SOUND INTENSITY
SESSIONS
State of the Art of Sound Intensity and Its Measurement
and Applications
Finn Jacobsen
Acoustic Technology, Ørsted DTU, Technical University of Denmark, Building 352, DK-2800 Lyngby, Denmark
The advent of sound intensity measurement systems in the beginning of the 1980s has had a significant influence on noise control engineering. Today sound intensity measurements are routinely used in the determination of the sound power of machinery
and other sources of noise in situ. Other important applications of sound intensity include the identification and rank ordering
of partial noise sources, visualisation of sound fields, determination of the transmission losses of partitions, and determination
of the radiation efficiencies of vibrating surfaces, and recent work suggests the possibility of measuring sound absorption in situ
using sound intensity. This paper summarises the basic theory of sound intensity and its measurement and gives an overview of
the state of the art in the various areas of application, with particular emphasis on recent developments.
INTRODUCTION
The most important acoustic quantity is certainly
the sound pressure. However, sources of sound emit
sound power, and sound fields are also energy fields
in which potential and kinetic energies are generated,
transmitted and dissipated. In spite of the fact that the
radiated sound power is a negligible part of the energy
conversion of almost any sound source, energy considerations are of enormous practical importance in
acoustics. In ‘energy acoustics’ sources of noise are
described in terms of their sound power, acoustic materials are described in terms of the fraction of the incident sound power that is absorbed, and the sound
insulation of partitions is described in terms of the
fraction of the incident sound power that is transmitted, the underlying assumption being that these properties are independent of the particular circumstances.
This is not strictly true, but it is usually a good approximation in a significant part of the audible frequency range, and alternative methods based on linear
quantities are vastly more complicated.
Sound intensity is a measure of the flow of acoustic energy in a sound field. More precisely, the sound
intensity I is a vector quantity defined as the time average of the net flow of sound energy through a unit
area in a direction perpendicular to the area. Although
acousticians have attempted to measure this quantity
since the early 1930s it took almost fifty years before
reliable measurements could be made. Commercial
sound intensity measurement systems came on the
market in the beginning of the 1980s, and the first international standards for measurements using sound
intensity and for instruments for such measurements
were issued in the middle of the 1990s. A description
of the history of the development of sound intensity
measurement is given in Frank Fahy’s monograph
Sound Intensity [1].
The advent of sound intensity measurement systems in the 1980s has had a significant influence on
noise control engineering. Sound intensity measurements make it possible to determine the sound power
of sources without the use of costly special facilities
such as anechoic and reverberation rooms, and today
sound intensity measurements are routinely used in the
determination of the sound power of machinery and
other sources of noise in situ. Other important applications of sound intensity include the identification and
rank ordering of partial noise sources, visualisation of
sound fields, determination of the transmission losses
of partitions, and determination of the radiation efficiencies of vibrating surfaces.
It is more difficult to measure sound intensity than
to measure sound pressure. The difficulties are mainly
due to the fact that the accuracy of sound intensity
measurements with a given measurement system depends very much on the sound field under study. Another problem is that the distribution of the sound intensity in the near field of a complex source is far
more complicated than the distribution of the sound
pressure, indicating that sound fields can be much
more complicated than earlier realised. The problems
are reflected in the extensive literature on the errors
and limitations of sound intensity measurement and in
the fairly complicated international standards for
sound power determination using sound intensity, ISO
9614-1 and ISO 9614-2.
The purpose of this paper is to give an overview of
the status of sound intensity measurement, with emphasis on developments from the past few years.
SESSIONS
MEASUREMENT PRINCIPLES AND
THEIR LIMITATIONS
There are essentially three methods of measuring
sound intensity: i) one can combine two pressure
transducers (or more), ii) one can combine a pressure
transducer with a particle velocity transducer, and iii)
one can use a microphone array. Each method has its
own limitations, but in addition there are some fundamental sources of error that do not depend on the measurement principle [2, 3]. Array methods are used for
reconstructing sound fields, including sound intensity.
This topic is outside the scope of this overview; see
ref. [4].
The two-microphone principle
All sound intensity measurement systems in commercial production today are based on the ‘two-microphone’ (or ‘p-p’) principle, which makes use of two
closely spaced pressure microphones and relies on a
finite difference approximation to the sound pressure
gradient, from which the particle velocity is determined. The IEC 1043 standard on instruments for the
measurement of sound intensity deals exclusively with
this measurement principle.
The finite difference approximation obviously implies an upper frequency limit. However, because of
the fact that the finite difference approximation error
will be partially cancelled by a resonance if the length
of the spacer equals the diameter of the microphones,
the upper limit of the frequency range can be extended
to twice as much as suggested by the finite difference
error. Thus the resulting upper frequency limit of a
sound intensity probe composed of half-inch microphones separated by a 12-mm spacer is 10 kHz [5],
which is an octave above the limit determined by the
finite difference error when the interference of the
microphones on the sound field is ignored [1]. Another way of extending the frequency range is to apply
a ‘sensitivity correction’ for the finite difference error.
Such a correction has recently been suggested for 3-D
intensity probes (without spacers) [6]; this also seems
to double the upper frequency limit.
Even with the best sound intensity measurement
systems available today the range of measurement is
limited by small deviations between the phase responses of the two measurement channels. Above a
few hundred hertz the main source of phase mismatch
is a difference between the resonance frequencies of
the two microphones, which gives rise to a phase error
that is proportional to the frequency. For example, one
must, with state-of-the-art equipment, allow for phase
mismatch errors ranging from 0.05° below 250 Hz to
1° at 5 kHz, which is slightly less than the maximum
values specified in the IEC standard.
It can be shown that the effect of a given phase
error is a bias error that is proportional to the mean
square pressure [7]. This implies that there is a limit to
the amount of noise from sources outside the measurement surface that can be tolerated in sound power determination using sound intensity, because such noise
will invariably increase the surface integral of the
mean square pressure and thus the bias error.
Another limitation of the measurement principle is
the restricted dynamic range at low frequencies: because the electrical noise of the microphones at low
frequencies is amplified by the time integration needed in determining the particle velocity from the
pressure gradient, one cannot in practice measure
sound intensity levels below, say, 50 dB re 1 pW/m2
below 100 Hz [8].
Other sources of error include the effect of airflow. Airflow gives rise to correlated noise signals
that contaminate the signals from the two microphones. The result is a ‘false’ additive sound intensity
at low frequencies [9].
The p-u measurement principle
Attempts to develop sound intensity probes based
on the combination of a pressure transducer and a particle velocity transducer have occasionally been described in the literature [1]. A recent example is based
on the ‘microflown’ particle velocity transducer [10],
which is a micro machined hot wire anemometer. One
problem with any particle velocity transducer, irrespective of the measurement principle, is the strong
influence of airflow. Another unresolved problem is
how to determine the phase correction that is needed
when two fundamentally different transducers are
combined (even though the influence of p-u phase
mismatch will usually be less serious that the influence of p-p phase mismatch). Whereas one can compensate for p-p phase mismatch by employing a simple
correction determined with a small coupler [7], there
is no similarly simple way of determining a correction
for p-u phase mismatch.
The measurement principle has the important advantage in sound power measurements that the performance is not affected by the global pressure-intensity
index on the measurement surface, so in principle a
larger amount of noise from sources outside the surface can be tolerated with p-u sound intensity probes
[11]. However, any such advantage has yet to be demonstrated experimentally.
SESSIONS
APPLICATIONS
Visualisation of Sound Fields
Some of the most important applications of the
sound intensity method are now discussed.
Visualisation of sound fields, helped by modern
computer graphics, contributes to our understanding
of radiation and propagation of sound and of diffraction and interference effects [16] and has stimulated
much research on sound fields [4]. Visualisation of
the flow of sound energy is particularly useful, with
obvious applications in noise source identification,
and identifying and ranking noise sources and transmission paths is the first step in practically any noise
reduction project. Near fields can be very complicated,
however. Johansson has examined various methods of
decomposing such near field intensity data into statistically independent components, using principal component analysis and partial least squares regression
[17]. He concludes that a decomposition into principal
components enhances the identification of noise
sources.
One particular problem in using sound intensity in
identification of partial sources on vibrating structures
is the circulation of sound energy that occurs when the
flexural waves on the structure travel slower than the
speed of sound. When the wavelength on the structure
is shorter than the wavelength in air the waves do not
radiate to the far field but may generate a circulatory
flow of sound power, out of the structure and back
into the structure again in adjacent regions. To overcome the problems with such complicated near field
phenomena Williams has introduced the concept of
‘supersonic intensity’ [4]. This quantity is defined as
the normal component of the part of the sound intensity associated with supersonic waves on the structure,
that is, waves that travel faster than the speed of
sound. Eliminating the subsonic wave components
does not affect the total radiated sound power of the
structure, and, in the words of the author, ‘the credibility of the concept of supersonic intensity lies in the
fact that power is conserved.’ The supersonic intensity
is obtained by filtering out the subsonic wavenumber
components from near field holographic data decomposed into wavenumber components, and in practice
this is only possible for structures of planar or cylindrical geometry. Pascal and Li have developed a related concept with similar properties called ‘irrotational acoustic intensity’ (since it has zero curl), defined as the gradient of the potential of the intensity
[18]. Thivant and Guyader’s prediction tool ‘the intensity potential approach’ [19] is closely related to the
irrotational intensity. One cannot determine the supersonic intensity or the irrotational intensity with an ordinary sound intensity measurement system; a
holographic reconstruction method is needed.
Sound Power Determination
One important application of sound intensity is the
determination of the sound power of operating machinery in situ. Sound power determination using intensity involves integrating the normal component of
the intensity over a surface that encloses the source.
The integral is usually approximated by scanning manually or with a robot over the surface. Two international standards are available, and a third one (for precision measurement) is on its way.
A few years ago Paine and Simmons described the
results of an investigation of the uncertainties in the
determination of sound power levels according to the
measurement procedures specified in various standards, including ISO 9614-1 [12]. The reproducibility
standard deviation was found to be around 1 dB, and
no indication of a systematic difference between the
standards was found. Nevertheless, Jonasson recently
maintained that ‘it has long been suspected that this
standard....gives lower sound power levels than the
other standards in the ISO 3740 series’ [13]. However,
the only factor known to give a negative bias error in
intensity-based sound power measurements is the absorption of the source under test, which reduces the
net sound power output. In fact, Jonasson’s observation was based on preliminary results that have later
been found to be affected by the absorption of the test
source [14].
Measurement of Transmission Loss
The intensity-based method of measuring transmission losses of partitions was introduced in the early
1980s as an alternative to the conventional method,
which requires a diffuse sound field in the receiving
room. It has often been reported that the intensity
method gives lower values of the sound transmission
index than the conventional method at low frequencies
and higher values at high frequencies. However, a critical examination of the literature has revealed that in
most cases of poor agreement the receiving rooms
have been very small [15], and a recent investigation
published in the same paper has demonstrated that
excellent agreement can be obtained in a fairly wide
frequency range, from 80 Hz to 6.3 kHz, provided that
the measurements take place in adequate facilities and
are carried out with state-of-the-art equipment.
SESSIONS
Other Applications
Ideally the ‘emission sound pressure level’ of machines at specified positions should be measured under hemi-anechoic conditions; and corrections for
background noise and for the effects of reflections
should be applied when the environment differs from
ideal conditions. Jonasson has proposed to determine
emission sound pressure levels by measuring the
sound intensity instead of the sound pressure, the idea
being that the sound intensity would be less sensitive
to reflections from the surroundings than the sound
pressure, which means that the troublesome environmental corrections could be avoided [20]. An international standard is under development. Jonasson’s proposal is based on the assumption that the reflections
that occur in a real room can be regarded as diffuse
noise, since ideal diffuse noise would not contribute to
the sound intensity. Hübner and Gerlach have compared several measurement procedures and seem to
favour determining the emission pressure level from
the maximum value of the sound intensity vector from
three perpendicular components [21].
Finally it should be mentioned that recent work
suggests the possibility of measuring sound absorption
in situ using sound intensity [22].
UNRESOLVED PROBLEMS AND FUTURE RESEARCH DIRECTIONS
There are still a number of unresolved problems in
sound intensity measurement.
As described above, there are essentially two factors that limit the range of measurement with p-p
sound intensity probes, phase mismatch and electrical
noise from the microphones and preamplifiers. It
should be possible to improve the performance of the
conventional sound intensity probes in these respects.
If particle velocity transducers of sufficient stability can be developed, the alternative p-u measurement
principle would have a significant advantage in sound
power determination in the presence of strong background noise, as pointed out above. Such probes
would probably be too sensitive to airflow for many
practical applications, but might on the other hand be
a very useful supplement.
REFERENCES
1. F.J. Fahy, Sound Intensity (2nd edition). E & FN Spon,
London, 1995.
2. F. Jacobsen, J. Sound Vib. 128, 247-257 (1989).
3. F. Jacobsen, Applied Acoustics 42, 41-53, (1994).
4. E.G. Williams, Fourier Acoustics. Academic Press, San
Diego, 1999.
5. F. Jacobsen, V. Cutanda and P.M. Juhl, J. Acoust. Soc.
Am. 103, 953-961 (1998).
6. H. Suzuki, S. Ogura and T. Ono, J. Acoust. Soc. Jpn. (E)
21, 259-265 (2000).
7. F. Jacobsen, Applied Acoustics 33, 165-180 (1991).
8. F. Jacobsen, J. Sound Vib. 166, 195-207 (1993).
9. F. Jacobsen, ‘Intensity measurements in the presence of
moderate airflow’ in Proc. Inter-Noise 94, Yokohama,
Japan, 1994, pp. 1737-1742.
10. W.F. Druyvesteyn and H.E. de Bree, J. Audio Eng. Soc.
48, 49-56 (2000).
11. F. Jacobsen, J. Sound Vib. 145, 129-149 (1991).
12. R.C. Paine and D.J. Simmons, ‘Measurement uncertainties in the determination of the sound power level of machines’ in Proc. Inter-Noise 96, Liverpool, England,
1996, pp. 2713-2716.
13. H.G. Jonasson, ‘Noise emission measurements - Systematic errors’ in Proceedings of Inter-Noise 99, Fort
Lauderdale, FL, USA, 1999, pp. 1461-1466.
14. G. Hübner and V. Wittstock, ‘Further investigations to
check the adequacy of sound field indicators to be used
for sound intensity determining sound power level’ in
Proc. Inter-Noise 99, Fort Lauderdale, FL, USA, 1999,
pp. 1523-1528.
15. M. Machimbarrena and F. Jacobsen, Building Acoustics
6, 101-111 (1999).
16. J. Tichy, ‘Visualization of sound energy flow and sound
intensity in free space and enclosures’ in Proc. International Symposium on Simulation, Visualization and
Auralization for Acoustic Research and Education, Tokyo, Japan, 1997, pp. 185-192.
17. Ö. Johansson, ‘Sound intensity vector fields in relation
to different reference signals’ in Proc. Fifth International Congress on Sound and Vibration, Adelaide, Australia, 1997, pp. 2255-2262.
18. J.-C. Pascal and J.-F. Li, ‘Irrotational acoustic intensity:
A new method for location of sound sources’ in Proc.
Sixth International Congress on Sound and Vibration,
Copenhagen, Denmark, 1999, pp. 851-858.
19. M. Thivant and J.-L. Guyader, ‘The intensity potential
approach to predict sound propagation through partial
enclosures’ in Proc. Inter-Noise 2000, Nice, France,
2000, pp. 3509-3514.
20. H.G. Jonasson, ‘Measurement of sound pressure levels
with intensity technique’ in Proc. Inter-Noise 93,
Leuven, Belgium, 1993, pp. 363-366.
21. G. Hübner, and A. Gerlach, ‘Further investigations on
the accuracy of the emission sound pressure levels determined by a three-component intensity measurement procedure’ in Proc. Inter-Noise 99, Fort Lauderdale, FL,
USA, 1999, pp. 1517-1522.
22. F. Jacobsen and L. Zamboni, ‘In-situ measurement of
sound absorption using sound intensity’ in Proc. Eighth
International Congress on Sound and Vibration, Hong
Kong, 2001.
SESSIONS
Irrotational Acoustic Intensity and Boundary Values
J.-C. Pascala and J.-F. Lib
a
Université du Maine, (LAUM UMR 6613 & ENSIM), rue Aristote, 72085 Le Mans, France
b
Visual VibroAcoustics, 51 rue d'Alger, 72000 Le Mans, France
The irrotational intensity corresponds to the non-vortex component of the active sound intensity. With the curl component
eliminated, it can be used to characterize exactly `sources' and `sinks' regions on a complex radiator and to give more
comprehensive behaviours of complex situations. By the use of processing data in wave number domain, the irrotational
intensity can be computed from the 3-D standard sound intensity, which can be obtained by numerical calculations or from the
technique of acoustical holography. Effects of source interferences on the field of the active intensity are evaluated and
boundary values are studied. The use of the potential intensity field to predict sound fields from sources is discussed.
INTRODUCTION
The field of the sound intensity vector can be
decomposed into an irrotational component and a
curl component. The advantage of the irrotational
intensity [1] is the suppression of the well-known
vortices in near field of radiation structures [2]. The
distribution of the normal component of the
irrotational intensity on a vibrating surface can
provide us with the exact images of the sound power
of sources without sign fluctuations (positive and
negative) due to the interferences among sources.
Another advantage of the irrotational component is
that it is the gradient of a scalar potential field, which
is the only quantity that depends on the sound power
of the sources. This property makes it possible to
compute the scalar potential of the active intensity
from the acoustic power of sources using software of
calculating the diffusion by numerical methods (heat
transfer FE). However, it is complicated to perform
predictions of acoustic field from energy quantities
because of the interferential nature of a sound field.
A general Helmholtz decomposition of a vector field
can be applied to sound intensity and structural
intensity fields [3]
(1)
From this relation, a field vector function A can be
introduced such that [4]
)
I = -Ñ 2 A = -Ñ(1
A ) + Ñ ´ (1
Ñ2
×3
Ñ2
´A
3
f
C
I(K ) = ò I(r ) e j K ×r dr
(3)
D
Performing this operation on the left and right hand
sides of Eq. (2) yields
A (K ) = I (K ) K 2 with K 2 = K x2 + K y2 + K z2 (4)
The scalar potential and the vector potential can
respectively be expressed in terms of the field vector
A such that f = - j K × A (K ) and C = - j K ´ A (K ) .
IRROTATIONAL INTENSITY
ON PLANE BOUNDARIES
The behaviours of the standard and irrotational
intensity on a plane surface can better be understood
with the help of the field vector A.
Radiation of vibrating surfaces - The scalar potential
in a field point r can be written by
ACTIVE INTENSITY FIELDS
I = If + I C = -Ñf + Ñ ´ C , with Ñ × C = 0
The field vector A (r ) of the active intensity contains
all information on the sound intensity field. It can be
calculated in wave number domain by the use of 3-D
Fourier transform on the standard acoustic intensity
in a whole volume [1]
(2)
f (r ) =
1
2p
òò
S
I f (r0 ) × nˆ
R
dS ,
with R = r - r0
(5)
Thus I f (r0 ) × nˆ corresponds exactly to the acoustic
power density of the sources on S. For a vibrating
plate, this quantity is always positive whereas the
normal component of the standard intensity presents
changes of sign, which do not correspond to a correct
image of the sources.
The difference between the normal components of
the standard intensity and the irrotational intensity
SESSIONS
h
FIGURE 1. From left to right: standard sound intensity, field vector A, scalar potential f and out of plane
component of vector potential C of the sound field produced by a point source located at a distance h from the
rigid plane S at kh = p .
can be seen when both of them are expressed in
terms of the field vector A (r )
I × nˆ = -Ñ 2 Az = -
¶ 2 Az
¶x 2
-
¶ 2 Az
¶y 2
-
¶ 2 Az
¶z 2
2
¶ 2 Ax ¶ A y ¶ 2 Az
¶
I f × nˆ = - (Ñ × A ) = ¶z
¶x¶z
¶y¶z
¶z 2
(6)
Sources above a reflection plane - Consider a point
source of a distance h from a rigid boundary. The
interferences of the sound field depend on the value
of kh (k is the wave number). In Figure 1 is shown
the field vector A (r ) calculated from the standard
intensity I(r ) by using Eq. (4). Calculating Ñ × A
yields the scalar potential f of which spatial
distribution is independent of kh (thus of the
frequency). The kh value modifies only the initial
power W0 of the source. In addition, the scalar
potential can be expressed by
1 ö
æ sin 2kh ö æ 1
+
f (r ) = W0 ç1 +
÷
֍
2kh ø è 4pR 4pR ¢ ø
è
(7)
where R and R ¢ are respectively the distances of
the source and its image from a field point. The
irrotational intensity calculated from f corresponds
quantitatively to the radiated energy and does not
show the spatial fluctuations due to interferences
(Figure 2). Contrary to f , the vector potential (there
is only one tangential component because of axial
symmetry) is very sensitive to kh values. It is
evident that the curl intensity, which is derived from
C , contains the information of directivity of the field
(Figure 2). Since the field can be built by using an
image source, introduced symmetry leads to
I n = If × nˆ = I C × nˆ = 0 on S
(8)
This shows that the effects of interferences due to the
reflection plane cannot be described by these
boundary conditions on S.
FIGURE 2. Irrotational (left) and curl (right)
intensities of the same source as in Figure 1.
DISCUSSIONS
Experimental method of the acoustic holography
makes it possible to compute the distribution of the
irrotational intensity on a vibrating surface to obtain
the exact image of the sound power distribution on
sources [1,3]. Another application relates to the
calculation of the scalar potential by solving the
Poisson equation Ñ 2f = - w , where w is the
distribution of source power. The prediction of the
sound field obtained by calculating the irrotational
intensity gives good quantitative results, in particular
by using finite element methods [5]. However, local
phenomena due to the interferences cannot be
represented without a complete definition of the field
vector A.
REFERENCES
1. J.C. Pascal, J.F. Li, "Irrotational acoustic intensity: a
new method for location of sound sources", 6th Int.
Con. Sound Vib., Copenhagen, 5-8 July 1999, 851-858.
2. F.J. Fahy, Sound intensity (2nd ed.), E./F.N. Spon, 1995.
3. J.C. Pascal, "Sources and potential fields of power flow
in acoustics and vibration", NOVEM, Lyon, 31 aug.-2
sept. 2000.
4. P.M. Morse, H. Feshbach, Methods of theoretical
physics, McGraw-Hill, 1953.
5. M. Thivant, J.L. Guyader, "Prediction of sound using
the intensity potential approach: comparison with
experiments", NOVEM, Lyon, 31 aug.-2 sept. 2000.
SESSIONS
Introduction of sound absorbing materials in the Intensity
Potential Approach
M. Thivant, J.L. Guyader
Laboratoire Vibrations Acoustique, INSA de Lyon, 69621 Villeurbanne France
In industrial noise control issues, energy methods are often worth being used in the mid and high frequency range [1]: CPU costs
are much lower than using integral methods, solutions are more robust, and one gets energy paths, which can indicate relevant
design modifications. The Intensity Potential Approach derives from the local energy balance. A thermal analogy makes it
possible to use standard Heat Transfer Solvers. The irrotational part of active intensity [2,3] is obtained, as well as the acoustic
power through the apertures and the pressure level in the far free field. Previous work [3] showed the capabilities of this method
for predicting sound propagation through rigid partial enclosures. This paper completes the Intensity Potential Approach by
taking into account local absorption phenomena. Absorbing materials are described by thermal convection factors related to
acoustic impedance. This model has been tested on several cases and shows good agreement with IBEM solutions. The method
is also being applied to trucks engine noise prediction.
1.
INTENSITY POTENTIAL
APPROACH
The Intensity Potential Approach aims to predict
acoustic energy transfer through apertures, as well as
its dissipation into acoustic shielding. Helmholtz
decomposition (1) of the active intensity vector field
into rotational and irrotational parts, together with
power balance relation (2), leads to exact Poisson’s
equation (3) for the Intensity Potential ϕ.
r r r
r
r r
I = Iϕ + IC =−gradϕ +ro tC
r
div I (M)=q0δ(M,M0)
∆ϕ(M)=q0δ(M,M 0)
(1)
(2)
(3)
Where q0 is a point power source located in M0 and δ is
the Dirac delta function. Previous studies [2,3] have
shown that irrotational intensity Iϕ is related to far field
energy transfer and to energy sources and sinks. On the
other hand, rotational intensity IC describes local loops
and directivity patterns, but it does not affect global
energy balance. ϕ and Iϕ are fully determined once the
correct boundary conditions are defined. (4) holds for a
source of power πs distributed on a surface S. Far free
field radiation can be taken into account through (5),
written on a Rl-radius hemisphere. Rigid obstacle are
described by (6). This paper focuses on the validation
of (7), which stands for absorbing obstacles.
∂ϕ πS
=
∂n s
∂ϕ
=0
∂n
(4)
(6)
∂ϕ
= − 1 ϕ ( R l)
∂n
Rl
−
at every points Q on the obstacle. G(Q) is the potential
-temperature- obtained when replacing all absorbing
boundaries by rigid ones (6). Then, G(Q) must be
divided by the total injected power to get the
normalized blocked potential g(Q). In the thermal
analogy, (7) can be viewed as a convection relation.
The surface convection factor H=εg(Q) depends on
acoustic properties of the boundary via ε, and on
location of both the sources and the obstacle Q via
g(Q). From a practical point of view, such varying
convection coefficients can be easily defined in heat
transfer solvers.
2. ESTIMATION OF ABSORBING
COEFFICIENT ε
A theoretical expression can be derived for ε, based on
the solutions of equivalent thermal and acoustic
problems. Consider a non-directional acoustic source S
of power π0(S) radiating above a partially absorbing
plane characterized by its impedance:
(8)
Z=ρ0c(R + jX)
Where ρ0 is the mass density of the medium, c is the
speed of sound, and R and X are respectively the real
and imaginary parts of the normalized impedance of
the plane.
s0(S)
θ
(5)
∂ϕ
(Q)=εgϕ(Q,S)ϕ(Q) (7)
∂n
ε is a coefficient related to acoustic impedance of the
boundary surface. Theoretical expression for ε is
proposed in §II. Boundary condition (7) requires
preliminary calculation of the blocked potential G(Q)
Z
I n (Q )
Fig 1 - Intensity absorbed by an infinite plane
The net intensity absorbed by the plane at point Q can
be written, using plane wave approximation:
2
(9)
In(Q)= π 0 2 (1− ℜ(θ) )cosθ
4πrQ
SESSIONS
Where rQ is the distance between the source S and the
point Q, and |ℜ(θ)| is the modulus of the reflection
coefficient defined by (10).
ℜ(θ)=
Z cosθ − ρ0c
Z cosθ + ρ0c
3. COMPARISON WITH I-BEM
(10)
An equivalent thermal problem can be defined, taking
a heat source with the same power π0(S) and an image
source |ℜ(θ)|²π0. We get the potential -or temperatureat Q, as well as the intensity -or heat flux:
2
ϕ(Q)= π0 (1+ ℜ(θ) )
4πrQ
−
2
∂ϕ
(Q)= π0 2 (1− ℜ(θ) )cosθ
∂n
4πrQ
(11)
(12)
For a rigid plane, ℜ(θ)=1, thus
ϕ(Q)=G(Q)= π0
2πrQ
G(Q) 1
g(Q)=
=
π0 2πrQ
(13)
(14)
Then ε derives from (7), (11), (12) and (14):
2
(1− ℜ(θ) )
ε(θ, Z)=
2πcosθ
2
(1+ ℜ(θ) )
(15)
This coefficient ε, defining absorption condition (7) in
the equivalent thermal problem, depends only on
acoustic properties of the plane via ℜ, and on
incidence angle θ. For diffuse fields one can use an
average value of ε with respect to θ:
π2
 tan −1( R² + X²) 
(16)

ε θ = ∫ ε(θ)sin θdθ= 4πR 1−
R
²
X
²
+
R² + X² 
0

For X=0, (16) becomes:
tan −1(R) 
(17)
ε θ = 4π 1−
R
R 
Figure 2 shows both <ε>θ and the usual diffuse field
absorption coefficient <α>θ, defined by (18), versus
the real part of normalized impedance. Amplitudes are
quite different because <α>θ links absorbed intensity to
incident intensity, where as <ε>θ links absorbed
intensity to potentials g(Q) and ϕ(Q). Nevertheless
they have similar evolution and reach their maximum
for very closed values of R -respectively 1.51 and 1.56.
Fig. 3 - Intensity map on the source and at the aperture - IPA.
Figure 3 shows IPA results for a constant velocity
piston source radiating in a box with uniform
absorption inside. On figure 4 is plotted the power
through the aperture computed by IBEM and by IPA.
The power of this source in the free field is plotted in
bold. Variations between both methods stand within
6dB. One should notice that source velocity is the input
data for IBEM, which leads to important uncertainty on
the injected power. However both methods predict
similar changes of output power when either
absorption or geometry is changed.
Fig 4 Output power versus frequency - left: <α>θ=0.4 - right:
<α>θ=0.944 - bold: free field. solid: IPA - dot: IBEM
4. CONCLUSION
A model is proposed for absorption in the Intensity
Potential Approach. Comparison of IPA with IBEM
shows good agreement. Computing time is much lower
with IPA than with IBEM. On the other hand, positions
of sources, absorbing materials and apertures can be
taken into account quite precisely. Therefore IPA
seems to be an interesting tool for vehicle and
machinery outdoor noise prediction.
5. ACKNOWLEDGMENTS
This work is supported by Renault V.I. and the French
Ministry of Research and Technology.
6. REFERENCES
Fig. 2 - Solid: <ε>θ. Bold: <α>θ versus R -log scale.
{
}
α θ = 8 R +2 − 2 ln(R +1)
R R +1 R
(18)
[1] Guyader, J.L., State of the art of energy methods used for
vibroacoustic prediction, In Proceedings of 6th International
Congress On Sound And Vibrations, Copenhagen, Denmark,
1999, pp. 59-84.
[2] Pascal, J. C., Structure and patterns of acoustic intensity
fields, In Proceedings of 2nd International Congress on
Acoustic Intensity, CETIM, Senlis, France, 1985, pp. 97-104.
[3] Thivant, M., Guyader, J.L., Prediction of sound
propagation using the Intensity Potential Approach:
comparison with experiments, In Proceedings of NOise and
Vibration Energy Methods int. congress, Lyon, France, 2000.
SESSIONS
A Method to Minimize the Effect of Random Error when
Measuring Intensity Vector Fields in Narrow Bands
Ö. Johansson
Engineering Acoustics, Division of Environment Technology,
Luleå University of Technology, S-97187 Luleå, Sweden
The objective of this paper is to validate a method that minimises the time required for narrow band measurements of
sound intensity vector fields. An experiment was performed in a reverberation chamber. The source under investigation
was a radial fan and its outlets. Sound intensity was measured over a rectangular grid that contained 84 sub-areas. The
measurement was repeated four times using different procedures for spatial integration. The measurement results were
then post processed by principal component analysis The idea is that the information contained in a complete set of
data, can be used for minimising the effects of random error in a specific position in the vector field. The result was
analysed at several frequencies having different global pI-indexes. Comparison between measured and refined vector
field confirmed that the method reduces the effect of random error and improves interpretation of the result.
INTRODUCTION
Measuring sound intensity vector fields in narrow
frequency bands over a large surface is time
consuming. To reduce random error in vector
estimates long averaging times are required especially
where there are large pressure intensity differences [1].
For source characterisation, however, the information
contained in spectral data of high resolution is needed.
Several methods for source location have been
presented. Non of the methods, however, is as
straightforward, robust and refined as the sound
intensity method. The objective of this paper is to
validate a method that minimises the time required for
narrow band measurements of sound intensity vector
fields. The method is based on principal component
analysis (PCA) and has been validated experimentally
by studies on a relatively simple source in good
acoustical conditions [2]. For further validation an
experiment was performed in a reverberation chamber
(T60=6-8s). The source under investigation was a radial
fan plus its outlets. Intensity was measured in “realtime” over a rectangular grid that contained 84 subareas. The measurement was repeated four times using
various BT-products, at two different distances from
the source. The intensity estimates were further post
processed by PCA, which is an iterative procedure
based on singular value decomposition [2,3]. PCA is
used to detect correlated patterns in intensity
measurement data, both in frequency and spatial
domains, e.g. areas of sound radiation. The idea is that
the information contained in a complete set of data can
be used for minimising the effects of random error in a
specific position in the vector field. The validation was
made by visual comparison between measured and
refined vector field at several frequencies, having
different global pI-indexes.
EXPERIMENTAL PROCEDURE
The sound intensity [4] was measured over a planar
grid in front of the cover of a radial fan and its out lets.
Four measurements were conducted using, 30 or 10cm distances, and 4 Hz or 16 Hz resolution. The
intensity was measured in the frequency range 0 3600 Hz, by a 3D-probe (B&K 0447) connected to an
FFT-based 8-channel measurement system (LMS
CADA-X, front-end HP-Paragon). The probe was
calibrated using an acoustic coupler B&K 3541. The
residual pressure-intensity index varied between 18 24 dB (200–3600 Hz). To ensure true real-time
performance, the six microphone signals of the 3dprobe were sampled during the measurement
sequence. The intensity vector field was determined by
post processing the sampled pressure signals
(Hanning, 75% overlap). The time averaged active
intensity spectrum (t=1.5s) was estimated for each
sub-area (0.01 m2) in the plane grid (7x12). The spatial
SESSIONS
position of the probe was logged on a seventh channel.
Finally, the intensity data was processed by PCA.
By using an appropriate number of principal
components, the remaining non-systematic varying
data can be removed, i.e. reducing the effect of random
error [2]. The final result is visualised as principal
intensity vector fields (PIV) for different frequencies.
RESULTS AND DISCUSSION
The following discussion is based on measurement
results at 30-cm distance from the fan cover. The
results at 10-cm distance will be analysed and
discussed in the extended paper version.
The averaged sound intensity spectrum (normal
component) has predominant peaks at 380, 756 and
1136 Hz. At these frequencies, the cover and its
outlets, is forced to vibrate by pressure fluctuations
related to the fan blade frequency and its harmonics.
Remaining intensity is related to structural modes
excited by the turbulent flow and noise from the
electric motor.
Changing the resolution from 4 to 16 Hz is
expected to reduce random error by 50%, if averaging
time is kept constant. The difference in random error
was most easily seen as sign shifts of the intensity
component normal to the surface (from negative to
positive at 4 positions). Generally, the intensity vectors
at the most important frequencies indicated large
random errors. The effect of random error can be
predicted on the basis of BT-product and the pI-index
[1]. For example, at 380 Hz the global pI-index is 10
dB (756 Hz, 5dB; 1136 Hz, 6dB). This indicates
severe measurement conditions, since the pI-index
may be very large at specific positions. The predicted
normalised standard deviation at 380 Hz, based on the
BT-product (B=4Hz, T=1.5 s) and the global pI-index
(10dB), is 1.65 locally and 0.18 in relation to the total
averaging time (T=126s).
The proposed method was tested on the
measurement results based on 4 Hz resolution. Before
PCA, measurement data was transformed, to obtain
nearly normal distributed intensity data in both spatial
and frequency domain. The absolute value of each
amplitude estimate was power transformed (|I|0.25).
Sign information was stored and added after PCA.
Another important requirement is that each variable
(frequency component) is normalised (mean centred).
PCA resulted in eight significant principal
components, which describe 59% of the variance in
the transformed data matrix. However, at 380 Hz two
principal components explained 95% of the variance
(756 Hz, 4 comp, 70%; 1136Hz, 4 comp, 23%). The
comparison between measured intensity and the
refined vector field, indicates a reduction of random
error. However, the result is not as good as in the
previous study [2], due to the large number of negative
signs in the normal intensity components. Negative
intensity could be expected in a highly reverberant
field, but in several positions it is erroneous estimates.
These errors may be detected by analysis of the
residues between the measured and refined vector
field. By this procedure all sign-shifts found using 16
Hz resolution, were detected. Correction of detected
sign errors improved the result and gave a better
definition of the sound radiating surfaces. However,
measurements far from the source in a reverberation
room are not an ideal condition and make it difficult to
interpret the vector fields. The results at 10-cm
distance, which also will be compared with results
using very long averaging times, will be used as a final
validation of the method. These results will be
presented and discussed at the conference and in the
extended paper.
SUMMARY
Rapid measurements of 3D acoustic intensity
vector fields require real-time performance of an FFTbased measurement system. Real-time measurements
were achieved by post processing the pressure signals,
sampled during scanning. The effect of increased
random error due to short averaging time was partly
reduced by principal component analysis. The
averaging time may be reduced by 75 % and still good
estimates of the intensity vector field may be obtained.
The problem is to know the minimum time required
for an acceptable measurement quality. Ideally the
procedure may be done automatically. The most
important benefits of the described method are the
limited time in front of the source, and the true realtime measurement of the sound intensity vector field.
REFERENCES
1.
[1] F. Jacobsen Prediction of Random Errors in Sound
Intensity Measurement. International Journal of Acoustics and
Vibration, 5, No 4, 173-178 (2000).
2.
Ö. Johansson “Rapid Measurements of Acoustic Intensity
Vector Fields”, Proc. of the 7th International Congress on
Sound and Vibration, Garmisch Partenkirchen 7-10 july 2000.
3.
A. Höskuldsson, Prediction methods in science and
technology, Vol 1. Basic theory. Thor Publish., Denmark, 1996
4.
F. Fahy, Sound intensity, 2nd edition,. E & FN Spon, London,
1995
SESSIONS
$Q LQGLFDWRU EDVHG RQ PRGDO LQWHQVLW\ IRU WHVWLQJ WKH LQÀXHQFH RI
WKH HQYLURQPHQW RQ WKH VRXQG SRZHU RI DQ DFRXVWLF VRXUFH
Domenico Stanzial4, Davide Bonsi
7KH 0XVLFDO DQG $UFKLWHFWXUDO $FRXVWLFV /DERUDWRU\ 6W *HRUJH 6FKRRO )RXQGDWLRQ &15 ,VROD GL 6*LRUJLR
0DJJLRUH FR )RQGD]LRQH &LQL 9HQH]LD ,WDO\ KWWSZZZFLQLYHFQULW
A new indicator very sensitive to the room resonances excited by an acoustic source is here introduced and used as an environmental
parameter for certifying the sound ¿eld condition when measuring the acoustic power of the source by the intensity method. This
quantity will be called UHVRQDQFH LQGLFDWRU Thanks to its experimentally proved properties a de¿nitive insight of the physical
meaning of the “oscillating intensity” as introduced in previous papers is accomplished. The results of a set of measurements of
the sound power of a reference source in different acoustical environments will be presented and discussed using the introduced
indicator.
,1752'8&7,21
The de¿nition of the acoustic power of sources is based
on the well-known acoustic energy corollary which
is demonstrated to be consistent with the requirement
of energy conservation in a Àuid to second order [1].
Within this theoretical frame, the power of an acoustic source can be expressed in terms of the stationary
time-averaged radiant (active) intensity D+{, as
] ]
V
D+{, q g5 {
(1)
where the integral is taken over a closed spatial surface V bounding the volume Y which usually contains
the source. is a real quantity measured in Z @
M v4 whose sign can be positive for a radiating source,
negative for a dissipative or absorbing source (a sink)
and must be equal to zero when the integral of Eq.
1 is derived from the homogeneus continuity equation
zw . u m @ 3, where z+{> w, and m+{> w, stand respectively for the instantaneous sound energy density and intensity. In this paper only the case 3 will be considered.
When a source is enveloped by V the continuity
equation takes the inhomogeneous form zw . u m @
€+{> w, where € is the SRZHU GHQVLW\ depending in general ERWK on the source and the environment. It is worth
noting that zw vanishes when a stationary time-average
is performed on it so that also its integral over any volume (contained or not in V) is zero. This fact, often
misunderstood, has an important physical meaning because tells us that when stationary energy conditions are
reached the time rate of the energy emission from the
source (i.e. the source power) must be equal to the time
rate of the energy absorption by the boundaries. Due

to the equality kzw l 3 the rate of energy absorption,
in turn, must be equivalent to the integral of k€+{> w,l
over Y (stationary energy balance). Unfortunately, neither mathematical nor experimental handling de¿nition
for €+{> w, has been formulated yet, so the only way for
evaluating the acoustic power of a source must be based
on sound intensity: can be experimentally evaluated
determining the time average vector ¿eld D+{, at some
discrete points { 5 V and then approximating the integral (see [2]). In this paper a careful experimental investigation about the inÀuence of ¿eld boundaries condition
on the measured power of a reference acoustic source
has been carried out and a new ¿eld indicator accounting for the sound energy “trapped-in-the-average” in the
standing waves (normal modes of the environment) excited by the source itself has been usefully employed as
an environmental parameter for acoustically certifying
the source power measurement.
7KH UHVRQDQFH LQGLFDWRU
Following a reasoning similar to the one which led to the
ghi
de¿nition of the sound radiation indicator @ D@f kzl
the resonance indicator will be here introduced as
ghi
@
P
f kzl
(2)
ghi s 5
ku l (apart from a numerical factor) is
where P @
the Hilbert-Schmidt norm of the sound intensity polarization tensor [3]. In other words is de¿ned as the
effective value of the PRGDO LQWHQVLW\ P normalized to
f kzl thus representing the fraction of the energy density
which is locally associated in the average to the normal
modes excited by the source. A preliminary analysis on
a similar indicator can be found in Ref. [4]
Also with Cemoter-CNR, Ferrara, Italy
SESSIONS
Figure 1 shows the reference source and the intensity
probe used for measuring the sound power in two different rooms. The ¿rst one is very “dead” due to the
absorbing upholstery (W93 ? 434 v) while the second
one is a normal room of the same volume and characterized by W93 * 4 v. The input signal to the source (white
noise) and the enveloping surface (a sphere centered at
the top of the source having a radius of 4=8 p) were the
same for the two measurement sets.
sources are needed, the two indicators and should be
employed for certifying the inÀuence of the environment
on the measured ¿eld. This should be done optimally at
every point where data for the power computation are
collected.
80
78
76
Power (dB)
0($685(0(17 $1' 5(68/76
74
72
70
68
66
125
250
500
1000
freq. (Hz)
2000
4000
),*85( : Values of sound power (squares: dead
room, circles: normal room).
),*85( : The source and a measurement point.
Once ¿xed the equatorial plane parallel to the Àoors,
the radial component of sound intensity in two positions
78 North and 78 South belonging to 7 meridians <3
apart for a total of ; points was measured. These data
were then used for the power computation. We collect
data for evaluating at only one measurement point in
the equatorial plane at a distance of 5 p from the centre
of the sphere.
Figure 2 reports the calculated power of the source in
the two rooms for third-octave bands ranging from 433
to 8333 K}. Figure 3 reports the comparison between
the indicators and , measured at one point in front of
the source in the two rooms.
&21&/86,21
According to the precision of our measurements the
power of the reference source results slightly greater
(4 5 dB) for all bands when the source is radiating
in the absorbing room. This can be considered an experimental evidence of the fact that the power of an acoustic
source depends inescapably from the boundaries conditions or, stated in other words, that when the acoustic
¿eld feeds back to the source, the source itself becomes
a boundary condition for the acoustic ¿eld. For this
reason, when accurate power measurements of acoustic
1
0.9
µ2
0.8
µ1
0.7
0.6
0.5
η1
0.4
0.3
0.2
η2
0.1
0
125
250
500
1000
freq. (Hz)
2000
4000
),*85( : Values of and (squares: dead room,
circles: normal room).
5()(5(1&(6
1.
A.D. Pierce, $FRXVWLFV, pp. 36-39, ASA-AIP, 1989.
2.
F. Fahy, 6RXQG LQWHQVLW\, Chapter 9, Chapman & Hall,
London, 1995.
3.
D. Stanzial, N. Prodi, D. Bonsi, J. Sound Vib., 232(1),
2000, 193-211.
4.
D. Bonsi, Ph. D. dissertation, University of Ferrara,
1999.
SESSIONS
Time-dependent Sound Intensity Measurement Methods
H. Suzuki
Department of Network Science, Chiba Institute of Technology
2-17-1 Tsudanuma, Narashino-shi, Chiba-ken 275-0016 Japan
The most commonly used sound intensity is the time-averaged sound intensity. This method, however, is not suitable for the
analysis of transient sounds. In this paper, two types of time-dependent sound intensities are introduced. The first one is the
envelope intensity. The envelope intensity is obtained from the multiple of the analytic (with zero spectra at negative
frequencies) time-dependent pressure and particle velocity. The envelope intensity gives a rather slowly varying intensity like the
envelope of the signal. Since it has the directional information, it can be used for the localization of transient sound sources in the
three dimensional space. The second method is the application of cross Wigner distribution to the sound intensity. This gives the
sound intensity distribution on the time-frequency plane. Some application examples of the above two methods will be
introduced.
INTRODUCTION
xa (t ) = x(t ) - jx h (t )
Information on the intensity and the direction of a
transient noise is important in order to understand its
generation mechanism and contrive a method to reduce
or eliminate it. For example, a large glass window
occasionally generates a transient noise when the
ambient temperature changes drastically. It is difficult,
however, to estimate from which part of the window
the noise is generated since the phenomenon is
unrepeatable. Dozens of similar cases can be easily
listed.
In this paper, two sound intensity methods that can be
used for the analysis of the transient sounds are
reported. The first is the envelope intensity method[1]
and the second is the "instantaneous sound intensity
spectrum" method[2].
ENVELOPE INTENSITY
where xh(t) is the Hilbert transform of x(t).
A three-dimensional sound intensity probe shown in
Figure1 was used for the measurement of the envelope
intensity. The microphones are located at the apexes of
a tetrahedron. The pressure is the average of the four
pressure signals and the particle velocity components
for x, y, and z axes are also calculated from linear
combinations of the four microphone signals[3].
FIGURE 1. Three-dimensional sound intensity probe
The instantaneous intensity is given by
i (t ) = p (t )v(t )
(1)
where p(t) and v(t) are the sound pressure and the
particle velocity in the direction of the measurement.
Even for a monotone sound, the instantaneous intensity
changes with twice of the original frequency. This is
the reason why the time-averaged intensity method is
used for most of practical applications, which is
unfortunately not suitable for transient sounds. The
envelope intensity was developed with a purpose of
keeping the time dependency and still avoiding the
radical change of the instantaneous intensity. It is
defined by
I e (t ) = Re[ p a* (t )va (t )] / 2
(3)
An example of the envelope intensity measurement
is shown in Figure 2. An impact sound generated by
hitting a 900x900x9mm plywood from behind by a
small metal hammer was recorded in an anechoic room
by use of the probe shown in Figure 1. The top left and
right charts show the horizontal and vertical plane
displays of the envelope intensity vector loci,
respectively. The analysis frequency band was from
125Hz to 1kHz. Figure 2 shows that the direction of the
maximum intensity coincides with that of the impact
shown by the dotted line in the horizontal plane. The
middle and the bottom charts show x-, y-, and z-axis
intensity components and the pressure of microphone 1
as functions of time.
(2)
where pa(t) and va (t) are the analytic functions of p(t)
and v(t), respectively. The analytic function of x(t) is
defined by
SESSIONS
FIGURE 3. Experimental setup for the measurement of a
direct and reflected transient sounds
0
time(ms)
A result is shown in Figure 4. The horizontal and
vertical axes represent the time and frequency,
respectively. Arrows in the charts show the directions
of the energy flow and the magnitudes as well as the
time and frequency dependence. Several words shown
in the box show possible reflectors for each wave
packets. As this figure shows, it is possible to
graphically represent frequency components of an
instantaneous intensity by use of the cross-Wigner
distribution.
160
FIGURE 2. Envelope intensities and a pressure waveform of
an impact sound generated by hitting a plywood
INSTANTANEOUS SI SPECTRUM
The cross-Wigner distribution between p(t) and v(t) is
defined by
+¥
W pv (t , f ) = ò p * (t - t / 2)v(t + t / 2)e - 2p
-¥
ft
dt
(4)
If p(t) and v(t) are given, respectively, by
p(t ) = [ p1 (t ) + p 2 (t )] / 2
(5)
t
1
v(t ) = [ p 2 (t ' ) - p1 (t ' )]dt ' (6)
rd ò-¥
with pressure signals of two microphones of a p-p type
one-dimensional intensity probe, p1(t) and p2(t), the
instantaneous sound intensity spectrum (ISIS) is given
by
I (t , f ) = - Re[W p1 p 2 (t , f )] / 2 p fr d (7)
Figure 3 shows an experimental setup used for testing
a usefulness of ISIS. A loudspeaker generated a 4-cycle
10 kHz sinusoidal tone burst and the direct and
reflected sounds were recorded at the cross point of
two arrows inside the region surrounded by three
wooden boards.
FIGURE 4. ISIS obtained from the setup shown in Fig.3.
REFERENCES
1. Suzuki, H., Oguro, S., Anzai, M., and Ono, T, Inter-noise
91 Proceedings, Vol.2, 1037-1040 (1991).
2. Tohyama, M., Suzuki, H., and Ando, Y., The Nature and
Technology of Acoustic Space, Academic Press, 1995, 143145.
3. Suzuki, H., Oguro, S., Anzai, M., and Ono, T,
J.Acoust.Soc.Jpn, 16, 4, 233-238 (1995).
SESSIONS
Investigations into the correlation between field indicators
and sound absorption
Gerhard Hübnera, Volker Wittstockb
a
ITSM, Universität of Stuttgart, Pfaffenwaldring 6, D-70550 Stuttgart, Germany, huebner@itsm.uni-stuttgart.de
b
PTB, Bundesallee 100, D-38116 Braunschweig, Germany, volker.wittstock@ptb.de
The sound power of a sound source measured by the intensity technique over an enveloping surface S may differ
significantly from its true value if, in the presence of absorption within S, stronger directly incident background noises flow
through this surface. The magnitude of this effect can be determined by a second measurement where the source under test is
switched off and the background noise remains unchanged. This second measurement yields a negative sound power which
directly describes the error of the first measurement if background noise and noise from the source under test are not
correlated.
Another approach is to check the presence of relevant sound absorption within S using the well-known sound field
indicators. In principle, the global unsigned indicator and the global signed indicator, especially their difference, the sound
field non-uniformity indicator and the difference between global indicators and the instrument‘s dynamic capability react to
absorption inside S as they do to other influences. The paper deals with theoretical and experimental results of the correlation
between different indicators and absorption within S.
1 INTRODUCTION
Intensity-based sound power determinations are
influenced by an additional error when an absorption
is included in the measurement surface and stronger
extraneous noise sources are present. When the source
is switched off, with the background noise remaining
unchanged, the absorbed sound power Pabs can be
determined which can be used to correct the measured
sound power
(1)
Ptrue = Pmeas + Pabs
assuming that the source under test and the
background noise are not correlated.
Under practical conditions, however, it is
sometimes not possible to switch off the source under
test without changing the background noise. An
example of this is the sound power determination of
one specific part of a machinery set. In such cases, it is
of interest whether one of the well-known sound field
indicators [1] or a new one can be used to indicate and
limit significant sound absorption effects.
2 THEORETICAL CONSIDERATIONS
The effect of absorption situated within the space
enveloped by the measurement surface S depends on
the magnitude of the background noise during the
measurements. Here, two different cases can be
distinguished:
a) the background noise enters directly from a strong
parasitic sound source in close proximity to the
measurement surface, or
b) the background noise randomly superposes the
original sound field at the measurement positions.
As is known [1], sound field situations can be
described by certain sound field indicators which are
in particular the unsigned pressure intensity indicator
Fp I n = L p − L I n − K 0 ,
(2)
the signed pressure intensity indicator
FpI n = L p − LIn − K 0
(3)
and the field non-uniformity indicator
FS =
1
In
(
1 N
∑ I n,i − I n
N − 1 I =1
)
2
.
(4)
Definitions of the several symbols are given for
example by the Draft International Standard ISO
9614-3. The bar indicates the average value on the
measurement surface. The error due to absorption is

P 
P
∆ abs = meas =  1 − abs 
(5)
Ptrue 
Ptrue 
where Ptrue is the sound power of the source
determined in the absence of absorption and Pabs is the
absorbed sound power determined by the “zero test“,
where the source under test is switched off and the
background noise remains unchanged. Exclusion of all
other error sources and introduction of the sound
power Pmeas, measured in-situ, using eq. (1), yields
−1

P 
∆ abs = 1 + abs  .
P
meas 

(6)
With
SESSIONS
I n,zero S
Pabs
0.1 ( FpIn + Fzero ) /dB ,
=
= 10
Pmeas I n,meas S
(7)
the level of the absorption error follows to be
0.1 ( FpI n + Fzero ) /dB 

(8)
L∆,abs = −10lg 1 + 10
 dB


with the zero test indicator

I n,zero 
 dB .
9
Fzero = 10lg  ρ c
2


%
p
meas


The magnitude of the indicator FpI comprises
n
different influences such as the near field error,
environmental corrections and the noise-to-signal
ratio. This indicator cannot therefore be used for
detecting the presence of background noise or of
absorption errors, only. Eq. (8) shows that for the
determination of L∆,abs both indicators, FpI and Fzero,
n
levels between 0 and 10 dB. The absorption was
caused by a plate (chipboard, 40 mm thick) forming
one of the boundaries of the measurement surface.
For this practical approach the indicators
and FS are regarded which can be
FpI − Fp I
n
situations – for instance for semi-reverberant environments – other limiting values can be expected.
20
F abs/dB
criterion
10 limit
0
are necessary. Only if the measurement surface is
situated in a reverberant field region ( Fzero → −∞ ),
-10
L∆,abs is equal to zero. This means for measurements
outside the reverberant radius that the absorption
inside S is simply added to the total room absorption.
Furthermore, alternatively to Fzero, the indicator
P
Fabs = 10 lg abs dB .
(10)
Pmeas
can be introduced which directly indicates the ratio of
the absorbed sound power to that measured. Based on
the formula given above, the error due to the
absorption effect such as LÄ,abs ≤ 0.41 dB or ≤ 0.14 dB
is limited by
Fabs = Fp In − Fzero ≤ −10dB or ≤ −15 dB .
n
determined under conditions where performance of the
zero test is not possible. Figures 1 and 2 show the
dependence of the absorption indicator on these two
indicators. For this specific example, the criterion
from eq. (11) is fulfilled if FS < 3 or < 1 or
Fp In − Fp In < 4 dB or < 0 dB. For other acoustical
-20
0,1
1
10
100
FS
1000
Figure 1 Indicator Fabs as a function of FS
F abs/dB
20
criterion
limit
10
0
-10
(11)
If the zero test cannot be carried out for practical
reasons, L∆,abs can be determined or limited only
approximately.
3 PRACTICAL APPROACH
An absorption error can of course be excluded if
the noise-to-signal ratio vanishes. This can be tested
by determination of the difference
(12)
FpIn − Fp In .
If this difference is zero, no significant background
noise is present.
Under in situ conditions, however, where a certain
background noise level and an unknown absorption
exist, a procedure capable of limiting the absorption
error is of interest. The data base of an earlier
investigation [2] can be used for such an approach.
These investigations were carried out in a semianechoic room with extraneous noise superelevation
-20
0
5
10
15
20
25
30
(F p lI nl-F pI n)/dB
Figure 2 Indicator Fabs as a function of Fp I − Fp I
n
n
4 CONCLUSIONS
In order to check the significance of absorption effects,
the zero test should preferably be carried out. Results
of this test allow sound power corrections or calculations of the amount of the absorption error to be
carried out. If the zero test cannot be carried out,
certain other criteria allow the error to be assessed
only approximately.
REFERENCES
1.
2.
G. Hübner, INTER-NOISE 84 Proceedings, pp.
1093-1098
G. Hübner, V. Wittstock, INTER-NOISE 99
Proceedings, pp. 1523-1528
SESSIONS
ISO standards on sound power determination
by sound intensity method
H. Tachibanaa and H. Suzukib
a
b
Institute of Industrial Science, University of Tokyo, Komaba 4-6-1, Meguroku, Tokyo 153-0041, Japan
Dept. of Network Science, Chiba Institute of Technology, Tsudanuma 2-17-1, Narashino, Chiba 275-0016, Japan
The outline of ISO 9614-3 “Acoustics-Determination of sound power levels of noise sources using sound intensity-Part 3:
Precision method for measurement by scanning” is introduced by comparing with the former two standards, ISO 9614-1 and 2.
INTRODUCTION
For the determination of sound power levels of
sound sources using sound intensity method, we
have two ISO standards under the general title
“Acoustics - Determination of sound power levels
of noise sources using sound intensity”; ISO 9614
- Part 1 “Measurement at discrete points” and ISO
9614 - Part 2 “Measurement by scanning”. As the
third standard of this series, ISO 9614 - Part 3:
“Precision method for measurement by scanning”
has just been drafted as an ISO/FDIS (as of
February 2001).
Evaluate the temporal variability
indicator and determine TFT <0, 6
Define the measurement surface,
partial surfaces and scanning path
Choose one partial surface
Determine the scanning time
TS = N S ⋅ TFT <0, 6
TS practicable ?
Is
MEASUREMENT UNCERTAINTY
No
Take action A
Yes
See TABLE1.
Perform two scans
TABLE 1. Measurement uncertainty of ISO 9614-3
1/3 octave band
Upper values of
center frequencies
standard deviation of
[Hz]
reproducibility [dB]
50 to 160
2,0
200 to 315
1,5
400 to 5 000
1,0
6 300
2,0
A-weighted*
1,0
* Calculated from third-octave bands from 50 Hz to 6,3
kHz.
Crt. 1
LIn (1) − LIn ( 2 ) ≤ s / 2
Yes
Crt. 2
Ld ≥ F pIn
No
For all partial
surfaces
No
F pIn − Fp In ≤ 3
Take action C
Take action D or E
Yes
Crt. 3
Take action B
No
MEASUREMENT PROCEDURE
The measurement surface is defined around the
source under test as shown in Fig.2. It is divided
into several partial surfaces on which partial sound
power is determined by each scanning.
In the measurement by scanning, time-series of
intensity and squared sound pressure are
continuously measured and stored using
instantaneous mode of the instrument. From these
results, four kinds of field indicators, temporal
variability indicator: FT , field non-uniformity
indicator: FS , unsigned pressure intensity
indicator: F p I n , and signed pressure intensity
Yes
Scanning
already increased?
No
Crt. 4
FS ≤ 2
Take action G
Yes
Crt. 5
0,83 ≤ FS(1) / FS( 2 ) ≤ 1,2
No
No
Yes
Take action
F or G
Yes
Final result
FIGURE 1. Flow of the measurement in ISO 9614-3
SESSIONS
partial
surface
partial
surface
1
Partial
surface
surface (Crit.1), for the adequacy of the measurement
equipment (Crit.2), for the presence of strong
extraneous noise (Crit.3), for the field non-uniformity
(Crit.4) and for the adequacy of the scan-line density
(Crit.5) are made as shown in Fig.1. If these
requirements are not satisfied, actions to increase the
grade of accuracy are indicated.
segment
∆y
NORMALIZED SOUND POWER LEVEL
0,83 ≤ ∆x / ∆y ≤ 1,2
2
To standardize the value of sound power level of a
source, “normalized sound power level” LW 0 is
defined as the sound power level under the reference
meteorological condition (temperature θ 0 =23 °C ,
barometric pressure B0 =101,325 kPa), given by:
scanning path
∆x
FIGURE 2. Measurement surface
indicator: F pIn
shown in Table 2 are evaluated.
296,15 
 B
dB
LW 0 = LW − 15 lg 

 101325 273,15 + θ 
where, θ is the air temperature in °C during the
actual measurement, B is the barometric pressure in
(Terminology in the former two standards, Part 1 and 2,
is inconsistent and reconsidered in Part 3.)
In order to guarantee the upper limits for measurement
uncertainties, check for the adequacy of the averaging
time, for the repeatability of the scan on a partial
Pa during the actual measurement.
Table 2. Field indicators used in 9614-1, 2, and 3
Indicator
ISO 9614-1
Temporal
variability indicator
F1 =
1
In
Field
non-uniformity
indicator
F4 =
1
In
ISO 9614-2
1 M
∑ ( I nk − I n ) 2
M − 1 k =1
N
1
N −1
∑(I
ni
− I n )2
F2 = Lp − L I n
L p = 10 lg(
1
N
Signed
FT =
1
In
1 M
( I n − I n )2
M − 1 m =1 m
―
FS =
1
In
1
J −1
Negative partial power
indicator
F+ / − = 10 lg[
N
∑10
0,1L pi
)
i =1
i
P=
J
∑(I
j =1
n j
− I n )2
Unsigned pressure-intensity
indicator
∑ P /∑P]
Pi = I ni Si
∑
Fp I n = L p − L I n
i
N
∑P
L p = 10 lg(
i
i =1
1
J
J
∑p
2
j
/ p02 )
j =1
* F+ / − = F3 − F2 in a special
case.
L I = 10 lg(
Negative partial power
indicator
Sound field pressure-intensity
index
Signed pressure-intensity
indicator
F3 = L p − LI n
FpI = [ L p ] − LW + 10 lg( S / S 0 )
FpIn = L p − LI
1
[ L p ] = 10 lg 
S
LI = 10 lg
L I n = 10 lg(
Pressure
intensity
indicator
―
i =1
Surface pressure intensity
indicator
Unsigned
ISO 9614-3
LI n = 10 lg(
1
N
1
N
N
∑( I
ni
/ I0 )
i =1
N
∑I
i =1
ni
/ I0 )
S=
N
∑S
i
N
∑ ( S 10
0 ,1L pi
i
i =1
n

)

n
1 J
∑ In / I0 )
J j =1 j
1
J
n
J
∑I
j =1
n j
/ I0
, S 0 = 1 m2
i =1
* FpI = F3 in a special case.
SESSIONS
Sound Intensity Detection for Multi-sound Field Using
Microphone Array System Based on Wavelength Constant
Matrix
K. Kuroiwaa and O. Hoshinob
Faculty of Engineering, Oita University, 700 Dannoharu Oita 870-1192, Japan
a
e-mail: kuroiwa@cc.oita-u.ac.jp hoshino@cc.oita-u.ac.jp
b
One problem with the traditional method of sound intensity is that it cannot give the individual sound intensity of interest in a
multi-sound field. The objective of this study is to develop a sound intensity detection system for a multi-sound field. The
proposed system consists of three-dimensionally spaced microphone array, sound detection filters based on wavelength
constant matrix and sub-system of sound intensity estimation. A wavelength constant matrix is exactly formulated in terms of
sound directions so that it can give a relation between the microphone outputs and the respective sound existing in a multisound field. Feeding microphone outputs to the filters, the system can issue the respective sound at the specified virtual points
of our choice. Therefore, it can give three-dimensional vector of the respective sound intensity. Experiments of the incident
and reflected sound intensities near the reflective panel for oblique incidence from 0 to 60 degrees in an anechoic chamber
showed that the proposed system is fundamentally valid in sound intensity detection. The mean errors were 4.3 percents in
amplitude and 5.1 degrees in direction in comparison with the original.
DETECTION OF SOUND INTENSITY
The block diagram of sound detection system is
illustrated in FIGURE 1. In a multi-sound field, it is
assumed that there are N sorts of sound sources. The
nth of sound source, Sn , shown with ‘filled circle’ with
coordinates (rn, qn, jn) radiates its original sound pn(t)
and the relevant sound intensity, In, shown with arrow
towards a microphone array located at the origin. The
mth microphone, Mm, with coordinates (dm, zm, jm)
receives all arriving sounds and issues the output of
qm(t) that alters in phase and magnitude depending
upon the mutual situation given by the position vector
Am of Mm with m=1 M, Bn of Sn with n=1 N and
position vector V of such virtual points as Mx and M-x
indicated with ‘gray circle’ on XYZ axes.
Letting the kth sample of the original sound pn(V,t)
assumed at the virtual point expressed by position
vector V and the microphone output qm(t) be pn(k) and
qm (k), and also letting the ith frequency component of
the DFT spectrum of them be Pn(V,i) and Qm(i),
th
respectively, then the n sound Pn(V,i) at V can be
detected by using the following equation [1]:
(
)(
æ j2p iDf A m - V × Bn - V
c mn (V, i) = expçç
Bn - V
è U
(
)
)ö÷
÷
ø
(4)
in which, U, Df and raised dot denote sound velocity,
frequency resolution in DFT spectrum and inner
product of two position vectors, respectively. Cv
implies the wavelength constant matrix having the size
of N ´ M.
Substituting the position vector VDx of Mx for V in
Equation (1) and also V-Dx of M-x, then the nth sound Pn
(VDx,i) for Mx and Pn(V-Dx,i) for Mx can be estimated,
respectively. Therefore, we can obtain the x-axis
component of nth sound intensity In,x [2]. It has the
form of
In, x =
i2
1
2pr V Dx
-
V- Dx
å
i= i 1
(
*
Im Pn (V - Dx , i) Pn (V Dx , i)
i
)
(5)
where r denotes the density of air. In the same manner
as for In,x, the y-axis component In,y and the z-axis
component In,z are estimated. Consequently, the nth
sound intensity In can be obtained as follows:
M
Pn (V, i) =
åH
nm
(V ,i) Qm (i)
(1)
In = i I n,x + j I n,y + k I n,z
(6)
m =1
where i, j and k imply unit vector of XYZ-axes,
respectively.
where, the FIR filter Hnm(V,i) is given as follows:
(H nm (V, i))= C .
C = (cmn(V, i) )
-1
V
V
(2)
(3)
SESSIONS
FIGURE 2
Schematic diagram of the experiment.
FIGURE 1. Block diagram of the sound intensity
detection system.
EXPERIMENTAL RESULTS
The schematic diagram of the experiment in an
anechoic chamber is shown in FIGURE 2. It is the
purpose of this experiment to detect the incident sound
intensity, Ii (I1), coming from a loud-speaker S1 with
the coordinates ( 1.5 m, 90 deg., ji deg. ) and reflected
sound intensity, Ir (I2), from the reflector. For this case,
second sound source, S2 , with the coordinates ( 1.5 m,
90 deg., 180-ji deg. ) implies the mirror image source
of S1 with respect to the reflector.
Results of the sound intensity detection for acryl
resin hard board and urethane foam board are
illustrated in FIGURE 3. The filled arrows directed
leftwards and rightwards imply the detected incident
sound intensity, Ii, and the reflected sound intensity, Ir,
respectively. The detected sound intensity is expressed
with ‘^’ on it in this section. To evaluate the detection
accuracy, the incident sound intensity, Ii, measured by
removing reflector is also illustrated with gray arrow.
It is shown that the proposed system worked well over
the incident angle from 0 to 60 degrees. Mean errors
were 4.3 % in magnitude and 5.1 degrees in direction.
FIGURE 3
Experimental results of the sound
intensity detection.
REFERENCES
1) K.Kuroiwa, “Multi-microphone system for sound
detection based on wavelength constant matrix”, J.
Acoust. Soc . Jpn.(E) 19, 289-292 (1998).
2) F.J.Fahy, 'Measurement of acoustic intensity using
cross-spectral density of two microphone signals',
J.Acoust.Soc. Am. 62, 1057-1059 (1977).
SESSIONS
Free Field Measurements in a Reverberant Room Using
the Microflown Sensors
W.F. Druyvesteyn, H.E. de Bree and R. Raangs
University of Twente, postbus 217, 7500 AE Enschede, The Netherlands; w.f.druyvesteyn@el.utwente.nl
Free field measurements in a reverberant room using the microflown sensors are described. The microflown is a sensor which
does not measure the pressure, but the particle velocity in a sound field.
INTRODUCTION
The microflown is a sensor that measures the
particle velocity in an acoustic disturbance [1]. It
consists of two heated wires; the particle velocity is
determined from the differential resistance changes
[1]. When a microflown sensor is combined with a
pressure microphone an intensity probe is obtained.
Experimentally such a (simple) p-u intensity probe
has been tested and found to be as good as the
existing p-p intensity probes [2]. Main advantages of
the p-u probe above the p-p probe as intensity probe
are that errors (or uncertainties) in the phases do not
lead to large errors in the estimated intensity, and
that one probe configuration can be used for a broad
frequency region (the p-p probe use different
configurations for the low- and high frequency
region). A disadvantage of the p-u probe can be that
the calibration of the two sensors with respect to
each other is more difficult.
An important property of the microflown is its
directional characteristic. Denote the orientation of
the microflown by a vector n[x,y,z], which is defined
as follows. The vector n is in the plane through the
two conducting wires of the microflown and
perpendicular to the length of the wires (the
maximum sensitivity of the microflown is when the
particle velocity is parallel with n). When the
direction of the particle velocity makes an angle q
with n, the measured output of the microflown is
proportional to cos(q), see [1] and [2]. This property
makes it possible to obtain free field ( no contribution
of reflections) properties of a sound field in a
reverberant environment.
One microflown
Consider the case that a sound source ( a loudspeaker
to which random noise is applied) is placed in a
reverberant room. When the microflown is oriented
such that n is in the direction of the free field particle
velocity of the sound source the r.m.s. value follows
from the square root of the squared free field- and
reverberant particle velocity:
u//2 =ud2 + 1/3urev.2 The factor 1/3 comes from the
integral over all angles in a sphere, representing all the
image sources, contributing to the diffuse reverberant
p
field: (1 / 4p ) 2p sin(q ). cos(q ). cos(q ) dq = 1 / 3 . When
ò
0
in a second experiment the microflown is oriented such
that n is perpendicular to the free field velocity of the
sound source, the r.m.s. value is given by: u^2 =
1/3urev.2. So by subtracting u//2 - u^2 the free field is
obtained. Two recordings have been made of speech in
a reverberant room, one corresponding to u// and the
other corresponding to u^. When reproducing these
recordings a clear difference is heard: the recording of
u^ contains only the reverberant sound field. The
recordings will be reproduced during the presentation at
the conference.
Two microflowns perpendicular to each
other
The contribution of a pure diffuse reverberant sound
field to the cross-correlation of two microflowns
perpendicular to each other, vanishes. To show this,
consider the two dimensional case with two
microflowns, having n-vectors n1 = [1,0] and n2 = [0,1].
Suppose two uncorrelated sound sources with power
strength s1 and s2 are oriented in perpendicular
directions, e.g. one under an angle q with the x-axis,
and the second one having an angle q + p/2 with the xaxis. When the output signals of the two microflowns
are u1(t) and u2(t), the long term value of the crosscorrelation Ru1.u2(0) is written as:
T
Ru1.u2(0) = (1 / T ) ò u1(t ).u 2 * (t ).dt
0
; the symbol (0) in Ru1.u2(0) refers to the fact that no
time difference is taken between u1(t) and u2*(t). The
contribution of source s1 is proportional to –cos(q)*(sin(q)) and of source s2 to sin(q)*(-cos(q)); a positive
signal is assumed when the components of the
connection vector r from source to sensor are in the
positive x- or positive y-direction. When the strength of
the two uncorrelated sources are equal, s1 = s2, the
SESSIONS
contributions to Ru1.u2(0) are equal but with different
sign, thus Ru1.u2(0) vanishes. To verify this
experiments have been done with two sound sources
(loudspeakers) perpendicular to each other and with q
= 45 0. The loudspeakers were excited with two
uncorrelated noise signals within a frequency band of
900 – 1100 Hz. The results of the experiments are
shown in figure 1, where is plotted R*u1.u2(0) as a
function of the ratio’s of the power strengths s2/s1;
R*u1.u2(0) being the normalized Ru1.u2(0) value with
respect to the Ru1.u2(0) value at s2 = 0. The crosses (+)
are the experimental results. For s2/s1 =1 the cross
correlation is zero.
1.5
measuring the different sound quantities at almost the
same point. A photograph of such a device is shown in
figure 2.
p
uy
ux
uz
1mm
1
Figure 2. A photograph of the ux-uy-uz-p sensor.
0
R
*
u1.u2
0.5
-0.5
-1
-1.5
0
0.5
1
s 2/s 1
1.5
2
2.5
Figure 1. The value of the cross correlation Ru1.u2(0)
normalized to the value at s2 =0 as a function of the
ratio of the power strengths s2/s1 of the two sound
sources.
In a pure diffuse reverberant sound field the
contribution of an image source to Ru1.u2(0) is always
compensated by the contribution of another image
source in a perpendicular direction. Intensity
measurements have also the property that the
contribution of a diffuse sound field vanishes. So free
field measurements in a reverberant environment can
be done using an intensity p-u (or p-p) probe or using
two microflowns oriented in perpendicular directions.
An important difference is of course that with two
microflowns perpendicular to each other one
measures the product of two perpendicular
components of the free field particle velocity vector,
while with the intensity the free field product of p and
u, being the energy flow.
Three microflowns combined with
a pressure microphone
A more sophisticated device is to combine one
pressure microphone with three microflowns oriented
in perpendicular directions, e.g. n1 = [1,0,0], n2 =
[0,1,0] and n3 = [0,01]. The four sensors are quite
small and are positioned close to each other, thus
Information of the direction and the strength of a sound
source in an arbitrary direction can be found from the
determination of the six possible cross-correlations. As
was explained above the diffuse reverberant sound field
does not contribute to each of the possible six crosscorrelations. On the other hand the reverberant field
does contribute in the four auto-spectra’s. Some
experimental results are shown in table 1. The
abbreviations Ix refers to the found intensity in the xdirection, Rux.uy to the cross-correlation between the
signals from microflown with nx = [1,0,0] and ny =
[0,1,0]. The direction of the sound source was
[0.43,0.41,0.50].
Table 1. Results using the ux-uy-uz-p sensor.
Experimental Calculated
Ix/Iy
1.05
1.04
Ix/Iz
1.05
0.85
Rux.uy/Rux.uz
1.11
0.82
Ruy.uz/Rux.uz
0.98
0.96
ACKNOWLEDGEMENTS
The authors whish to thank the Dutch Technology
Foundation STW for the financial support.
REFERENCES
1.H.E. de Bree, The Microflown (book) ISBN 9036515793;
www.Microflown.com (àR&DàBooks).
2. W.F. Druyvesteyn, H.E. de Bree, Journal Audio Eng. Soc. 48, 4956 (2000); R. Raangs, W.F. Druyvesteyn and H.E. de Bree 110th
AES Convention Amsterdam 2001, preprint 5292.
SESSIONS
Experimental Study on the Accuracy of Sound Power
Determination by Sound Intensity Using Artificial Complex
Sound Sources
H. Yanoa, H. Tachibanab and H. Suzukic
a
Department of Computer Science, Chiba Institute of Technology, Tsudanuma 2-17-1,Narashino,275-0016,Japan
b
Institute of Industrial Science, Univ. of Tokyo, Komaba 4-6-1, Meguro-ku, Tokyo, 153-8505, Japan
c
Department of Network Science, Chiba Institute of Technology, Tsudanuma 2-17-1,Narashino,275-0016,Japan
In order to examine the measurement accuracy of the scanning sound intensity method for the determination of sound power
levels of sound sources, a basic experiment was performed using an artificial complex sound source. As a result, it has been
found that the scanning method specified in ISO/FDIS 9614-3 provides almost the same results as those by the discrete method
specified in ISO 9614-1 and the field indicators can be evaluated during the scanning using a running signal processing
technique.
INTRODUCTION
As the third standard of ISO 9614 series, the
precision method for the determination of sound power
levels of sound sources by scanning sound intensity
methods
is
now
being
drafted
in
ISO/TC43/SC1/WG25 (DIS stage as of June, 2001).
For this work, some experimental studies were
performed on the measurement accuracy. In this paper,
an experimental study on the accuracy of sound power
determination by sound intensity under the condition
where strong extraneous sources exist is introduced
and the validity of the field indicators are discussed.
EXPERIMENTS
As the sound source under test, the following three
kinds of sources were used and driven by different
random noises, respectively.
Source-1 “CAN”: a steel box
to which an electric-shaker
is attached.
source 1:CAN
0.75 m
room (4m x 6.9m x 7.6m), in which the sound sources
were placed as shown in Figure 1 at the center position
on the reflective floor. As shown in the figure, the
measurement surfaces for each sound sources were set
and sound intensity measurement was performed in 1/3
octave bands from 100 Hz to 5k Hz by the scanning
method and the discrete point method (12.5 cm mesh).
The measurement by manual scanning was performed
in two scanning patterns on each partial surface by
keeping the scanning speed at about 25 cm/s. In a
successive scanning on each partial surface, both data
of sound pressure and sound intensity were obtained at
the same time at a sampling rate of 0.5 s. As the
instrumentation, a pair of 1/2 in. condenser
microphone (B&K 4181) with 12 mm spacing and
digital real-time analyzer (B&K 2133) were used. The
field indicators specified in ISO/DIS 9614-3 were
calculated automatically in a desk-top computer.
source 2:BK4205
0.75 m
source 3: 8SP
1m
Source-3 “8SP”: a wooden
box in which 8 loudspeakers
are equipped.
1m
Source-2 “BK 4205”: a
loudspeaker-type reference
sound source.
The
measurement
was
performed in an hemi-anechoic
FIGURE 1. Configuration of artificial sound sources and measurement surfaces
SESSIONS
80
scan, simultaneous
scan, individual
discrete, simultaneous
125
250
500 1000 2000
frequency (Hz)
PWL (dB)
60
PWL (dB)
PWL (dB)
BK4205
70
50
80
80
CAN
70
60
50
4000
70
60
scan, simultaneous
scan, individual
disctrete, simultaneous
125
scan, simultaneous
scan, individual
discrete, simultaneous
8SP
50
250 500 1000 2000 4000
frequency (Hz)
125 250 500 1000 2000 4000
frequency (Hz)
discrete, simultaneous
500 1000 2000
frequency (Hz)
4000
3
2
1
0
-1
discrete, simultaneous
125
14
12 8SP
10
8
6
4
2
0
125
scan, simultaneous
scan, individual
250
scan, simultaneous
scan, individual
discrete,
simultaneous
FpIn-Fp|In|
scan, simultaneous
scan, individual
5
4 BK4205
FpIn-Fp|In|
FpIn-Fp|In|
FIGURE 2. Measured results of PWL
5
4 CAN
3
2
1
0
-1
125 250
500 1000 2000 4000
frequency (Hz)
250 500 1000 2000 4000
frequency (Hz)
FIGURE 3. Measured results of field indicator: F pI n - F p I
5
5
CAN
F
4
scan, simultaneous
discrete, simultaneous
scan, simultaneous
BK4205
3
Fs
Fs
3
scan, simultaneous
discrete, simultaneous
scan, individual
2
25
20
15
2
10
1
1
5
0
0
0
125 250
500 1000 2000 4000
frequency (Hz)
scan, simultaneous
discrete,simultaneous
scan, individual
8SP
Fs
4
n
125
250
500 1000 2000
frequency (Hz)
4000
125
250
500 1000 2000
frequency (Hz)
4000
FIGURE 4. Measured results of field non-uniformity indicator:Fs
RESULTS
Figure 2 shows the sound power levels of the three
sound sources measured by the scanning method and
the discrete point method under the two operation
conditions; “individual”- each source was driven
individually and “simultaneous”- all sources were
driven
simultaneously.
In
the
results
of
“simultaneous”, it is seen that the results obtained by
the scanning method and the discrete point method are
in good agreement over all frequency bands. To
compare the results measured under the two source
operation conditions, the result for “individual” and
“simultaneous” are fairly in good agreement in every
sound source not being influenced by the mutual
intervention.
the value of FS exceeded the criteria (2 dB) in almost
all frequency bands under “simultaneous” condition,
the measurement result of sound power level is in good
agreement with the result measured under the
“individual” condition.
CONCLUSION
As a result of the basic experimental study
mentioned above, it has been found that the intensity
scanning method provides an accurate sound power
level even under the adverse condition with strong
extraneous noises.
REFERENCES
1.
Figures 3 and 4 show the values of F pI n - F p I
n
( F3 - F2 in 9614-1) and FS (F4 in 9614-1),
respectively, measured under the two source operation
conditions. In the case of the source 8SP, although the
value of F pI n - F p I exceeded the criteria (3 dB) and
n
ISO 9614-1, -2 and DIS 9614-3
2. H. Yano, H. Tachibana and H. Suzuki, Proc. of 16th.
I.C.A., 1998, pp.1461-1462
3. H. Tachibana and H. Suzuki, “ISO standards on sound
power determination by sound intensity method”, Proc.
of 17th. I.C.A., 2001
SESSIONS
Approximate Calculation of Sound Intensity
Radiated through an Aperture from a Source in a Room
Y. Furue, K. Miyake and K. Konya
Department of Architecture, Fukuyama University, Hiroshima 729-0292, JAPAN
The purpose of this work is to develop a technique to calculate approximately the sound intensity field by acoustic radiation
through an aperture from a sound source in a room. In this method the sound field outside of the room is assumed to consist of
diffracted waves by the aperture of the direct sound from the source and of the diffused one in the room. Both Kirchhoff’s diffraction formula for the sound pressure and its derivative form for the particle velocity are applied to calculate the sound intensity
owing to the direct sound from the source, while for the diffused sound in the room it is supposed for the sound to radiate from
the aperture according to Lanbert’s cosine law. Numerical examples are presented with the measurements.
INTRODUCTION
In order to calculate exactly the sound intensity field
by acoustic radiation through an aperture of a room in
which a sound source is located, we have to solve a
certain integral equation whose unknown function is
the velocity potential or sound pressure on the boundary surface. Such an integral equation method should
be applied to calculate the sound field diffracted by any
object whose dimension is comparable to the wavelength concerned. [1]
In a very large room as compared to the wavelength,
the integral equation method to calculate the sound
field is not applicable because of limitations in computer capacity or difficulties of determining the boundary conditions.
In this case an approximate approach based on
Kirchhoff’s diffraction formula [2] becomes more
practical. This formula requires both the velocity potential and its normal derivative on the aperture, which
are unknown. These unknowns are assumed to consist
of the direct wave from the source and the diffused
waves. The latter ones are assumed to make a surface
source at the aperture which radiates the sound to the
outside of the room according to Lanbert’s cosine law.
THEORETICAL BASIS
We consider the situation where an n omnidirectional
point source (Ps) is located in a room with an aperture
and a receiving point (P) is outside of the room as
shown in Fig.1. Under Kirchhoff’s boundary conditions, the velocity potential at point P, j(P) , is given by
j(P)=
1
4p
ì
¶ exp(ikr) ö ¶j(q) æ exp(ikr) öü
֍
÷ýds (1)
r ø ¶nq è r øþ q
q
òò íîj(q) ¶n æçè
A
q
q
nq
s
Ps
A
P
A
r
P
Ps
FIGURE 1. Model room with an aperture (left) and calculation model for direct component(right)
where j(q) is the velocity potential at point q on A, r is
the distance from point P to point q and nq is the outward normal at q. Bothj(q) and ¶j (q) /¶n q in Eq.(1)
are generally unknown. They must consist of the direct
wave from the source and reflected ones by the room
surfaces. Those waves might be assumed incoherent
with each other.
The sound intensity at P, I(P), could be divided into
two parts, the direct component Idir(P) and the reflected one Iref(P). I(P) is the sum of those vectors.
I (P) = I dir (P) + I ref (P)
(2)
Calculation of the direct component
The velocity potential and its normal derivative at
point q on the aperture are given by
and
j (q) = exp(iks) / s
(3)
¶j (q) æ iks -1 ö
= ç 2 ÷ exp(iks) cos(s,n q) , (4)
¶n q
è s ø
where s is the distance from Ps to q.
The direct component of the velocity potential at P is
expressed by:
SESSIONS
j dir (P ) =
1
4p
òò
ì exp(iks) ¶ æ exp(ikr ) ö
ç
÷
í
A
ø
s
¶n q è
r
î
æ iks -1 ö
æ exp(ikr) öü
-ç 2 ÷ exp(iks) cos(s,n q) ç
÷ý dsq (5)
r
è s ø
è
øþ
The particle velocity at P, u(P), is given by:
u(P) = -grad{j dir (P)}
æ ¶j (P) ¶j dir (P) ¶j dir (P) ö
(6)
= ç- dir
,,÷
¶x
¶y
¶z ø
è
where
ì¶j (q) ¶ æ exp(ikr) ö
¶j dir (P)
1
ç
=
÷
í
A
¶x
ø
4p
r
î ¶x ¶nq è
òò
¶ 2j (q) æ exp(ikr) öü
ç
÷ý dsq
(7)
øþ
¶x¶n q è
r
Let Idir,x the sound intensity for x-direction at P,
ì
æ ¶j (P) öü
I dir,x = Re í p(P) × ç- dir
(8)
÷ý ,
¶ x øþ
è
î
where r denotes the air density, c the speed of sound
and k wave number. Then the intensity of normal direction at point q (In(q)) is given by means of geometrical acoustic theory as follows,
W (1 - a )
(10) where a is the
A
average absorption coefficient and A the absorption
area of the room.
Let Iref,x(P) the sound intensity for x-direction at
point P by above surface source, we obtain
cos q
I ref ,x (P) = I n( q)
cos(r , x)ds ,
(11) also
A
r2
Iref,y(P) and Iref,z(P) similarly.
I n (q) =
òò
NUMERICAL SAMPLES
AND MEASUREMENTS
-
where p(P) = - iwrj dir (P) is the complex conjugate of
the sound pressure at P.
Similarly we can calculate Idir,y and Idir, z and we obtain the sound intensity of the direct component by
summing up those vectors.
Calculation of the reflected component
We suppose that the sound in a room is diffused after
the first reflection and a surface sound source is made
at the aperture from which the sound is radiated according to Lambert’s cosine law.
A model room which has a rectangular aperture
(45cm x 45cm) was set in an anechoic chamber. The
room which size is 600 x 600 x 900cm is made of plywood board 12mm thick. A third octave band noise
was used as the source signal and the location of the
source is the center of the room.
Sound intensity was measured by means of 4microphone system (ONO SOKKI Co.) and measurement points were set near the aperture.
The numerical examples and the corresponding
measurements at 2 kHz are shown in Figure 4 in sound
intensity level.
The approximate calculation method of the sound
intensity field mentioned above might be applicable.
Calculated
Measured
0
A
-10
-20
dS q
Į
I nq
1
(
r, x )
Aperture
61
73 85
97 109 121 132
Measurement point
FIGURE 3. Calculation model for reflected component
When the volume velocity of an omnidirectional
point source is unity, its acoustic power (W ) is given:
2
37 49
x
P
W = 4p r c k
13 25
(9)
FIGURE 4. Calculated and measured example.
REFERENCES
1. Y. Furue, Applied Acoustics 31, 133-146, (1990)
2. M.Born and E.Wolf, Principles of Optics, 4th ed.,
Pergamon Press, Oxford, (1970)
SESSIONS
A Code for Computation and Visualization of Sound Power
Measurements based on the
ISO 9614-1 International Standard
E. Pianaa, G. Pianab and A. Armanic
a
Dipartimento di Ingegneria Meccanica, Università degli Studi di Brescia
Via Branze 38, I-25123 Brescia, Italy, e-mail: piana@ing.unibs.it
b
Analisys, Via Monte Rosa 1, I-25069 Villa Carcina, Brescia, Italy, e-mail: analisys@tiscalinet.it
c
SPECTRA srl, Via Ferruccio Gilera 110, I-20043 Arcore, Italy, e-mail: aarmani@intercom.it
Acoustic Intensity Data Analysis software has been developed with the two fold objective to calculate sound power of a source
from intensity measurements and to allow a 3D representation of both the source outer surface and the map of interpolated results on the measuring grid. The code complies with the prescriptions of ISO 9614-1 and is particularly helpful when one of the
field indicators specified in the ISO standard does not satisfy the necessary requirements so that measurements must be repeated
by successive refinements of the measurement grid and, therefore, the computation may be rather time consuming. The code imports intensity data from several types of analyzers and computes the sound power level for each surface and each frequency
complying with the ISO 9614-1 flowchart. The calculation procedure includes the treatment of one-octave as well as one-thirdoctave band data. It also performs a three dimensional visualization of pressure and intensity fields on the measurement surface
by colour contour plots. The possibility of drawing three dimensional objects inside the measurement surface and the real time
acquisition of FFT intensity spectra makes the program suited for research and development purposes.
INTRODUCTION
3D EDITING AND REPRESENTATION
F.J. Fahy has proved that sound intensity measurement is one of the most effective techniques to determine sound power level and to perform a mapping of
the major sources contained within a measurement surface, as discussed in [1]. The international standard
ISO 9614-1 [2] provides practical guidelines to achieve
this aim and requires the calculation of different indicators giving information about the quality of the determination and hence the grade of accuracy: F1 (temporal variability indicator), F2 (pressure-intensity indicator), F3 (negative partial power indicator) and F4
(non-uniformity indicator). If one of the field indicators does not meet the requirements, the test procedure
should be modified. Moreover the Standard suggests
also to check two criteria related to the adequacy of the
measurement equipment and to the adequacy of the array of measurement positions. All the tests must be
performed for every single set of measurements on
each measurement surface used. Finally a table and a
flowchart specify the actions to undertake if the chosen
measurement surface does not comply with the standard requirements. The data to be collected are the
pressure and sound intensity levels in octave or 1/3
octave bands, whether they come from a CPB or from
an FFT analysis.
In order to start a new project, the first step is to define the geometry of the measurement surface and the
structures contained inside and outside it. Basic tools
for editing the geometry are already implemented in
Acoustic Intensity Data Analysis (AIDA) package. The
program can represent two different kinds of objects:
frames and measurement surfaces. Frames are the rectangular elements used to represent the source and the
environment around it: equipment, bench, floor, walls,
etc. Measurement surfaces are the rectangular elements
used to represent the set of segments surrounding the
sound source. Each segment defines a measurement
position. The user can draw several surfaces and put
them together to build a more complex one. The program provides a 3D graphical representation of the
sound power level over each surface (Figure 1). It also
provides a numerical listing and ranking of the sound
power levels coming from each one of them (Figure 2).
The user can represent pressure and sound intensity
levels in three dimensions for both CPB and FFT
analysis (up to 30 bands). Another important feature is
the simultaneous visualization of more than one 3D
surface, so that the user can check the source behaviour
in different frequency bands at the same time. Different
easily customizable colour scales are available.
SESSIONS
CALIBRATION
The program performs several types of calibration
procedures to set up the gain of the microphones and to
determine the pressure - residual intensity index. At the
same time it is also possible to perform a phase correction for the entire apparatus. A phase correction is useful for FFT based measurements, when the operator
wants to make a software correction of the equipment
response. All the data coming from the calibration procedure are stored in a database where the user can retrieve the correct probe type and verify the calibration
history of the transducers. The last calibration performed is automatically taken into account to check the
ISO 9614 – 1 field indicators and criteria.
REAL-TIME PERFORMANCES
The program can show pressure and intensity levels
from the OROS 25 power pack. In this case octave and
1/3 octave bands are synthesized directly from the FFT
spectra. The real-time performance of the system enables the program to show and import directly pressure
and intensity data from the analyzer, thus allowing
real-time computation of the sound power level radiated by the sound source through the probe remote
control. Real-time analysis and visualization of data
can be performed for the pressure levels, for the intensity levels and for the pressure – intensity index. Of
course the user can set up directly the analyzer from
the AIDA software choosing between linear and exponential averaging, number of averages, coupling, frequency span, etc.
During the measurements, the program computes the
sound power level for each surface and each frequency
complying with the Standard flowchart, and provides
FIGURE 2. Sample of numerical and graphical output
showing measurement surface ranking and 1/3 octave band
analysis.
instant suggestions of the necessary actions the user
should consider in order to meet the Standard requirements. The real-time acquisition procedure enables the
user to choose the sequence he wants to use to perform
the measurements. Alternatively, a direct access to any
point of the measurement surface is also possible.
Moreover, a table gives a comprehensive insight of the
field indicators computed for each measurement surface. The possibility of drawing three dimensional objects inside the measurement surface and the linking
with virtually any FFT analyzer makes the program
suited also for research and development purposes.
SOUND POWER DETERMINATION
If the ISO 9614-1 requirements are fulfilled, then the
final result is within the selected grade of accuracy.
The Sound Power levels are calculated and displayed
for each surface as well as for the overall measurement
surface with or without “A” weighting. It is also possible to exclude some frequencies from the calculation
procedure selecting them directly from a list. Figure 2
shows the numerical and graphical outputs for a 1/3
octave band analysis associated with the overall measurement surface represented in Figure 1. The histogram
depicts, for each frequency band, the sum of both linear and “A” weighted sound power levels coming from
the five measurement surfaces enveloping the source.
REFERENCES
1. F.J. Fahy, Sound Intensity, Elsevier Applied Science,
London, 1989.
2. ISO 9614-1:1993, Acoustics - Determination of Sound
Power Levels of Noise Sources Using Sound Intensity Part 1: Measurement at Discrete Points.
FIGURE 1. 3D representation of the measurement surface.
SESSIONS
Sound Level Meter Software Implementation with
LabVIEW
J.M. Lópeza,b, M. Ruiza,b and M. Recuerob,c
a
Departamento de Sistemas Electrónicos y de Control. Universidad Politécnica de Madrid Cta. De Valencia Km
7, Madrid 28031 Spain
b
INSIA. Instituto Universitario de Investigación del Automóvil. Universidad Politécnica de Madrid
c
Departamento de Ingeniería Mecánica y Fabricación. Universidad Politécnica de Madrid
This paper demonstrates how a sound level meter can be developed in LabVIEW with the help of a toolkit from National
Instruments. An implementation in real time is also presented using Welch algorithm to avoid.
INTRODUCTION
LabVIEW is a graphical programming language from
National Instruments. Its easy of use, combined with
high its high power and a good variety of toolsets
available, have made LabVIEW become one of the
most important tools in many engineering disciplines.
One of the available toolsets in LabVIEW is the Sound
and Vibration Toolset. This toolset provides a library
of Virtual Instruments (VIs), LabVIEW subroutines,
that implement basic operations such as fractional
octave band frequency analysis, measuring Leq, dada,
etc... With the help of these VIs, the design of a sound
level meter (SLM) has been accomplished using
LabVIEW and a data acquisition board. The sound
level meter has been tested in accordance to IEC 651
standards to verify its quality and functionality[1].
Special attention has been pay to the execution time of
the different routines in order to study the possibility
of implementing a real time sound level meter with
one-third-octave band frequency analysis capability.
The following resources were used to develop these
tests:
•
•
•
•
•
National Instruments high precision data
acquisition board PCI-4451 with 16 bits
resolution.
Microphone model M53 from Linear X.
PC de Pentium 166 and 64Mb RAM
LabVIEW version 5.1
Sound and Vibration toolset V1.0
TESTS
Several tests had been developed to validate the
quality of the algorithms provided in the Sound and
Vibration toolset. These tests use sinusoidal signals
synthesized with LabVIEW and therefore they do not
use the data acquisition board in order to avoid errors
from the acquisition process. Different VIs where
developed to test several aspects from IEC651 and
IEC804 standards[2]. These tests are presented in table
1..
Table 1. Sound and Vibration toolset tests developed.
Nombre del test
IEC references
Time Weighting (F,S,I)
IEC 651: 4.5, 7.2-7.5, 9.4.1,
9.4.3, 9.4.4
Time Averaging
IEC 804: 4.5, 6.1, 9.3.2
Rms Detector
IEC 804: 7.5, 9.4.2
The following VIs from the Sound and Vibration
toolset were tested[3]:
Exp avg sound level
Equivalent Continuous Sound Level
Acceptable results were obtained from these tests so it
was concluded that they could be used to implement a
virtual sound level meter.
Another important aspect related with the Sound and
Vibration toolset to be analysed was execution time.
Execution time of the abovementioned VIs was
measured using the profiler tool provided with
LabVIEW. After several trials, the average execution
time of the different VIs can be estimated in the
numbers showed in table 2.
Table 2. Execution time.
VI
Exp avg sound level
Equivalent Continuous Sound Level
Average Time (ms)
42
17
SESSIONS
These execution times are good enough as to
implement a SLM in real time.
Another important block of VIs from the Sound and
Vibration toolset are the ones related to octave and
one-third-octave band frequency analysis. These VIs
were tested applying a white noise synthesized in
LabVIEW to the IEC-Third-octave Analysis VI. From
a quality point of view, the results obtained in this test
were acceptable. On the other hand, excessively high
execution times (777ms average) were measured. This
made this VI not suitable for a real time
implementation.
VIRTUAL SOUND LEVEL METER
A data acquisition board from National Instruments,
mod PCI-4451, and a microphone with its preamplifier
from LinearX, mod M53, were used to develop a
virtual SLM. To cover an analysis range up to 20 kHz
the sample frequency must be over 40 kHz. A sample
frequency of 51.200 Hz was chosen for this
application. After several trials, it was observed that,
with a 16K samples buffer size, the acquisition could
be accomplished in real time but the results could be
not displayed. If only one block out of each four is
analysed, then the complete process can be done in
real time. Figure 1 shows the Virtual Sound Level
Meter user interface and its code in LabVIEW.
FIGURE 1. Virtual Sound Level Meter user interface and LabVIEW code
REAL TIME FREQUENCY ANALYSIS
In order to include in the SLM one-third-octave band
frequency analysis capabilities in real time, a VI was
specifically developed based on the Welch
algorithm[4] show in figure 2. This VI has an
execution time of 80 ms, making it possible to
developed a real time implementation of the SLM
with one-third-octave band frequency analysis
capabilities.
Unfortunately, from a quality point of view, this VI
has a worst response in low frequencies due to the
reduce number of samples available.
ACKNOWLEDGMENTS
The authors would like to acknowledge National
Instruments Spain for all the help provided and
Lorenzo Barba and Carlos Diaz for their
contributions.
REFERENCES
1. IEC 651, Specification for Sound Level Meter,
International Electrotechnical Commision.
2. IEC 804, Integrating-Averaging Sound Level Meter.,
International Electrotechnical Commision
3. National Instruments, Sound and Vibration Toolset
Reference Manual, Agoust 1999.
FIGURE 2 Welch algorithm.
4. Oppenheim, A.V., Discrete-Time Signal processing;
Prentice Hall 1989
SESSIONS
A method for the Estimation of Loudness Based
on Fuzzy Theory
M. Ferri, J. Ramis, J. Alba and J.A. Martínez.
Departamento de Física Aplicada, Escuela Politécnica Superior de Gandia, Universidad Politécnica de Valencia
Carretera Nazaret-Oliva sn, 46730 Gandia, Spain
In this work, we present a method for the estimation of loudness, as a combination of normalised levels (A,B,C),
given in integrative sound level meters3. Usually, the acoustic signals have been pondered with one of these normalised nets
depending on its sound pressure level. However, given a source with certain spectral distribution of energy, there isn’t a
common criteria about which net should be applied. To correct this problem, we propose a method, based on fuzzy sets, that
provides a single sound level as a combination of both three classical pondered sound levels. A comparation is shown, given a
group of signals with relatively uniform spectra, of the sound levels obtained by the proposed method with the normalised
loudness levels calculated by Stevens method4.
INTRODUCTION
Integrative sound level meters, do not usually
perform frequency analysis of the sound signals;
thus, is absolutely impossible determinate the
loudness level of some acoustic event. Generally the
sound pressure level A-weighted (dBA) is accepted
as the best approximation to evaluate loudness.
Anyway, different signals with the same SPL (dBA)
may have obviously very different loudness levels
(Zwicker or Stevens). To solve this point, Fuzzy
Logic based model1,2 -containing information about
non-linearities in frequency and level response of
human hearing- has been used.
divided in 3³=27 fuzzy hyper-sets conformed by the
fuzzy logic condition “and”
if LpA is A11 and LpB is A21 and LpC is A31
(Fuzzy inference)
... then
Lpfuz k = a k 1 × LpA + a k 2 × LpB + a k 3 × LpC
(consequence)
Prediction (Fuzzy level): As the fuzzy inferences are
not classical logic inferences {0,1}, they have some
membership function to the truthfulness in the
interval (0,1], so each consequence is as true as its
inference. Thus the predicted level is
Lpfuz = å
The Model And Its Application Field
As the normalised nets A ,B, y C, are designed to be
used respectively on small, medium & high levels,
we apply a weighted average of these normalised
levels, increasing the value of the coefficient of level
A if the sound level is considered small, and
respectively, if it is medium or high. The algorithm
used to determinate the value of the coefficients is
based on Fuzzy inferences & Fuzzy sets
The Model
Entry variables: LpA, LpB, LpC
Variables considered on Fuzzy inferences: LpA,
LpB, and LpC
Fuzzy sets: Three sets related to each three variable;
small (Ai1), medium (Ai2), & high (Ai3)
Fuzzy inferences and consequences: In the threedimensional space, three fuzzy sets to each
dimension have been defined, so the whole space is
trk × Lpfuzk
å trk
being trk the truthfulness of the k-esim fuzzy
inference, calculated as follows1
(
trk = min m A1l ( LpA), m A2 m ( LpB ) m A3 n ( LpC )
mA11(LpA)
Ai1: “small”
m1
0
mA12(LpA)
Ai2: “medium”
m2
LpA
0
mA13(LpA)
)
Ai3: “big”
m3
LpA
0
LpA
Figure 1: Fuzzy sets relative to variable LpA
(Qualitative)
m A ( xo ) is the membership function of the value xo of x
to the fuzzy set A
(0< m A ( x o ) £1)
1
SESSIONS
Application field:
RESULTS
Fuzzy models are capable to predict situations
similar to that which were used to configure them, so
the model is to be used for continuous and with
relative uniform spectra sounds. We have chosen
these, caused for the impossibility to predict exactly
with three inputs (LpA LpB, LpC) the loudness of a
sound, that, in the easier case, needs a octave
analysis of the acoustic signal (Stevens). Anyway
these are the kind of signals we usually found in
environmental acoustic and industrial noise.
320 signals have been selected with SPL values
between 60 and 130 dB and slopes between –10
dB/oct and 10 dB/oct; comparing the Lpfuzzy with
de Stevens loudness level (ISO 532-1975).
In the next figure it is shown how fuzzy level curves
modify its shape “looking for” loudness level curves.
In the three-dimensional plot it is shown the
difference between fuzzy levels and LpA, compared
with the difference between loudness levels and LpA
for different spectra types and sound pressure levels.
LS-LA
(dB)
Lfuzzy-LA (dB)
Weighted levels (dBA,dBB,dBC) - sound pressure levels (dB)
20
Lfuzzy
10
0
LPC
SPL=130 dB
5 dB
-10
SPL=110 dB
-20
FITTING THE MODEL
-30
-40
The fuzzy model depends on parameters such as the
coefficients of the variables in the polynomial
functions consequence of each inference, and the
shape of the fuzzy sets (that can be defined with the
abscise of the slope discontinuities and the maximum
value of the membership function). These parameters
will be optimised to minimise the mean quadratic
error, considering as correct values the Stevens
loudness level
N
e=
å (Lpfuz
- LS i )
2
i
i =1
N
The polynomial functions have been prefixed,
limiting the model complexity, Here there is an
example of polynomial function
if LpA is “big” and LpB is “big” and LpC is “med”
then
1
1
1
Lpfuz k = ca × LpC + cb × LpC + cc × LpB
3
3
3
ca + cb + cc = 1
Notice that if some level is small, medium or big is
associated respectively to LpA, LpB or LpC.
There are only three set high considered: small sets
high Ai1, medium sets high Ai2 and big sets high Ai3.
Definitively, the model depends on thirty parameters
(two abscises of six sets, four abscises of three sets,
three ordenades, and the additional coefficients ca, cb
and cc). Heuristic algorithms have been used: the
gradient method, and different neural network ones.
LPB
SPL=130 dB
LS
SPL=110 dB
LPA
1
Spectrum type
40
120
100
SPL (dB) 80
40
60
Spectrum type
Figure 2. Comparing Lpfuzzy versus Loudness level
The mean quadratic error in our experimental bench
(320 signals) is e(Lpfuz)<3’6 dB2; e(LpA)>17 dB2;
e(LpB)>24 dB2; e(LpC)>92 dB2
CONCLUSIONS
Applying this fuzzy model, the ordinary sound level
meters can be used to evaluate (in first
approximation) the loudness of many different kinds
of acoustic events (continuous and with relative
uniform spectra), getting much better performance
that using any weighted sound pressure level.
REFERENCES
1) G. Cammarata, A. Fichera, S.Graziani and L.
Marletta, Fuzzy logic for urban traffic noise
prediction, Journal of Acoustical Society of
America 98, 1995, pp. 2607-2612
2) Y. Kato and S. Yamaguchi, A systemathical
study for psychological impression caused by
fluctuating random noise based on fuzzy set
theory, J. Acoust. Soc. Am. 91, 1992, pp 2748-55
3) UNE-EN 60651, Sonómetros, Asociación
Española de Normalización y Certificación
(AENOR), 1.996
4) ISO 532-1975: Method for calculating loudness
level, International Standard Organization, 1975
SESSIONS