and without panel diffusers, and the roomwith extra absorption on the floor with and without panel diffusers. The absorption
coefficient of the sample (10.8 m2 glass wool of thickness 100 mm and flow resistivity
12.9 kPa.s/m2) at 500 Hz is 1.04 (measured according to ISO 354 in the same reverberationchamber in the configuration with diffusers). A scanning robot UR5 (Universal Robot,
Odense, Denmark) is programmed to move a free-field microphone (Brüel & Kjær,Nærum, Denmark) and forms an array of 310 sequential measurement positions. The arrayconsists of two spherical layers with radii 0.25 m and 0.45 m respectively, each of themsampled at 144 positions chosen according to an equal-area grid on the sphere [19].
Additional sampling positions were used inside the spheres to achieve good stabilizationof the eigenfrequencies [20]. It should be remarked that the current measurement system
is not expected to provide valid results below 200 Hz, where the size of the arraycorresponds to about 10% of the wavelength in air (ka = 0.1), nor at high frequencies(above 3 kHz), where aliasing effects start to appear.

Figure 1 – Experimental setup
The room is excited by a built-in loudspeaker driven with exponential sweeps. An
impulse response is measured at each of the 310 positions using the ITA-Toolbox [21].
The measured impulse responses are considered over various time ranges to investigate
the temporal evolution of the sound decay and its isotropy. For each selected time rangeof the impulse response, the frequency response is calculated based on a DFT, and thecomplex coefficient vector x corresponding to the wavenumber spectrum is estimatedusing Eq. (5). A plane-wave basis of 2000 plane waves of unknown amplitudes is
considered, whose directions of propagation are distributed uniformly based on aThomson problem. Tikhonov regularization (i.e., a l 2 least-squares solution) is used forthe regularized inversion, along with the L-curve criterion as a parameter-choice method.
As for the spherical harmonic expansion in Eq. (2), only a limited number of sphericalharmonic orders can be used in practice. The spherical harmonic expans ion is heretruncated at N = 7. Although possible, adding more spherical harmonics in the expansionof the wavenumber spectrum does not contain relevant information.

4038

It can be remarked that one advantage of the method described in this work is that it
does not require a prescribed array geometry. It can be used for any given array, provided
that the basic sampling requirements are met [22,23]. In this paper, we use a sphericalarray as the method is meant to be compared in the future with another approach, basedon the same set of data (see Ref. 15, presented in the same proceedings).
Figure 2 compares the resulting wavenumber spectrum at 500 Hz in the damped room
without diffusers [left], and in the same room where 19 hanging panel diffusers have been
installed [right]. The wavenumber results are averaged over the third-octave band centered
at 500 Hz and include the arriving energy over the full impulse response (that is, theseresults correspond to the steady-state response of the room). It should be noted that, withthe chosen time convention, the wavenumber spectrum represents the direction of
propagation and not the direction of arrival (as apparent from Fig. 2). It can be seen thatin both cases, no waves are propagating in the positive z-direction, because no sound isbeing reflected by the absorbing sample ( α ≈ 1.04 at 500 Hz). In the room without
diffusers, a few dominant incident directions are detected, seemingly corresponding to thedirect radiation from the source and standing waves in the x-y plane (tangential plane tothe absorber). When panel diffusers are added to the room, the distribution of incident
energy is somewhat more uniform, as seen in Fig. 2 (right).

Figure 2 – Magnitude of the wavenumber spectrum in the damped room without
diffusers (left) and with diffusers (right). Frequency: 500 Hz (1/3 oct.). t0 = 0s.
Figure 3(a) compares the magnitude σnm = -nȁAmn ȁ of the moments from the sphericalharmonic expansions in the undamped and damped room without (black) and with (blue)
diffusers, for the third-octave band centered at 500 Hz. In order to analyze the changes inisotropy at different times, and as described in section 2.3, the wavenumber spectr a andcorresponding spherical harmonic expansions are calculated using windows starting at t0= 0s, 50ms, 100ms, … 1.4s, although Fig. 3(a) displays results for t0 = 0s, 0.5s, and 1s
only. For t0 = 0.5s and 1s, the analysis of isotropy is restricted to t > 0.5s and t > 1s,
respectively.
In the undamped room, the monopole moment dominates the spherical harmonic
expansion of the wavenumber spectrum (both in the empty room and in the room withadded panel diffusers), indicating a fairly isotropic sound field. Addition of panel
diffusers essentially reduces the contribution of the 2 nd and 4th order spherical harmonic
moments, which account for standing waves in the x-, y- and z-directions (this is seen in
the corresponding wavenumber spectra, not shown for conciseness). These moments areless prominent in the late decay (see t > 1 s, without diffusers), suggesting that the sound4039

field is composed of multiple waves of similar amplitude (stronger early re flections are,in fact, excluded).
In the damped room, due to the presence of the absorbing sample, the sound field isless omnidirectional and higher-order moments are required to describe the wavenumber
spectra. This indicates that the wave field is less isotropic than in the undamped case, inagreement with previous findings [16]. In particular, in the room without diffusers at t >0.5 s and t > 1 s, the wavenumber spectrum is best described by the 1 st, 2 nd and 4th order
spherical harmonic moments. Inspection of the wavenumber spectra (not shown for
conciseness) reveals persisting tangential components in the late decay, corresponding tostanding waves in the x-y plane. This is in agreement with the general behavior of soundpropagating in a rectangular room containing a highly absorptive surface and no scattering
objects, as described in e.g. Ref. 24, where it is shown that the grazing part of the soundfield is less affected by the absorption than the non-grazing field. The results show noevidence of such persisting waves when panel diffusers are installed (see Fig. 3(a),damped room with diffusers), suggesting that the panels re-direct the sound waves
successfully.
Figure 3(b) shows the isotropy indicator in the four configurations as a function oftime, for the successive time windows corresponding to t > 0 s, t > 50 ms, t > 100 ms, …
t > 1.4 s. The results confirm that the sound field is less isotropic in the damped room thanin the undamped room (0.27 < ι < 0.47 in the damped room without diffusers, against
0.70 < ι < 0.78 in the undamped room), and that addition of panel diffusers influences
positively the isotropy of the wave field (in the steady-state, isotropy increases by 11.4 %in the undamped room, and by 12.1 % in the damped room). Yet, the sound field in theundamped room with diffusers is not perfectly isotropic (ιmax = 0.84), which is to be
expected as perfect isotropy is difficult to achieve in a room.
In the damped room without diffusers, the sound field is highly anisotropic in thesteady-state (ι = 0.27), due to the direct radiation from the source and a few interferingmodes (see Fig. 2). As time progresses, the results indicate increasing isotropy initially (ιincreases from 0.27 to 0.47 between t > 0 s and t > 0.55 s) as there is no influence of thesource and early reflections, followed by increasing anisotropy (ι drops from 0.47 to 0.42
between t > 0.55 s and t > 1.4 s), seemingly caused by persisting sound waves propagatingback and forth in the x-y plane, after sound waves in the other directions have beenabsorbed (i.e., sound in the non-grazing field is absorbed more quickly than in the grazing
field [24]). In the damped room with diffusers, room reflections seem to build up faster,in a way that increases isotropy earlier (ι = 0.53 at t > 0.2 s). It can further be noticed thatin the undamped room without diffusers, a comparable buildup of room reflections is
detected, where the isotropy indicator reaches a plateau at t > 0.9 s, presumably due tomixing. In this case, a longer processing time would be required to examine the behaviorof the sound field in the later decay.
4. CONCLUSIONS
An experimental method to evaluate sound field isotropy in the decaying state has been
proposed, based on measurements with an array of microphones. By considering measured
impulse responses over various time windows, it is possible to examine the temporal
evolution of the wavenumber spectrum, which determines the magnitude of the soundwaves arriving from definite directions at the observation point. The extent of sound fieldisotropy as a function of time is subsequently assessed by expanding the wavenumber
spectrum in the spherical harmonics domain for each time window.
Experimental results obtained in four reverberation chamber configurations indicatethat the method is suitable for evaluating the isotropy of the reverberant field in the4040

decaying state. The analysis not only confirms the anisotropy of the sound fields, but also
reveals asymmetries in energy distribution both in time and direction. It is anticipated thatthe proposed framework will be of value in diagnosing biases in laboratory configurations,as well as in analyzing the deviations encountered across standardized laboratories.
(b)Damped room

Undamped room(a)Figure 3 – (a) Spherical harmonic expansion at 500 Hz (1/3 oct.) in the undamped (top)and damped (bottom) room without and with diffusers. The results are presented forseveral time windows of the measured impulse response. (b) Isotropy indicator at 500
Hz (1/3 oct.) calculated in the undamped and damped room without a nd with diffusersfor (t > 0 s; t > 50 ms; t > 100 ms; … t > 1.4 s). Truncation order: N = 7.
ACKNOWLEDGEMENTS
This work is supported by the Innovation Fund Denmark, under the individual
postdoctoral Grant No. 8054-00042B. The authors would like to thank Samuel A.
Verburg, for help with the experimental arrangement.
REFERENCES
1. ISO 354:2003: Measurement of Sound Absorption in a Reverberation Room
(International Organization for Standardization, Geneva, Switzerland, 2003).
2. P. M. Morse and K. U. Ingard, Theoretical Acoustics (McGraw-Hill, New York, 1968),
Sec. 9.5.
3. C. W. Kosten, “International comparison measurements in the reverberation room,”
Acustica 10, 400-411 (1960).
4. R. E. Halliwell, “Inter-laboratory variability of sound absorption measurement,” J.
Acoust. Soc. Am. 73(3), 880-886 (1983).
5. C. G. Balachandran, “Random sound field in reverberation chambers,” J. Acoust. Soc.
Am. 31(10), 1319-1321 (1959).
6. K. E. Randall and F. Ward, “Diffusion of sound in small rooms,” Proc. Inst. Elect. Eng.
107(35), 439-450 (1960).
7. C. G. Balachandran and D. W. Robinson, “Diffusion of the decaying sound field,”
Acustica 19(5), 245-257 (1967/68).

4041

8. R. W. Guy and A. Abdou, “A measurement system and method to investigate thedirectional characteristics of sound fields in enclosures,” Noise Control Eng. J. 42, 8-18
(1994).
9. J. Merimaa, T. Lokki, T. Peltonen, and M. Karjalainen, “Measurement, analysis, andvisualization of directional room responses,” Convention Paper 5449, 11 th Convention of
the Audio Engineering Society, New York, September 2001.
10. M. Nolan and E. Fernandez-Grande, “Volumetric reconstruction of acoustic energy
flows in a reverberation room,” J. Acoust. Soc. Am. 145(3), EL203-EL208.
11. G. Venzke and P. Dämmig, “Measurement of diffuseness in reverberation chamberswith absorbing material,” J. Acoust. Soc. Am. 33(12), 1687-1689 (1961).
12. E. Meyer and R. Thiele, “Raumakustische Untersuchungen in zahlreichen
Konzertsälen und Rundfunkstudios unter Anwendung neuerer Messverfahren,” Acustica,
6-425 (1956).
13. B. N. Gover, J. Ryan, and M. Stinson, “Measurements of directional properties ofreverberant sound fields in rooms using a spherical microphone array,” J. Acoust. Soc.
Am. 116(4), 2138-2148 (2004).
14. M. Berzborn and M. Vorländer “Investigations on the directional energy decay curves
in reverberation rooms,” in Proceedings of Euronoise 2018, Heraklion, Crete.
15. M. Berzborn, M. Nolan, E. Fernandez-Grande, and M. Vorländer “On the directional
properties of energy decay curves,” in Proceedings of ICA 2019, Aachen, Germany.
16. M. Nolan, E. Fernandez-Grande, J. Brunskog, and C.-H. Jeong, “A wavenumber
approach to quantifying the isotropy of the sound field in reverberant spaces,” J. Acoust.
Soc. Am. 143(4), 2514-2526.
17. E. G. Williams, Fourier Acoustics: Sound Radiation and Near-Field Acoustical
Holography (Academic Press, New York, 1999).
18. P. C. Hansen, Discrete Inverse Problems: Insight and Algorithms (Fundamental of
Algorithms, Vol. 7) (SIAM, Philadelphia, PA, 2010).
19. P. Leopardi, “A partition of the unit sphere into regions of equal area and small
diameter,” Electronic Transactions on Numerical Analysis 25(12), 309-327 (2006).
20. G. Chardon, W. Kreuzer, and M. Noisternig, “Design of spatial microphone arrays forsound field interpolation,” IEEE Journal of Selected Topics in Signal Processing 9(5),780-790 (2015).
21. ITA Toolbox, “A Matlab Toolbox for Acoustics,” http://www.ita-toolbox.org (Last
viewed June 1, 2019).
22. P. D’Antonio, C.-H. Jeong, and M. Nolan, “Design of a new test chamber to measureabsorption, diffusion, and scattering coefficients,” J. Acoust. Soc. Am. 144(3), 1814(2018).
23. M. Nolan, S. A. Verburg, J. Brunskog, and E. Fernandez-Grande, “Experimental
characterization of the sound field in a reverberation room,” J. Acoust. Soc. Am. 145(4),2237-2247 (2019).
24. F. V. Hunt, L. L. Beranek, and D. Y. Maa, “Analysis of sound decay in rectangularrooms,” J. Acoust. Soc. Am. 11(1), 80-94 (1939).

4042