Patrizia

SCATTERING IN ROOM ACOUSTICS

Scattering in Room Acoustics and the Related Activitiesin ISO and AES

Jens Holger Rindel

Ørsted•DTU, Acoustic Technology, Technical University of Denmark, Building 352, Ørsted Plads,DK-2800 Kgs. Lyngby, Denmark

Scattering from the surfaces in a room is known to be of great importance for the acoustic quality of the room. In order to dealwith this in a scientific way there is a need for definitions of concepts to characterise the scattering and for related measuringmethods. Through the co-ordinated efforts of two working groups under ISO and AES a basis for scattering has been established.ISO has derived a method for measuring the random-incidence scattering coefficient in a reverberation chamber. This measurecan be used to characterise the degree of scattering due to the roughness of a surface. AES has derived a method for measuringthe diffusion coefficient of a surface as a function of the angle of incidence. This measure can be used to characterise theuniformity of the scattering from a surface. The practical aspects of the measuring methods and some of the inherent problemsare discussed.

INTRODUCTION

Together with the sound absorption characteristics,the acoustic scattering from surfaces is very importantin all aspects of room acoustics, e.g. in concert halls,sound studios, industrial halls and reverberationchambers. Already W. C. Sabine knew this by intuitionmore than one hundred years ago when he wasinvolved in the design for Boston Symphony Hall. Scattering of sound can be achieved by singleelements like columns, statues etc. or by surfaces witha sufficiently rough structure like QRD diffusers or bydiffraction effects along the edges of panels. Until nowthe work has focused on the surface scattering, whereassingle diffusers have not been dealt with. Two different measures are introduced withsomewhat different applications: One measure givesthe quantity of scattered reflections, which may besufficient for use in room acoustic computer models.The other measure is intended mainly for evaluation ofthe quality of sound diffusers and in situations where itis important to achieve scattered first-order reflections,e.g. in sound studios. The main features of the two newmeasures are described in the following.

THE SCATTERING COEFFICIENT

A working group under ISO has prepared a draftstandard, which was accepted with some comments bythe ISO member bodies in the fall 2000 [1]. Themethod describes how to measure the random-incidence scattering coefficient in a reverberant roomand it is based on [2]. The scattering coefficient is defined as one minus theratio between the specularly reflected acoustic energy

and the total reflected acoustic energy. When measuredin an approximate diffuse sound field it is called therandom-incidence scattering coefficient with thesymbol s and values between 0 and 1. The scatteringcoefficient describes the degree of scattering due to theroughness or irregularity of a surface – the scatteringdue to diffraction from the edges is not included.

The Measurement Method

The measurements are performed in a reverberationchamber like a double set of absorption measurements.The test sample must be circular and it is placed on ahighly reflecting base plate with the same diameter.The reverberation time is measured with and withoutthe test sample on the base plate, and the random-incidence absorption coefficient αs is determined. The base plate with the test sample is mounted on aturntable, and a new pair of measurements is madewith continuously rotating test sample/base plate.Alternatively the turntable is turned in 72 steps of 5°for each new measurement. With different positions ofthe test sample the scattered energy is not correlated.The impulse responses are measured and bysynchronised averaging only the correlated (specular)part of the impulse responses is obtained and thereverberation time is derived. Thus, the so-calledspecular absorption coefficient αspec is determined. Inthis measurement the scattered energy is treated as anadditional energy loss that is included in the specularabsorption coefficient. Thus αspec ≥ αs. Finally, the random-incidence scattering coefficient sis calculated from:

)1()1(1 sspecs αα −−−= (1)

Some Limitations

The preferred area of the test sample depends on thevolume of the room, but the minimum is 10 m2 for afull size measurement. However, for practical reasonsit is often advantageous to use a physical scale model.The absorption of the test sample should be low inorder to obtain acceptable measurement accuracy. Theair attenuation may cause a reduced accuracy at highfrequencies in the case of scale measurements. The structural dept of the test sample should notexceed d / 16, where d is the diameter of the sample.Still, the variation of height along the perimeter of thesample can cause additional edge scattering that maylead to a measured scattering coefficient higher thanunity. In general the scattering coefficient is frequencydependent with low values at low frequencies andhigher values at high frequencies.

THE DIFFUSION COEFFICIENT

A working group under AES has prepared aninformation document [3] that describes how tomeasure the surface scattering uniformity in a free fieldenvironment. The considerations that have lead to theproposed diffusion coefficient are described in [4]. The diffusion coefficient is derived from theautocorrelation function of the polar responsemeasured on a semicircle or a hemisphere and it hasthe symbol d. It can take values between 0 and 1, and d= 1 means that the polar response is completelyuniform.

The Measurement Method

The measurements are performed in an anechoicroom with the receivers on an arc with a preferredradius of 5 m and an angular resolution of max. 5°. Thesource positions should be in a distance of 10 m. Oneproblem is that the polar response depends on the sizeof the test specimen and the distance to the source andreceivers. Thus, it is important to follow the describedprocedure as closely as possible. The directional diffusion coefficient dθ is derived forone particular source position and θ is the angle ofincidence. If pi denotes the sound pressure measured atreceiver i and there is a total of n receivers, then dθ is

∑

∑∑

−

−

=

ii

ii

ii

pn

pp

d4

422

)1(

)(

θ (2)

The random-incidence diffusion coefficient d is themean result for a number of different source positionswith an angular separation of max. 10°.

Some Limitations

The measurement method does not distinguishbetween surface scattering and edge scattering - bothare included. Thus it is important that the test sample islarge enough so that surface effects rather than edgeeffects are prominent in the scattering.

COMPARISONS BETWEEN THE TWOMEASURES

There is no direct relationship between the scatteringcoefficient s and the diffusion coefficient d. A highvalue of d implies that s is also high, whereas a highvalue of s can be combined with any value of d. Forexample a saw tooth profile can redirect the reflection,so s is high, but the scattering is far from uniform, so dis low. Both measures can be calculated from the reflectiondirectivity. However, for calculation of the scatteringcoefficient it is also necessary to know the reflectiondirectivity from a flat reference surface with the samearea as the test sample, see [5].

CONCLUSION

Through the co-ordinated efforts of two workinggroups under ISO and AES a basis for scattering hasbeen established. It has been agreed that there is a needfor two different measures in order to characterise theacoustic scattering from surfaces. The measurementmethods may be used for test samples either in fullscale or in an appropriate scale ratio. It is believed thatthe new measures of surface scattering will improvethe basis for good design in room acoustics.

REFERENCES1. ISO/CD 17497:2000. Acoustics – Measurement of the random-

incidence scattering coefficient of surfaces.2. M. Vorländer & E. Mommertz. Definition and measurement of

random-incidence scattering coefficients. Applied Acoustics 60(2000) 187-199.

3. AES-4id-2001. AES information document for room acousticsand sound reinforcement systems – Characterization andmeasurement of surface scattering uniformity. J. Audio Eng.Soc. 49 (2001) 148-165.

4. T.J. Hargreaves, T.J. Cox, Y.W. Lam & P. D’Antonio. Surfacediffusion coefficients for room acoustics: Free-field measures.J. Acoust. Soc. Am. 108 (2000) 1710-1720.

5. E. Mommertz. Determination of scattering properties from thereflection directivity of architectural surfaces. AppliedAcoustics 60 (2000) 201-203.

Contrasting Surface Diffusion and Scattering Coefficients

Trevor J. Cox a,b, Peter D’Antonio b

a School of Acoustics and Electronic Engineering, University of Salford, Salford, UK.b RPG Diffusor Systems Inc., 651-C Commerce Drive, Upper Marlboro, MD 20774, USA.

In recent years, several coefficients to measure the degree of scattering or diffusion from a surface have been developed. Thesehave been developed to meet the needs of diffuser manufacturers, room designers and geometric room acoustics modellers. Thecoefficients give a frequency dependent measure analogous to the absorption coefficient. Two of these coefficients are, or areabout to be, enshrined in international standards. Very little has been published, however, directly comparing the coefficientsand so their relative merits are difficult to judge. Consequently, this paper will contrast the different coefficients, demonstratingtheir strengths and weaknesses. A simple prediction model for the free field Mommertz and Vorländer scattering coefficient forSchroeder-style diffusers will be given; this leads to a diffuser design methodology. In general, it appears that this scatteringcoefficient gives significantly larger values than a diffusion coefficient based on polar response uniformity. Prediction andmeasurement results on other surfaces, such as curved surfaces and rectangular battens, will enable guidance to be given as tothe applicability of these coefficients. None of the coefficients appears to be perfect and it will be shown that the correct choiceis application dependent.

INTRODUCTION

The first ever international standard on thecharacterisation of surface scattering/diffusion waspublished this year [1]. AES-4id-2001 characterisesdiffusion as the uniformity of polar distributions.Characterisation is done in terms of the autocorrelationfunction [2]. In this respect, it is meant to be aquantifier of diffusion quality for diffuser designersand installers. Current ISO WG25 is working onstandardising a random-incidence scatteringcoefficient method [3]. The scattering coefficient usesthe variance of the sound field when the test surface ismoved. It is intended to be used in geometric roomacoustic models to improve the prediction accuracy.Also discussed in this paper is the approach ofMommertz [4], who suggested a method for obtainingscattering coefficients from polar responses. Theabove methods have tended to be developed inisolation. The intention of this paper is a comparisonof these different coefficients to highlight their someof their advantages and disadvantages.

SCHROEDER DIFFUSERS

Schroeder diffusers can be approximately modelledas variable impedance surfaces using a simple Fourierapproach. This then forms a convenient test bed for thedifferent scattering and diffusion coefficients.Consider the ISO technique using surface movements.For this test bed, a free field formulation is used andsurface translation rather than rotation is implemented.Therefore the philosophy of ref [3] is preserved, but

not the exact method. Under these conditions, thescattering coefficient δ is:

2

1

211 �−−≈

Njkdne

Nδ (1)

Where N is the number of wells per period, k thewavenumber and dn the depth of the nth well. This is asum of the reflection coefficients, no reference to wellposition is given because the receiver is fixed in thespecular reflection. Furthermore, the scatteringcoefficient according to this formulation isindependent of incident angle. The best scatterers have

0 1000 2000 3000 40000

0.2

0.4

0.6

0.8

1

f (Hz)

Diff

usio

n/sc

atte

ring

ISOAES PRDAES planeMommertz

Figure 1. Schroeder diffuser test bed formodified Primitive Root Diffuser (PRD)

a vanishing sum of the reflection coefficients, which isachieved if the reflection coefficients are evenlyspaced around the unit circle. This is achieved formodified versions of the primitive root diffuser (PRD)[5,6]. Figure 1 shows some typical results using aPRD. Comparing the “ISO” scattering and AESdiffusion coefficients, it can be seen that a numericallysmaller value is found for the AES coefficient. Thereis good agreement, however, as to the frequency atwhich the PRD starts dispersing sound (200Hz) andthe flat plate frequency (3kHz). At many frequencies,the PRD is scattering considerable sound energy fromthe specular zone, but not evenly. This is why the AEScoefficient is significantly less than the ISOcoefficient, and why the AES coefficient is a better testof diffuser quality.

SCATTERING COEFFICIENTS FROMPOLAR DISTRIBUTIONS

The ISO and AES coefficients satisfy two differentneeds. The former to give a number required for moreaccurate computer modelling, the later to allow properevaluation of diffuser quality. It would be useful to beable to translate between the two coefficients, evenknowing that there will be pathological cases that willcause problems. Mommertz forwarded a method [4],using the correlation between the pressures scatteredfrom a test sample and a plane surface. This allows thescattering coefficient to be obtained from polarresponses. In the Schroeder diffuser test bed, Fig. 1,the Mommertz coefficient matches the “ISO”coefficient well at high frequencies, but less so at lowfrequencies. This has also been tested for a variety ofsurfaces in measurement and with BEM predictions. InFigure 2 the scattering coefficient for a N=7, 1D

quadratic residue diffuser is shown. It shows that

Mommertz’s method of getting scattering coefficientsfrom polar responses yields values similar to random-incidence measurements following the proper ISOprocedure. Results on other surface types have alsoshown that this may be a route to enable the random-incidence scattering coefficient to be predicted usingfree-field predictions models such as BEMs.

CONCLUSIONS

The ISO and AES coefficients are contrastedbelow. They have different philosophies and thereforedifferent uses. Efforts to enable translating between thecoefficients are on-going.

ISO scatteringcoefficient

AES diffusioncoefficient

Uses For computermodellingLarge surfaces, smallirregularities

Assessing diffuserqualitySmall surfaces, largeirregularities

Advantage Quick measurement Can be predictedLimitations Redirection seen as

dispersionProblemsinterpreting singleplane scatterers witha hemisphericalcoefficientValues > 1

Ignores phasedispersionUndefined meaningof intermediatevaluesAchieving Far fieldmeasurementdifficult

ACKNOWLEDGEMENTS

Thanks to Janina Fels and Marcio Gomes fromInstitut für Technische Akustik, RWTH Aachen forcarrying out the measurements.

REFERENCES

1. AES-4id-2001. J.Audio.Eng.Soc. 49(3) 149-165. (2001).

2. T J Hargreaves, T J Cox, Y W Lam and P D'Antonio.J.Acoust.Soc.Am. 108 (4), 1710-1720 (2000).

3. M Vorländer and E Mommertz. Applied Acoustics. 60(2)187-200 (2000).

4. E Mommertz. Applied Acoustics. 60(2) 201-204 (2000).

5. Feldman, E. J.Acoust.Soc.Am. 98(1) 623-634 (1995).

6. T J Cox and P D’Antonio. Applied Acoustics. 60(2).167-186. June 2000.

0

0.2

0.4

0.6

0.8

1

1.2

100

160

250

400

630

1000

1600

2500

4000

f (Hz)

Scat

terin

g co

effic

ient

MommertzISO

Figure 2 Flutterfree™ scattering coefficients

Comparison between measurements of the scattering and diffusion coefficients

A. Farina1, L. Tronchin2

1IED, University of Parma, 43100 Parma, Italy 2DIENCA-CIARM, University of Bologna, 40136 Bologna, Italy

In this paper the results of a wide comparative experiment are presented. 14 different small-sized scattering panels were tested making use of the new method based on the Wave Field Synthesis [1]. From the experimental raw results, with proper processing, both the scattering coefficient [2] and the diffusion coefficient [3] can be derived. The octave-band spectra of these two coefficients are compared for each panel. As expected, it resulted that it is generally difficult to find a stable relationship between the two coefficients. The WFS method proved of consequence its value, being capable of yielding both coefficients from the same measurement results. On two panels, the measured values were cross-checked with other measurement methods. It resulted that the diffusion coefficient is almost the same as the value obtained by the AES-standard method, whilst the scattering coefficient is much less correlated with the value measured, on a large continuous surface, with the Mommertz/Vorlander method. This difference was explained considering that the WFS method is applied to a single panel, where border effects are predominant (particularly at low frequency), and these effects are instead minimized making use of a large continuous surface.

DIFFUSION AND SCATTERING

COEFFICIENTS

The acoustical scattering properties of uneven surfaces are judged to be very important for proper numerical simulation of the sound propagation in enclosed spaces. Furthermore, many diffusing panels are on the market nowadays, and it is not easy for the acoustical designer to specify what kind of panels is optimal for a given case. There are no standardized measurement methods. Furthermore, two different approaches are followed by AES and ISO Committees, which are confusing in some way the meaning of diffusion and scattering properties.

THEORY:1 MOMMERTZ-VORLÄNDER METHODOLOGY

The first method of measuring scattering was developed by Mommertz and Vorländer and it’s called free-field scattering measurements method. The scattering coefficient is defined as:

diffspec

diff

tot

diff

EEE

EE

s+

==

therefore, after FFT post-processing of the measured IR (obtained rotating the panel at 5° steps in 72 angles), extracting the spectrum of reflected IR, for each frequency the absorption coefficient is:

2

2

2

2

(

(1;

(

(1

ref

specspec

ref

tottot

IRFFT

IRFFT

IRFFT

IRFFT−=−= αα

from which the scattering coefficient is obtained:

tot

totspec

tot

spec

E

Es

α

αα

−

−==

1

The direct and reflected sound can be simply obtained by time-windowing the IRs, as in the following figure

FIGURE 1. Time-windowing of direct and reflected sound

THEORY: 2 WAVE FIELD SYNTHESIS METHODOLOGY

Since the scattered wavefronts have more curvature than the specularly reflected one, the WFS approach can easily separate the two wavefronts. The data acquisition setup consists of a soundfield microphone moving along a straight line instead of a hemi-circumference. For each microphone position, the total diffused energy coming from the panel is calculated.

( )∫ ∫−= −=

⋅⋅⋅π⋅

⋅α−⋅

⋅π⋅

⋅=

b

by

a

ax2

2

loc3

1

cdiff dydx

r2

s1

r4

zWI

whilst the specular reflected intensity coming from the panels and measured only in those positions of the microphone within the specular zone, is:

( ) ( )( )[ ]2

r2

rc

locspec

xzz24

s11WI

+−⋅⋅π⋅

−⋅α−⋅=

The estimation of ISO scattering coefficient is therefore obtained simply minimizing the difference between numerical calculations and experimental measurements.

FIGURE 2. Setup of the acoustical apparatus for the measurements with the Vorlander-mommertz free field method

ANALYSIS AND RESULTS FROM MEASUREMENTS

Different panels were analyzed, and the scattering coefficient was measured as reported. Specialized software was developed, and all the 255 IRs measured with WFS approach were fitted in only one waveform. Direct and reflected sound were separated by software, and analyzed.

FIGURE 3. Polar plot: Whole IRs, (left) direct (center),

reflected sound (right)

From experimental data, polar plot were obtained. The spectra of both direct and reflected sound from numerical formulation were compared with measurements; absorption and scattering coefficient were obtained minimizing differences between numerical and experimental data, by meaning of a worksheet solver. The procedure was repeated for octave frequencies, from 125 to 16000 Hz.

Scattering Coefficients

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

125 250 500 1000 2000 4000 8000 16000

Frequencies

Sca

tter

ing

Co

effi

cien

t (s

)

absorb 0 absorb 45 absorb 90

FIGURE 4. Scattering coefficient for different angles on incident sound

ACKNOWLEDGMENTS

The authors whish to thank Michele Zanolin for his help during the acoustical measurements and subsequent data analysis.

REFERENCES

1. A. Farina - "A new method for measuring the scattering coefficient and the diffusion coefficient of panels", Acustica/Acta Acustica, vol. 86, n. 6 pp. 928-942, ISSN 1436-7947, December 2000

2. E. Mommertz, M. Vorländer, "Measurement of scattering coefficients of surfaces in the reverberation chamber and in the free field" - Proc. 15th ICA, Trondheim, 1995, pp. 577-590.

3. P. D’ Antonio, T. Cox, “Two Decades of Sound Diffusor Design and Development, Part 2: Prediction, Measurement and Characterization” Journal of AES vol. 46, n° 12, pp. 1075-1091 (December 1998).

4. A. Farina, M. Zanolin, E.Crema – “Measurement of sound scattering properties of d iffusing panels through the Wave Field Synthesis approach”, Pre-prints of the 108th AES Convention, Paris, 19-22 February 2000

Application of Scattering Surface in Absorber Design

Y.W. Lam and T. Wu

School of Acoustics and Electronic Engineering, University of Salford, Brindley Building, Meadow Road Site,Salford M7 9NU, UK. E-mail: y.w.lam@salford.ac.uk

It has been known for some time that a Schroeder diffuser can provide unusually high absorption over a wide frequency band.This is due to the multiple resonances and the surface impedance discontinuity created by the different well depths in the diffuser.It has also been shown theoretically and verified by experiments that the absorption can be enhanced considerably by placing aresistive layer on the surface of the diffuser. This paper describes an exploitation of this absorption mechanism under an ECfunded project, RANNTAC, to develop a honeycomb based absorber design. In the simplest form, a resistive facing sheet can beplaced on top of a honeycomb cavity structure to provide a one degree-of-freedom resonant absorber. The honeycomb cavitiescan be modified by means of inserts to create well depth sequences that are similar to those of Schroeder diffusers to extend thefrequency range of high absorption. The particular problems faced by this application were the very small size of the honeycombcells (around 0.375” and a depth of 2”), and the 2 dimensional hexagonal arrangement of the cells. A computer model wasdeveloped to optimise the depth sequence to provide near –zero reactance over a wide frequency band. Samples of the designwere tested at normal and oblique incidences to verify the prediction. It was found that the design can extend significantly thefrequency range of the absorber. The small reactance is well suited for room acoustic applications, and it is also possible toextend the low frequency performance by means of hollow inserts.

INTRODUCTION

Absorptive intake liners are one of the mostimportant means for controlling aircraft noise. Thedesign of conventional 1 degree-of-freedom (DOF)liners has been established for many years. Howevercurrent and future demands on noise control requireliner performances that are beyond the capability ofconventional liners, and innovative techniques arebeing sought to significantly improve linerperformance. The enhanced absorption provided by ascattering surface was first observed in studies onSchroeder diffusers, and particularly in the quadraticresidue diffusers (QRDs). The reasons for this highabsorption produced by variable depth scatteringsurfaces has been thoroughly investigated andprediction model was developed and verified againstvarious configurations of QRD and similar scatteringsurfaces typical in audio applications [1]. This paperreports on a study to adapt this variable depth designinto an intake liner.

ABSORPTION MECHANISMS

A schematic representation of the liner concept fora period N=7 design is shown in Figure 1. The increasein sound absorption comes from two mechanisms:• Each cavity resonates at a different frequency

from other cavities because of the different depths,giving a wide frequency band width of highabsorption and a smooth frequency response.

• The differences in the wave responses betweenadjacent cavities increases energy flow between

cavities. Therefore a resistive layer placed at theentrance to the cavities significantly increases theabsorption (resistance).

ADAPTATION TO LINER DESIGN

The particular problems faced by this applicationare the very small size of the honeycomb cells, whichmeans that boundary losses are significant, and thehexagonal arrangement of the cells. One of the biggestpractical difficulty of the variable depth design is thecreation of the different depth in the honeycombcavities of a liner. In a typical liner there are thousandsof cavities per m2. Inserts were considered the mostpractical solution to create the different depth. In theoriginal design, it was considered necessary tominimise potential gaps between the inserts and thehoneycomb walls in order to eliminate unwantedacoustic effects. Hence hexagonal inserts with a cross-sectional shape identical to the cavity cells was used.

z

�ePPe

w

L

b

0 1 2 ..... ...... N-1T

xln

Figure 1 Variable depth scattering surface

Unfortunately this also means that the fit between theinserts and the cavities is very tight and it is extremelydifficult to put in the inserts without damaging thehoneycombs. The problem is particularly severe for thelarge samples as there are thousands of inserts to putin. To reduce this construction difficulty, roundcylindrical inserts were used to construct the 1mx1msample. This created small gaps between the insertsand the cavity walls. Nevertheless investigations on theeffect of perforation in cells suggested that these gapsare likely to improve rather than reduce absorption.

The measured normal incidence impedance of a1mx1m liner sample, which used the round inserts, andthat of a small impedance tube sample, which used thehexagonal inserts, are compared together withprediction in Figure 2. Both samples are of typical linerdimensions. The depth sequence was identical in bothsamples, optimised to give minimum reactance overthe frequency range of the design.

It can be seen that the two measured impedancespectra agree very well with each other and there areno obvious problems created by the use of the roundinserts. It should be noted that the impedance tubemeasurement was limited to 3000Hz which is not highenough to cover the main operational frequency rangeof the design, which extends beyond 5000Hz.Nevertheless the 1mx1m sample measurement, whichproduced results up to 5000z, does agree fairly wellwith the prediction by the computer model, whichassumed no gaps between the inserts and the cavitywalls, up to about 3.5kHz. Importantly the imaginarypart of the impedance (reactance) remains close to zeroas designed at the higher frequencies.

Since the variable depth sequence creates non-uniform impedance distribution on the surface that istheoretically dependent on the angle of incidence,oblique incidence measurements were also performed

on the large 1mx1m sample in an anechoic chamber.The frequency trend of the impedance at up to 30˚incidence, which was the limit of the measurement set-up, remains similar to that at normal incidence andagrees well with prediction.

ASSESSMENT OF PERFORMANCE

A liner design with reactance optimised to stayclose to zero over the frequency band of interest wassubmitted to our partners in the RANNTAC project forevaluation against engine noise reduction criteria. Theevaluation result suggested that the final variable depthsequence design is good at low frequency and has asmooth/flat frequency spectrum. However it still hastwo significant drawbacks: a) the resistance value isgenerally too low, and b) the reactance values wereoptimised to near-zero values which is good for oneflight condition but not for others. This is mainly dueto the fact that the variable depth liner design used alinear resistive facing sheet and as a result could notmatch the different values of the optimum impedancerequired for different flight conditions.

CONCLUSION

The theoretical optimisation of the variable depthsequence liner concept was successfully verified byexperimental tests on liner samples. The result of theoptimisation process has produced a liner design thatshows significant improvement over conventionalliners at low frequencies and in the consistence(smoothness) of the mid to high frequencyperformance. However the current design onlyprovides significant improvements in on flightcondition and is worse than an optimised 1 DOF linerin the mid to high frequency range in other flightconditions. The poorer performance in these conditionsis mainly due to the lower resistance value and thesmall reactance of the liner. The latter is a consequenceof the optimisation process that was aimed to producenear-zero reactance values, and that the liner wasdesigned to be linear and therefore impedance does notchange with operation conditions.

ACKNOWLEDGMENTS

This work is funded by the EC Brite–Euram projectRANNTAC. Support from all partners in the projectare kindly acknowledged.

REFERENCES

1. T. Wu, T. J. Cox, Y. W. Lam, J.Acoust.Soc.Am. 108(2),pp.643-650, 2000.

Normal incident

-2

-1

0

1

2

0 1000 2000 3000 4000 5000

Frequency,Hz

Impe

danc

e

Prediction:R X Anechoic-room:R X Impedance-tube:R X

Figure 2 Impedance of a variable depth liner design.

Investigations on the ISO measurement method forscattering coefficient in the reverberation room

Márcio H. A. Gomesa, b, Michael Vorländera, Samir N. Y. Gergesb

a Institute of Technical Acoustics, Technical University of Aachen, Germany - Templergraben 5552056 Aachen – Tel.: (0241) 809 7985/86 -E -mail: mav@akustik.rwth-aachen.de, mvo@akustik.rwth-aachen.de

b Laboratory of Acoustics and Vibration, Federal University of Santa Catarina, Brazil

The correlation method for measuring the scattering coefficient in a reverberation chamber is most likely to be internationallystandardised in ISO. In this method the impulse response of a reverberation chamber is measured, first without and next with thesample placed with different orientations. As one sums these impulse responses, one gets as result an impulse response related tothe specular reflection only, with a decay faster than compared with a single measured impulse response decay. From theseresults the scattering coefficient is determined. An ISO working group discusses and develops the referred method. Somequestions still are to be answered, for instance, on the edge effects and the corresponding demands on scale factor and samplesize. In this paper a preliminary investigation about the edge effects over the results and a comparison with results obtained withanother method is reported.

INTRODUCTION

The correlation method for measuring the scatteringcoefficient [1] can be applied either in free or diffusesound field. In the last years more attention was givento the application of the method in the diffuse field andan ISO working group prepares the release of astandard for measuring the scattering coefficient in thereverberation chamber. In the diffuse field onlyscattering coefficients due to random incidence aremeasured and the procedure is similar to themeasurement of the random incidence absorptioncoefficient. Here an additional absorption coefficientrelated to the specular component of the reflection hasto be determined. This is done by evaluating thereverberation time calculated from averaged impulseresponses measured with the sample oriented in asufficient number of different orientations (completingone rotation). As the scattered components of thereflection are cancelled out, the resulting impulseresponse contains only information about the specularcomponent of the reflection. The absorptioncoefficient and the “specular” absorption coefficientare then calculated through the following relations:

αs = 55,3

−

12 T1

T1

cSV (1)

αspsec = 55,3

−

34 T1

T1

cSV (2)

Here T2 and T1 are the reverberation times measured inthe chamber with and without the sample, respectively.T4 and T3 correspond to the reverberation timesobtained from the averaged impulse responses, again

with and without the sample. The scattering coefficientis calculated through the relation:

s

sspec

1s

α−α−α

= (3)

Some aspects of this method are under investigation,such as the influence of edge effects and air absorptionon the results (since it is usual to perform themeasurements in scale models). Next a firstinvestigation related to the influence of edge effectswill be briefly addressed. The results of ameasurement performed with this method and thoseobtained from a method which may be standardised bythe Audio Engineering Society (AES), a free fieldmeasurement, were also compared through theapplication of the Paris formula [2].

Edge effects

When measuring the scattering coefficient in thereverberation chamber, the set-up consists of soundsource, receiver and a round base plate placed over aturn table. For practical reasons the samples areconstructed sometimes over a square plate. It isknown, however, that the square plate itself scattersthe incident sound. Three measurements wereperformed in order to have an idea of how thisadditional scattering affects the results and if thiseffect can be corrected. First the scattering coefficientsof small hemispheres randomly spread over a squarearea were measured (directly over the round baseplate). The same was performed, but now with thehemispheres spread over a square plate (60 x 60 x 1,6cm, placed over the round base plate). Finally, the

scattering coefficients of the square plate alone weremeasured. The results are shown in Fig. 1.

FIGURE 1. Scattering coefficients from hemispheresspread over a square area, hemispheres spread over asquare plate and only the square plate. “a” is theaverage distance between the hemispheres (5,4 cm)and “λ”, the wave length of sound in air (20°C).

The scattering from the square plate self cansignificantly influence the results and its scatteringcoefficients are excessively high above a/λ = 0,63. Anattempt to correct the results was done, considering theset-up to contain also the square plate over the roundbase plate. This is equivalent to substitute T1 and T3 inthe measurement of the hemispheres spread over thesquare plate by T2 and T4 obtained in the measurementof the square plate alone. The corrected results areshown in Fig. 2, compared with the results for thehemispheres spread over the square area (without thesquare plate). The correction works well, in this case, up toa/λ = 0,80, but above this frequency the differencebetween corrected results and the measurementperformed with the hemispheres spread over thesquare area are larger than expected.

Comparison with other methods

A sample from the RPG Flutter Free diffuser wasmeasured by the RPG with the AES method (in thefree field) and the results were compared to thoseobtained using the correlation technique in the diffusefield. At RPG the scattering coefficient was measuredfor three angles of incidence (0, 30 and 60 degrees). Inorder to correct these values for random incidence, theParis formula [2] was used (see Figure 3). The discrepancies at lower frequencies are probablyrelated to the influence of the edges. At middle andhigher frequencies the errors may be also related to thefact that the measurements in the free field wereperformed only for three angles of incidence. To havea better approximation for the random incidence

scattering coefficient, it is necessary to have moreterms weighted by the Paris formula. The generalaspect of the comparison, however, is consideredsatisfactory.

FIGURE 2. Correction attempt for the influence of thesquare plate over the measurements of the scatteringcoefficient.

FIGURE 3. Comparison between the scattering coefficientsof a RPG Flutter Free diffuser, measured with the ISOcorrelation technique in the diffuse field and the AEStechnique in the free field. “a” is the length of one profile ofthe Flutter Free.

The topics here discussed are objects of currentresearch. Useful results are expected from numericalcalculations of the scattering coefficient of simplesurfaces and further comparisons with measurementsperformed in other laboratories.

ACKNOWLEDGEMENTS

The authors would like to kindly acknowledgeJanina Fels, Dr. Peter D’Antonio and Dr. Trevor Coxfor the data sent from RPG.

REFERENCES

1. Vorländer, M.., Mommertz, E. Applied Acoustics 60, 187-199 (2000).

2. Kuttruff, H., Room Acoustics, Spon Press 2000, pg. 49.

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terin

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correction attempt

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ITA - correlation technique

RPG - AES technique - Paris

A Standard Method for Assessing Diffusor Quality

P. D’Antonio a, T. J. Cox b

a RPG Diffusor Systems Inc., 651-C Commerce Drive, Upper Marlboro, MD 20774, USA.b School of Acoustics and Electronic Engineering, University of Salford, Salford, UK.

One of the challenges in room acoustics is to develop diffusing surfaces that complement contemporary architecture in theway that statuary, columns and relief ornamentation complemented classic architecture. A key element to enable holisticdiffusor design is a diffusion coefficient to enable the evaluation of surface scattering. The diffusion coefficients developedhave facilitated design via numerical optimisation. This has enabled the palette of surfaces with defined acousticperformance to be expanded. The diffusion coefficient is a measure of the uniformity of the free field scattered polardistribution. The polar responses can be readily measured using MLS techniques or predicted using boundary elementmethods. An autocorrelation approach tests the spatial similarity of the polar distributions. This autocorrelation approach issummarised in a simple equation that enables the diffusion coefficient to be readily calculated. The method has recentlybeen published in an Audio Engineering Society standard information document AES-4id-2001, which summarises thefindings of the working group sc-04-02. In this paper, the rationale behind the diffusion coefficient standard will be given.The use of the coefficient in diffusor design and the role it plays in performance specifications will be reviewed.

INTRODUCTION

It is understood that surface diffusion can play animportant part in determining the sound quality ofmany rooms. The surface diffusion may come fromincidental features of the room such as statuary,columns and relief ornamentation, or from especiallydesigned diffusors. Whether diffusion comes by designor not, it is important to evaluate the diffusivity of thereflections, to better understand the role of diffusion inroom acoustics. Working under the auspices of theAudio Engineering Society, the authors have beenchair and vice chair of working group sc-04-02, whichhas been developing standard methods for evaluatingthe diffusivity of reflections. This has culminated in astandard information document [1] being publishedearlier this year detailing a method. This paper intendsto outline the rationale behind AES-4id-2001, and toshown how it can be used in diffusor design andperformance specifications.

DIFFUSION COEFFICIENT

The test surfaces are evaluated in terms of theirability to disperse sound energy uniformly over a polarresponse, whether this is measured in a single plane orover a whole hemisphere. It is assumed that completediffusion occurs when every measurement position onthe polar response receives the same scattered energy.Examples of measured polar balloons are shown inFigure 1, which shows that the skyline is better at

diffusing the reflected sound. The standardinformation document details measurement methodsrequirements for getting these polar distributions. Themost straightforward method is to use a maximumlength sequence system and time gating to extract thescattered energy. Measurement in a single plane isrelatively straightforward, full 3D measurements aremore logistically difficult. Alternatively, predictionmodels can be used - it is well established thatBoundary Element Methods (BEMs) give remarkableprediction accuracy.

Within this evaluation philosophy, lie some of themethod’s weaknesses and strengths. The concept ofmeasuring the smoothness of energy polardistributions is straightforward. Yet, the issue ofwavefront phase is not considered. A single cylinderspatially disperses energy very efficiently, but doesthis without dispersing the phase. (There is anecdotalsubjective evidence that cylinders are not gooddiffusers for this reason). The other problem is that the

Figure 1. Measured 3D Polar balloons. Left2.5kHz, skyline, normal incidence. Right 1.25kHz,random battens, 60º incident sound.

polar distribution must be measured in the far field,and this extends a very long way for large surfaces.

Once the polar distribution is obtained, it is reducedto a single figure of merit. This is done using thecircular autocorrelation function, which measures thespatial similarity in the energy polar distribution. Anexhaustive test of many different surface types [2]showed that this was the best coefficient available.What could potentially be a complex calculationprocedure, fortunately reduces to a simple equation.The diffusion coefficient, d, is given by:

����

��

���

��

n

ii

n

ii EnEd

1

22

1

/ (1)

Where Eis are the n energies in the polar distribution(an even area sampling is assumed here). d isautomatically bounded between 1/n (specularreflection) and 1 (uniform diffusion). A simple scalingcan be done to make the bounding between 0 and 1.

APPLICATIONS

Acoustic designers are used to specifying therequired absorption coefficient of surfaces in roomdesigns. Yet, diffusion is often applied in a morehaphazard fashion, with no quantification of whethertoo little, or too much surface diffusion is being using.By creating a standard evaluation technique, it ishoped that a more scientific basis for the application ofsurface diffusion can be formed. For example, thetechnique outline here is already being used as thebasis of performance specifications for auditoria andother performance spaces.

This diffusion coefficient has also influenced thedesign of diffusers. Very few surface types havesimple design equations; exceptions are Schroederdiffusers and some simple concave arcs. Problems thenarise if these diffuser designs are visually unappealingto the architect. Consequently, an optimisation has tobe used to enable a design that is both visuallyappealing and meets acoustic requirements. Inoptimisation, an iterative search is undertaken to lookfor the best diffusor. For example, it is possible to taska computer to find a well depth sequence that givebetter performance than the quadratic residuesequence. During the trial and error process, thecomputer predicts polar responses from the surface,and then reduces the polar responses to a single figureof merit – the cost parameter. The figure of merit usedis the autocorrelation diffusion coefficient. This notonly enables better depth sequences to be obtained, italso makes it possible to make designs or arbitraryshape. The only restriction is that the shape must be

mathematically definable with a few shape parameters,and the polar response must be predictable. Figure 2(top) shows an example of a curved optimised surfacedesigned to reduce the focussing effect of a concavewall. Figure 2 (bottom) shows a stage canopy alsodesigned using optimisation and the autocorrelationdiffusion coefficient.

CONCLUSIONS

AES-4id-2001 sets out a diffusion coefficientwhich ranks the performance of diffusers correctly andis intended to evaluate diffuser quality. It is notperfect, but even the absorption coefficient has flaws.By ensuring that diffusers are tested against thisstandard, a better understanding of the role of diffusionin rooms can be developed, as well as ensuring thatdiffusers perform up to quantitative specifications.

REFERENCES

1. AES-4id-2001. AES Information document for room acousticsand sound reinforcement systems – Characterization andmeasurement of surface scattering uniformity J.Audio.Eng.Soc.49(3) 149-165. (2001).

2. T J Hargreaves, T J Cox, Y W Lam and P D'Antonio.J.Acoust.Soc.Am. 108 (4), 1710-1720 (2000).

Figure 2. Diffusers designed using optimisation

Determination of the Scattering Coefficient of StatisticalRough Surfaces

J.J. Embrechts

Department of Sound and Image Techniques, University of Liège, Sart-Tilman B28, B-4000 Liège 1, Belgium, e-mail : jjembrechts@ulg.ac.be

Abstract : The scattering coefficient of a diffusing surface accounts for the part of sound power which is “non-specularly”reflected. This coefficient is an essential parameter for the description of walls and surfaces in room acoustics problems. In thispaper, the scattering coefficient of random rough surfaces is calculated from the complete scattered sound pressure distribution.This distribution is evaluated using a Kirchhoff Approximation method. The results obtained for several rough surfaces arecompared with theoretical expressions of the scattering coefficient. These expressions show the influence of the angle ofincidence, the sound frequency and the geometrical parameters of the surface profile on the scattering coefficient. It is shownthat these theoretical expressions give reliable results as long as the Kirchhoff Approximation conditions hold.

INTRODUCTION

In room acoustics, the scattering coefficient of adiffusing surface is defined as the ratio of non-specularly reflected power to the total power reflectedby the surface. The value of this coefficient (and itsdependence on frequency) is essential for the users ofany modern room acoustics software, because most ofthem can now account for the effects of surfacediffusion. However, rather few data have already beencollected on surface scattering properties and only asmall part of these data have been published.Therefore, a great amount of work must still be donein order to better understand the mechanisms ofsurface scattering and to measure diffusion parameters.

The random-incidence scattering coefficient ofdiffusing surfaces can be measured in a reverberantroom, using a method developed by Vorlaender andMommertz [1]. This method is presently underinvestigation by an ISO working group. However, thiscommunication addresses the problem of finding thevalue of the scattering coefficient, not bymeasurements, but rather by theoretical developments.

METHOD SUMMARY

We consider finite size rough surfaces described bytheir elevation ),( yxz ξ= relative to a reference

plane. Assuming an incident plane wave, thedistribution of the scattered pressure can be calculatedwith the Kirchhoff Approximation method [2]. The

development of this method leads to the followingexpression for the complex pressure 1p in the

scattering direction sk :

)1().()( .1 ∫=

Sr

rvjs dydxCveKkp γ

In this equation, K is a constant complex number(depending on the strength of the incident wave andthe distance of the receiver), S is the area of the roughsurface projected onto the reference plane,

si kkv −= where ik is the incident vector (the

magnitude of both vectors k is λπ /2 ), r is theposition vector of the surface element at ),,( ξyx and

)1,,( ''yx ξξγ −−= is a vector perpendicular to the rough

surface at this surface element. rC is the local

reflection factor.

With this expression, it is possible to calculate (withinthe assumptions of the Kirchhoff Approximationwhich are not discussed here) the complete distributionof the complex scattered pressure. The scatteringcoefficient of the diffusing surface defined by

),( yxz ξ= can then be obtained with a formula

proposed by Mommertz [3]. This scattering coefficientis called in the following ..AKδ , since it is computed

using the more general formulation of the KirchhoffApproximation method.

RESULTS FOR GAUSSIAN ROUGHSURFACES

To solve (1), we of course need to define the profile ofa given rough surface. In this study, the analysis hasbeen focused on a typical class of random roughsurfaces, namely the gaussian surfaces.

For these particular surfaces, it is possible to derivefairly good approximations of the scattering coefficient

..AKδ . As will be seen in the following, these

approximations can be obtained without calculatingthe complete distribution of the scattered pressure.They are therefore more easily computed and they alsobetter illustrate the influences of sound frequency andgeometrical parameters on the scattering coefficient.The first approximation is :

)2(1

12

.. ∫=−=S

jvssFC dxdye

Srr zξδ

where C.F. means characteristic function, which is the

mathematical name for the average of ξzjve if ξ is a

random variable. The second approximation is :

)3(),(1

1 22cos4 222

∫=−= −

S

ksRMS dxdyyx

Sse i ξδ θ

In this expression, iθ is the angle of incidence of the

plane wave and s is the r.m.s. height of the randomrough surface.

Both approximations have been compared with thetheoretically exact value of the scattering coefficient

..AKδ derived from equation (1). The characteristicfunction model (2) always leads to a very goodcorrelation with ..AKδ , as long as the conditions of

validity of the Kirchhoff Approximation are satisfied.

The rms height model (3) states that the scatteringcoefficient of gaussian rough surfaces only depends onthe r.m.s. height of the surface (relative to thewavelength) and on the angle of incidence. Thecorrelation with ..AKδ is not as good as in the previous

model, but the deviations are not really significantunless the scattering coefficient reaches high values(δ>0.8). This is illustrated for a particular class ofgaussian surfaces in figure 1.

On the other hand, the rms height model is mucheasier to calculate, and it gives a very clear

interpretation of the influence of all parametersaffecting the scattering coefficient.

FIGURE 1. Scattering coefficients ..AKδ and RMSδcomputed from expression (1) and by the rms height model(3) respectively, for 200 rigid ( 1=rC ) gaussian rough

surfaces characterized by a correlation length of 5λ. Theangle of incidence is 20°.

Figure 1 gives only some examples of the manygaussian surfaces which have been considered in thisstudy. After analysing all these results, it has beenfound that the approximations (2) and (3) are in factvalid in many situations, including many angles ofincidence ( °≤ 60iθ ), surface dimensions, correlation

lengths ( λ≥T ) and r.m.s. heights. We even found thatthe approximations also hold for non-rigid surfaces.

It turns out that, at least for gaussian rough surfaces,the key parameters (concerning sound diffusion) seemto be the ratio of r.m.s. height to the wavelength andthe angle of incidence. Further similar studies on othersurfaces with deterministic profiles will certainly bringnew information to this theory.

REFERENCES

1. Vorlaender, M., and Mommertz, E., Applied Acoustics,60, 187-199 (2000).

2. Thorsos, E.I., J. Acoust. Soc. Am., 83(1), 78-92 (1988).

3. Mommertz, E., Applied Acoustics, 60, 201-203 (2000).

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Studies of Scattering from Faceted Room Surfaces

R. R. Torresa, M. Vorländerb, U. P. Svenssonc, and M. Kleinerd

aProgram in Architectural Acoustics, Rensselaer Polytechnic Institute, 110 8th St., Troy, NY 12180, USAbInstitute of Technical Acoustics, Technical University Aachen, D-52056 Aachen, Germany

cAcoustics Group, Dept. of Telecommunications, Norwegian Inst. of Technology, NO-7491 Trondheim, NorwaydChalmers Room Acoustics Group, Chalmers Univ. of Tech., SE-412 96 Gothenburg, Sweden

To improve room acoustics modeling and design, studies of edge diffraction and scattering are performed. Examples of roomcomponents where modeling of edge diffraction is important include reflector panel arrays, grazing incidence over seats,orchestra pits, and other surfaces whose dimensions are comparable to or less than the wavelength of interest. Here, scale-model measurements of edge diffraction are performed to illustrate that complex diffraction interferences cannot be accuratelymodeled with simple, energy-based Lambert models of diffusion. Hence, the natural spectral “coloration” in the early impulseresponse would be likewise inaccurately represented in numerical simulations and auralization.

INTRODUCTION

The scattering behavior of faceted and rough surfacesis of particular importance in room acoustics, as itdetermines the spatial coverage and spectral colorationof early reflections to listeners. In this paper, thebehavior of faceted surfaces is investigated.

MEASUREMENTS

Previous studies modeled edge diffraction from a stagehouse, considered as an assemblage of wedges [1-3].For increased understanding, we study here thescattering from a small number of facets. A commonexample in architectural acoustics is the use ofoverhead reflector arrays to optimize the delay times,strengths, and frequency spectra of early reflections inthe room impulse response (RIR).

We first measure the scattering from various arrayscomposed of 1 to 35 facets (i.e., panels). Figure 1depicts a square-panel array (or, alternatively, longrectangular panels). We examine the effect of the arraydensity on the reflected component and thedependence of the “specular zone” on frequency. Thesource is an 11.5-cm dodecahedron loudspeakeremitting maximum-length sequences (MLS), whichhave advantages over spark sources. The source andmicrophone are 50 cm below the array. Measurements on reflector arrays for roomacoustics have been documented sporadically duringthe past few decades. One of the earliest systematicstudies was performed by Leonard et al. [4], whomeasured the sound pressure at the receiver andcompared it with that obtained for spherical soundspreading. They discussed frequency ranges where thereflection coefficient was negative and also confirmed

intuitive predictions of the diffraction behavior at thelow- and high-frequency limits. The work here studiesspectral changes in the scattering and examines itsspatial spread relative to the specular reflection zone.

A1

G3

G1

A3

G5A5

A4

A2 G2

G4

C.L .

85cm

FIGURE 1. Example square-panel array. The solidcircle is the source; the open circle, the microphone.

RESULTS

One interesting part of this study is the spreading ofthe energy around the geometrical, specular reflectionzone (the middle region in Fig. 2). The wavelengthsfor the octave bands (500-4 kHz) in Fig. 2 are similarto or greater than the panel widths (12 cm). Figure 2plots the normalized (for spherical spreading), squaredamplitude of the scattering from a rectangular panel(top) and from a row of seven square panels. In the topfigure, the scattering for the 1 and 2 kHz bands peakswhen the incidence angle to the panel is closer tonormal (greater projected panel area) and the specularreflection point is within 1/4 to 1/2 wavelength of the

edge. Here the scattering depends more on incidenceangle and projected reflecting area, and the “specularzone” seems to be less meaningful.

Second, in the lower figure, it is interesting that thereflected amplitude of the 4-kHz band no longerdominates over the 2-kHz band, although one mightexpect that higher frequencies are reflected much moreeffectively, especially near the specular zone (as it isfor the rectangular panel). At 4 kHz, however, thewavelength is on the order of the panel width andseparation, which allows inter-panel interferences thatmay cause the total reflected amplitude to decrease.Perceptually, this may change the perceived“coloration” because the spectral content of thisreflection changes. This could be investigated in futuresubjective studies. One may note [5] that interferenceeffects such as these (on the order of 5 dB here) are notmodeled explicitly by energy-based scattering

approximations (such as Lambert’s Law), whichwould possibly result in lower accuracy simulations.

NUMERICAL MODELING

Parameter studies are in progress, where edgediffractions from facets are computed using a modelby Svensson et al. [2], based on the Biot-Tolstoy andMedwin (BTM) approach [3]. The edges are dividedinto sources with analytically derived strengths. Onesample of the impulse response h is then (Eq. 35, [2]):

� �z

mlh RS

i ����,,,,

4�����

�

�(1)

where z� is the length of the source at zi along theedge, � describes the wedge angle, m and l are sourceand receiver distances, and � is an analytical edge-source directivity-function. Although the model hasbeen validated for several cases, one must examinehow it is best applied to more complex facetedsurfaces. For example, when several separate facets liein the same plane, certain propagation paths betweenedges would theoretically undergo second or third-order diffraction. Computing all of these paths may notbe necessary to achieve a desirable degree of accuracy.

ACKNOWLEDGEMENTSThe authors thank Nikitas Gianni, Stefan Goertz, andthe workshop of the Institute of Technical Acousticsfor their assistance.

REFERENCES

1. R. R. Torres, U. P. Svensson, M. Kleiner, “Computationof edge diffraction for more accurate room acousticsauralization” J. Acoust. Soc. Am. 109, 600-610 (2001).

2. U. P. Svensson et al., “An analytic secondary sourcemodel of edge diffraction impulse responses,”J. Acoust.Soc. Am. 106, 2331-2344 (1999).

3. H. Medwin, “Shadowing by finite noise barriers,” J.Acoust. Soc. Am. 69, 1060-1064 (1981).

4. R.W. Leonard et al., “Diffraction of sound by an arrayof rectangular reflective panels, ” J. Acoust. Soc. Am.36(12), 2328-2333 (1964).

5. R. R. Torres, M. Kleiner, B.-I. Dalenbäck, “Audibilityof ‘diffusion’ in room acoustics auralization: an initialinvestigation,” Acustica (Special Issue on RoomAcoustics) 86(6), 919-927 (2000).

FIGURE 2. Scattering from rectangular panel andmultiple square panels. The specular zone is themiddle third. Inter-panel interferences seem to lowerthe relative reflected amplitude of the 4 kHz band.

Boundary Element Analysis as a Design Tool for Panel Arrays

T. E. Gulsrud

Kirkegaard Associates, Consultants in Architectural Acoustics, 954 Pearl Street, Boulder, Colorado, USA The boundary element method (BEM) is an established computational technique for engineering problems involving acoustic scattering and radiation. We discuss the application of the BEM to a frequently encountered design problem in room acoustics: scattering from arrays of panels suspended over a concert platform. The BEM is based on wave theory and does not depend on high frequency or far-field approximations, enabling it to predict low frequency scattering for a variety of source and receiver positions that ray tracing and other techniques cannot. We present a series of graphics to demonstrate the capability of the BEM to provide useful information to the designer of a panel array and offer some insight into the trends encountered in the data. The computations do not require a high performance workstation, placing this information within the grasp of the owner of a modest desktop or laptop computer.

INTRODUCTION

A panel array is frequently provided for music performance or rehearsal spaces when other architecture in the room is not able to provide early sound reflections to audience members or between musicians. While conventional ray tracing or other approximate methods [1] are available to the designer, these methods are limited by high frequency or far-field approximations. The boundary element method (BEM) permits low frequency reflections (wavelengths comparable to the dimensions of the panels) to be studied without restrictions on source or receiver locations.

Calculation Technique

The calculations are based on numerical solutions of the Helmholtz Integral Equation with the assumption of thin rigid panels, the development and validation of which can be found elsewhere [2,3]. The resulting surface integrals are approximated by subdividing the reflecting surfaces into N elements and setting the value of surface pressures constant across each element. The surface elements must be smaller than ¼ the wavelength of interest in order to accurately model sound waves across the surface. The resulting set of simultaneous equations are solved in two steps. The equations are first solved for ∆pj, the pressure difference across the panel at element j, due to the presence of a point source located below the panel array. This solution is then used to calculate the total scattered pressure ps from the panel array according to

)1(,)ˆˆ()1(41)( 2

1jjijij

ij

ikRijN

jjiS AikR

Repp nRx ⋅−∆= ∑

=π

where k is the wavenumber of the source, xi is the receiver location, Rij is the vector from element j to xi, nj is a unit vector normal to element j, and Aj the area of element j. The resulting total scattered pressure is referenced to the pressure from an image source, pi(xi), above the panel array. The result is ∆L, the level change in reflection due to the panel array compared to a reflection from a solid plane [1].

)2(.)()(

10 10

=∆

ii

is

xpxp

LogL

Any variation from zero in ∆L indicates the presence of diffraction from the finite size of the panels or interference from scattering from multiple panels.

EXAMPLES

To demonstrate the technique we present a series of calculation results for a sample 5x4 array of 6ft square panels illustrated in Figure 1. The on-center spacing of the panels is 8ft in both directions and the source is located 17ft below the center of the array.

FIGURE 1. Plan view of the sample panel array. Array is located 20ft above the floor. The source is at (0,0,3)ft.

FIGURE 2. Distribution of ∆L across a grid located 3.1ft above the floor. The resolution is 1 point per 0.5ft.

Effect of Source Frequency

Plots on the left side of Figure 2 show the change in coverage pattern of the array with source frequency. Even at 125Hz (λ ~ 9ft) considerable energy is scattered from the array of 6ft panels. At 250Hz (λ ~ 4.5ft) scattering from individual panels and the resulting interference pattern is apparent. Results from a larger parameter study indicate that interference between waves scattered from individual panels cause dips in the steady-state frequency response of the panel array [4].

Effect of Panel Size

Plots on the right side of Figure 2 show the effect of panel size, maintaining the panel spacing in Figure 1 and the source frequency at 250Hz. Decreasing the panel size by only 0.5ft increases the depth of interference minima in the coverage pattern and introduces a new dip at the source position. Larger panels provide a smoother coverage pattern but still cause a dip around the source position.

CONCLUSIONS

Parameters within the designer’s control can significantly change the coverage pattern of the array, and the BEM allows the designer to evaluate a variety of panel sizes, shapes, and orientations. The method is limited primarily by the large number of elements (and computer memory) required for very large arrays or high frequencies. Further work is required to evaluate whether the diffraction patterns shown in Figure 2 are audible when the direct sound and reflections from all room surfaces are taken into account.

REFERENCES

1. Rindel, J.H., Applied Acoustics 34, 7-17, (1991).

2. Cox, T.J., and Lam, Y.W., Applied Acoustics 40, 123-140, (1993).

3. Terai, T., J. Sound and Vibration 69, 71-100, (1980).

4. Gulsrud, T.E., “Acoustical Properties of Hanging Panel Arrays in Performance Spaces”, M.Sc. thesis, Dept. of Physics, Univ. of Colorado, (1999).

Shape and Diffusion in the Design of Music Spaces

R. J. Orlowski

Arup Acoustics, St Giles Hall, Pound Hill, Cambridge CB3 0AE, UK

The design of diffusers based on boundary element methods has enabled much greater freedom in the geometrical design ofmusic spaces whilst providing precise diffusion requirements. An example is presented of the design of a small recital hallwhich has used this technique.

INTRODUCTION

We are well aware that heavy ornamentation innineteenth century concert halls gives rise to a highdegree of sound diffusion and that this is evidentlybeneficial for providing good acoustic quality. Thereduction of ornamentation in twentieth centuryarchitecture has led to other geometries and devicesbeing used to provide diffusion. However, design ofthese various diffusing elements has had only tenuousconnections with the theory of sound scattering.Design guidelines typically recommend that diffusingelements should protrude between 300 mm and600 mm or alternatively that they should be convexlycurved in plan and section.

This rather general approach was dramatically changedwhen Schroeder proposed a method for providingsound diffusion using mathematical number theorysequences [1]. The idea was quickly taken up byMarshall and Hyde who installed Schroeder-typediffusers, based on a quadratic residue sequence, in thenew Michael Fowler Centre concert hall inWellington, New Zealand.

The acoustic result appeared to be successful andmany installations followed in concert halls and also inrecording studios.

However, Schroeder diffusers have not becomeuniversally popular because their aesthetic, whichgenerally consists of a series of parallel slots ofdifferent depths, does not particularly appeal toarchitects and designers of auditoria.

A NEW APPROACH

Research by Cox and D’Antonio [2] has producedaccurate prediction methods for scattering fromdiffusers based on boundary element methods. Thishas enabled diffuser design to break away from the

constraints of Schroeder diffusers to provide a muchgreater range of stepped and curved diffuser shapes.

Using numerical optimisation techniques andprediction methods, an architect or designer can selecta desired shape or motif and this can be tailored toprovide a diffuser with the required diffusion.

This technique has been applied by Arup Acoustics tothe design of a new recital hall, the Edwina PalmerHall, for the Benslow Music Trust near London. Forarchitectural reasons, a concave form was developedfor the hall which obviously gave rise to concernsabout focusing.

Curve-shape optimisation was used to minimisefocussing by the concave wall using a geometricalmotif based on an amplitude modulated wave; theconcave wall and optimised curve are shown inFigure 1. A comparison of the sound scattering by theoptimised surfaces is shown in Figure 2.

FIGURE 1. Concave wall and optimised amplitudemodulated diffuser

The optimised ’wavy’ wall satisfied the architect’sdesign intentions as well as meeting the acoustician’srequirement to neutralise focusing and providediffusion. Figure 3 shows a plan of the rehearsal hall

and Figure 4 shows a view of the diffusing wall in thecompleted hall.

FIGURE 2. Comparison of sound scattering at 2 kHz byoptimised and concave surfaces with source on axis andreceiver at focal point

The wall was formed from 25 mm thick mediumdensity fibreboard (MDF) with a paint finish althoughglass reinforced gypsum (GRG) was considered as analternative material.

It can be seen from Figure 4 that the wall steps back athigh level. Also, thin vertical strips have been addedto the curved surface. Both these changes to theoptimised curve were implemented by the architect toenhance the visual aspects of the wall in the context ofthe overall space. Neither is considered to have asignificant effect on the overall scattering performanceof the wall.

FIGURE 3. Plan of Rehearsal Hall

Subjective listening tests to piano and clarinet music inthe hall indicated a very uniform sound field with noevidence of focusing. Furthermore both instruments

produced an expansive sound with a very good balancebetween clarity and reverberance. Both musiciansfound the hall easy to play in. Objectivemeasurements relating to diffusion will be presented inthe spoken version of this paper.

The success of the Benslow project has led ArupAcoustics to consider the curve optimisation techniquefor providing diffusion for other projects. Currently,designs are being developed where the modulations ofthe walls occur in both vertical and horizontal planes.

FIGURE 4. View of diffusing wall in completed hall

ACKNOWLEDGMENTS

Thanks are due to the Benslow Music Trust forallowing reference to their building design, and toRPG Diffuser Systems Inc who collaborated on theproject. The Institute of Acoustics is acknowledgedfor permitting the reproduction of excerpts fromAcoustics Bulletin.

REFERENCES

1. M. R. Schroeder, J. Acoust. Soc. Am. 57, 149-150(1975)

2. P. D’Antonio and T. Cox, J. Audio Eng, Soc. 46 (11),955-976 (1998)

A Method of Evaluating Surface Diffusivity of RoomsUsing a Rectangular Scale Model

C. H. Haana and T. J. Coxb

aDepartment of Architectural Engineering, Chungbuk National University, Chongju, 361-764, KoreabSchool of Acoustics and Electronic Engineering, University of Salford M5 4WT, UK

The present study aims to express a new insitu method to measure the global diffusivity of surfaces in rooms. Thepresent study verifies the results of the previous investigations using computer modeling and field measurements. Thenew criterion (∆x) of surface diffusivity is the difference in the room acoustical parameters between two microphonesof which the first microphone (M1) is facing the stage while the other microphone (M2) is facing the rear wall of theroom. The experiments were undertaken using a rectangular scale model with different surface diffusivity conditions.SPL, EDT, C80, were measured at six measurement positions and the ∆x of acoustic parameters calculated andanalyzed. The results show that the difference in the acoustic parameters in a room with high surface diffusion, issmaller than the difference in a room with little surface diffusion. This was true at every frequency and every positionmeasured.

METHOD OF EVALUATING SURFACEDIFFUSIVITY

A new method of evaluating global surface diffusivityof rooms was introduced using computer modeling andfield measurements [1,2]. This was done by a pair ofmicrophones, one microphone facing the front and onethe rear wall of room respectively. A speciallydesigned microphone holder panel is used to shadowthe microphones appropriately.

If the sound level at one position can be measured,eliminating the direct sound and early specularreflections, the amount of diffused sound could bepredicted by comparing it with the value of soundlevels in a non-diffused condition. This could beapplied to any other acoustic parameters, so that wemay know how much diffused sound affects the valueof acoustic parameters. In this way, it can be shownthat the difference of acoustic parameters at oneposition is depend on the surface condition of room.The value of acoustic parameters at the firstmicrophone (M1), facing the front of room, isnormally expected to be larger than the value of thesecond microphone (M2), facing the rear wall of room.This occurs because M2 cannot capture the directsound. The difference in the values between the twomicrophones changes depending on the surfacediffusivity. The more diffusing elements on walls andceiling the more sound that reaches M2 compared to aroom with no diffusing elements.

It is expected that the difference of values betweentwo microphones will get smaller depend on the

surface diffusivity of room. Therefore, an evaluationcriterion, ∆x, of surface diffusivity was suggested asfollows: ∆x = x (M1 ) – x (M2 ). Where, x(M1) is thevalue of acoustic parameter obtained from microphonefacing the stage of hall and x(M2) is the value ofacoustic parameter obtained from microphone facingthe rear wall of hall.

SCALE MODEL EXPERIMENT

In order to investigate the effect of surface diffusivityon the acoustics of rooms, two different surfaceconditions were employed to make different surfacediffusivity in the scale model. The surface condition ofscale models was changed using 2 and 3-dimensionaldiffusers including quadratic residue and primitive rootdiffusers. The diffusers were installed on the bothsidewalls. The floor and rear wall was covered withpyrosorb foam to simulate absorptive areas typicallyfound in real halls. The rectangular scale model haddimensions of 3.5 m length, 2m width and averageheight of 2.0m. A sound source was located at thecenter of the stage and six measurement positions wereselected in the scale mode. 1/10 scaled omnidirectional speaker was used to radiate MLS signalsfive times at each measurement.

In the scale model experiment, a microphoneholder panel was employed to measure the ∆x. Thereare two holes for insertion of microphones. Thediameter of the microphone holder panel is 20 cmwhich is correspond with the wave length of 1700 Hz.

The effect of the microphone holder panel on thefrequency range was investigated in an anechoicchamber changing the angle of incident sound to thepanel by every 30 degree. Through the FFT analysis, itwas detected that diffractions occur around 1800-2000Hz depending on the angle of incident. Themicrophone holder panel was installed at eachmeasurement position 12cm above the floor. Themeasurements of acoustic parameters were undertakenusing two microphones at the same time in order toenhance the reliability of the measurements.

RESULTS

Measurements were undertaken for acoustic measuressuch as sound level (SPL), early decay time (EDT),clarity index (C80) at each measurement position. Also,the difference of acoustic values between twomicrophones was calculated and analyzed for bothfrequencies and measurement position. Fig.1 showsthe average difference of sound level (∆SPL) obtainedfrom two microphones at each surface diffusivityconditions. The values below 2000 Hz were eliminateddue to the diffraction caused by the size of themicrophone holder panel. Fig.1 represent that ∆SPL ofdiffused room (D) and is smaller than that of non-diffused (N-D) room. This means that more soundreflections arrive at the receiving position in late soundfield as well as in early sound field in the hall withhigh surface diffusivity.

0

2

4

6

8

10

12

14

2.5 3.1 4 5 6.3 8 10 12.6 15.9 20

Frequency (KHz)

¥ÄSPL (dB

N-D

D

Figure 1. Average difference of sound level (�SPL)versus frequency for each diffuse condition.

Also, the average differences of sound level, ∆SPL, ofeach receiver position are illustrated in Fig.2 in orderto show the difference of acoustic values depend onthe position. Fig.2 shows that the smaller difference of∆SPL is evenly distributed in the room of high surfacediffusivity regardless of the positions.

In the same way, the difference of clarity index andearly decay time were analyzed. Fig.3 shows theaverage difference of clarity index (∆C80) between twomicrophones for each diffuse condition. The figure

displayed same result as shown in Fig.2. Thedifference of ∆C80 of room D is smaller than that ofroom N-D at every octave band above 2000Hz.

0

1

2

3

4

5

6

7

8

9

r1 r2 r3 r4 r5 r6M easurem ent Position

¥ÄSPL (d

N-D

D

Figure 2. Difference of sound level (�SPL) of eachmeasurement position.

0

1

2

3

4

5

6

7

2.5k 3.1k 4.0k 5.0k 6.3k 8.0kFrequency (Hz)

¥ÄC (dB N-D

D

Figure 3. Average difference of clarity index (�C80)versus frequency for each diffuse condition.

DISCUSSION AND CONCLUSION

The results from the present paper confirm theprevious findings from the computer modeling andfield measurements. It must be mentioned that thereare some needs to know the objective scale of themeasured values in order to evaluate the surfacediffusivity of room. However, it was clearly shownthat the difference of acoustic parameters, ∆x, workswell as a possible criterion of evaluating the surfacediffusivity of room.

ACKNOWLEDGEMENT

The present work was supported by the KoreaResearch Foundation Grant (KRF-2000-2A0148).

REFERENCES

1. C.H. Haan, and. K.W. Kwon., Applied Acoustics, to bepublished, (2001).

2. C.H.Haan, Proc. of Institute of Acoustics Conference,Liverpool, 323-330 (2000).

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Influence of Different Kind of Diffusers on the Degree ofDiffusion of a Room

J Redondo, J. Ramis, J. Alba

Departmento de Física Aplicada, Escuela Politécnica Superior de Gandía, Universidad Politécnica de ValenciaCarretera Nazaret-Oliva, SN. 46730 Gandía, Spain

The last years have been devoted to study how diffuse reflections affect sound. However, at this moment, there is no model thatpermits to predict the diffusion degree in a room from the values of absorption and diffusion coefficients of the walls in a room.So diffusion seems a problem more complicated that reverberation in the sense that reverberation can be predicted if one knowsthe absorption coefficients of the walls. In the present work we perform a review of some of the classical methods to evaluatethe diffusion degree in a room. Some important mismatches between these methods are shown.

CHARACTERIZATION OF THEDIFFUSORS

First step in our work has been to evaluate the acousticparameters that characterize the scattering due to adiffuser in an anechoic chamber. As a starting point forour work we use a Diffractal� diffuser. This kind ofdiffusers consists on a repetition of the generatingsequence at to scales. This fact enlarges the bandwidthof the scattered sound an also avoid particularreflection directions.

We have chose free field methods, so, the diffuser hasbeen mounted on a turntable at a distance of 4’5meters to a loudspeaker inside an anechoic chamber.Finally an omnidirectional microphone has beenplaced in a goniometer. All of the reflection angles forall of the incidence angles have been measured form90 to 90 degrees with steps of 5 degrees. The impulseresponse measurement was obtained with a Clio�Card operating with a 14MLS sequence. Appropriatetime windowing was used in order to separate directand reflected sound. Thus, we have obtained thereflected to incident intensity ratio, i.e.

� � )1(10 10/LIincidentdLIreflecteirI �

��

From that, one can obtain the diffusion coefficientscommonly used in order to evaluate the scattering dueto the diffuser [1]:

)2(EnergyTotalEnergyDiffuse

��

� � � �� �

)3(1 2

22

�����

����

ir

iriruc IN

II�

�uc evaluates the “uniformity” of the scattered soundwhile � compares the acoustic energy reflected in adiffuse way (out from the specular zone defined by theSnell law) and the total reflected energy (i.e.diffuse+specular).

The relationship between these two coefficientsdepends on the form of the scattering pattern and onthe distance between microphones and the diffuser. Itis usually a nonlinear one of the form:

)4(nuc �� �

with n greater than one. In our experiment n�2.

00,25

0,50,75

1

250 500 1000 2000 4000

f(Hz

�UC

flat paneldiffuser

FIGURE 1. Comparative plot of the �uc for thediffuser and for a flat panel of the same width.

EXPERIMENTAL SET-UP

After having evaluated the scattering coefficients ofthe diffuser we have performed an experiment in orderto study the increase of the diffusion degree of theacoustic field inside a room. For this purpose we havemeasured some of the parameters related with thediffusion in a room with and without diffusers insideof it.

Diffuser

1’6 m

Rear wall (metallic)

Loudspeakers

1 mMeasurement

positions Glazed wall

Absorbentmaterial

FIGURE 2. Evaluated room. The floor is made ofwood and the ceiling is made of plaster.

The particular room evaluated is a rectangular onewith large unhomogeneitys in absorption coefficientson its walls. (See Fig. 2) Two loudspeakers wereplaced at 5 meters of a metallic wall (covered withdiffusers in the second part of the experiment), with adistance between then of 2 meters. We measured theimpulse response of the system with severalmicrophones. From the impulse response we haveobtained a list of parameters related with diffusion. Forthe sake of brevity we present here the results for only4 of them:

)5(33'0

33'01 LEFd LEF�

��

where LEF stands for Lateral Energy Fraction and it iscalculated as the quotient between the sound pressuremeasured with a 8-patern-directional microphone andan omnidirectional one integrated from 5 to 80 ms.(Evaluated from 250Hz to 1000Hz) (0’33 is theexpected value of LEF for a perfectly diffuse field).Another parameter evaluated is the dArea defined as thequotient between the area below the line of thebackward integrated decay in the plot of sound levelpressure vs time, and the area below a straight line.(Spring and Randall[2])

)7(80

521 dtppd

ms

msfallings � ��

where pi are the acoustic pressures measured atdifferent points (7 cm between then). The lastparameter is the decrease of coloration obtained fromthe standard deviation of the power spectra.

)8()(1 spectrapowerdcoloration ���

dLEF , dArea and dcoloration take a value of 1 in perfectdiffuse fields while dfallings goes to 0 in that case.

ESPERIMENTAL RESULTS

Table 1 shows the results for one of the positionsevaluated.

Table 1. Experimental resultsParameter Without diffusers With diffusers

dLEF 0’88 0’82dArea 0’98 0’96

dfall normalized 1 0’53dcoloration 0’71 0’65

The results for dcoloration and dfallings show a growth ofthe diffusion degree when the diffusers are present inthe room while dLEF and dArea show a decrease of it.This apparent contradiction between these parameterscan be interpreted as follows: the diffusers enforce thelateral reflections making the sound field lessisotropic. However another effect may be expected.The diffuse reflections redistribute the sound in timemaking the fall from stationary more gradual.Nevertheless this fact does not justify the decrease ofdArea.

ACKNOWLEDGMENTS

We gratefully acknowledge fruitful discussions withX. Meynial (University of Lemans).

REFERENCES

[1]. A. Farina, M. Zanolin, E. Crema. Measurement of soundscattering properties of diffusing panels through theWave Field Synthesis approach. 108th AES Convention,Paris. 18-22 February 2000

[2].N.F. Spring, K.E. Randall. The measurement of thesound diffusion index in rooms. BBC ResearchDepartment Report (1969) 16

Control of sound fields in reverberating rooms : Timereversal and inverse filter

S. Yon, M. Tanter and M. Fink.

Laboratoire Ondes et Acoustique, CNRS UMR 7587, ESPCI, 10 rue Vauquelin 75005 Paris, France.

The ability of controlling sound in a given zone of a room can be assimilated to the capacity of focusing acousticalenergy, both temporally and spatially. Focusing sound in rooms is a complicated problem, essentially because of themultiple reflections on obstacles and walls that may occur during propagation. In order to obtain a good focusingquality, the two well known ultrasound techniques of time reversal and inverse filter have been applied to this specificproblem. Compared to classical focusing techniques such as delay law focusing, time reversal and inverse filteringappear to considerably improve quality of both temporal and spatial focusing. This phenomenon is due to the ability ofthese techniques to compensate for the different sound paths between the emitting antenna and the focal point. Thosesystems also prove to be robust towards small modifications in the medium, such as moving people.After a short introduction to those techniques, experimental results are presented. Time reversal and inverse filter arecompared, showing the respective advantages of each technique; and future applications are discussed.

INTRODUCTION

Control of sound fields is an interesting domain ofapplication for room acoustics, with many applicationssuch as noise control, or creation of virtual 3-dimensionnal sound (auralization). Many techniqueshave been developed in order to achieve this soundfield control, such as Wave Field Synthesis [1];however, most of them are unable to compensate forthe restitution room, leading to noise due reflectionsagainst walls.

SOUND CONTROL TECHNIQUES

Two techniques used in ultrasound to focusacoustical energy in complex media can be used toobtain a sound field control in a reverberating room.Both are based upon the acquisition of a ‘Propagationoperator’ hmj(t), between a set of J emitters

(loudspeakers in our situation) and a set of M controlpoints located in the zone where sound field has to becontrolled. Thus, when a set of signals ej(t) are emittedby the loudspeakers, signals measured on the controlpoints can be written:

1) ( ) ( )

J

m mj jtjf t h t e t

== ∗( (1)

Both techniques of time reversal [2] and Inversefiltering [3] aim to obtain a set of signals to emit ej(t),for a given objective sound field 0 ( )mf t . To obtain thesignal, it is necessary to invert eq (1). On the one hand,time reversal provides a simple way to achieve thisinversion: when the propagation occurs withoutattenuation, and when spatial reciprocity is respected,hjm(-t) provide a good approximation of the inverseoperator.

-70

-60

-50

-40

-30

-20

-10

0

500 1000 1500 2000 2500 3000 3500 4000

2

4

6

8

10

12

14

16-70

-60

-50

-40

-30

-20

-10

0

Hz 500 1000 1500 2000 2500 3000 3500 4000

2

4

6

8

10

12

14

16

I (dB)

Sin

gula

r val

ue n

umbe

r

FIGURE 1: Singular values space for free space (left), and reverberating room (right).

Σ

On the other hand, it is possible to obtain a directapproximation of the inverse operator by inverting thematrix in Fourier domain Hmj(ω) for each frequency.As for every inverse problem, for this operation to bedone properly, one must separate the physicallysignificant part of the operator from measurementnoise. This separation is obtained by achieving asingular value decomposition (SVD) of the operatorfor each frequency. Indeed, this decomposition letappear singular values with different weights, asshown by fig. 1.Interesting results can be observed with such arepresentation: indeed, physically relevant singularvalues represent the number of available degrees offreedom in the reconstruction of the desired soundfield. Fig 1 Shows that in the circumstances of theexperiment, a reverberating room is a more interestingpropagation medium than free space, because with anidentical configuration, more complicated sound fieldcan be created. The main problem is to recreate thissound field.

EXPERIMENTAL RESULTS

Both time reversal and inverse filtering provide a wayto use reverberation in the room to our advantage, andthus creating the desired sound field with precision. Inorder to evaluate the performance of the twotechniques, the objective sound field is chosen to be afocal spot of width 20 cm @ -6dB, in the spatialdimension, and a impulsion with a 100-5000 Hzfrequency range in the time domain.Results obtained with this objective field are presentedin figures 2 and 3. From both spatial and temporalpoint of view, Inverse filtering appears to give betterresults than time reversal. Especially, as shown infigure 3, temporal recompression is a lot more efficientfor inverse filtering.

-50 -40 -30 -20 -10 0 10 20 30 40 50-45

-40

-35

-30

-25

-20

-15

-10

-5

0

distance to focal point (cm)

I (dB

)

IF, M=5

IF, M=25

IF, M=1

RT

FIGURE 2: Focusing obtained with the twotechniques for a spatial impulse response.

The main improvement coming from the use of inversefiltering is indeed the fact that the spectrum variationsare compensated during the process. Thus, temporalresponses are optimized both in their width and theirlevel of sidelobe level.However, figure 2 show that improvement in terms ofspatials focusing are not very strong, especially whenthe overall energetic efficiency is considered: SPLlevel obtained at focal point with time reversal is 10dB above level obtained with inverse filtering.

0 5 10 15 20 25 30 35 40-60

-40

-20

0Time reversal

0 5 10 15 20 25 30 35 40-60

-40

-20

0

t (ms)Inverse filtering

FIGURE 3: Quality of temporal compression for thetwo techniques.

CONCLUSION

Reverberant rooms are very interesting media to obtaina shaping of the sound field. Use of reverberations ishowever tricky, and imply the use of powerful soundshaping techniques. Of the two described techniques,inverse filter is more efficient for audio applications,because of its ability to compensate spectrumvariations. However, time reversal is easier toimplement in practical systems, especially when not alot of computing power is available.

REFERENCES

1. Berkhaut, De Vrie and Vogel, “Acoustic control by wavefield synthesis”, J. Acous. Soc. Am. 93(5), pp. 2764-2778 (1993)

2. M. Fink, “Time reversed acoustics”, Physics Today, pp.34-40, march 1997.

3. Tanter, Aubry, Gerber, Thomas & Fink, “optimalfocusing by spatio-temporal inverse filter part I.: Basicprinciples”, J. Acous. Soc. Am, Accepted for publication.