Patrizia

SOUND INTENSITY

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 con-trol engineering. Today sound intensity measurements are routinely used in the determination of the sound power of machineryand other sources of noise in situ. Other important applications of sound intensity include the identification and rank orderingof partial noise sources, visualisation of sound fields, determination of the transmission losses of partitions, and determinationof the radiation efficiencies of vibrating surfaces, and recent work suggests the possibility of measuring sound absorption in situusing sound intensity. This paper summarises the basic theory of sound intensity and its measurement and gives an overview ofthe state of the art in the various areas of application, with particular emphasis on recent developments.

INTRODUCTION

The most important acoustic quantity is certainlythe sound pressure. However, sources of sound emitsound power, and sound fields are also energy fieldsin which potential and kinetic energies are generated,transmitted and dissipated. In spite of the fact that theradiated sound power is a negligible part of the energyconversion of almost any sound source, energy con-siderations are of enormous practical importance inacoustics. In ‘energy acoustics’ sources of noise aredescribed in terms of their sound power, acoustic ma-terials are described in terms of the fraction of the in-cident sound power that is absorbed, and the soundinsulation of partitions is described in terms of thefraction of the incident sound power that is transmit-ted, the underlying assumption being that these prop-erties are independent of the particular circumstances.This is not strictly true, but it is usually a good ap-proximation in a significant part of the audible fre-quency range, and alternative methods based on linearquantities are vastly more complicated.

Sound intensity is a measure of the flow of acous-tic energy in a sound field. More precisely, the soundintensity I is a vector quantity defined as the time av-erage of the net flow of sound energy through a unitarea in a direction perpendicular to the area. Althoughacousticians have attempted to measure this quantitysince the early 1930s it took almost fifty years beforereliable measurements could be made. Commercialsound intensity measurement systems came on themarket in the beginning of the 1980s, and the first in-ternational standards for measurements using soundintensity and for instruments for such measurementswere issued in the middle of the 1990s. A description

of the history of the development of sound intensitymeasurement is given in Frank Fahy’s monographSound Intensity [1].

The advent of sound intensity measurement sys-tems in the 1980s has had a significant influence onnoise control engineering. Sound intensity measure-ments make it possible to determine the sound powerof sources without the use of costly special facilitiessuch as anechoic and reverberation rooms, and todaysound intensity measurements are routinely used in thedetermination of the sound power of machinery andother sources of noise in situ. Other important applica-tions of sound intensity include the identification andrank ordering of partial noise sources, visualisation ofsound fields, determination of the transmission lossesof partitions, and determination of the radiation effi-ciencies of vibrating surfaces.

It is more difficult to measure sound intensity thanto measure sound pressure. The difficulties are mainlydue to the fact that the accuracy of sound intensitymeasurements with a given measurement system de-pends very much on the sound field under study. An-other problem is that the distribution of the sound in-tensity in the near field of a complex source is farmore complicated than the distribution of the soundpressure, indicating that sound fields can be muchmore complicated than earlier realised. The problemsare reflected in the extensive literature on the errorsand limitations of sound intensity measurement and inthe fairly complicated international standards forsound power determination using sound intensity, ISO9614-1 and ISO 9614-2.

The purpose of this paper is to give an overview ofthe status of sound intensity measurement, with em-phasis on developments from the past few years.

MEASUREMENT PRINCIPLES ANDTHEIR LIMITATIONS

There are essentially three methods of measuringsound intensity: i) one can combine two pressuretransducers (or more), ii) one can combine a pressuretransducer with a particle velocity transducer, and iii)one can use a microphone array. Each method has itsown limitations, but in addition there are some funda-mental sources of error that do not depend on the mea-surement principle [2, 3]. Array methods are used forreconstructing sound fields, including sound intensity.This topic is outside the scope of this overview; seeref. [4].

The two-microphone principle

All sound intensity measurement systems in com-mercial production today are based on the ‘two-micro-phone’ (or ‘p-p’) principle, which makes use of twoclosely spaced pressure microphones and relies on afinite difference approximation to the sound pressuregradient, from which the particle velocity is deter-mined. The IEC 1043 standard on instruments for themeasurement of sound intensity deals exclusively withthis measurement principle.

The finite difference approximation obviously im-plies an upper frequency limit. However, because ofthe fact that the finite difference approximation errorwill be partially cancelled by a resonance if the lengthof the spacer equals the diameter of the microphones,the upper limit of the frequency range can be extendedto twice as much as suggested by the finite differenceerror. Thus the resulting upper frequency limit of asound intensity probe composed of half-inch micro-phones separated by a 12-mm spacer is 10 kHz [5],which is an octave above the limit determined by thefinite difference error when the interference of themicrophones on the sound field is ignored [1]. An-other way of extending the frequency range is to applya ‘sensitivity correction’ for the finite difference error.Such a correction has recently been suggested for 3-Dintensity probes (without spacers) [6]; this also seemsto double the upper frequency limit.

Even with the best sound intensity measurementsystems available today the range of measurement islimited by small deviations between the phase re-sponses of the two measurement channels. Above afew hundred hertz the main source of phase mismatchis a difference between the resonance frequencies ofthe two microphones, which gives rise to a phase errorthat is proportional to the frequency. For example, onemust, with state-of-the-art equipment, allow for phase

mismatch errors ranging from 0.05° below 250 Hz to1° at 5 kHz, which is slightly less than the maximumvalues specified in the IEC standard.

It can be shown that the effect of a given phaseerror is a bias error that is proportional to the meansquare pressure [7]. This implies that there is a limit tothe amount of noise from sources outside the measure-ment surface that can be tolerated in sound power de-termination using sound intensity, because such noisewill invariably increase the surface integral of themean square pressure and thus the bias error.

Another limitation of the measurement principle isthe restricted dynamic range at low frequencies: be-cause the electrical noise of the microphones at lowfrequencies is amplified by the time integration need-ed in determining the particle velocity from thepressure gradient, one cannot in practice measuresound intensity levels below, say, 50 dB re 1 pW/m2

below 100 Hz [8].Other sources of error include the effect of air-

flow. Airflow gives rise to correlated noise signalsthat contaminate the signals from the two micro-phones. The result is a ‘false’ additive sound intensityat low frequencies [9].

The p-u measurement principle

Attempts to develop sound intensity probes basedon the combination of a pressure transducer and a par-ticle velocity transducer have occasionally been de-scribed in the literature [1]. A recent example is basedon the ‘microflown’ particle velocity transducer [10],which is a micro machined hot wire anemometer. Oneproblem with any particle velocity transducer, irre-spective of the measurement principle, is the stronginfluence of airflow. Another unresolved problem ishow to determine the phase correction that is neededwhen two fundamentally different transducers arecombined (even though the influence of p-u phasemismatch will usually be less serious that the influ-ence of p-p phase mismatch). Whereas one can com-pensate for p-p phase mismatch by employing a simplecorrection determined with a small coupler [7], thereis no similarly simple way of determining a correctionfor p-u phase mismatch.

The measurement principle has the important ad-vantage in sound power measurements that the perfor-mance is not affected by the global pressure-intensityindex on the measurement surface, so in principle alarger amount of noise from sources outside the sur-face can be tolerated with p-u sound intensity probes[11]. However, any such advantage has yet to be dem-onstrated experimentally.

APPLICATIONS

Some of the most important applications of thesound intensity method are now discussed.

Sound Power Determination

One important application of sound intensity is thedetermination of the sound power of operating ma-chinery in situ. Sound power determination using in-tensity involves integrating the normal component ofthe intensity over a surface that encloses the source.The integral is usually approximated by scanning man-ually or with a robot over the surface. Two interna-tional standards are available, and a third one (for pre-cision measurement) is on its way.

A few years ago Paine and Simmons described theresults of an investigation of the uncertainties in thedetermination of sound power levels according to themeasurement procedures specified in various stan-dards, including ISO 9614-1 [12]. The reproducibilitystandard deviation was found to be around 1 dB, andno indication of a systematic difference between thestandards was found. Nevertheless, Jonasson recentlymaintained that ‘it has long been suspected that thisstandard....gives lower sound power levels than theother standards in the ISO 3740 series’ [13]. However,the only factor known to give a negative bias error inintensity-based sound power measurements is the ab-sorption of the source under test, which reduces thenet sound power output. In fact, Jonasson’s observa-tion was based on preliminary results that have laterbeen found to be affected by the absorption of the testsource [14].

Measurement of Transmission Loss

The intensity-based method of measuring transmis-sion losses of partitions was introduced in the early1980s as an alternative to the conventional method,which requires a diffuse sound field in the receivingroom. It has often been reported that the intensitymethod gives lower values of the sound transmissionindex than the conventional method at low frequenciesand higher values at high frequencies. However, a crit-ical examination of the literature has revealed that inmost cases of poor agreement the receiving roomshave been very small [15], and a recent investigationpublished in the same paper has demonstrated thatexcellent agreement can be obtained in a fairly widefrequency range, from 80 Hz to 6.3 kHz, provided thatthe measurements take place in adequate facilities andare carried out with state-of-the-art equipment.

Visualisation of Sound Fields

Visualisation of sound fields, helped by moderncomputer graphics, contributes to our understandingof radiation and propagation of sound and of diffrac-tion and interference effects [16] and has stimulatedmuch research on sound fields [4]. Visualisation ofthe flow of sound energy is particularly useful, withobvious applications in noise source identification,and identifying and ranking noise sources and trans-mission paths is the first step in practically any noisereduction project. Near fields can be very complicated,however. Johansson has examined various methods ofdecomposing such near field intensity data into statis-tically independent components, using principal com-ponent analysis and partial least squares regression[17]. He concludes that a decomposition into principalcomponents enhances the identification of noisesources.

One particular problem in using sound intensity inidentification of partial sources on vibrating structuresis the circulation of sound energy that occurs when theflexural waves on the structure travel slower than thespeed of sound. When the wavelength on the structureis shorter than the wavelength in air the waves do notradiate to the far field but may generate a circulatoryflow of sound power, out of the structure and backinto the structure again in adjacent regions. To over-come the problems with such complicated near fieldphenomena Williams has introduced the concept of‘supersonic intensity’ [4]. This quantity is defined asthe normal component of the part of the sound inten-sity associated with supersonic waves on the structure,that is, waves that travel faster than the speed ofsound. Eliminating the subsonic wave componentsdoes not affect the total radiated sound power of thestructure, and, in the words of the author, ‘the credi-bility of the concept of supersonic intensity lies in thefact that power is conserved.’ The supersonic intensityis obtained by filtering out the subsonic wavenumbercomponents from near field holographic data decom-posed into wavenumber components, and in practicethis is only possible for structures of planar or cylin-drical geometry. Pascal and Li have developed a re-lated concept with similar properties called ‘irrota-tional acoustic intensity’ (since it has zero curl), de-fined as the gradient of the potential of the intensity[18]. Thivant and Guyader’s prediction tool ‘the inten-sity potential approach’ [19] is closely related to theirrotational intensity. One cannot determine the super-sonic intensity or the irrotational intensity with an or-dinary sound intensity measurement system; aholographic reconstruction method is needed.

Other Applications

Ideally the ‘emission sound pressure level’ of ma-chines at specified positions should be measured un-der hemi-anechoic conditions; and corrections forbackground noise and for the effects of reflectionsshould be applied when the environment differs fromideal conditions. Jonasson has proposed to determineemission sound pressure levels by measuring thesound intensity instead of the sound pressure, the ideabeing that the sound intensity would be less sensitiveto reflections from the surroundings than the soundpressure, which means that the troublesome environ-mental corrections could be avoided [20]. An interna-tional standard is under development. Jonasson’s pro-posal is based on the assumption that the reflectionsthat occur in a real room can be regarded as diffusenoise, since ideal diffuse noise would not contribute tothe sound intensity. Hübner and Gerlach have com-pared several measurement procedures and seem tofavour determining the emission pressure level fromthe maximum value of the sound intensity vector fromthree perpendicular components [21].

Finally it should be mentioned that recent worksuggests the possibility of measuring sound absorptionin situ using sound intensity [22].

UNRESOLVED PROBLEMS AND FU-TURE RESEARCH DIRECTIONS

There are still a number of unresolved problems insound intensity measurement.

As described above, there are essentially two fac-tors that limit the range of measurement with p-psound intensity probes, phase mismatch and electricalnoise from the microphones and preamplifiers. Itshould be possible to improve the performance of theconventional sound intensity probes in these respects.

If particle velocity transducers of sufficient stabil-ity can be developed, the alternative p-u measurementprinciple would have a significant advantage in soundpower determination in the presence of strong back-ground noise, as pointed out above. Such probeswould probably be too sensitive to airflow for manypractical applications, but might on the other hand bea 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 uncertain-

ties in the determination of the sound power level of ma-chines’ in Proc. Inter-Noise 96, Liverpool, England,1996, pp. 2713-2716.

13. H.G. Jonasson, ‘Noise emission measurements - System-atic errors’ in Proceedings of Inter-Noise 99, FortLauderdale, FL, USA, 1999, pp. 1461-1466.

14. G. Hübner and V. Wittstock, ‘Further investigations tocheck the adequacy of sound field indicators to be usedfor sound intensity determining sound power level’ inProc. Inter-Noise 99, Fort Lauderdale, FL, USA, 1999,pp. 1523-1528.

15. M. Machimbarrena and F. Jacobsen, Building Acoustics6, 101-111 (1999).

16. J. Tichy, ‘Visualization of sound energy flow and soundintensity in free space and enclosures’ in Proc. Interna-tional Symposium on Simulation, Visualization andAuralization for Acoustic Research and Education, To-kyo, Japan, 1997, pp. 185-192.

17. Ö. Johansson, ‘Sound intensity vector fields in relationto different reference signals’ in Proc. Fifth Interna-tional Congress on Sound and Vibration, Adelaide, Au-stralia, 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 potentialapproach to predict sound propagation through partialenclosures’ in Proc. Inter-Noise 2000, Nice, France,2000, pp. 3509-3514.

20. H.G. Jonasson, ‘Measurement of sound pressure levelswith intensity technique’ in Proc. Inter-Noise 93,Leuven, Belgium, 1993, pp. 363-366.

21. G. Hübner, and A. Gerlach, ‘Further investigations onthe accuracy of the emission sound pressure levels deter-mined by a three-component intensity measurement pro-cedure’ in Proc. Inter-Noise 99, Fort Lauderdale, FL,USA, 1999, pp. 1517-1522.

22. F. Jacobsen and L. Zamboni, ‘In-situ measurement ofsound absorption using sound intensity’ in Proc. EighthInternational Congress on Sound and Vibration, HongKong, 2001.

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, Franceb 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 componenteliminated, it can be used to characterize exactly `sources' and `sinks' regions on a complex radiator and to give morecomprehensive behaviours of complex situations. By the use of processing data in wave number domain, the irrotationalintensity can be computed from the 3-D standard sound intensity, which can be obtained by numerical calculations or from thetechnique of acoustical holography. Effects of source interferences on the field of the active intensity are evaluated andboundary 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 bedecomposed into an irrotational component and acurl component. The advantage of the irrotationalintensity [1] is the suppression of the well-knownvortices in near field of radiation structures [2]. Thedistribution of the normal component of theirrotational intensity on a vibrating surface canprovide us with the exact images of the sound powerof sources without sign fluctuations (positive andnegative) due to the interferences among sources.

Another advantage of the irrotational component isthat it is the gradient of a scalar potential field, whichis the only quantity that depends on the sound powerof the sources. This property makes it possible tocompute the scalar potential of the active intensityfrom the acoustic power of sources using software ofcalculating the diffusion by numerical methods (heattransfer FE). However, it is complicated to performpredictions of acoustic field from energy quantitiesbecause of the interferential nature of a sound field.

ACTIVE INTENSITY FIELDS

A general Helmholtz decomposition of a vector fieldcan be applied to sound intensity and structuralintensity fields [3]

CIII C �������� �� , with 0��� C (1)

From this relation, a field vector function A can beintroduced such that [4]

� � � �������

CAAAI �������������

�

2 (2)

The field vector � �rA of the active intensity containsall information on the sound intensity field. It can becalculated in wave number domain by the use of 3-DFourier transform on the standard acoustic intensityin a whole volume [1]

� � � � rrIKI rK d��

�

Dje (3)

Performing this operation on the left and right handsides of Eq. (2) yields

� � � � 2KKIKA � with 222zyx KKK ���

2K (4)

The scalar potential and the vector potential canrespectively be expressed in terms of the field vectorA such that � �KAK ��� j� and � �KAKC ��� j .

IRROTATIONAL INTENSITYON PLANE BOUNDARIES

The behaviours of the standard and irrotationalintensity on a plane surface can better be understoodwith the help of the field vector A.

Radiation of vibrating surfaces - The scalar potentialin a field point r can be written by

� �� �

���

�

SS

RI

dˆ

21 0 nr

r �

�� , with 0rr ��R (5)

Thus � � nr ˆ0 ��I corresponds exactly to the acousticpower density of the sources on S. For a vibratingplate, this quantity is always positive whereas thenormal component of the standard intensity presentschanges of sign, which do not correspond to a correctimage of the sources.The difference between the normal components ofthe standard intensity and the irrotational intensity

FIGURE 1. From left to right: standard sound intensity, field vector A, scalar potential � and out of planecomponent of vector potential C of the sound field produced by a point source located at a distance h from therigid plane S at ��kh .

can be seen when both of them are expressed interms of the field vector � �rA

� � 2

222

2

2

2

2

2

2

ˆ

ˆ

zA

zyA

zxA

z

zA

yA

xAA

zyx

zzzz

�

��

��

��

��

�����

�

����

�

��

�

��

�

�������

AnI

nI 2

�

(6)

Sources above a reflection plane - Consider a pointsource of a distance h from a rigid boundary. Theinterferences of the sound field depend on the valueof kh (k is the wave number). In Figure 1 is shownthe field vector � �rA calculated from the standardintensity � �rI by using Eq. (4). Calculating A��

yields the scalar potential � of which spatialdistribution is independent of kh (thus of thefrequency). The kh value modifies only the initialpower 0W of the source. In addition, the scalarpotential can be expressed by

� � ��

���

�

���

�

���

��

RRkhkhW

���

41

41

22sin10r (7)

where R and R� are respectively the distances ofthe source and its image from a field point. Theirrotational intensity calculated from � correspondsquantitatively to the radiated energy and does notshow the spatial fluctuations due to interferences(Figure 2). Contrary to � , the vector potential (thereis only one tangential component because of axialsymmetry) is very sensitive to kh values. It isevident that the curl intensity, which is derived fromC , contains the information of directivity of the field(Figure 2). Since the field can be built by using animage source, introduced symmetry leads to

SIn on0ˆˆ ����� nInI C� (8)

This shows that the effects of interferences due to thereflection plane cannot be described by theseboundary conditions on S.

FIGURE 2. Irrotational (left) and curl (right)intensities of the same source as in Figure 1.

DISCUSSIONSExperimental method of the acoustic holographymakes it possible to compute the distribution of theirrotational intensity on a vibrating surface to obtainthe exact image of the sound power distribution onsources [1,3]. Another application relates to thecalculation of the scalar potential by solving thePoisson equation w��� �2 , where w is thedistribution of source power. The prediction of thesound field obtained by calculating the irrotationalintensity gives good quantitative results, in particularby using finite element methods [5]. However, localphenomena due to the interferences cannot berepresented without a complete definition of the fieldvector A.

REFERENCES

1. J.C. Pascal, J.F. Li, "Irrotational acoustic intensity: anew 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.-2sept. 2000.

4. P.M. Morse, H. Feshbach, Methods of theoreticalphysics, McGraw-Hill, 1953.

5. M. Thivant, J.L. Guyader, "Prediction of sound usingthe intensity potential approach: comparison withexperiments", NOVEM, Lyon, 31 aug.-2 sept. 2000.

h

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.

CtordagrIII C

r

rr

rrr

+−=+= ϕϕ (1)

)M,M(q)M(Idiv 00δ=r

(2)

),()( 00 MMqM δϕ =∆ (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.

snSπ=∂

∂ϕ (4) )R(

R1

nl

lϕ−=∂

∂ϕ (5)

0=n∂

∂ϕ (6) )Q()S,Q(g)Q(n

ϕε=∂∂ϕ− ϕ (7)

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)

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:

)jXR(cZ 0 +ρ= (8)

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.

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:

θθππ cos))(1(

4)(

2

20 ℜ−=Q

nr

QI (9)

Z

θ s0(S)

)(QIn

�

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).

cZcZ

0

0

coscos)( ρθ

ρθθ +−=ℜ (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 temperature- at Q, as well as the intensity -or heat flux:

))(1(r4

)Q(2

Q

0 θℜ+ππ=ϕ (11)

θθℜ−ππ=∂

∂ϕ− cos))(1(r4

)Q(n

2

2Q

0 (12)

For a rigid plane, ℜ(θ)=1, thus

Q

0

r2)Q(G)Q( π

π==ϕ (13)

Q0 r21)Q(G)Q(g π=π= (14)

Then ε derives from (7), (11), (12) and (14):

θπθℜ+

θℜ−=θε cos2

))(1(

))(1()Z,( 2

2

(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:

+

+−+π=θθθε=ε −

π

θ ∫ ²X²R

)²X²R(tan1

²X²RR4d)sin(

12

0

(16)

For X=0, (16) becomes:

−π=ε −

θ R)R(tan1

R4 1 (17)

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. 2 - Solid: <ε>θ. Bold: <α>θ versus R -log scale.

( ){ }1RlnR2

1R2R

R8 +−+

+=αθ

(18)

3. COMPARISON WITH I-BEM

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 [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.

A Method to Minimize the Effect of Random Error whenMeasuring 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 ofsound intensity vector fields. An experiment was performed in a reverberation chamber. The source under investigationwas a radial fan and its outlets. Sound intensity was measured over a rectangular grid that contained 84 sub-areas. Themeasurement was repeated four times using different procedures for spatial integration. The measurement results werethen post processed by principal component analysis The idea is that the information contained in a complete set ofdata, can be used for minimising the effects of random error in a specific position in the vector field. The result wasanalysed at several frequencies having different global pI-indexes. Comparison between measured and refined vectorfield confirmed that the method reduces the effect of random error and improves interpretation of the result.

INTRODUCTION

Measuring sound intensity vector fields in narrowfrequency bands over a large surface is timeconsuming. To reduce random error in vectorestimates long averaging times are required especiallywhere there are large pressure intensity differences [1].For source characterisation, however, the informationcontained in spectral data of high resolution is needed.

Several methods for source location have beenpresented. Non of the methods, however, is asstraightforward, robust and refined as the soundintensity method. The objective of this paper is tovalidate a method that minimises the time required fornarrow band measurements of sound intensity vectorfields. The method is based on principal componentanalysis (PCA) and has been validated experimentallyby studies on a relatively simple source in goodacoustical conditions [2]. For further validation anexperiment was performed in a reverberation chamber(T60=6-8s). The source under investigation was a radialfan plus its outlets. Intensity was measured in “real-time” over a rectangular grid that contained 84 sub-areas. The measurement was repeated four times usingvarious BT-products, at two different distances fromthe source. The intensity estimates were further postprocessed by PCA, which is an iterative procedurebased on singular value decomposition [2,3]. PCA isused to detect correlated patterns in intensity

measurement data, both in frequency and spatialdomains, e.g. areas of sound radiation. The idea is thatthe information contained in a complete set of data canbe used for minimising the effects of random error in aspecific position in the vector field. The validation wasmade by visual comparison between measured andrefined vector field at several frequencies, havingdifferent global pI-indexes.

EXPERIMENTAL PROCEDURE

The sound intensity [4] was measured over a planargrid in front of the cover of a radial fan and its out lets.Four measurements were conducted using, 30 or 10-cm distances, and 4 Hz or 16 Hz resolution. Theintensity was measured in the frequency range 0 -3600 Hz, by a 3D-probe (B&K 0447) connected to anFFT-based 8-channel measurement system (LMSCADA-X, front-end HP-Paragon). The probe wascalibrated using an acoustic coupler B&K 3541. Theresidual pressure-intensity index varied between 18 -24 dB (200–3600 Hz). To ensure true real-timeperformance, the six microphone signals of the 3d-probe were sampled during the measurementsequence. The intensity vector field was determined bypost processing the sampled pressure signals(Hanning, 75% overlap). The time averaged activeintensity spectrum (t=1.5s) was estimated for eachsub-area (0.01 m2) in the plane grid (7x12). The spatial

position of the probe was logged on a seventh channel. Finally, the intensity data was processed by PCA.By using an appropriate number of principalcomponents, the remaining non-systematic varyingdata can be removed, i.e. reducing the effect of randomerror [2]. The final result is visualised as principalintensity vector fields (PIV) for different frequencies.

RESULTS AND DISCUSSION

The following discussion is based on measurementresults at 30-cm distance from the fan cover. Theresults at 10-cm distance will be analysed anddiscussed in the extended paper version.

The averaged sound intensity spectrum (normalcomponent) has predominant peaks at 380, 756 and1136 Hz. At these frequencies, the cover and itsoutlets, is forced to vibrate by pressure fluctuationsrelated to the fan blade frequency and its harmonics.Remaining intensity is related to structural modesexcited by the turbulent flow and noise from theelectric motor.

Changing the resolution from 4 to 16 Hz isexpected to reduce random error by 50%, if averagingtime is kept constant. The difference in random errorwas most easily seen as sign shifts of the intensitycomponent normal to the surface (from negative topositive at 4 positions). Generally, the intensity vectorsat the most important frequencies indicated largerandom errors. The effect of random error can bepredicted on the basis of BT-product and the pI-index[1]. For example, at 380 Hz the global pI-index is 10dB (756 Hz, 5dB; 1136 Hz, 6dB). This indicatessevere measurement conditions, since the pI-indexmay be very large at specific positions. The predictednormalised standard deviation at 380 Hz, based on theBT-product (B=4Hz, T=1.5 s) and the global pI-index(10dB), is 1.65 locally and 0.18 in relation to the totalaveraging time (T=126s).

The proposed method was tested on themeasurement results based on 4 Hz resolution. BeforePCA, measurement data was transformed, to obtainnearly normal distributed intensity data in both spatialand frequency domain. The absolute value of eachamplitude 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 principalcomponents, which describe 59% of the variance inthe transformed data matrix. However, at 380 Hz twoprincipal components explained 95% of the variance(756 Hz, 4 comp, 70%; 1136Hz, 4 comp, 23%). Thecomparison between measured intensity and the

refined vector field, indicates a reduction of randomerror. However, the result is not as good as in theprevious study [2], due to the large number of negativesigns in the normal intensity components. Negativeintensity could be expected in a highly reverberantfield, but in several positions it is erroneous estimates.These errors may be detected by analysis of theresidues between the measured and refined vectorfield. By this procedure all sign-shifts found using 16Hz resolution, were detected. Correction of detectedsign errors improved the result and gave a betterdefinition of the sound radiating surfaces. However,measurements far from the source in a reverberationroom are not an ideal condition and make it difficult tointerpret the vector fields. The results at 10-cmdistance, which also will be compared with resultsusing very long averaging times, will be used as a finalvalidation of the method. These results will bepresented and discussed at the conference and in theextended paper.

SUMMARY

Rapid measurements of 3D acoustic intensityvector fields require real-time performance of an FFT-based measurement system. Real-time measurementswere achieved by post processing the pressure signals,sampled during scanning. The effect of increasedrandom error due to short averaging time was partlyreduced by principal component analysis. Theaveraging time may be reduced by 75 % and still goodestimates of the intensity vector field may be obtained.The problem is to know the minimum time requiredfor an acceptable measurement quality. Ideally theprocedure may be done automatically. The mostimportant benefits of the described method are thelimited time in front of the source, and the true real-time measurement of the sound intensity vector field.

REFERENCES

1. [1] F. Jacobsen Prediction of Random Errors in SoundIntensity Measurement. International Journal of Acoustics andVibration, 5, No 4, 173-178 (2000).

2. Ö. Johansson “Rapid Measurements of Acoustic IntensityVector Fields”, Proc. of the 7th International Congress onSound and Vibration, Garmisch Partenkirchen 7-10 july 2000.

3. A. Höskuldsson, Prediction methods in science andtechnology, Vol 1. Basic theory. Thor Publish., Denmark, 1996

4. F. Fahy, Sound intensity, 2nd edition,. E & FN Spon, London,1995

�� �������� ���� �� ���� ��������� �� ������� ��� �������� ����� ������ ��� �� ��� ����� ���� �� �� �������� �����

Domenico Stanzial�, Davide Bonsi

��� ����� �� ���������� �������� ��������� ��� ������ ����� ��������� � ���� ��� �� ���������������� ��� ���� ���� ����� !��� �� ���� ���"#��$$$������%��������

A new indicator very sensitive to the room resonances excited by an acoustic source is here introduced and used as an environmentalparameter for certifying the sound �eld condition when measuring the acoustic power of the source by the intensity method. Thisquantity will be called �������� ��������� Thanks to its experimentally proved properties a de�nitive insight of the physicalmeaning of the “oscillating intensity” as introduced in previous papers is accomplished. The results of a set of measurements ofthe sound power of a reference source in different acoustical environments will be presented and discussed using the introducedindicator.

������������

The de�nition of the acoustic power of sources is basedon the well-known acoustic energy corollary whichis demonstrated to be consistent with the requirementof energy conservation in a �uid to second order [1].Within this theoretical frame, the power � of an acous-tic source can be expressed in terms of the stationarytime-averaged radiant (active) intensity ���� as

� �� �

�

���� � � ��� (1)

where the integral is taken over a closed spatial sur-face � bounding the volume � which usually containsthe source. � is a real quantity measured in � �

����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�� � � � � , where ���� �� and ���� �� stand respec-tively for the instantaneous sound energy density and in-tensity. In this paper only the case � � will be con-sidered.

When a source is enveloped by � the continuityequation takes the inhomogeneous form �� � � � ����� �� where � is the "�$�� ������� depending in gen-eral ���� on the source and the environment. It is worthnoting that �� vanishes when a stationary time-averageis performed on it so that also its integral over any vol-ume (contained or not in �) is zero. This fact, oftenmisunderstood, has an important physical meaning be-cause tells us that when stationary energy conditions arereached the time rate of the energy emission from thesource (i.e. the source power) must be equal to the timerate of the energy absorption by the boundaries. Due

to the equality ���� � the rate of energy absorption,in turn, must be equivalent to the integral of ����� ���over � (stationary energy balance). Unfortunately, nei-ther mathematical nor experimental handling de�nitionfor ���� �� has been formulated yet, so the only way forevaluating the acoustic power of a source must be basedon sound intensity: � can be experimentally evaluateddetermining the time average vector �eld ���� at somediscrete points � � � and then approximating the inte-gral (see [2]). In this paper a careful experimental inves-tigation about the in�uence of �eld boundaries conditionon the measured power of a reference acoustic sourcehas been carried out and a new �eld indicator account-ing for the sound energy “trapped-in-the-average” in thestanding waves (normal modes of the environment) ex-cited by the source itself has been usefully employed asan environmental parameter for acoustically certifyingthe source power measurement.

��� �������� ��������

Following a reasoning similar to the one which led to the

de�nition of the sound radiation indicator ����� � ���

the resonance indicator will be here introduced as

�����

� ��� (2)

where ����

����� (apart from a numerical factor) isthe Hilbert-Schmidt norm of the sound intensity polar-ization tensor [3]. In other words � is de�ned as theeffective value of the &�� ��������� normalized to� ��� thus representing the fraction of the energy densitywhich is locally associated in the average to the normalmodes excited by the source. A preliminary analysis ona similar indicator can be found in Ref. [4]

� Also with Cemoter-CNR, Ferrara, Italy

!�"��! !�� ��� �!"�#�"

Figure 1 shows the reference source and the intensityprobe used for measuring the sound power in two dif-ferent rooms. The �rst one is very “dead” due to theabsorbing upholstery (��� � ��� �) while the secondone is a normal room of the same volume and character-ized by ��� � � �. The input signal to the source (whitenoise) and the enveloping surface (a sphere centered atthe top of the source having a radius of ��� �) were thesame for the two measurement sets.

$�%��! &: The source and a measurement point.Once �xed the equatorial plane parallel to the �oors,

the radial component of sound intensity in two positions �� North and �� South belonging to meridians ��

apart for a total of � points was measured. These datawere then used for the power computation. We collectdata for evaluating � at only one measurement point inthe equatorial plane at a distance of � � from the centreof the sphere.

Figure 2 reports the calculated power of the source inthe two rooms for third-octave bands ranging from �to � ��. Figure 3 reports the comparison betweenthe indicators � and �, measured at one point in front ofthe source in the two rooms.

����#�"���

According to the precision of our measurements thepower of the reference source results slightly greater(� � dB) for all bands when the source is radiatingin the absorbing room. This can be considered an exper-imental evidence of the fact that the power of an acousticsource depends inescapably from the boundaries condi-tions or, stated in other words, that when the acoustic�eld feeds back to the source, the source itself becomesa boundary condition for the acoustic �eld. For thisreason, when accurate power measurements of acoustic

sources are needed, the two indicators � and � should beemployed for certifying the in�uence of the environmenton the measured �eld. This should be done optimally atevery point where data for the power computation arecollected.

125 250 500 1000 2000 400066

68

70

72

74

76

78

80

freq. (Hz)

Pow

er (

dB)

$�%��! ': Values of sound power (squares: deadroom, circles: normal room).

125 250 500 1000 2000 40000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

freq. (Hz)

µ2

η1

η2

µ1

$�%��! (: Values of � and � (squares: dead room,circles: normal room).

�!$!�!��!"

1. A.D. Pierce, ��������, pp. 36-39, ASA-AIP, 1989.

2. F. Fahy, ����� ���������, 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.

Time-dependent Sound Intensity Measurement MethodsH. Suzuki

Department of Network Science, Chiba Institute of Technology2-17-1 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 theanalysis of transient sounds. In this paper, two types of time-dependent sound intensities are introduced. The first one is theenvelope intensity. The envelope intensity is obtained from the multiple of the analytic (with zero spectra at negativefrequencies) time-dependent pressure and particle velocity. The envelope intensity gives a rather slowly varying intensity like theenvelope of the signal. Since it has the directional information, it can be used for the localization of transient sound sources in thethree dimensional space. The second method is the application of cross Wigner distribution to the sound intensity. This gives thesound intensity distribution on the time-frequency plane. Some application examples of the above two methods will beintroduced.

INTRODUCTION

Information on the intensity and the direction of atransient noise is important in order to understand itsgeneration mechanism and contrive a method to reduceor eliminate it. For example, a large glass windowoccasionally generates a transient noise when theambient temperature changes drastically. It is difficult,however, to estimate from which part of the windowthe noise is generated since the phenomenon isunrepeatable. Dozens of similar cases can be easilylisted. In this paper, two sound intensity methods that can beused for the analysis of the transient sounds arereported. The first is the envelope intensity method[1]and the second is the "instantaneous sound intensityspectrum" method[2].

ENVELOPE INTENSITY

The instantaneous intensity is given by)()()( tvtpti � (1)

where p(t) and v(t) are the sound pressure and theparticle velocity in the direction of the measurement.Even for a monotone sound, the instantaneous intensitychanges with twice of the original frequency. This isthe reason why the time-averaged intensity method isused for most of practical applications, which isunfortunately not suitable for transient sounds. Theenvelope intensity was developed with a purpose ofkeeping the time dependency and still avoiding theradical change of the instantaneous intensity. It isdefined by

� 2/)]()(Re[)( a*ae tvtptI � (2)

where pa(t) and va (t) are the analytic functions of p(t)and v(t), respectively. The analytic function of x(t) isdefined by

)(j)()( ha txtxtx �� (3)where xh(t) is the Hilbert transform of x(t). A three-dimensional sound intensity probe shown inFigure1 was used for the measurement of the envelopeintensity. The microphones are located at the apexes ofa tetrahedron. The pressure is the average of the fourpressure signals and the particle velocity componentsfor x, y, and z axes are also calculated from linearcombinations of the four microphone signals[3].

FIGURE 1. Three-dimensional sound intensity probe

An example of the envelope intensity measurementis shown in Figure 2. An impact sound generated byhitting a 900x900x9mm plywood from behind by asmall metal hammer was recorded in an anechoic roomby use of the probe shown in Figure 1. The top left andright charts show the horizontal and vertical planedisplays of the envelope intensity vector loci,respectively. The analysis frequency band was from125Hz to 1kHz. Figure 2 shows that the direction of themaximum intensity coincides with that of the impactshown by the dotted line in the horizontal plane. Themiddle and the bottom charts show x-, y-, and z-axisintensity components and the pressure of microphone 1as functions of time.

0 time(ms) 160

FIGURE 2. Envelope intensities and a pressure waveform ofan impact sound generated by hitting a plywood

INSTANTANEOUS SI SPECTRUM The cross-Wigner distribution between p(t) and v(t) isdefined by

����� detvtpftW f

pv 2)2/()2/(*),( �

��

��

��� �(4)

If p(t) and v(t) are given, respectively, by 2/)]()([)( 21 tptptp �� (5)

���

���

tdttptp

dtv ')]'()'([1)( 12

� (6)

with pressure signals of two microphones of a p-p typeone-dimensional intensity probe, p1(t) and p2(t), theinstantaneous sound intensity spectrum (ISIS) is givenby dfftWftI pp 2/)],(Re[),( 21 ���� (7) Figure 3 shows an experimental setup used for testinga usefulness of ISIS. A loudspeaker generated a 4-cycle10 kHz sinusoidal tone burst and the direct andreflected sounds were recorded at the cross point oftwo arrows inside the region surrounded by threewooden boards.

FIGURE 3. Experimental setup for the measurement of adirect and reflected transient sounds

A result is shown in Figure 4. The horizontal andvertical axes represent the time and frequency,respectively. Arrows in the charts show the directionsof the energy flow and the magnitudes as well as thetime and frequency dependence. Several words shownin the box show possible reflectors for each wavepackets. As this figure shows, it is possible tographically represent frequency components of aninstantaneous intensity by use of the cross-Wignerdistribution.

FIGURE 4. ISIS obtained from the setup shown in Fig.3.

REFERENCES

1. Suzuki, H., Oguro, S., Anzai, M., and Ono, T, Inter-noise91 Proceedings, Vol.2, 1037-1040 (1991).

2. Tohyama, M., Suzuki, H., and Ando, Y., The Nature andTechnology of Acoustic Space, Academic Press, 1995, 143-145.

3. Suzuki, H., Oguro, S., Anzai, M., and Ono, T,J.Acoust.Soc.Jpn, 16, 4, 233-238 (1995).

Investigations into the correlation between field indicatorsand sound absorption

Gerhard Hübnera, Volker Wittstockb

a ITSM, Universität of Stuttgart, Pfaffenwaldring 6, D-70550 Stuttgart, Germany, huebner@itsm.uni-stuttgart.deb 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 differsignificantly from its true value if, in the presence of absorption within S, stronger directly incident background noises flowthrough this surface. The magnitude of this effect can be determined by a second measurement where the source under test isswitched off and the background noise remains unchanged. This second measurement yields a negative sound power whichdirectly describes the error of the first measurement if background noise and noise from the source under test are notcorrelated.

Another approach is to check the presence of relevant sound absorption within S using the well-known sound fieldindicators. In principle, the global unsigned indicator and the global signed indicator, especially their difference, the soundfield non-uniformity indicator and the difference between global indicators and the instrument‘s dynamic capability react toabsorption inside S as they do to other influences. The paper deals with theoretical and experimental results of the correlationbetween different indicators and absorption within S.

1 INTRODUCTIONIntensity-based sound power determinations are

influenced by an additional error when an absorptionis included in the measurement surface and strongerextraneous noise sources are present. When the sourceis switched off, with the background noise remainingunchanged, the absorbed sound power Pabs can bedetermined which can be used to correct the measuredsound power

true meas absP P P= + (1)

assuming that the source under test and thebackground noise are not correlated.

Under practical conditions, however, it issometimes not possible to switch off the source undertest without changing the background noise. Anexample of this is the sound power determination ofone specific part of a machinery set. In such cases, it isof interest whether one of the well-known sound fieldindicators [1] or a new one can be used to indicate andlimit significant sound absorption effects.

2 THEORETICAL CONSIDERATIONSThe effect of absorption situated within the space

enveloped by the measurement surface S depends onthe magnitude of the background noise during themeasurements. Here, two different cases can bedistinguished:a) the background noise enters directly from a strong

parasitic sound source in close proximity to themeasurement surface, or

b) the background noise randomly superposes theoriginal sound field at the measurement positions.

As is known [1], sound field situations can bedescribed by certain sound field indicators which arein particular the unsigned pressure intensity indicator

n n 0pp I IF L L K= − − , (2)

the signed pressure intensity indicator

n n 0pI p IF L L K= − − (3)

and the field non-uniformity indicator

( )2n, n

n 1

1 1.

1

N

S iI

F I INI =

= −− ∑ (4)

Definitions of the several symbols are given forexample by the Draft International Standard ISO9614-3. The bar indicates the average value on themeasurement surface. The error due to absorption is

absmeasabs

true true1

PPP P

∆ = = −

(5)

where Ptrue is the sound power of the sourcedetermined in the absence of absorption and Pabs is theabsorbed sound power determined by the “zero test“,where the source under test is switched off and thebackground noise remains unchanged. Exclusion of allother error sources and introduction of the soundpower Pmeas, measured in-situ, using eq. (1), yields

1abs

absmeas

1 .P

P

−

∆ = +

(6)

With

( )zeron0.1 /dBn,zeroabs

meas n,meas

10 pIF FI SPP I S

+= = , (7)

the level of the absorption error follows to be( )zeron

0.1 /dB,abs 10lg 1 10 dBpIF F

L+

∆ = − +

(8)

with the zero test indicator

n,zerozero 2

meas

10lg dB .I

F cp

ρ = %

9

The magnitude of the indicator npIF comprises

different influences such as the near field error,environmental corrections and the noise-to-signalratio. This indicator cannot therefore be used fordetecting the presence of background noise or ofabsorption errors, only. Eq. (8) shows that for thedetermination of L∆,abs both indicators,

npIF and Fzero,

are necessary. Only if the measurement surface issituated in a reverberant field region ( zeroF → −∞ ),

L∆,abs is equal to zero. This means for measurementsoutside the reverberant radius that the absorptioninside S is simply added to the total room absorption.

Furthermore, alternatively to Fzero, the indicator

absabs

meas10 lg dB .

PF

P= (10)

can be introduced which directly indicates the ratio ofthe absorbed sound power to that measured. Based onthe formula given above, the error due to theabsorption effect such as Ä,abs 0.41dBL ≤ or 0.14 dB≤

is limited by

nabs zero 10dB or 15 dB .pIF F F= − ≤ − ≤ − (11)

If the zero test cannot be carried out for practicalreasons, L∆,abs can be determined or limited onlyapproximately.

3 PRACTICAL APPROACHAn absorption error can of course be excluded if

the noise-to-signal ratio vanishes. This can be testedby determination of the difference

n n.pI p IF F− (12)

If this difference is zero, no significant backgroundnoise is present.

Under in situ conditions, however, where a certainbackground noise level and an unknown absorptionexist, a procedure capable of limiting the absorptionerror is of interest. The data base of an earlierinvestigation [2] can be used for such an approach.

These investigations were carried out in a semi-anechoic room with extraneous noise superelevation

levels between 0 and 10 dB. The absorption wascaused by a plate (chipboard, 40 mm thick) formingone of the boundaries of the measurement surface.

For this practical approach the indicators

n npI p IF F− and FS are regarded which can be

determined under conditions where performance of thezero test is not possible. Figures 1 and 2 show thedependence of the absorption indicator on these twoindicators. For this specific example, the criterionfrom eq. (11) is fulfilled if FS < 3 or < 1 or

nn p Ip IF F− < 4 dB or < 0 dB. For other acoustical

situations – for instance for semi-reverberant environ-ments – other limiting values can be expected.

Figure 1 Indicator Fabs as a function of FS

Figure 2 Indicator Fabs as a function of nn p Ip IF F−

4 CONCLUSIONSIn order to check the significance of absorption effects,the zero test should preferably be carried out. Resultsof this test allow sound power corrections or calcula-tions of the amount of the absorption error to becarried out. If the zero test cannot be carried out,certain other criteria allow the error to be assessedonly approximately.

REFERENCES1. G. Hübner, INTER-NOISE 84 Proceedings, pp.

1093-10982. G. Hübner, V. Wittstock, INTER-NOISE 99

Proceedings, pp. 1523-1528

-20

-10

0

10

20

0,1 1 10 100 1000F S

F abs/dB criterion limit

-20

-10

0

10

20

0 5 10 15 20 25 30(F p lI nl-F pI n)/dB

F abs/dB criterion limit

ISO standards on sound power determination by sound intensity method

H. Tachibanaa and H. Suzukib

aInstitute of Industrial Science, University of Tokyo, Komaba 4-6-1, Meguroku, Tokyo 153-0041, Japan bDept. 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).

MEASUREMENT UNCERTAINTY

See TABLE1.

TABLE 1. Measurement uncertainty of ISO 9614-3

1/3 octave band center frequencies

[Hz]

Upper values of standard deviation of reproducibility [dB]

50 to 160 200 to 315

400 to 5 000 6 300

2,0 1,5 1,0 2,0

A-weighted* 1,0 * Calculated from third-octave bands from 50 Hz to 6,3 kHz.

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: TF , field non-uniformity

indicator: SF , unsigned pressure intensity

indicator:nIpF , and signed pressure intensity FIGURE 1. Flow of the measurement in ISO 9614-3

Yes

For all partial surfaces

Crt. 5

Crt. 4

Crt. 3

Crt. 2

Crt. 1

Yes

No Yes

No

Yes

No

Yes

Yes

No

No

Yes

No

No

Evaluate the temporal variability indicator and determine 6,0<TFT

Define the measurement surface, partial surfaces and scanning path

Choose one partial surface

Determine the scanning time 6,0SS <⋅=

TFTNT

Perform two scans

Take action A

Take action B

Take action C

Take action D or E

Take action G

Take action F or G

nd pIFL ≥

3nn ≤− IppI FF

Scanning already increased?

2,1/83,0 )2(S)1(S ≤≤ FF

2/)2(n)1(n

sLLII

≤−

Is ST practicable ?

2S ≤F

Final result

indicator: npIF shown in Table 2 are evaluated.

(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

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.

NORMALIZED SOUND POWER LEVEL

To standardize the value of sound power level of a source, “normalized sound power level” 0WL is

defined as the sound power level under the reference meteorological condition (temperature 0θ =23 C° ,

barometric pressure 0B =101,325 kPa), given by:

+−=

θ15,273

15,296

101325lg150

BLL WW dB

where, θ is the air temperature in C° during the actual measurement, B is the barometric pressure in Pa during the actual measurement.

Table 2. Field indicators used in 9614-1, 2, and 3

Indicator ISO 9614-1 ISO 9614-2 ISO 9614-3

Temporal variability indicator ∑

=

−−

=M

kk II

MIF

1

2nn

n

1 )(1

11

― ∑=

−−

=M

mmT II

MIF

1

2nn

n

)(1

11

Field non-uniformity indicator

∑=

−−

=N

ii II

NIF

1

2nn

n

4 )(1

11

― ∑

=

−−

=J

jj II

JIF

1

2nn

n

S )(1

11

Un- signed

Surface pressure intensity indicator

n2 Ip LLF −=

)101

lg(101

1,0∑=

=N

i

Lp

pi

NL

∑=

=N

iiI II

NL

10n )/(

1lg(10

n

Negative partial power indicator

]/lg[10/ ∑∑=−+ ii PPF

iii SIP n= ∑=

=N

iiPP

1

* 23/ FFF −=−+ in a special

case.

Unsigned pressure-intensity indicator

nn IpIp LLF −=

)/1

lg(10 20

1

2 ppJ

LJ

jjp ∑

=

=

)/1

lg(101

0nn

∑=

=J

jjI

IIJ

L

Pressure intensity indicator

Signed

Negative partial power indicator

n3 Ip LLF −=

)/1

lg(101

0n∑=

=N

iiI II

NL

n

Sound field pressure-intensity index

)/lg(10][ 0SSLLF WppI +−=

= ∑

=

N

i

Lip

piSS

L1

1,0)10(

1lg10][

∑=

=N

iiSS

1

, 10 =S m2

* 3FFpI = in a special case.

Signed pressure-intensity indicator

nn IppI LLF −=

∑=

=J

jjI II

JL

10n /

1lg10

n

2,1/83,0 ≤∆∆≤ yx

FIGURE 2. Measurement surface

2

1

Partial surface

scanning path

partial surface

partial surface

segment

x∆

y∆

Sound Intensity Detection for Multi-sound Field UsingMicrophone Array System Based on Wavelength Constant

MatrixK. Kuroiwaa and O. Hoshinob

Faculty of Engineering, Oita University, 700 Dannoharu Oita 870-1192, Japan e-mail: kuroiwa@cc.oita-u.ac.jpa hoshino@cc.oita-u.ac.jpb

One problem with the traditional method of sound intensity is that it cannot give the individual sound intensity of interest in amulti-sound field. The objective of this study is to develop a sound intensity detection system for a multi-sound field. Theproposed system consists of three-dimensionally spaced microphone array, sound detection filters based on wavelengthconstant matrix and sub-system of sound intensity estimation. A wavelength constant matrix is exactly formulated in terms ofsound directions so that it can give a relation between the microphone outputs and the respective sound existing in a multi-sound field. Feeding microphone outputs to the filters, the system can issue the respective sound at the specified virtual pointsof our choice. Therefore, it can give three-dimensional vector of the respective sound intensity. Experiments of the incidentand reflected sound intensities near the reflective panel for oblique incidence from 0 to 60 degrees in an anechoic chambershowed that the proposed system is fundamentally valid in sound intensity detection. The mean errors were 4.3 percents inamplitude and 5.1 degrees in direction in comparison with the original.

DETECTION OF SOUND INTENSITY

The block diagram of sound detection system isillustrated in FIGURE 1. In a multi-sound field, it isassumed that there are N sorts of sound sources. Thenth of sound source, Sn , shown with ‘filled circle’ withcoordinates (rn, �n, �n) radiates its original sound pn(t)and the relevant sound intensity, In, shown with arrowtowards a microphone array located at the origin. Themth microphone, Mm, with coordinates (dm, �m, �m)receives all arriving sounds and issues the output ofqm(t) that alters in phase and magnitude dependingupon the mutual situation given by the position vectorAm of Mm with m=1�M, Bn of Sn with n=1�N andposition vector V of such virtual points as Mx and M-xindicated with ‘gray circle’ on XYZ axes. Letting the kth sample of the original sound pn(V,t)assumed at the virtual point expressed by positionvector V and the microphone output qm(t) be pn(k) andqm (k), and also letting the ith frequency component ofthe DFT spectrum of them be Pn(V,i) and Qm(i),respectively, then the nth sound Pn(V,i) at V can bedetected by using the following equation [1]:

Pn (V, i) � Hnm(V ,i)

m�1

M

� Qm(i) (1)

where, the FIR filter Hnm(V,i) is given as follows:

Hnm (V, i)� ��CV�1.�� ��(2)

CV � cmn(V, i)� � (3)

cmn(V, i) � exp

j2� i� fU

Am � V� �� Bn � V� �Bn �V� �

��

����

�

������ (4)

in which, U, �f and raised dot denote sound velocity,frequency resolution in DFT spectrum and innerproduct of two position vectors, respectively. Cvimplies the wavelength constant matrix having the sizeof N � M. Substituting the position vector V�x of Mx for V inEquation (1) and also V-�x of M-x, then the nth sound Pn(V�x,i) for Mx and Pn(V-�x,i) for Mx can be estimated,respectively. Therefore, we can obtain the x-axiscomponent of nth sound intensity In,x [2]. It has theform of

In, x �

12��V

�x � V��x

Im Pn (V��x , i) Pn

* (V�x , i)� �i

i� i 1

i 2

� (5)

where � denotes the density of air. In the same manneras for In,x, the y-axis component In,y and the z-axiscomponent In,z are estimated. Consequently, the nth

sound intensity In can be obtained as follows:

In � i I n,x + j I n,y + k I n,z (6)

where i, j and k imply unit vector of XYZ-axes,respectively.

FIGURE 1. Block diagram of the sound intensitydetection system.

EXPERIMENTAL RESULTS

The schematic diagram of the experiment in ananechoic chamber is shown in FIGURE 2. It is thepurpose of this experiment to detect the incident soundintensity, Ii (I1), coming from a loud-speaker S1 withthe coordinates ( 1.5 m, 90 deg., �i deg. ) and reflectedsound intensity, Ir (I2), from the reflector. For this case,second sound source, S2 , with the coordinates ( 1.5 m,90 deg., 180-�i deg. ) implies the mirror image sourceof S1 with respect to the reflector. Results of the sound intensity detection for acrylresin hard board and urethane foam board areillustrated in FIGURE 3. The filled arrows directedleftwards and rightwards imply the detected incidentsound intensity, Ii, and the reflected sound intensity, Ir,respectively. The detected sound intensity is expressedwith ‘^’ on it in this section. To evaluate the detectionaccuracy, the incident sound intensity, Ii, measured byremoving reflector is also illustrated with gray arrow.It is shown that the proposed system worked well overthe incident angle from 0 to 60 degrees. Mean errorswere 4.3 % in magnitude and 5.1 degrees in direction.

FIGURE 2 Schematic diagram of the experiment.

FIGURE 3 Experimental results of the soundintensity detection.

REFERENCES

1) K.Kuroiwa, “Multi-microphone system for sounddetection based on wavelength constant matrix”, J.Acoust. Soc . Jpn.(E) 19, 289-292 (1998).2) F.J.Fahy, 'Measurement of acoustic intensity usingcross-spectral density of two microphone signals',J.Acoust.Soc. Am. 62, 1057-1059 (1977).

Free Field Measurements in a Reverberant Room Usingthe 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 whichdoes not measure the pressure, but the particle velocity in a sound field.

INTRODUCTION

The microflown is a sensor that measures theparticle velocity in an acoustic disturbance [1]. Itconsists of two heated wires; the particle velocity isdetermined from the differential resistance changes[1]. When a microflown sensor is combined with apressure microphone an intensity probe is obtained.Experimentally such a (simple) p-u intensity probehas been tested and found to be as good as theexisting p-p intensity probes [2]. Main advantages ofthe p-u probe above the p-p probe as intensity probeare that errors (or uncertainties) in the phases do notlead to large errors in the estimated intensity, andthat one probe configuration can be used for a broadfrequency region (the p-p probe use differentconfigurations for the low- and high frequencyregion). A disadvantage of the p-u probe can be thatthe calibration of the two sensors with respect toeach other is more difficult.An important property of the microflown is itsdirectional characteristic. Denote the orientation ofthe microflown by a vector n[x,y,z], which is definedas follows. The vector n is in the plane through thetwo conducting wires of the microflown andperpendicular to the length of the wires (themaximum sensitivity of the microflown is when theparticle velocity is parallel with n). When thedirection of the particle velocity makes an angle �with n, the measured output of the microflown isproportional to cos(�), see [1] and [2]. This propertymakes it possible to obtain free field ( no contributionof reflections) properties of a sound field in areverberant environment.

One microflown

Consider the case that a sound source ( a loudspeakerto which random noise is applied) is placed in areverberant room. When the microflown is orientedsuch that n is in the direction of the free field particlevelocity of the sound source the r.m.s. value followsfrom the square root of the squared free field- andreverberant particle velocity:

u//2 =ud2 + 1/3urev.

2 The factor 1/3 comes from theintegral over all angles in a sphere, representing all theimage sources, contributing to the diffuse reverberant

field: � �

�

������

0

3/1)cos().cos().sin(2)4/1( d . When

in a second experiment the microflown is oriented suchthat n is perpendicular to the free field velocity of thesound source, the r.m.s. value is given by: u�2 =1/3urev.

2. So by subtracting u//2 - u�2 the free field isobtained. Two recordings have been made of speech ina reverberant room, one corresponding to u// and theother corresponding to u�. When reproducing theserecordings a clear difference is heard: the recording ofu� contains only the reverberant sound field. Therecordings will be reproduced during the presentation atthe conference.

Two microflowns perpendicular to eachother

The contribution of a pure diffuse reverberant soundfield to the cross-correlation of two microflownsperpendicular to each other, vanishes. To show this,consider the two dimensional case with twomicroflowns, having n-vectors n1 = [1,0] and n2 = [0,1].Suppose two uncorrelated sound sources with powerstrength s1 and s2 are oriented in perpendiculardirections, e.g. one under an angle � with the x-axis,and the second one having an angle � + �/2 with the x-axis. When the output signals of the two microflownsare u1(t) and u2(t), the long term value of the cross-correlation Ru1.u2(0) is written as:

��

T

dttutuT0

).(*2).(1)/1( Ru1.u2(0)

; the symbol (0) in Ru1.u2(0) refers to the fact that notime difference is taken between u1(t) and u2*(t). Thecontribution of source s1 is proportional to –cos(�)*(-sin(�)) and of source s2 to sin(�)*(-cos(�)); a positivesignal is assumed when the components of theconnection vector r from source to sensor are in thepositive x- or positive y-direction. When the strength ofthe two uncorrelated sources are equal, s1 = s2, the

contributions to Ru1.u2(0) are equal but with differentsign, thus Ru1.u2(0) vanishes. To verify thisexperiments have been done with two sound sources(loudspeakers) perpendicular to each other and with �= 45 0. The loudspeakers were excited with twouncorrelated noise signals within a frequency band of900 – 1100 Hz. The results of the experiments areshown in figure 1, where is plotted R*u1.u2(0) as afunction of the ratio’s of the power strengths s2/s1;R*u1.u2(0) being the normalized Ru1.u2(0) value withrespect to the Ru1.u2(0) value at s2 = 0. The crosses (+)are the experimental results. For s2/s1 =1 the crosscorrelation is zero.

0 0.5 1 1.5 2 2.5-1.5

-1

-0.5

0

0.5

1

1.5

s2/s1

R* u1

.u2

Figure 1. The value of the cross correlation Ru1.u2(0)normalized to the value at s2 =0 as a function of theratio of the power strengths s2/s1 of the two soundsources.

In a pure diffuse reverberant sound field thecontribution of an image source to Ru1.u2(0) is alwayscompensated by the contribution of another imagesource in a perpendicular direction. Intensitymeasurements have also the property that thecontribution of a diffuse sound field vanishes. So freefield measurements in a reverberant environment canbe done using an intensity p-u (or p-p) probe or usingtwo microflowns oriented in perpendicular directions.An important difference is of course that with twomicroflowns perpendicular to each other onemeasures the product of two perpendicularcomponents of the free field particle velocity vector,while with the intensity the free field product of p andu, being the energy flow.

Three microflowns combined witha pressure microphone

A more sophisticated device is to combine onepressure microphone with three microflowns orientedin perpendicular directions, e.g. n1 = [1,0,0], n2 =[0,1,0] and n3 = [0,01]. The four sensors are quitesmall and are positioned close to each other, thus

measuring the different sound quantities at almost thesame point. A photograph of such a device is shown infigure 2.

ux

p

uz

uy

1mm

Figure 2. A photograph of the ux-uy-uz-p sensor.

Information of the direction and the strength of a soundsource in an arbitrary direction can be found from thedetermination of the six possible cross-correlations. Aswas explained above the diffuse reverberant sound fielddoes not contribute to each of the possible six cross-correlations. On the other hand the reverberant fielddoes contribute in the four auto-spectra’s. Someexperimental results are shown in table 1. Theabbreviations Ix refers to the found intensity in the x-direction, Rux.uy to the cross-correlation between thesignals 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 CalculatedIx/Iy 1.05 1.04Ix/Iz 1.05 0.85Rux.uy/Rux.uz 1.11 0.82Ruy.uz/Rux.uz 0.98 0.96

ACKNOWLEDGEMENTS

The authors whish to thank the Dutch TechnologyFoundation STW for the financial support.

REFERENCES

1.H.E. de Bree, The Microflown (book) ISBN 9036515793;www.Microflown.com (�R&D�Books).2. W.F. Druyvesteyn, H.E. de Bree, Journal Audio Eng. Soc. 48, 49-56 (2000); R. Raangs, W.F. Druyvesteyn and H.E. de Bree 110th

AES Convention Amsterdam 2001, preprint 5292.

Experimental Study on the Accuracy of Sound PowerDetermination by Sound Intensity Using Artificial Complex

Sound Sources

H. Yanoa, H. Tachibanab and H. Suzukic

aDepartment of Computer Science, Chiba Institute of Technology, Tsudanuma 2-17-1,Narashino,275-0016,JapanbInstitute of Industrial Science, Univ. of Tokyo, Komaba 4-6-1, Meguro-ku, Tokyo, 153-8505, Japan

cDepartment 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 powerlevels of sound sources, a basic experiment was performed using an artificial complex sound source. As a result, it has beenfound that the scanning method specified in ISO/FDIS 9614-3 provides almost the same results as those by the discrete methodspecified in ISO 9614-1 and the field indicators can be evaluated during the scanning using a running signal processingtechnique.

INTRODUCTION

As the third standard of ISO 9614 series, theprecision method for the determination of sound powerlevels of sound sources by scanning sound intensitymethods is now being drafted inISO/TC43/SC1/WG25 (DIS stage as of June, 2001).For this work, some experimental studies wereperformed on the measurement accuracy. In this paper,an experimental study on the accuracy of sound powerdetermination by sound intensity under the conditionwhere strong extraneous sources exist is introducedand the validity of the field indicators are discussed.

EXPERIMENTS

As the sound source under test, the following threekinds of sources were used and driven by differentrandom noises, respectively.

Source-1 “CAN”: a steel boxto which an electric-shakeris attached.

Source-2 “BK 4205”: aloudspeaker-type referencesound source.

Source-3 “8SP”: a woodenbox in which 8 loudspeakersare equipped.

The measurement wasperformed in an hemi-anechoic

room (4m x 6.9m x 7.6m), in which the sound sourceswere placed as shown in Figure 1 at the center positionon the reflective floor. As shown in the figure, themeasurement surfaces for each sound sources were setand sound intensity measurement was performed in 1/3octave bands from 100 Hz to 5k Hz by the scanningmethod and the discrete point method (12.5 cm mesh).The measurement by manual scanning was performedin two scanning patterns on each partial surface bykeeping the scanning speed at about 25 cm/s. In asuccessive scanning on each partial surface, both dataof sound pressure and sound intensity were obtained atthe same time at a sampling rate of 0.5 s. As theinstrumentation, a pair of 1/2 in. condensermicrophone (B&K 4181) with 12 mm spacing anddigital real-time analyzer (B&K 2133) were used. Thefield indicators specified in ISO/DIS 9614-3 werecalculated automatically in a desk-top computer.

FIGURE 1. Configuration of artificial sound sources and measurement surfaces

1 m0.75 m0.75 m

source 3: 8SPsource 1:CAN

1 m

source 2:BK4205

FIGURE 2. Measured results of PWL

FIGURE 3. Measured results of field indicator: nn IppI FF �

FIGURE 4. Measured results of field non-uniformity indicator:Fs

RESULTS

Figure 2 shows the sound power levels of the threesound sources measured by the scanning method andthe discrete point method under the two operationconditions; “individual”- each source was drivenindividually and “simultaneous”- all sources weredriven simultaneously. In the results of“simultaneous”, it is seen that the results obtained bythe scanning method and the discrete point method arein good agreement over all frequency bands. Tocompare the results measured under the two sourceoperation conditions, the result for “individual” and“simultaneous” are fairly in good agreement in everysound source not being influenced by the mutualintervention.

Figures 3 and 4 show the values of nn IppI FF �

( 23 FF � in 9614-1) and FS (F4 in 9614-1),respectively, measured under the two source operationconditions. In the case of the source 8SP, although thevalue of

nn IppI FF � exceeded the criteria (3 dB) and

the value of FS exceeded the criteria (2 dB) in almostall frequency bands under “simultaneous” condition,the measurement result of sound power level is in goodagreement with the result measured under the“individual” condition.

CONCLUSION

As a result of the basic experimental studymentioned above, it has been found that the intensityscanning method provides an accurate sound powerlevel even under the adverse condition with strongextraneous noises.

REFERENCES

1. 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 soundpower determination by sound intensity method”, Proc.of 17th. I.C.A., 2001

50

60

70

80 BK4205

scan, simultaneous scan, individual disctrete, simultaneous

frequency (Hz) 125 250 500 1000 2000 4000

PWL

(dB

)

discrete, simultaneous

frequency (Hz)

50

60

70

808SP scan, simultaneous

scan, individual

125 250 500 1000 2000 4000

PWL

(dB)

50

60

70

80CAN

scan, simultaneous

discrete, simultaneous scan, individual

frequency (Hz)125 250 500 1000 2000 4000

PWL

(dB

)

0

1

2

3

4

5 BK4205 scan, simultaneous

discrete, simultaneous

frequency (Hz)125 250 500 1000 2000 4000

scan, simultaneous

F s scan, individual

05

10

15

20

258SP scan, simultaneous

discrete,simultaneous

frequency (Hz) 125 250 500 1000 2000 4000

F s

0

1

2

3

4

5 CAN F

scan, simultaneous discrete, simultaneous scan, individual

frequency (Hz) 125 250 500 1000 2000 4000

F s

-1 0 1 2 3 4 5

BK4205 scan, simultaneous

discrete, simultaneous

scan, individual

frequency (Hz)125 250 500 1000 2000 4000

F pIn

-Fp |

In|

02468

101214

8SP scan, simultaneous

discrete, simultaneous

scan, individual

frequency (Hz)125 250 500 1000 2000 4000

F pIn

-Fp |

In|

-1 0 1 2 3 4 5

CAN scan, simultaneous

discrete, simultaneous

scan, individual

frequency (Hz)125 250 500 1000 2000 4000

F pIn

-Fp |

In|

Approximate Calculation of Sound IntensityRadiated 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 radiationthrough an aperture from a sound source in a room. In this method the sound field outside of the room is assumed to consist ofdiffracted waves by the aperture of the direct sound from the source and of the diffused one in the room. Both Kirchhoff’s dif-fraction formula for the sound pressure and its derivative form for the particle velocity are applied to calculate the sound intensityowing to the direct sound from the source, while for the diffused sound in the room it is supposed for the sound to radiate fromthe aperture according to Lanbert’s cosine law. Numerical examples are presented with the measurements.

INTRODUCTION

In order to calculate exactly the sound intensity fieldby acoustic radiation through an aperture of a room inwhich a sound source is located, we have to solve acertain integral equation whose unknown function isthe velocity potential or sound pressure on the bound-ary surface. Such an integral equation method shouldbe applied to calculate the sound field diffracted by anyobject whose dimension is comparable to the wave-length concerned. [1] In a very large room as compared to the wavelength,the integral equation method to calculate the soundfield is not applicable because of limitations in com-puter capacity or difficulties of determining the bound-ary conditions. In this case an approximate approach based onKirchhoff’s diffraction formula [2] becomes morepractical. This formula requires both the velocity po-tential and its normal derivative on the aperture, whichare unknown. These unknowns are assumed to consistof the direct wave from the source and the diffusedwaves. The latter ones are assumed to make a surfacesource at the aperture which radiates the sound to theoutside of the room according to Lanbert’s cosine law.

THEORETICAL BASIS

We consider the situation where an n omnidirectionalpoint source (Ps) is located in a room with an apertureand a receiving point (P) is outside of the room asshown in Fig.1. Under Kirchhoff’s boundary condi-tions, the velocity potential at point P, �(P) , is given by

�(P)�1

4��(q)

�

�nq

exp(ikr)r

��

����

��

������(q)�nq

exp(ikr)r

��

����

��

����

�����

�����A

�� dsq (1)

Ps P

��

q

� PsP

�

���

FIGURE 1. Model room with an aperture (left) and calcula-tion�model for direct component(right)

where�(q) is the velocity potential at point q on A, r isthe distance from point P to point q and nq is the out-ward normal at q. Both�(q) and�� (q) /�nq in Eq.(1)are generally unknown. They must consist of the directwave from the source and reflected ones by the roomsurfaces. Those waves might be assumed incoherentwith each other. The sound intensity at P, I(P), could be divided intotwo parts, the direct component Idir(P) and the re-flected 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 atpoint q on the aperture are given by

�(q) � exp(iks) / s (3)and

�� (q)�nq

�iks �1

s2��

����

��

���exp(iks) cos(s,nq) , (4)

where s is the distance from Ps to q. The direct component of the velocity potential at P isexpressed by:

� dir (P ) �1

4�exp( iks)

s�

�n q

exp( ikr )r

��

����

��

����

����A��

�iks �1

s2��

����

��

����exp(iks) cos(s,nq)

exp(ikr)r

��

����

��

��������

dsq (5)

The particle velocity at P, u(P), is given by:

u(P)� �grad � dir (P)� �

� ��� dir (P)

� x,��� dir (P)

� y,��� dir (P)

� z��

����

��

��� (6)

where

�� dir (P)

�x�

14�

�� (q)�x

�

�nq

exp(ikr)r

��

����

��

����

����A��

�� 2�(q)�x�n q

exp(ikr)r

��

����

��

��������

dsq (7)

Let Idir,x the sound intensity for x-direction at P,

I dir,x � Re p(P) � ��� dir (P)

� x��

����

��

��

���� �

������

, (8)

where p(P) ��i��� dir (P) is the complex conjugate ofthe sound pressure at P.�Similarly we can calculate Idir,y and Idir, z and we ob-tain the sound intensity of the direct component bysumming up those vectors.

Calculation of the reflected component

We suppose that the sound in a room is diffused afterthe first reflection and a surface sound source is madeat the aperture from which the sound is radiated ac-cording to Lambert’s cosine law.

����

InqdS q

Px

Į

�r, x �

��������

A

FIGURE 3. Calculation model for reflected component

When the volume velocity of an omnidirectionalpoint source is unity, its acoustic power (W ) is given:

W � 4� � c k2 (9)

where � denotes the air density, c the speed of soundand k wave number. Then the intensity of normal di-rection at point q (In(q)) is given by means of geomet-rical acoustic theory as follows,

In (q) �W (1���)

A (10) where �� is the

average absorption coefficient and A the absorptionarea of the room. Let Iref,x(P) the sound intensity for x-direction atpoint P by above surface source, we obtain

Iref ,x (P)� I n( q)cos�

r 2A�� cos(r , x)ds , (11) also

Iref,y(P) and Iref,z(P) similarly.

NUMERICAL SAMPLESAND MEASUREMENTS

A model room which has a rectangular aperture(45cm x 45cm) was set in an anechoic chamber. Theroom which size is 600 x 600 x 900cm is made of ply-wood board 12mm thick. A third octave band noisewas used as the source signal and the location of thesource is the center of the room. Sound intensity was measured by means of 4-microphone system (ONO SOKKI Co.) and measure-ment points were set near the aperture. The numerical examples and the correspondingmeasurements at 2 kHz are shown in Figure 4 in soundintensity level. The approximate calculation method of the soundintensity field mentioned above might be applicable.

-20

Calculated

������������ �

Measured

1 13 25 37 49 61 73 85 97 109 121 132

-10

0

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)

A Code for Computation and Visualization of Sound PowerMeasurements based on the

ISO 9614-1 International Standard

E. Pianaa, G. Pianab and A. Armanic

aDipartimento di Ingegneria Meccanica, Università degli Studi di BresciaVia Branze 38, I-25123 Brescia, Italy, e-mail: piana@ing.unibs.it

bAnalisys, Via Monte Rosa 1, I-25069 Villa Carcina, Brescia, Italy, e-mail: analisys@tiscalinet.itcSPECTRA 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 sourcefrom intensity measurements and to allow a 3D representation of both the source outer surface and the map of interpolated re-sults on the measuring grid. The code complies with the prescriptions of ISO 9614-1 and is particularly helpful when one of thefield indicators specified in the ISO standard does not satisfy the necessary requirements so that measurements must be repeatedby successive refinements of the measurement grid and, therefore, the computation may be rather time consuming. The code im-ports intensity data from several types of analyzers and computes the sound power level for each surface and each frequencycomplying with the ISO 9614-1 flowchart. The calculation procedure includes the treatment of one-octave as well as one-third-octave band data. It also performs a three dimensional visualization of pressure and intensity fields on the measurement surfaceby colour contour plots. The possibility of drawing three dimensional objects inside the measurement surface and the real timeacquisition of FFT intensity spectra makes the program suited for research and development purposes.

INTRODUCTION

F.J. Fahy has proved that sound intensity measure-ment is one of the most effective techniques to deter-mine sound power level and to perform a mapping ofthe major sources contained within a measurement sur-face, as discussed in [1]. The international standardISO 9614-1 [2] provides practical guidelines to achievethis aim and requires the calculation of different indi-cators giving information about the quality of the de-termination and hence the grade of accuracy: F1 (tem-poral variability indicator), F2 (pressure-intensity indi-cator), F3 (negative partial power indicator) and F4(non-uniformity indicator). If one of the field indica-tors does not meet the requirements, the test procedureshould be modified. Moreover the Standard suggestsalso to check two criteria related to the adequacy of themeasurement equipment and to the adequacy of the ar-ray of measurement positions. All the tests must beperformed for every single set of measurements oneach measurement surface used. Finally a table and aflowchart specify the actions to undertake if the chosenmeasurement surface does not comply with the stan-dard requirements. The data to be collected are thepressure and sound intensity levels in octave or 1/3octave bands, whether they come from a CPB or froman FFT analysis.

3D EDITING AND REPRESENTATION

In order to start a new project, the first step is to de-fine the geometry of the measurement surface and thestructures contained inside and outside it. Basic toolsfor editing the geometry are already implemented inAcoustic Intensity Data Analysis (AIDA) package. Theprogram can represent two different kinds of objects:frames and measurement surfaces. Frames are the rec-tangular elements used to represent the source and theenvironment around it: equipment, bench, floor, walls,etc. Measurement surfaces are the rectangular elementsused to represent the set of segments surrounding thesound source. Each segment defines a measurementposition. The user can draw several surfaces and putthem together to build a more complex one. The pro-gram provides a 3D graphical representation of thesound power level over each surface (Figure 1). It alsoprovides a numerical listing and ranking of the soundpower levels coming from each one of them (Figure 2). The user can represent pressure and sound intensitylevels in three dimensions for both CPB and FFTanalysis (up to 30 bands). Another important feature isthe simultaneous visualization of more than one 3Dsurface, so that the user can check the source behaviourin different frequency bands at the same time. Differenteasily customizable colour scales are available.

CALIBRATION

The program performs several types of calibrationprocedures to set up the gain of the microphones and todetermine the pressure - residual intensity index. At thesame time it is also possible to perform a phase correc-tion for the entire apparatus. A phase correction is use-ful for FFT based measurements, when the operatorwants to make a software correction of the equipmentresponse. All the data coming from the calibration pro-cedure are stored in a database where the user can re-trieve the correct probe type and verify the calibrationhistory of the transducers. The last calibration per-formed is automatically taken into account to check theISO 9614 – 1 field indicators and criteria.

REAL-TIME PERFORMANCES

The program can show pressure and intensity levelsfrom the OROS 25 power pack. In this case octave and1/3 octave bands are synthesized directly from the FFTspectra. The real-time performance of the system en-ables the program to show and import directly pressureand intensity data from the analyzer, thus allowingreal-time computation of the sound power level radi-ated by the sound source through the probe remotecontrol. Real-time analysis and visualization of datacan be performed for the pressure levels, for the inten-sity levels and for the pressure – intensity index. Ofcourse the user can set up directly the analyzer fromthe AIDA software choosing between linear and expo-nential averaging, number of averages, coupling, fre-quency span, etc. During the measurements, the program computes thesound power level for each surface and each frequencycomplying with the Standard flowchart, and provides

FIGURE 1. 3D representation of the measurement surface.

FIGURE 2. Sample of numerical and graphical outputshowing measurement surface ranking and 1/3 octave bandanalysis.

instant suggestions of the necessary actions the usershould consider in order to meet the Standard require-ments. The real-time acquisition procedure enables theuser to choose the sequence he wants to use to performthe measurements. Alternatively, a direct access to anypoint of the measurement surface is also possible.Moreover, a table gives a comprehensive insight of thefield indicators computed for each measurement sur-face. The possibility of drawing three dimensional ob-jects inside the measurement surface and the linkingwith virtually any FFT analyzer makes the programsuited also for research and development purposes.

SOUND POWER DETERMINATION

If the ISO 9614-1 requirements are fulfilled, then thefinal result is within the selected grade of accuracy.The Sound Power levels are calculated and displayedfor each surface as well as for the overall measurementsurface with or without “A” weighting. It is also possi-ble to exclude some frequencies from the calculationprocedure selecting them directly from a list. Figure 2shows the numerical and graphical outputs for a 1/3octave band analysis associated with the overall meas-urement surface represented in Figure 1. The histogramdepicts, for each frequency band, the sum of both lin-ear and “A” weighted sound power levels coming fromthe 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 SoundPower Levels of Noise Sources Using Sound Intensity -Part 1: Measurement at Discrete Points.

Sound Level Meter Software Implementation with LabVIEW

J.M. Lópeza,b, M. Ruiza,b and M. Recuerob,c

aDepartamento de Sistemas Electrónicos y de Control. Universidad Politécnica de Madrid Cta. De Valencia Km 7, Madrid 28031 Spain

bINSIA. Instituto Universitario de Investigación del Automóvil. Universidad Politécnica de Madrid cDepartamento 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 Average Time (ms)

Exp avg sound level 42 Equivalent Continuous Sound Level 17

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.

FIGURE 2 Welch algorithm.

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.

4. Oppenheim, A.V., Discrete-Time Signal processing; Prentice Hall 1989

A method for the Estimation of Loudness Basedon Fuzzy Theory

M. Ferri, J. Ramis, J. Alba and J.A. Martínez.

Departamento de Física Aplicada, Escuela Politécnica Superior de Gandia, Universidad Politécnica de ValenciaCarretera 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 netsdepending on its sound pressure level. However, given a source with certain spectral distribution of energy, there isn’t acommon criteria about which net should be applied. To correct this problem, we propose a method, based on fuzzy sets, thatprovides a single sound level as a combination of both three classical pondered sound levels. A comparation is shown, given agroup of signals with relatively uniform spectra, of the sound levels obtained by the proposed method with the normalisedloudness levels calculated by Stevens method4.

INTRODUCTION

Integrative sound level meters, do not usuallyperform frequency analysis of the sound signals;thus, is absolutely impossible determinate theloudness level of some acoustic event. Generally thesound pressure level A-weighted (dBA) is acceptedas 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, FuzzyLogic based model1,2 -containing information aboutnon-linearities in frequency and level response ofhuman hearing- has been used.

The Model And Its Application Field

As the normalised nets A ,B, y C, are designed to beused respectively on small, medium & high levels,we apply a weighted average of these normalisedlevels, increasing the value of the coefficient of levelA if the sound level is considered small, andrespectively, if it is medium or high. The algorithmused to determinate the value of the coefficients isbased on Fuzzy inferences & Fuzzy sets

The Model

Entry variables: LpA, LpB, LpCVariables considered on Fuzzy inferences: LpA,LpB, and LpCFuzzy sets: Three sets related to each three variable;small (Ai1), medium (Ai2), & high (Ai3)Fuzzy inferences and consequences: In the three-dimensional space, three fuzzy sets to eachdimension have been defined, so the whole space is

divided in 3³=27 fuzzy hyper-sets conformed by thefuzzy logic condition “and”

if LpA is A11 and LpB is A21 and LpC is A31(Fuzzy inference)

... then

LpCaLpBaLpAaLpfuz kkkk ������ 321

(consequence)

Prediction (Fuzzy level): As the fuzzy inferences arenot classical logic inferences {0,1}, they have somemembership function to the truthfulness in theinterval (0,1], so each consequence is as true as itsinference. Thus the predicted level is

�� �

�

k

kk

trLpfuztr

Lpfuz

being trk the truthfulness of the k-esim fuzzyinference, calculated as follows1

� �)()(),(321

LpCLpBLpAmintrnml AAAk ����

LpA

�A11(LpA) Ai1: “small”

m1

0 LpA

�A13(LpA) Ai3: “big”

m3

0LpA

�A12(LpA) Ai2: “medium”

m2

0

Figure 1: Fuzzy sets relative to variable LpA(Qualitative)

1 )( oA x� is the membership function of the value xo of x

to the fuzzy set A (0< )( oA x� �1)

Application field:

Fuzzy models are capable to predict situationssimilar to that which were used to configure them, sothe model is to be used for continuous and withrelative uniform spectra sounds. We have chosenthese, caused for the impossibility to predict exactlywith three inputs (LpA LpB, LpC) the loudness of asound, that, in the easier case, needs a octaveanalysis of the acoustic signal (Stevens). Anywaythese are the kind of signals we usually found inenvironmental acoustic and industrial noise.320 signals have been selected with SPL valuesbetween 60 and 130 dB and slopes between –10dB/oct and 10 dB/oct; comparing the Lpfuzzy withde Stevens loudness level (ISO 532-1975).

FITTING THE MODEL

The fuzzy model depends on parameters such as thecoefficients of the variables in the polynomialfunctions consequence of each inference, and theshape of the fuzzy sets (that can be defined with theabscise of the slope discontinuities and the maximumvalue of the membership function). These parameterswill be optimised to minimise the mean quadraticerror, considering as correct values the Stevensloudness level

� �

N

LSLpfuzN

iii�

�

�

�1

2

�

The polynomial functions have been prefixed,limiting the model complexity, Here there is anexample of polynomial function

if LpA is “big” and LpB is “big” and LpC is “med”

then

LpBccLpCcbLpCcaLpfuzk 31

31

31

������

1��� cccbca

Notice that if some level is small, medium or big isassociated respectively to LpA, LpB or LpC.There are only three set high considered: small setshigh 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, cband cc). Heuristic algorithms have been used: thegradient method, and different neural network ones.

RESULTS

In the next figure it is shown how fuzzy level curvesmodify its shape “looking for” loudness level curves.In the three-dimensional plot it is shown thedifference between fuzzy levels and LpA, comparedwith the difference between loudness levels and LpAfor different spectra types and sound pressure levels.

1 40-40

-30

-20

-10

0

10

20

SPL=130 dB

SPL=110 dB

SPL=130 dB

SPL=110 dB

LPA

LPB

LPC

Spectrum type

Weighted levels (dBA,dBB,dBC) - sound pressure levels (dB)

4060

80100

120

LS

Lfuzzy

Spectrum typeSPL (dB)

LS-LA (dB)Lfuzzy-LA (dB)

5 dB

Figure 2. Comparing Lpfuzzy versus Loudness level

The mean quadratic error in our experimental bench(320 signals) is �(Lpfuz)<3’6 dB2; �(LpA)>17 dB2;�(LpB)>24 dB2; �(LpC)>92 dB2

CONCLUSIONS

Applying this fuzzy model, the ordinary sound levelmeters can be used to evaluate (in firstapproximation) the loudness of many different kindsof acoustic events (continuous and with relativeuniform spectra), getting much better performancethat using any weighted sound pressure level.

REFERENCES1) G. Cammarata, A. Fichera, S.Graziani and L.

Marletta, Fuzzy logic for urban traffic noiseprediction, Journal of Acoustical Society ofAmerica 98, 1995, pp. 2607-2612

2) Y. Kato and S. Yamaguchi, A systemathicalstudy for psychological impression caused byfluctuating random noise based on fuzzy settheory, J. Acoust. Soc. Am. 91, 1992, pp 2748-55

3) UNE-EN 60651, Sonómetros, AsociaciónEspañola de Normalización y Certificación(AENOR), 1.996

4) ISO 532-1975: Method for calculating loudnesslevel, International Standard Organization, 1975