Calculation of temporal evolution of sound pressure levels in rooms, based on diffuse reflection

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1 Aoustis 08 Paris Calulation of temporal evolution of sound pressure levels in rooms, based on diffuse refletion R. Gamba, G. Cazard and C. Senat Gamba Aoustique, 2 rue de la déouverte, BP 63, 3676 Labege Cedex, Frane rene.gamba@aoustique-gamba.fr 348

2 Aoustis 08 Paris Abstrat This paper desribes a model whih enables the temporal evolution of sound pressure levels in rooms to be alulated. t is built on the assumption that sound waves are totally sattered when refleted by the walls. This model presribes the sampling of all the surfaes of the room and defines a proess of time disretization, proess whih enables the temporal evolution of energeti exhanges between eah sample to be known. Ultimately, the model allows the sound pressure level to be alulated in every point of the room for eah time sample (ehograms. Thanks to the ehograms, some useful riterias for room aousti studies an be evaluated : reverberation time, EDT, D50, C80,... Measurements and alulations have been arried out for different kinds of rooms. We will desribe them in the seond part of this paper.. ntrodution Room aousti studies need to take into aount many riteria whih the most used is the reverberation time (RT60. The reverberation time of a room is generally alulated from the ehograms. Our paper desribes a model whih enables the ehograms to be alulated aording to the assumption that sound waves are totally sattered when refleted by the walls. A first part desribes the formalism of the model and a seond part desribes the proess we have hosen to build an iterative formula based on wall surfaes and time sampling. Ultimately, alulations and measurements of reverberation times, arried out for different kinds of rooms, will be ompared. 2. Our model to alulate the temporal evolution of the aousti intensity The models for prediting the sound field based on speular refletion assumptions use infinitely smooth surfaes. However, in the ase of rough surfaes and dimensions less than the wavelength, experiments have shown that speular refletion of the sound no longer applies. Walls of the majority of the rooms are seldom smooth and flat : presene of furniture, shape of the building materials, floor luttering To take aount of these onditions, a diffuse refletion model has to be used. Our model is built on the assumption that the sound waves are totally sattered when refleted by the walls. nfluene of a surfae omponent on another one Our model assumes [], [2], [3], [4] that the intensity of the noise at any point onsists of two superposed omponents, a diret omponent onsisting of the intensity of the noise emitted diretly by the soure, and a omponent of noise reverberated from the walls. The first omponent, whih is easily determined, orresponds to the free field propagation of spherial waves, the theoretial model for whih is well known. The seond omponent (reverberated noise requires the assimilation of the walls as point soures (virtual the diretivity of whih takes aount of the diffusion assumption. The diretivity fator of the diffused refletion used by our model is: Eah omponent of a surfae whih reeives energy retransmits it towards all the surfae omponents. Let us examine two omponents ds and ds entred respetively on x and x. d x Fig. nfluene of a surfae omponent on another one These elements are both haraterized by their absorption oeffiients α(x and α(x. The surfae density of inident power on ds, noted d(t,x and indued by ds, is: os θ is the solid angle aording to whih ds is seen by the inoming flux ( θ 4osθ Q ( ( dxx t, x α ( πd ososds ² is the surfae density of inident power on ds, only the fration ( of whih is reemitted. n order to simplify the formula, we have grouped the geometrial terms within the same oeffiient, that we an all the influene oeffiient K(x,x. K( x, x The surfae density of inident power on ds indued by all the walls is therefore : xx os os πdxx ² (2 (3 xx x K( x, x ( t d, x S ( α ds (4 3482

3 Aoustis 08 Paris This expression would not be omplete if we did not take into aount the intensity of the soure reeived diretly by ds. This intensity is represented by d,x(t Therefore we obtain : xx ( ds x α + d ( t (5 ( t x K( x, x ( t d, x,, S ntensity at the reeptor The intensity reeived in every point has 2 omponents. The first one is the intensity diretly reeived from the soure (free field propagation. The seond one is the intensity oming from all the surfae omponents ds of the room. Thus, the surfae power density is expressed by : Setting up an iterative formula needs the sampling of the time axis. The time sample is noted k. The temporal shift ij ( t j t ( j Kij + di( t i j d dij m is sampled as following : dij Δt Δt is the temporal sampling step (expressed in seond. The «int» funtion gives the loser rounded integer for the quantity The equation (Eq. 8 is now written : α (8 ij int (9 dij Δ t. ( k ( j j( k mij Kij+ di( k i j α (0 Thus, the intensity in every point of the room is : Fig. 2. ntensity reeived by the reeptor The intensity reeived at any point of the room for eah time is: xr dsr( t rx ( t d, x α ( x d SR (t is the aousti intensity diretly reeived at the reeptor from the soure. The pressure level is obtained by : + ( R S πd 2 rx L ( t R( t 0log 0 2 osθ rxds (6 p (7 R d + ( k SR( k ( i i( k mri i osθ S ri i α ( πd 2 ri 4. Comparison between alulations and in situ measurements Reverberation times alulated with our model for 4 different rooms are given in this hapter. The rooms studied present various geometries and aoustial treatments. Those alulations will be ompared with in situ measurements. Our model allows the alulation of the ehogram. The reverberation time is dedued from it thanks to the slop of the late exponential derease. The figure below shows an ehogram alulated with our model. 3. Setting up an iterative formula n order to overome the integral and allow the equation to be solved numerially, the walls have to be disretised. Therefore the walls have to be broken down into surfae samples by onsidering that: the absorption oeffiient α is onstant for a same sample the surfae power density is onstant on all the surfae Si of the sample eah surfae sample will be identified by its entroid. Fig. 3. Ehogram at khz 3483

Aoustis 08 Paris Room # : Halle aux Grains - Samatan (32- Frane This is both a theatre and a onert hall.

Various absorbent materials are distributed on the eiling, on the walls and on the floor (sits.

ts volume is 24,660 m 3 and its floor surfae is 2,900 m².

The floor is luttered with many mahine-tools. Fig. 4.

table below : Frequeny (Hz 25 250 500 000 2000 4000 Measurement (s 0.77 0.64 0.67 0.69 0.78 0.8 Calulation (s 0.84 0.

Reverberation times Room #2 : Théâtre des 3 ponts - Castelnaudary (-Frane This is both a theatre and a onert hall.

modelisation of the fatory Calulated and measured values (for the same ouple of soure and reeptor are presented in

4 Aoustis 08 Paris Room # : Halle aux Grains - Samatan (32- Frane This is both a theatre and a onert hall. ts volume is,500 m 3 and its floor surfae is 250 m². Various absorbent materials are distributed on the eiling, on the walls and on the floor (sits. Frequeny (Hz Measurement (s Calulation (s Fig. 7. Reverberation times Room #3 : Cezus fatory - Montreuil (49- Frane This fatory is 20 m wide and 40 m long. ts volume is 24,660 m 3 and its floor surfae is 2,900 m². Various materials highly absorbent are distributed on the eiling and on the walls. The floor is luttered with many mahine-tools. Fig. 4. modelisation of the room Calulated and measured values (for the same ouple of soure and reeptor are presented in the table below : Frequeny (Hz Measurement (s Calulation (s Fig. 5. Reverberation times Room #2 : Théâtre des 3 ponts - Castelnaudary (-Frane This is both a theatre and a onert hall. ts volume is 2,40 m 3 and its floor surfae is 290 m². Various absorbent materials are distributed on the eiling, on the walls and on the floor (sits. Fig. 8. modelisation of the fatory Calulated and measured values (for the same ouple of soure and reeptor are presented in the table below: Frequeny (Hz Measurement (s Calulation (s Fig. 9. Reverberation times Room#4 : Havana Café Toulouse (3- Frane This is a onert hall used for loud musi. ts volume is 6,090 m 3 and its floor surfae is 870 m². Various absorbent materials are distributed on the eiling and on the walls. Fig. 6. Modelisation of the room Calulated and measured values (for the same ouple of soure and reeptor are presented in the table below : Fig. 0. modelisation of the room 3484

5 Aoustis 08 Paris Calulated and measured values (for the same ouple of soure and reeptor are presented in the table below: Frequeny (Hz Measurement (s Calulation (s Fig.. Reverberation times For those 4 rooms, alulated reverberation times are lose to in situ measurements. Moreover, the results show that the auray of the values given by our model is adapted for room aousti studies. 5. Conlusions The model presented in this paper allows the alulation of the ehograms of a room. t is built on the assumption that sound waves are totally sattered when refleted by the walls. Thanks to the ehograms, reverberation times an be deduted. Calulations and measurements had been arried out for different kinds of rooms and show that the auray of the values given by our model is adapted for room aousti studies. Other riteria would be alulated with the ehograms given by our model as, for example, EDT, C80, D50 or RAST. Referenes [] CAZARD G. : «Elaboration d une algorithmique permettant de déterminer la réponse temporelle d un loal en se basant sur l hypothèse de réflexion diffuse en paroi» - Rapport de DEA Aoustique. Univerité Paul Sabatier Toulouse (2002. [2] SEAT C. : «Prise en ompte de l enombrement, au sein d un modèle de prévision des niveaux de pression, reposant sur l hypothèse de réflexion diffuse sur les parois». Thèse de Dotorat spéialité aoustique de l Université Paul Sabatier Toulouse (992. [3] MALCURT C. : «Simulations informatiques pour prédire les ritères de qualifiation aoustique des salles. Comparaison des valeurs mesurées et alulées dans une salle à aoustique variable». Thèse de 3e yle spéialité aoustique. Université Paul Sabatier Toulouse (986. [4] ZULA P. : «Elaboration d un modèle prévisionnel fondé sur l hypothèse de réflexion diffuse en paroi». Rapport de DEA Aoustique. Université Paul Sabatier Toulouse (986. [5] PCAUT J., SMO L., POLACK J-D : «A mathematial model of diffuse sound field based on a diffusion equation». Aoustia Vol. 83 ( [6] KUTTRUFF H. : «Energeti sound propagation in rooms». Aoustia Vol. 83 ( [7] SCHROEDER M.R. : «terative alulation of reverberation time» Aoustia Vol. 45 (980 [8] GLBERT : «terative alulation of reverberation time» JASA 69 ( (

Theory. Coupled Rooms

Theory. Coupled Rooms Theory of Coupled Rooms For: nternal only Report No.: R/50/TCR Prepared by:. N. taey B.., MO Otober 00 .00 Objet.. The objet of this doument is present the theory alulations to estimate the reverberant