Experimental Assessment of the Absorption and the Sound Field of a Room Using Plane-Wave Decomposition

Abstract

A recently developed technique in acoustic signal processing is the plane wave decomposition of the sound field. It consists of a transform that describes the sound field in relation to a directional domain, the wavenumber (angular) domain. In practical terms, the calculation of this transform consists of solving an inverse and ill-posed problem using regularisation that has as input a set of sound pressures measured at a particular frequency. To that effect, a sequenced microphone array can sample the sound field in randomly generated positions. The result of this processing, the wavenumber spectrum, can then be used to calculate the absorption of an interest surface in installation conditions, to reconstruct the transfer function in positions not previously measured, and can be evaluated employing polar mapping. This paper describes an experiment in which the wavenumber spectrum of the sound field near an absorbing surface was measured using a robotic arm to scan the sound field. The room transfer function was reconstructed, and the reconstruction error was assessed, limited under 10 % for most of the analysed frequency range and increasing rapidly at frequencies over the 1600 Hz band. Next, the wavenumber spectrum was plotted in polar maps, and the absorption of the surface was calculated and compared to the absorption calculated from flow resistivity measurements and commercial values. The polar maps and the absorption coefficient display behaviour coherent with what is expected following the literature. The in situ technique displayed a systematic underestimation compared to the model values. Still, its results are more coherent with the model and the theory than commercial information.

(This preprint research paper has not been peer-reviewed.)

Full text

Graphical Abstract
Experimental assessment of the absorption and the sound field of
a room using plane-wave decomposition
Augusto Cesar Fantinelli de Carvalho, Eric Brand˜ao Carneiro, M´arcio
Henrique de Avelar Gomes, Hilbeth Parente Azikri de Deus, Lu´ıs Paulo Laus,
Paulo Henrique Mareze
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Highlights
Experimental assessment of the absorption and the sound field of
a room using plane-wave decomposition
Augusto Cesar Fantinelli de Carvalho, Eric Brand˜ao Carneiro, M´arcio
Henrique de Avelar Gomes, Hilbeth Parente Azikri de Deus, Lu´ıs Paulo Laus,
Paulo Henrique Mareze
•An industrial robot was used as a microphone array to sample the
sound field
•A simple experimental setup was used to excite a room for the
absorption measurement
•The wavenumber spectrum was calculated from the spatial sampling
near an absorber
•The error of the room transfer function reconstruction was quantified
•The diffuse field absorption was determined from an incidence-
dependent measurement
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Experimental assessment of the absorption and the
sound field of a room using plane-wave decomposition
Augusto Cesar Fantinelli de Carvalhoa,∗, Eric Brand˜ao Carneirob, M´arcio
Henrique de Avelar Gomesc, Hilbeth Parente Azikri de Deusa, Lu´ıs Paulo
Lausc, Paulo Henrique Marezeb
aPostgraduate Program of Mechanics and Materials Engineering, Federal University of
Technology of Paran´a, Rua Dep. Heitor Alencar
Furtado, Curitiba, 81280-340, Paran´a, Brazil
bAcademic Department of Mechanics, Federal University of Technology of Paran´a, Rua
Dep. Heitor Alencar Furtado, Curitiba, 81280-340, Paran´a, Brazil
cAcoustic Engineering, Federal University of Santa Maria, Avenida Roraima, Santa
Maria, 97105-900, Rio Grande do Sul, Brazil
Abstract
A recently developed technique in acoustic signal processing is the plane wave
decomposition of the sound field. It consists of a transform that describes
the sound field in relation to a directional domain, the wavenumber (angular)
domain. In practical terms, the calculation of this transform consists of
solving an inverse and ill-posed problem using regularisation that has as
input a set of sound pressures measured at a particular frequency. To
that effect, a sequenced microphone array can sample the sound field in
randomly generated positions. The result of this processing, the wavenumber
spectrum, can then be used to calculate the absorption of an interest surface
∗Corresponding author.
Email addresses: augustoc.fcarvalho@gmail.com (Augusto Cesar Fantinelli de
Carvalho), eric.brandao@eac.ufsm.br (Eric Brand˜ao Carneiro),
marciogomes@utfpr.edu.br (M´arcio Henrique de Avelar Gomes),
azikri@utfpr.edu.br (Hilbeth Parente Azikri de Deus), laus@utfpr.edu.br (Lu´ıs
Paulo Laus), paulo.mareze@eac.ufsm.br (Paulo Henrique Mareze)
Preprint submitted to Applied Acoustics September 21, 2023
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in installation conditions, to reconstruct the transfer function in positions
not previously measured, and can be evaluated employing polar mapping.
This paper describes an experiment in which the wavenumber spectrum of
the sound field near an absorbing surface was measured using a robotic arm
to scan the sound field. The room transfer function was reconstructed,
and the reconstruction error was assessed, limited under 10 % for most
of the analysed frequency range and increasing rapidly at frequencies over
the 1600 Hz band. Next, the wavenumber spectrum was plotted in polar
maps, and the absorption of the surface was calculated and compared to the
absorption calculated from flow resistivity measurements and commercial
values. The polar maps and the absorption coefficient display behaviour
coherent with what is expected following the literature. The in situ technique
displayed a systematic underestimation compared to the model values. Still,
its results are more coherent with the model and the theory than commercial
information.
Keywords: room acoustics, room impulse response, plane wave
decomposition, microphone array, sound field reconstruction
PACS: 43.55.Ev, 43.55.Mc
1. Introduction
Advancements in signal processing technology allow for new and improved
analyses of the sound field. A method that is being currently studied is
the plane-wave decomposition (PWD) of the sound field [1,2,3,4,5,6].
The plane-wave decomposition, also known as the wavenumber approach
[1,7,8], consists of a transform that describes the sound field in relation
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to a directional domain (by mean of a wavenumber vector k) instead of a
spatial domain (usually denoted by a vector r). This transform is a derivation
of a spatial Fourier transform in which the base functions are plane-wave
functions of the type e−jk·r.
In practical terms, the calculation of this plane-wave transform amounts
to the solution of an inverse problem which has as input data the sound
pressure sampled at several positions in space for a specific pure tone
frequency. To that effect, microphone arrays can be used. To improve
sampling flexibility [3], a sequenced microphone array can be used to measure
the sound field in randomly generated positions. A random sequenced
microphone array mitigates problems such as positioning errors [3] and
transducer mismatch [9].
A possibility proposed by Nolan [1,4] is to calculate the absorption
coefficient of a surface using the plane-wave decomposition. The main
advantage of using this method in contrast to other methods, such as using
Sabine’s equation, is that it does not require the existence of a diffuse
sound field nor well-distributed absorption [3], which circumvents systematic
errors that are usually identified when using these techniques. Thus, the
absorption coefficient calculation using the wavenumber approach could be
the foundation for developing a new in situ measurement procedure [1].
Therefore, this article describes an experiment to evaluate the application
of Nolan’s proposed absorption measurement technique that employs the
wavenumber (angular) spectrum. This paper also presents some topics on the
automation of the robotic arm used as a sequential array. The error between
measured and reconstructed room transfer functions (RTFs) is also quantified
3
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and analysed. From the wavenumber spectrum, the absorption coefficient of
a surface was calculated per third-octave band and then compared to values
calculated by Miki’s model [10].
The paper is structured as follows. Section 2will describe the theory
that supports the methods and analyses performed in this research and some
fundamental topics on robot kinematics. Section 3is used to further the
presentation of the utilised array and to describe some of the challenges
that the use of the 5 degrees of freedom presented. Section 4describes
the experimental setup that was used and the results of the sound field
reconstruction (Section 4.1), the sound energy mappings (Section 4.2), and
the absorption coefficient results (Section 4.3). Section 5closes the paper
with some considerations about the results obtained.
2. Theory
This section presents a review of techniques and concepts necessary and
valuable to understand the processing and analyses made in this paper.
2.1. Plane-wave decomposition
If a sound field is sampled at Mmicrophone positions, the sound
pressure at each measurement location can be written as a weighted sum
of plane waves {e−jk1·rm,...,e−jkL·rm}with unknown complex amplitudes
{e
P(k1), ..., e
P(kL)}, as in Equation (1)
p(rm) =
L
X
l=1 e
P(kl)e−jkl·rm, (1)
where the index l= 1, ..., L, represents different wavenumber vectors
{k1, ..., kl, ..., kL}, and the index m= 1, ..., M denotes different positions
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measured in the physical space {r1, ..., rm, ..., rM}. In this paper, the
wavenumber vectors considered in the base represent only propagating waves,
i.e. ∥k∥2=k2
x+k2
y+k2
zsuch that ∥k∥2≥k2
x+k2
y. This is because
the measurements are taken sufficiently far from any surface so that the
contribution of evanescent waves to the sound field can be disregarded (other
analyses such as those produced by Brand˜ao and Fernandez-Grande [11] take
evanescent waves into account, at the cost of obtaining inverse problems that
are harder to regularise).
Equation (1) relates a single measured sound pressure to a sum of plane
waves. For an array of microphones (with no shape or form restriction), a
matrix equation that relates all array points (in the vector p∈CM) with all
the weighing coefficients of the propagation directions (in the vector x∈CL)
can be formulated. The vector x=˜
P(k1). . . ˜
P(kL)Tis denominated
the wavenumber spectrum of the sound field. The referred matrix equation
is displayed in Equation (2) usually written in reduced form
p=Hx, (2)
where H∈CM×Lis called the sensing matrix [3].
To solve Equation (2) for x, one must resort to a regularisation technique.
In this work, the option for Tikhonov regularisation was made because it
is readily implemented [12,13] and, more importantly, because it provides
qualities that are desired when using this technique to represent environments
with predominant reverberant fields [1,2,5,6], such as smoothness and
solutions with minimum energy. The regularisation was done using Hansen’s
“Regularisation Tools Version 4.1 for Matlab 7.3” toolbox [14].
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2.2. Calculating the absorption coefficient
Once the wavenumber spectrum is calculated, a graphic representation
of |e
P(k)|over the radiation sphere can be produced (such as Figure 7
in Section 4.2). The lower hemisphere contains the directions where the
sound impinges on an absorbing surface. The upper hemisphere contains the
directions where the surface reflects the sound. Thus the elevation angle-
dependent sound absorption coefficient can be computed as Equation (3)
α(θ) = 1 −R2π
0|e
P(θ, ϕ)|2dϕ
R2π
0|e
P(π−θ, ϕ)|2dϕ. (3)
Nolan [4] suggests that the wavenumber spectrum be averaged in third-
octave bands to eliminate possible biases related to the discretization of the
frequency domain and non-isotropic energy distribution for pure tones in
rooms. Furthermore, to determine a diffuse field absorption coefficient, one
can use the Paris’ formula [15], given by
αs=Zπ/2
0
α(θ) sen(2θ) dθ. (4)
2.3. Reconstruction of a transfer function
The sound field can be reconstructed in arbitrary positions using the
wavenumber spectrum [1,3,6,16]. This can be employed to assess the
sound pressure and unobserved quantities, such as the surface impedance
and the sound intensity [1,3,6].
The reconstruction process is a direct problem. For sound pressure, it is
given by Equation (5)
pr=Hrˆx, (5)
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in which pr∈CKcontains the reconstructed sound pressures at Karbitrary
positions in the vicinity of the array, and Hr∈CK×Lis the reconstruction
matrix, calculated from the plane wave functions evaluated on the points
where the sound field will be reconstructed.
3. Array Programming and Measurements
The sampling of the sound field can be performed with several different
options. The application of multi-channel microphone arrays is extensively
documented [17,18], and its main advantages are that measurement times
are quick, leading to faster experiments. Moreover, Dupont, Melon and Berry
[19] state that array-based holographic techniques present superior qualities
for oblique incidence compared to model-matching methods. Thus, an array-
based approach was employed in this research. Multi-channel arrays have
three major downsides. One is the transducer mismatch errors [9], which are
prominent. Another is that the shape of the array in these cases is fixed,
which removes flexibility from the experiments regarding other array shapes
and parameters. Finally, the cost of equipment increases with the number
of microphones and channels. Conversely, sequential microphone arrays use
only one microphone that scans the sound field in specified positions, so
there is no transducer mismatch. The specified positions can be whichever is
designed for the experiment, such as a planar [20], spherical [21], or random
arrangement [4]; double or single layer [20]; and different array dimensions
[19]. Moreover, sequential arrays can operate with only one channel, so
simpler sound acquisition hardware can be used. The main downside of
sequential arrays is that the measurement times can be extended. Sequential
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arrays can be designed using either a robot [4,16] or a three-dimensional
positioning system [19,20]. Three-dimensional positioning systems usually
need to be built by the user to their specifications, while robots may be
available and adapted to the context of acoustic measurements.
A robot was used for this research due to its availability: a Mitsubishi
RV-M1 robot arm from the Flexible Manufacturing Systems laboratory
of UTFPR, Curitiba Campus. The Mitsubishi RV-M1 robot has 5
degrees of freedom (DOF) and is usually employed for numeric command
machine integration, mechanical assembly of parts, and didactic and training
purposes. The RV-M1 model has been discontinued and has limitations
compared to its current counterparts. Still, it can perform sequential
array measurements with relatively good repeatability (with positioning
errors limited to 0.3 mm), albeit within a smaller spatial range and follows
communication protocols compatible with current computers. It has a
maximum weight capacity of 1.2 kg, and a reach of approximately 482 mm
[22]. The sequencing and control of the robot were done using MATLAB,
sending commands to and receiving information from the robot through the
RS-232 serial communication protocol.
The designed array had 120 random positions determined inside a
400 mm ×170 mm ×100 mm rectangular volume. For the experiment
with fewer array positions (Section 4.1), points were removed from this
initial array. Aside from the 120 array points, another 30 points were
also measured to evaluate the accuracy of the reconstruction process. The
sequenced microphone array with the 120 pressure sampling points and the
30 evaluation points are depicted in Figure 1[16].
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Figure 1: (Colour online) Microphone positions [16]. Dots (green): points measured for the
inverse problem (random array). Triangles (magenta): points measured for reconstruction
error evaluation. Square (yellow): robot reference.
Regardless of their DOF, robots have physical restrictions on the positions
they can access. Robots like the one implemented in [4,21] have 6 DOF,
which means they can achieve more complex poses and have a larger region of
achievable positions than the Mitsubishi RV-M1 robot used in this research.
Thus, all the microphone positions had to be validated by verifying their
inverse kinematics, as it is difficult to limit a region where all points are
achievable. Inverse kinematics is the calculation of the joint angles that
the robot needs to reach to achieve a particular position (x, y, z) in the
space with a certain arrival angle (β, γ), βbeing the pitch angle (front
or back tilt) and γbeing the roll angle (lateral tilt). This combination of
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position and angle can be described in multiple joint combinations that can
be possible or impossible for the robot to achieve. It is not uncommon for
all the combinations to be unreachable. The inverse kinematics equations
are specific to each robot model, and deriving them can be challenging. The
inverse kinematics equations of the Mitsubishi RV-M1 for each joint angle
and the verification algorithm are defined on [23].
The points that form the sequential array were each randomly generated
and had their inverse kinematics tested. If a position was found unreachable,
it was discarded, and a new one was generated until a reachable position was
achieved. The sequential array that resulted from this process is shown in
Figure 1. The direction of the microphone was fixed to be perpendicular to
the surface.
For the positions to be informed to the robot, the microphone coordinates
need to be converted to their corresponding coordinates of the wrist of the
robot. The wrist is the flange in the last joint, where a tool — in this context,
a microphone holder — is attached to the robot. These values informed to
the robot’s control unit through serial communication shall be referred to as
“tool positions”. The conversion of the coordinates involves calculating the
direct kinematics relating the microphone holder to the wrist joint — this
process is thoroughly described in [23]. Because the microphone holder was
designed so that the centre of the microphone was conveniently aligned with
the centre of the robot’s wrist, the distance between the microphone centre
and the wrist centre can be informed to the robot, simplifying the kinematics
calculations. This was a design choice to simplify the calculations. Thus, the
tool position values informed to either the robot or the inverse kinematics
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verification function are the same as the randomly generated microphone
positions, and the only calculations needed are that of γ, and β, respectively
sin(γ) = √2(jmsin(θ1)−imcos(θ1)); (6)
tan(β) = km+jmcos(θ1)cos(θ1) + imcos(γ)sin(θ1)
jmcos(θ1) + imsin(θ1)−kmcos(γ); (7)
in which im,jmand kmare the microphone directions, and θ1is the joint
angle of joint #1 (the robot’s waist). Then, the verification of the inverse
kinematics is done by calculating all the remaining joint angles θ2,θ3,θ4
and θ5and verifying if they result in values inside the operating range of
the robot. Figure 2, adapted from [22], displays a schematic of the robot
emphasising each joint and its operating range.
θ4 = [-90o,+90o]
θ2 = [-30o,+100o]
θ3 = [-110o,+0o]
Joint #1
Joint #4 Joint #3
θ5 = [-180o,+180o]
Joint #5
Joint #2
θ1 = [-150o,+150o]
Figure 2: Mitsubishi RV-M1 joints (adapted from Mitsubishi Electric Corporation [22]).
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4. Experimental Setup and Results
The room utilised for the experiments was a typical laboratory room (EK
- 019), with no acoustic treatment done, at the Ecoville Campus of UTFPR.
The enclosure of the room was approximately 11.8 m ×5.3 m ×3.3 m, and
inside were pieces of furniture and equipment resembling in situ conditions.
All impulse responses were measured using the ITA Toolbox [24]
developed by the Institute for Hearing Technology and Acoustics at RWTH
for MATLAB. The selected measurement technique was linear deconvolution.
An omnidirectional sound source (model GROM DDC-100) was used to
excite the room with an exponential frequency sweep, and a GRAS 40AE free-
field microphone was used to record the room’s response. It should be stated
that the response level [dB] versus frequency [Hz] curve of this microphone is
flat until about 10000 Hz (with the correction for direct incidence) [25]. One
advantage of the proposed method is that there is no need for a structure
to position a speaker in several incidence angles, as the omnidirectional
source is expected to excite the room and, through interaction with its many
surfaces, generate incidence from many directions. The sampling frequency
was 96000 Hz, amounting to a sweep lasting 5.46 s.
In all the measured positions, the microphone was placed beyond the
calculated critical distance of the room (approximately 1.2 m, measured with
reference to the source). The shortest reverberation time of the room is 0.5 s.
A glass wool sample of 10.8 m2with 40 kg/m3of density and 50 mm
thickness was placed in one of the walls of the room. Being a porous material
[15], this type of mineral wool is expected to be more absorbent at higher
frequencies.
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The array’s closest measurement point to the sample’s surface was 0.3 m
from the glass wool overlay. Figure 3shows the glass wool panels and the
Mitsubishi RV-M1 robot.
Figure 3: (Colour online) Glass wool absorbing surface and Mitsubishi RV-M1 robotic
arm.
The sound field was sampled near the absorber using the sequential array
shown in Figure 1. Because the coordinates of the array were generated
with respect to the robot’s coordinate system, an adjustment to describe the
coordinates with a reference on the absorption plane was needed.
The plane-wave decomposition was calculated for 1000 directions
uniformly distributed on a sphere [1]. The solution to the Thomson problem
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achieves this distribution [26,27].
4.1. Sound Field Reconstruction
The reconstruction of the RTF was done in the 30 evaluation points
cited in Section 3. The relative error between the reconstructed transfer
function and the measured transfer function was calculated following the
metric defined in Equation (8) [6,16],
ε%= 100 ×
pR(f)−pM(f)
pM(f); (8)
where pR(f) and pM(f) are the values of the reconstructed and measured
RTF at a frequency f, respectively. The value ε%is averaged across the 30
reconstruction points and inside the third-octave bands. The reconstruction
relative errors were calculated using 80, 100, and 120 microphone positions.
The average relative errors from the 100 Hz to the 2500 Hz third-octave
bands are shown in Figure 4(circles, with the dashed line added to avoid
parallax).
It can be promptly noticed that the reconstruction at lower frequencies
is more accurate than at higher frequencies. The reconstructions for all
cases perform well between the 100 Hz and the 630 Hz band, with a
relative percentage error limited to under 6.0 %. The differences in the
errors start to be visible from the 800 Hz band onward. Generally, the
120 positions microphone array presents the smallest average reconstruction
errors. Another thing worth noticing is that at the 1600 Hz band, the
error for the 80 microphone case is marginally smaller than that for the
100 microphone case. This can be because of the different microphone
dispositions that influence the numerical aspects of the regularisation, such
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Figure 4: (Colour online) Average relative reconstruction error from 100 Hz band to 2500
Hz band. Eighty positions: red circles. One hundred positions: green circles. One hundred
twenty positions: blue circles.
as condition number and column redundancy [17], which in turn affect the
calculated angular spectrum. The smallest percentage error value for the 120
microphones case is in the 500 Hz band, at 2.1 %. Between the 400 Hz band
and the 630 Hz, the error for the 120 microphones case is limited to under
3.0 %.
Another point of notice is the increased errors with the increase in
frequency. From 2000 Hz onward, the error is always above 10.0 %, and it
rises following the increase in frequency, comparable to the measured values.
This can happen due to a series of factors, such as the number of plane wave
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directions in the wavenumber sphere or a badly discretized region of the space
in the array, i.e. areas in which the microphone spacing is too large because
the robot cannot reach the in-between positions. Evidently, for the number
of directions in the wavenumber domain, a set of factors must be taken into
account between the desired reconstruction error, the processing capabilities
of available hardware, and the numerical errors that can increase due to
the Hmatrix being ill-conditioned. As the frequency increases, the radius
of the wavenumber sphere grows, and the spacing between the directions
of reconstruction also grows. Nolan [28] presents that there is a limit to
how much the number of directions can make a model better. For 1 kHz,
1000 plane waves can represent the sound field without a significant loss of
information. However, it should be pointed out that the analysis in [28] was
carried out for a single frequency. The inverse deviance information criterion
(IDIC) analysis for a more extensive range of frequencies is still a future
consideration.
Figure 5shows the reconstruction (blue solid line) and the measured
transfer function (green dashed line) at the evaluation position number 30,
between 89.1 Hz and 2818 Hz (the lower limit of the 100 Hz third-octave band
and the upper limit of the 2500 Hz third-octave band). This reconstruction
was done using the 120 positions array. The detail shows the reconstruction
between 2510 Hz and 2560 Hz, making some errors more evident.
It can be seen that in lower frequencies, there is excellent concordance
between the measured and the reconstructed transfer functions. The
reconstruction still preserves concordance with increased frequency but
generally undershoots the transfer functions at the anti-resonance
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Figure 5: (Colour online) Comparison between the reconstructed RTF and the measured
RTF at evaluation point number 30. Blue solid line: reconstruction. Green dashed
line: measurement. Detail: reconstructed and measured RTF from 2510 Hz to 2560 Hz,
displaying a discontinuity error.
frequencies. The errors in the reconstructed modes also rise less intensely.
Alongside the errors at the anti-resonance frequencies, errors caused by
discontinuities can also be detected, as seen in the detail in Figure 5. These
errors are also more prominent in higher frequencies. Errors of this nature
likely occur due to the lack of restrictions on the existence of derivatives
and second derivatives in the regularisation process, which in turn carry over
when solving the direct problem. Because these errors occur more often
in higher frequencies, a higher modal density will mean that the average
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reconstruction error will also increase because of the accumulation of errors.
4.2. Sound Energy Mappings
Since the wave number vectors are comprised solely of propagating plane
waves distributed on the surface of a sphere, one can plot the magnitude of
the plane waves as a function of the travel directions. The graphics were
made using Eckert III projections to avoid excessive distortion on the poles,
and the elevation (θangle) is measured with relation to the xy plane. For
this, the averaged spectra needed to be interpolated into an evenly-spaced
grid. The interpolation was done with a resolution of π/60 rad. The maps
for the 400 Hz, 1000 Hz, 1500 Hz, and 2500 Hz are on the left side of
Figure 7, respectively. The map on Figure 7b also indicates some general
directions, such as the ceiling, the ground, the source and the room’s sides.
The energy dynamic range was limited to 15 dB to visualise the incidence
and the reflections better. In that context, although it may seem that there
is no energy reflected on the maps on Figures 7c and 7d, the attenuation
caused by the absorber makes it so that the energy is small enough to be
under the defined threshold.
Some artefacts can be seen near the (ϕ, θ) = (180o,0o) positions. These
are caused because the interpolation originally is calculated for angles nearing
−180oto angles approaching 180o, leaving the azimuth limits unfilled.
Notwithstanding, it can be noticed that, with an increase in frequency, the
specular energy reflected by the absorber (“northern” hemisphere) decreases
in level, indicating that the surface is absorbing the sound irradiated by the
omnidirectional source. Moreover, it can be seen that with the increase in
frequency, the lobes that denote the incidence and specular reflection get
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more minor, such as is expected for this technique [11].
Furthermore, even with the averaging on the third-octave bands,
reflections that are not related to the surface of the absorber can be noticed
in all displayed maps. These reflections from other directions of the room
contribute to calculating an absorption coefficient that depends on the
direction of incidence and, subsequently, the diffuse incidence absorption
coefficient calculated by the Paris formula, even in a small room with a
sound field that is not very diffuse. This corresponds to an advantage of
this technique, as with a relatively simple excitation setup composed of an
omnidirectional source conveniently positioned beyond the critical distance,
it is possible to obtain incidence from several directions from only one
measurement process.
4.3. Absorption Determination
The absorption of the glass wool surface was calculated following the
theory proposed in Subsection 2.2. For that, the absolute value of the
wavenumber spectrum needed to be averaged in third-octave bands to remove
biases that could occur in specific discrete frequencies [4]. Then, the
incidence-dependent absorption was calculated for each third-octave band.
For comparison, the incidence-dependent absorption of the glass wool sample
was also determined using the Delany-Bazley-Miki model [10] — henceforth
referred to as “Miki’s model”, for short. The flow resistivity of the sample
was measured at the Federal University of Santa Maria (UFSM), with a
value of 11821 kgm−3s−1. The measured and the model incidence-dependent
absorption for the 400 Hz, 1000 Hz, 1600 Hz, and 2500 Hz are shown on the
right side of Figure 7.
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An interesting occurrence happens in the 250 Hz band. Figure 6shows
the map and the directional absorption of that band. It should be noted that
θis the elevation measured from the zaxis varying from 0oto 180o, while the
incidence angle ϑgoes from 0o(direct incidence) to 90o(parallel incidence).
Azimuth f?g[o]
Elevation f3g[o]
45° 90° 135° 180° 225° 270° 315°
-60°
-30°
0°
30°
60°
-15 -12 -9 -6 -3 0
Energy level [dB]
(a) Energy map.
:=6:=3:=2
Incidence angle f#g[rad]
-0.4
-0.2
0.0
0.2
0.4
0.6
Absorption f,(3)g[-]
In situ measurement
Miki's model
(b) Directional absorption.
Figure 6: (Colour online) Energy map (Figure 6a) and Directional absorption (Figure 6b)
for the 250 Hz band.
It is noticeable that the map shows energy coming from the reflected
region (left side of the upper hemisphere) with the same intensity as the
incident energy from the source direction. This makes it so that in the 250 Hz
band (Figure 6b), the absorption values are negative after approximately
51o(17 π
60 rad). This corresponds to one of the downsides of this technique.
Nolan [4] states that at lower frequencies, the absorption values at angles of
incidence almost parallel or parallel to the surface are underestimated due
to what the referred author calls “leakage” of the incident energy into the
reflected energy region (“northern” hemisphere). In extreme cases, such as
the one in Figure 6, the absorption values calculated can reach impossible
negative values. This can happen due to limitations on the wavenumber
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resolution provided by the array [4], the smoothing of the solution caused by
the Tikhonov regularisation process, or the model not considering evanescent
waves. Moreover, in any case, the largest deviance from the model values
seems to happen consistently at angles almost parallel to the surface. The
values of the in situ at incidences between 0 π6 rad are very close to the
values predicted by Miki’s model [10], which can also be explained by the
same leakage effect.
Figure 8shows the diffuse incidence absorption calculated with
Equation (4) for both the in situ measurement (circles) and the absorption
calculated by Miki’s model (asterisks), from the 315 Hz to the 2500 Hz third-
octave bands.
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Azimuth f?g[o]
Elevation f3g[o]
45° 90° 135° 180° 225° 270° 315°
-60°
-30°
0°
30°
60°
-15 -12 -9 -6 -3 0
Energy level [dB]
:=6:=3:=2
Incidence angle f#g[rad]
0.0
0.2
0.4
0.6
0.8
1.0
Absorption f,(3)g[-]
In situ measurement
Miki's model
(a) 400 Hz.
Azimuth f?g[o]
Elevation f3g[o]
45° 90° 135° 180° 225° 270° 315°
-60°
-30°
0°
30°
60°
-15 -12 -9 -6 -3 0
Energy level [dB]
:=6:=3:=2
Incidence angle f#g[rad]
0.0
0.2
0.4
0.6
0.8
1.0
Absorption f,(3)g[-]
In situ measurement
Miki's model
(b) 1000 Hz.
Azimuth f?g[o]
Elevation f3g[o]
45° 90° 135° 180° 225° 270° 315°
-60°
-30°
0°
30°
60°
-15 -12 -9 -6 -3 0
Energy level [dB]
:=6:=3:=2
Incidence angle f#g[rad]
0.0
0.2
0.4
0.6
0.8
1.0
Absorption f,(3)g[-]
In situ measurement
Miki's model
(c) 1600 Hz.
Azimuth f?g[o]
Elevation f3g[o]
45° 90° 135° 180° 225° 270° 315°
-60°
-30°
0°
30°
60°
-15 -12 -9 -6 -3 0
Energy level [dB]
:=6:=3:=2
Incidence angle f#g[rad]
0.0
0.2
0.4
0.6
0.8
1.0
Absorption f,(3)g[-]
In situ measurement
Miki's model
(d) 2500 Hz.
Figure 7: (Colour online) Sound energy mappings (left side) and directional
absorption (right side) for the 400 Hz (Figure 7a) band, the 1000 Hz (Figure 7b)
band, the 1600 Hz (Figure 7c) band, the 2500 Hz (Figure 7d) band.
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500 1000 2000
Frequency ffg[Hz]
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Absorption f,sg[-]
In situ measurement
Miki's model
Figure 8: Diffuse field absorption coefficient per third-octave band, calculated from third-
octave averages of interpolated wavenumber spectra (circles), reference absorption values
calculated using Miki’s model (asterisks).
The graphic displays a general trend of rise in absorption following the
increase in the frequency, which is coherent with what is expected for this
type of material [15]. Due to the severe underestimation in the 250 Hz band
in the proposed in situ method, the displayed frequencies were chosen to
be above that. The in situ values also display coherence with the diffuse
incidence absorption coefficients calculated by applying the Paris formula.
Both curves display a similar evolution between 315 Hz and 2500 Hz, with
the in situ experimental values being, on average, 0.16 smaller than the
23
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model values. The differences between the in situ and the values calculated
by Miki’s model can be attributed to the fact that the Paris formula is
an approximation, as it considers the existence of a perfectly diffuse sound
field [29] and an infinite absorbing sample. Another factor that can be
responsible for these errors is the measurement distance between the array
and the sample. Richard and Fernandez-Grande [17] show that, even though
the spherical array has a better condition number than the planar array,
the planar array displays better results because the constraints related
to the hardware allow for a measurement closer to the absorbing surface
resulting in more accurate surface impedance measurements. Moreover, when
combined, the relative underestimations of the proposed technique result in
an underestimation of the result of the Paris formula. Combining these
factors with other errors from calculating the average absorption concerning
the incidence angle can lead to an accumulation of errors when calculating
the diffuse field absorption coefficient.
5. Conclusion
The experiment described in this paper was the calculation and polar
mapping of the wavenumber spectrum, the reconstruction of the sound field
using the wavenumber spectrum, and the evaluation of the absorption of a
surface of interest in installation conditions. In this experiment, the room had
one surface covered with 10.8 m2of 40 kg/m3glass wool, a porous absorbing
material, and 150 RTF measurements in random positions were performed —
120 for the sequential array and 30 for comparison with the reconstructions.
The calculation of the inverse problem’s solution and the reconstruction of
24
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30 RTFs were evaluated for a range between the 100 Hz and 5000 Hz third-
octave bands for arrays of 80, 100, and 120 positions. The case using 120
positions displayed the smaller errors, limited to under 10 % until the 1600 Hz
band. For higher frequencies, the errors become even more pronounced —
over 10 % — for all cases, most likely due to the excessive spacing between
plane-wave directions. The processing was deemed successful.
The sound field mapping for each band in the proposed frequency range
showed increased absorption when the frequency rises. The mapping was also
important from an operational standpoint when implementing Equation (3).
Moreover, some phenomena related to excessive smoothing of the solution
or array characteristics in the wavenumber domain could be noticed on the
250 Hz band. This type of problem cannot be noticed only by analysing
reconstruction errors, as the solution can solve the direct problem with
relatively small errors but is a misrepresentation of the sound field in the
angular domain.
The plane-wave decomposition allowed the identification of the direct
incidence from the sound source over the absorbing sample, the reflection
of the surface of interest, and spurious reflections caused by the other
regions of the room and influenced by the position of the sound source.
This is important because the lack of hemispherical-isotropic incidence
over the sample can be one of the causes of the underestimation of the
absorption coefficient values obtained, along with factors influenced by the
array characteristics. When compared to values calculated with Miki’s
model [10], the in situ absorption coefficients display a similar behaviour
in relation to the frequency, with an apparent systematic underestimation,
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between the 315 Hz and the 2500 Hz bands. For bands of frequencies lower
than the mentioned, phenomena related to the underestimation at 250 Hz
made it so that the results before and including this band were considered
untrustworthy. Nevertheless, the absorption values show an overall behaviour
consistent with models present in the literature [15] for that type of material
and verified by comparing the energy distributions for increasing frequencies.
Overall, the validity of the algorithms for plane wave decomposition and
subsequent operations was confirmed. Ideas for future works also stem from
the experiments described in this paper, such as the application of plane wave
decomposition using other regularisation parameters for in situ calculation of
absorption, implementations of early sound incidence windowing to control
spurious reflections, the assessment of the decomposition and reconstruction
errors of the processing, the investigation on optimal array configurations
for absorption measurements and the influence of the plane-wave domain in
the reconstruction errors. The criteria for defining a clear frequency band
in which the technique works properly are not very clear at this time and
should also be approached in future research.
Acknowledgements
The authors would like to thank the National Research Council of Brazil
(CNPq - Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ogico;
Projeto Universal: n◦402633/2021-0) and the Coordination for the
Improvement of Higher Education Personnel (CAPES - Coordena¸c˜ao de
Aperfei¸coamento de Pessoal de N´ıvel Superior; CAPES-DS grant: n◦
88887.644633/2021-00) for the partial financial support to this research
26
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paper. The authors would also like to thank Prof. Luiz Rodrigues (UTFPR)
for allowing access to the Mitsubishi RV-M1 robot.
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