PREDICTION OF REVERBERATION TIME IN RECTANGULAR ROOMS WITH NON UNIFORMLY DISTRIBUTED ABSORPTION USING A NEW FORMULA

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1 PREDICTION OF REVERBERATION TIME IN RECTANGULAR ROOM WITH NON UNIFORMLY DITRIBUTED ABORPTION UING A NEW FORMULA PAC REFERENCE: Br Neubauer, Reihard O. Ig.-Büro Neubauer VDI Theresiestr. 8 D Igolstadt Germay Tel.: ++49 (0) Fax: ++49 (0) @reihard-neubauer-vdi.de ABTRACT I this paper it will be preseted a ew formula, i.e. a modified Fitzroy-Kuttruff equatio, for estimatig the reverberatio time i rooms with o uiformly distributed soud absorptio. Calculated results are preseted comparig differet up-to-date formulae for the estimated reverberatio time as well as for measured data. A modificatio of Fitzroy s equatio is discussed i this paper ad some practical examples are preseted which compare predicted ad measured values of reverberatio time i real rooms. INTRODUCTION It is geeral kow that abie s or Eyrig s formula seriously i error if the soud absorptio of the room is uevely distributed. I 1988 Higii Arau [1] published a paper reportig a ew formula for calculatio of the reverberatio time i rectagular rooms with o uiformly distributed soud absorptio. Relatig to the work of Fitzroy [] it will be show i this paper that the empirical Fitzroy equatio ca be modified applyig Kuttruff s correctio [3] origially related to Eyrig s formula, yieldig reverberatio times which are close to measured values [4]. A suggested modificatio of Fitzroy s equatio was first preseted i [4] ad was further discussed i [5]. Differeces betwee results derived from Fitzroy s, abie s ad Eyrig s equatio, as well as from Arau s formula ad others, are compared to those obtaied from measuremets i real rooms. A CORRECTION TO FITZROY FORMULA Fitzroy s equatio [] assumes the Eyrig cocept [6] ad cosiders the reverberatio time of the room aalogous to a area-weighted arithmetical mea of the reverberatio time of the three room directios. O the other had, Kuttruff [3, 7] itroduced a correctio to the Eyrig formula ad could show that his correctio to the Eyrig formula ca easily be applied to the case where -1 surfaces have early the same reflectio coefficiet ad oe surface, amely the th surface, e.g. the floor where the audiece sits over, a differet reflectio coefficiet shows. His preseted results showig a very good agreemet with computer simulated results [7]. Págia 1 de 7

2 With the assumptio that i real rooms the mai absorptio is always o the floor or at the ceilig or o both oe ca modify the Fitzroy equatio ito a more coveiet equatio for practical use. It is proposed to deal with the cocept of a Kuttruff correctio origially desiged to correct the Eyrig formula. Tha it is possible to split the Kuttruff correctio ito two parts amely the part of ceilig-floor ad the part of the walls. THE MODIFIED FITZROY EQUATION For the importat practical case where either the ceilig ad/or the floor is highly absorptive Fitzroy s equatio may be writte i a modified maer. Kuttruff preseted i [3] the Eyrig correctio usig the cocept of the reflectio coefficiet: ρ = 1 - ad preseted a expressio yieldig a absorptio expoet. Uder the assumptio that the absorptio coefficiet ad hece ρ are idepedet of the agles Kuttruff made use of Lambert s law of diffuse reflectio. By focussig o the overall reverberatio time, eglectig details of the decay process ad uder the assumptio of a expoetial law for the time depedece of the irradiatio stregth he defied a absorptio expoet, which is valid uder the aforemetioed assumptio. The assumptio of a expoetial law is reasoable sice, at least i rectagular rooms, the decay process of the soud eergy will decrease expoetially [7, 8]. Kuttruff derived a correctio to Eyrig s formula ad showed that the absorptio expoet would assume its Eyrig value if the irradiatio stregth were costat [7] The absorptio expoet is Eyrig = - l ρ = -l (1- ) = l 1 ρ + l ρ ρ 1+ ρ ρ Σ ρ where ρ = 1, deotes the average reflectio coefficiet of surface area is the total surface area of the room i m². I most cases the secod term i the deomiator is much smaller tha the first ad hece ca be eglected [7]. Expadig the secod logarithm ito a power series ad eglect all terms of higher tha first order gives Eyrig + ρ ρ ρ ρ ice the Eyrig formula has to be modified it is reasoig modifyig the Fitzroy equatio i a similar maer. This modificatio leads to the modified Fitzroy equatio (Fitzroy- Kuttruff equatio). We may rewrite Fitzroy s equatio ad itroducig Kuttruff s modified correctio yieldig the New Formula Págia de 7

3 T 60 = 0, 3 V h( l + w) + lw where V, = Volume i m³ ad total surface area of the room i m² h, l, w = room dimesios height, width ad legth i m ; = average effective absorptio expoet of walls, ceilig + floor with ρ = l( 1 ) + ( ρ ρ ) ( ρ ) ρ = l( 1 ) + ( ρ ρ ) ( ρ ) ad is the arithmetic mea of the surface averaged absorptio coefficiet ρ = (1-) is the reflectio coefficiet = lw; is the surface area of ceilig ad floor, i m² WW = lh + hw; is the surface area of the walls, i m² total = [h(l + w) + lw)]; is the total surface area of ceilig, floor ad walls, i m² COMPARION OF CALCULATED VALUE AND MEAURED REULT A compariso of calculated results usig differet reverberatio formulae uder ivestigatio are show i the i Figure 1. The formulae uder ivestigatio i Figure 1 are abie, Eyrig [6], Fitzroy [], Milligto-ette [9, 10], Tohyama [11], Nilsso [1] ad Arau [1]. NB: I this paper all preseted figures are related to a mid frequecy of 500 Hz ad eglectig soud absorptio of air. It is iterestig i some aspects to compare calculated reverberatio time usig differet calculatio formulae. I Figure 1 the calculated reverberatio time is show for differet room volumes ad overall surface absorptivity (averaged mea absorptio coefficiet) of 0,-0,3. Compariso of predicted results with abie s reverberatio time shows that Fitzroy, Tohyama, Arau ad Nilsso always yield greater values. Whereas Eyrig, Milligto-ette ad Fitzroy- Kuttruff reveal smaller values tha calculated by abie. The Nilsso model (pren ) predicts, at least for the used date, urealistic high reverberatio time values, eve loger values tha predicted by Fitzroy. At this stage, it could ot attaied ay explaatio why Nilsso s model produces such high values. Págia 3 de 7

4 = 0,0 = 0,40 (Floor 0,60; Ceilig 0,0) Reverberatio Time Fitzroy Arau Room Volume i m³ abie Eyrig Fitzroy Milligto-ette Tohyama et al. Arau Fitzroy-Kuttruff Nilsso (pren ) Figure 1. Calculated reverberatio time for differet reverberatio time formulae. I order to cofirm the New Formula it is eeded to compare with measured reverberatio time values. Due to the fact that i practice, i.e. i real rooms, the soud absorptio coefficiet of the surfaces are ot kow it is assumed that from the measured reverberatio time the appropriate values of the soud absorptio of the idividual surfaces ca be matched usig the theory of abie. This was doe by fittig the calculated reverberatio time usig abie s formula ad comparig the obtaied results with the measured reverberatio time. The idividual soud absorptio coefficiets where the used calculatig the respective reverberatio time usig differet formulae uder ivestigatio. The real rooms where empty with some absorbet surfaces. No air absorptio is take ito accout for computig reverberatio time. The results are preseted i Figure. It tured out for these data that the measured RT is always greater tha the Eyrig values. Fitzroy s equatio yields about 75% greater ad Milligto-ette s equatio as well as the Fitzroy-Kuttruff equatio yield about 96% smaller values compared to the measured values. Usig Arau s equatio yield about 93% ad Tohyama s about 89% greater values tha measured. Nilsso s model as give i pren yields always greater values tha measured ad for the used data it tured out to be at least 95% greater tha 50% ad about 65% greater tha 100% of the measured values. I the followig table the differeces usig Arau ad New Formula compared with measured date are preseted. N = 9 Usig Arau Usig Fitzroy-Kuttruff Results differ from measured RT ± 0% 58 % 6 % RT < measured RT 7 % 96 % RT > measured RT 6 % 3 % Págia 4 de 7

5 Reverberatio Time i s 7,0 6,5 6,0 5,5 5,0 4,5 4,0 3,5 3,0,5,0 1,5 1,0 0,5 0,0 Compariso of Measured ad Calculated Reverberatio Time Room Volume i m³ Measured RT Eyrig New Formula Fitzroy Milligto-ette Tohyama Arau Nilsso Figure. Calculated ad measured reverberatio time for differet reverberatio time formulae. As a result of the foregoig data the differece betwee Arau s ad Fitzroy-Kuttruff s value differ from each other i a rage of ± 0% by about 44%. This meas, the New Formula yields about 3% lower ad about 93% greater values tha usig Arau s formula. For about 3% of the data there have bee idetical values obtaied. I Figure 3 it will be focused graphically o the compariso of measured results ad predicted values usig the formula of Arau ad the New Formula. Measured RT Calculated ad Measured Reverberatio Time 4,0 3,5 3,0,5,0 1,5 1,0 0,5 0,0 0,0 0,5 1,0 1,5,0,5 3,0 3,5 4,0 Calculated Reverberatio Time Arau Fitzroy-Kuttruff Figure 3. Compariso of calculated ad measured reverberatio times. From Figure 3 it is see that the measured reverberatio time teds to higher values tha the computed reverberatio time usig the New Formula (Fitzroy-Kuttruff) by use of fitted Págia 5 de 7

6 soud absorptio coefficiets, whereas Arau s value teds to provide greater values tha measured. Arau s formula produced a greater differece for the measured value tha it does usig Fitzroy-Kuttruff s equatio. The biggest differece of measured ad predicted reverberatio time usig Arau s equatio was a value of 44% greater ad more tha about 41% lower tha measured. This rage was observed similar but slightly smaller usig the New Formula. I most cases, however, the differece usig the Fitzroy-Kuttruff equatio was less tha about 0% of the measured value. A similar result is observed usig the Eyrig-Kuttruff equatio. ice for both equatios the Kuttruff-correctio should be applied, it is worthwhile usig the Fitzroy-Kuttruff equatio. UMMERY I this paper some of the most kow formulae to predict the reverberatio time are compared with calculated ad measured values. It has bee show that the reverberatio time accordig to a modified Fitzroy equatio (Fitzroy-Kuttruff equatio) is useful where the soud absorptio at opposite sides are substatially higher tha o the rest of the room surfaces. This is e.g. typically for offices where the assumptio of diffuse field coditios for applyig abie s theory are ot i agreemet with the existig absorber distributio. Compariso of calculated abie ad measured reverberatio times revealed always too high values of computed reverberatio times. I cotrast, if oe compares computed results usig Eyrig s formula ad the Fitzroy-Kuttruff equatio usig fitted soud absorptio coefficiets, i.e. adequate absorptio coefficiets to match the measured reverberatio time, revealed that Eyrig formula teds to predict, at least i this ivestigatio, about 10-0% higher values tha does usig the Fitzroy-Kuttruff equatio. I most practical cases where a fitted soud absorptio coefficiet o the basis of abie s theory is used to predict the reverberatio time, i compariso with measured values, the differece betwee measuremet ad the New Formula (Fitzroy-Kuttruff equatio) is less tha 0%. Additioally, Arau s formula was compared which revealed that the New Formula provides values which are more tha about 51% closer to the measured reverberatio times. BIBLIOGRAPHY [1] Arau-Puchades, H. A Improved Reverberatio Formula, Acustica, Vol. 65, No. 4, (1988) [] Fitzroy, D. Reverberatio formulae which seems to be more accurate with o-uiform distributio of absorptio, The Joural of the Acoustical ociety of America, Vol. 31, (1959) [3] Kuttruff, H. Nachhall ud effektive Absorptio i Räume mit diffuser Wadreflexio, Acustica, Vol. 35, No 3, (1976) [4] Neubauer, R.O. Predictio of Reverberatio Time i Rectagular Rooms with a Modified Fitzroy Equatio, IEM 99, 8 th Iteratioal ymposium o oud Egieerig ad Masterig, Gdask, Polad, (1999) [5] Neubauer R.O. Estimatio of Reverberatio Time i Rectagular Rooms with No Uiformly Distributed Absorptio Usig a Modified Fitzroy Equatio. ICV 7 Cogress July 000, Garmisch-Partekirche, Germay. [6] Eyrig, C.F. Págia 6 de 7

7 Reverberatio Time i Dead Rooms, The Joural of the Acoustical ociety of America, Vol. 1, (1930) [7] Kuttruff, H. Room Acoustics, 3 rd Editio 1991, Elsevier ciece Publishers LTD [8] Miles, R.N. J. oud Vibr., 9, No. (1984), 03 [9] Milligto, G. A Modified Formula for Reverberatio, The Joural of the Acoustical ociety of America. Vol. 4, (193), pp 69-8 [10] ette, W.H. A New Reverberatio Time Formula, The Joural of the Acoustical ociety of America. Vol. 4, (1933), pp [11] Tohyama, M., uzuki, H., ad Ado Y. The Nature ad Techology of Acoustic pace, Academic Press, Lodo, (1995) [1] Europea tadard pren February 000, Aex D. (Nilsso s Model) Buildig Acoustics. Estimatio of acoustic performace of buildigs from the performace of elemets, Part 6: oud absorptio i eclosed spaces.cen/tc16/wg N16 Págia 7 de 7