Searching for a theoretical relation between reverberation and the scattering coefficients of surfaces in a room

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1 Proeeing of he Aoui 212 Nne Conferene April 212, Nne, Frne erhing for heoreil relion eeen revererion n he ering oeffiien of urfe in room J-J Emreh Univeriy of Liege - Aoui Lo, Cmpu u r-tilmn, B28, B- Liege 1, Belgium jjemreh@ulge 2397

2 23-27 April 212, Nne, Frne Proeeing of he Aoui 212 Nne Conferene The revererion ime of room i rele o i glol orpion properie hrough he ine or Eyring formul urfe ering lo influene he revererion Hoeve even if he phyil priniple governing hi influene re lrey knon, ler relion i miing eeen he "quniy" of ering, he urfe ering oeffiien n revererion Thi oul help o peify he ering properie of urfe in room oui proje Thi pper oe no give oluion o hi prolem, u i i rher refleion e on exiing reerh I lo propoe n pproh e on he oui riive rnfer (or rnpor equion Thi equion i fir of ll expree for room oui prolem, epeilly onerning he ounry oniion hih expliily inlue he urfe ering oeffiien I i hen pplie o iffue oun fiel n o ome oher imple onfigurion, o illure he poiiliie of he meho 1 Inrouion I i ell knon h urfe ering influene he revererion in room [1,2,3] Hoeve ler relion i miing eeen he "quniy" of ering, he ering oeffiien of urfe n revererion, omehing like he ine or Eyring formul hih link orpion n revererion Thi oul help o peify he ering properie of urfe in room oui proje A urvey of he ienifi lierure revel h only fe uie hve een eie o hi uje Kuruff [] uie he revererion of room fie ih lmerin urfe (ering oeffiien 1 u hi uhor i no onier oher vlue of he ering oeffiien everl uhor [,5] hve nlye he relive iriuion of peulr n iffue reflee energie, funion of he (uniformly iriue ering oeffiien Bu he influene on he revererion ime no inveige A enive pproh h een me y Hnyu [5] o efine he verge ering oeffiien in room Like he verge orpion oeffiien, he propoe imple urfe-eighe verge Thi uhor lo efine he iffuion ime he ime hen he rio of peulrly o ol reflee energy eome -6B Hoeve hee efiniion n relionhip rely on he umpion of exponenil energy ey n, herefore, hey nno re ll kin of revererion In priul here no influene of he verge ering oeffiien on he revererion ime in Hnyu uy Beie heoreil evelopmen, ome uhor hve lo ollee experimenl reul hrough ompuer imulion Ong n Rinel [6] hve nlye he effe of he ering ue y uiling fe on he oun level n revererion ime in ree nyon They ho h in lo-fe ree he revererion ime i eermine y he um of orpion oeffiien n ering oeffiien In high-fe ree, he revererion ime eenilly epen on orpion, exep for very lo ering oeffiien In noher pper y umr-pvlovi n Miji [7], he revererion ime T3 h een ompue in 52 room, ll of hem ih uniform ering oeffiien n uniform orpion oeffiien (α1 Thee uhor hoe h T3 vlue epen on he room hpe In priul non exponenil ey ere preen for ome room, for exmple room ih lrge prllel urfe n lo ering ( More inereing, hey efine four group of room hih preen ifferen verge relionhip eeen T3 n he ering oeffiien One group in priulr i hrerie y ignifin vriion hoing minimum T3 roun 3 n mximum The ompue vlue of T3 orrepon ih he Eyring vlue in he ompleely iffue e (1, exep for one group of room for hih T3 vlue n exee Eyring vlue y ( mo 2 peren Clerly, hee reul n e oniere e e, if verifie They gin illure for ome room he influene of ering hih i le o ignifinly eree he revererion ime, even ih < The onfigurion in hih hi migh our re hoever no lerly ienifie, nor i he y o omine ifferen ering oeffiien in he me room In hi ppe e propoe o pply he oui riive rnfer (or rnpor equion o hi prolem A fir enive i me o erive heoreil relionhip eeen he revererion ime n ering In eion 2, he riive rnfer equion i preene n formulion of i ounry oniion i propoe, inluing he ering oeffiien In eion 3, he umpion of perfely iffue fiel i nlye in relion ih hi equion In eion, he equion i olve for room ih lmerin urfe In eion 5, e onier he e of mixe refleion, u he oluion i only evelope uner peifi umpion me on he lou of imge oure 2 The oui riive rnfer equion In geomeril oui, he oun energy n e erie y oun prile n heir energy iriuion funion N( r, ˆ, in (Jm -3 r -1, epening on he poiion r in he room, he ireion of propgion ŝ (uni veor n he ime [8,9] Thi funion ifie ifferenil equion lle he riive rnfer (or rnpor equion In hi pper e ue he formulion of hi equion erie y Nvrro e l [8] If he iriuion funion i inegre for ll poile ireion of propgion roun he poiion r, hen e oin he oun energy eniy ( r, in (Jm -3 Thi funion oey o he folloing equion [8]: ( J m ( q in hih m i he ir orpion enuion in (m -1, i he oun eleriy (m -1 n q i he oun poer eniy (Wm -3 genere y he oure he poiion r The oun energy flo veor J i efine y [8] : J ˆ N( ˆ, Ω π No, if e efine ( he volume-verge energy eniy, inegring (1 on he hole volume of he room give: (1 (2 2398

3 Proeeing of he Aoui 212 Nne Conferene April 212, Nne, Frne ( J ( r, ˆ n m ( W ( In hi l expreion, W( i he ol poer genere in he room ime, i he urfe enloing he volume of he room n i he uni veor norml o hi urfe he poiion erior of r (3 n iree or he Bounry oniion re neery o olve hee equion n inroue he orpion n ering properie of urfe The more generl form of hee ounry oniion i expree y Eq (17 of [8]: for ˆ Ω in : N( r, ˆ, Ω R F ( r ; ˆ', ˆ N( r, ˆ', ( ˆ'ˆ n Ω' ( The ol oli ngle roun poiion r h een ivie ino Ω in n Ω onining he uni veor ŝ iree or he inerior n he erior of he volume of he room, repeively Equion ( herefore repreen he energy rnpore y he oun prile hih re reflee y he urfe poiion r, in he ireion ŝ R F i he urfe refleing funion ih uni of r -1 On he oher hn, equion (2 give : J ( r, Ω Ωin ( ˆ'ˆ n N( r, ˆ', Ω' ( ˆˆ n N( r, ˆ, Ω The fir erm on he righ repreen he ol flux (Wm -2 inien on he urfe poiion r, hile he eon erm i he ol reflee flux muliplie y (-1 We n herefore rerie (5 : J ( r, φ φ φ ( r, in refl (5 (6 We n furher efine o group of inien oun prile: hoe hih hve unergone le one iffue refleion (N n hoe hih hve no (N Then, from (5 n (6, he inien flux φ in i ielf eompoe ino peulr inien flux φ in, n iffue inien flux φ in, The peulr reflee flux i ree y he peulr inien flux only, hile he iffue reflee flux i ompoe of he non-ore pr of φ in, n he iffuely reflee pr of φ in, : φ φ refl, refl, ( r, (1 ( r (1 α( r φ ( r, (1 α( r ( r, [ φ ( r, ( r φ ( r, ] in, in, in, (7 Wih hi moel of refleion, e ume like Kuruff [] h he onverion of iffue energy ino peulr energy never our n α repreen he ering n orpion oeffiien repeively Finlly, inrouing (7 ino (6 for mixe refleion le o ( J J J : J ˆ ( r, n α ( r φin, ( r, J ( r, α( r φin ( r, α ( r φ α( r φ ( r, ( r (1 α( r φ in, in, in, ih he peulr orpion oeffiien efine : α ( r α( r ( r ( 1 α( r ( r, ( r, (8 (9 3 The iffue oun fiel In perfely iffue oun fiel, he iriuion funion N( r, ˆ, oul no epen on he ireion ŝ n he oun energy eniy ( r, oul e he me ll poiion in he room Thi oul imply h J in (2 n, herefore, ( r, in (1 oul no epen on he urfe orpion properie, u only on ir orpion Thi imple oervion prove h he iffue oun fiel moel i no irely ompile ih he riive rnfer equion, exep if ll urfe in he room re perfely refleing (α I ill e hon in he folloing eion h he iffue oun fiel n e oniere n ympoi reul of he iffue refleion umpion Diffue refleion Room ih iffuely refleing ll o no ly ree iffue oun fiel [, 5] Hoeve everl uie hve hon h in revererion experimen, hen he oun oure i u off, he reuling ime-energy ey in hee room i loe o n exponenil funion [, 7, 9, 1] Moreove he lope i nerly ienil ifferen poiion in he room, hough omeime ignifinly ifferen from he one preie y ine or Eyring formul The olue vlue of he energy ey n lo iffer from one poiion o noher In revererion experimen, if he oure i u off, e herefore ume h: ( W ( r e e ( W ( r We > (1 The exponenil ey i generlly elihe fer n iniil rnien perio Aeque ounry oniion n e eue from he evelopmen of he prile iriuion funion N( r, ˆ, fir orer pheril hrmoni expnion ll poiion in he room, inluing on i urfe [8] A fir orer expnion i juifie in room ih iffuely 2399

4 23-27 April 212, Nne, Frne refleing ounrie ine he oun fiel i no very fr from iffue fiel Thi evelopmen give [8]: α J ( r, r, 2 2 ( r α( r (, χ( r ( r (11 Noe h χ ( r i funion of he urfe orpion oeffiien poiion r Applying he pproximion (1 n (11 in (3 n oniering h W( for > in revererion experimen le o: γ m χ ( r W ( r W (12 Thi expreion i more generl hn he perfely iffue fiel moel ine i llo ifferen olue imeey ifferen poiion r The iffue oun fiel moel i reovere if he oun energy eniy en o ( W uniformiy W ( r lo ( ( α( χ r r n he urfe orpion i In he n eion, e ill nee o preie he relion eeen W n W, he poer of he oure in revererion experimen, if he equion (1-12 re ene up o ( In hi iuion, ( W γ repreen he ol ore poer in he room ju efore uoff in -, hih mu equl he ol emie poer in ionry fiel: W W γ ( if (13 Expreion (1 i he oluion of he rnfer equion (3 ih J r (, given y (11 n he oure onriuion W( W for n W( for > Therefore, he oluion of he me rnfer equion for Dir pule exiion W( δ( i given y: δ δ W ( r γ e W > < 5 A moel for mixe peulr n iffue refleion (1 A eh poiion in he room n every inn, e efine o group of oun prile: hoe hih hve lrey unergone le one iffue refleion (lle he iffue prile n hoe hih hve no (lle he peulr one: r (, ( ( ( (15 We lo ume h, uring eh refleion of group of oun prile ime, given perenge (1-α of he inien flux of peulr prile i rnforme ino iffue reflee flux, hile he iffue inien flux never give rie o reflee peulr flux (ee eion 2 I n e hon h oh omponen of he oun energy eniy ify equion (1 n (3 eprely Noe h one emie y he oure, he oun prile re iniilly peulr, ine hey hve no unergone ny iffue refleion ye Therefore, he oure onriuion q in (1 n W in (3 re inlue in he peulr equion The peulr omponen oul e expree y he um of he onriuion of ll imge oure, if heir poiion n poer ere lule inepenenly In hi eion, e reri our moel o hoe room for hih he lou of imge oure i pproximely ioropi, le from erin ine from he rel oure Thi implie h he peulr prile iriuion funion N n e evelope fir orer pheril hrmoni expnion n h (11 hol for he peulr flo veor J Furhermore, e onier only he room for hih he lou of imge oure i pproximely he me ll reepor poiion in he room, mening h: ( r, ( (16 Wih hee o umpion on he lou of imge oure, he expreion of he revererion ey n e oine from (3, (11 n (16: ( γ γ m ( χ ( r, > α χ 1 α / 2 (17 A n e een in (17, he vliiy of he o umpion i rerie o hoe imge oure hih re in from he rel oure y le The oluion of (17 i given y: γ ( ( e ( > (18 The iffue omponen n e oine y he folloing nlyi Fi e ill onier h he iffue prile iriuion funion N n e evelope fir orer pheril hrmoni expnion ine N n N verify hi pproximion, o i heir um Therefore, expline in eion, i pproxime y (11 eonly, oniering h J J J J J le o: ( χ( r ( r, χ ( r ( r, ( χ( r ( r, ( χ ( r χ( r ( r, Proeeing of he Aoui 212 Nne Conferene (19 2

5 Proeeing of he Aoui 212 Nne Conferene The iffue energy eniy ( herefore ifie he folloing equion: ( χ ( r, m ( ( χ χ ( r, K ( r r (2 The oluion of (2 i h of iffue oun fiel ih oure onriuion of he ype W ( K ( Applying he impule oluion (1 give: K ( τ δ τ τ (21 In he folloing, he oluion of hi inegrl i evelope in he e of rpily elihe exponenil ey in he iffue fiel, ie For > >, pplying (13 n (1 le o: ( ( e K K ( e ( τ e ( τ e ( τ ( τ τ τ (22 The fir inegrl n e neglee if >>, hih finlly le o: ( ( e K ( γ γ ( ( ( e e (23 Aing (18 n (23 give he verge oun energy eniy ( he um of o exponenil ey: ( ( μ ( γ γ ( ( ( μ e (1 μ e ( ( > W ( r χ( r 1 W (2 The influene of he ering oeffiien on he oun energy ey n he revererion i onine in he prmeer μ, hrough γ n he rio ( ( urfe in he room re iffue refleor ( 1 / If ll, hen hi l rio n μ en or zero n he moel expree y (1 n (12 i rerieve If hoever ll urfe re peulr refleo ( μ 1 n he moel expree y (18 i rerieve The prmeer μ in (2 mu ly e omprie eeen n 1, oherie ome unexpee ehviour oul pper in he grph of ( (negive vlue or inreing egmen The influene of ering lo epen on he uniformiy of he oun fiel hen he room i limie y ( W W iffuely refleing ounrie: if ( r hen ( μ n oh exponenil ey hve he me lope In h e, e rerieve he iffue fiel reul Therefore, oring o (2, he ering oeffiien n hve n influene on he lope of he ey n he revererion ime only in room here he oun energy eniy long he ounrie i ifferen from he verge energy eniy, if heir urfe re iffue refleor Rememer lo h (2 hol for room hih hve n ioropi n uniform lou of imge oure To verify hee fir onluion, ome ompuion hve een performe ih ry-ring progrm [11] in ui room, hih i knon o pproximely fulfil hi l oniion ou he imge oure iriuion Figure 1 illure he reul oine in ui room ih L1m, α1 for he ix ll, poin oure i ple he enre of he ue n 12 reepor re uniformly iriue in he volume of he room (in f, he hole volume h een ivie ino 125 ienil mller ue, ih reepor he enre of eh of hem, exep he oure poiion 1 7 ry hve een emie y he oure uh h he revererion ime T3 re ignifin up o o eiml ple (eime rnom error 1 Figure 1 ho he men vlue of T3 oine y verging for ll reepor In hi ui room, he revererion ey re pproximely liner (in B n he T3 vlue re nerly ienil eh reepor poiion (ifferene le hn 2 T3( ering oeffiien April 212, Nne, Frne Ry ring ine Eyring Moel eq (2 Figure 1: Revererion ime T3 ompue y four moel in ui room ih L1m n α1 The ering oeffiien i he me for ll urfe n i vlue i one of he folloing {, 3, 65, 1} The ry ring urve in figure 1 ho oninuou ignifin eree of T3 from 1 o 3 (he ering oeffiien i he me for ll urfe Then, he revererion ime uenly rie for Thi ehviour i imilr (hough in le en o he reul preene y umr-pvlovi n Miji [7] An explnion for hi uen inree for lo ering oeffiien i propoe in he folloing To ompue T3 vlue ih he moel (2, e nee ome more pproximion For exmple, he rio W ( r / W hih pper in he expreion of γ in (12 n μ in (2 ill e pproxime in he ui room y, 21

6 23-27 April 212, Nne, Frne ienifying in he ry ring reul he PL ompue hoe reepor hih re he loe from he ue envelope Anoher exmple i he rio ( ( / hih mu e eime funion of he ering oeffiien (pe i miing o evelop n erie hi eimion in hi pper The moel (2 prei oninuou eree of T3 from 1 o in he ui room ih α1 The 1 vlue i eermine y he exponenil onn γ in (12 The verge vlue of he rio W ( r / W on he urfe equl 97, hih le o revererion ime T3262 in figure 1 Thi preiion i i loer o he ry ring vlue hn ine n Eyring vlue The vlue i eermine y he exponenil onn γ in (17 I oul lo e goo preiion of he ry ring reul if he T3 vlue ere ill oninuouly ereing eeen 3 n Hoeve he preiion of he moel (2 re no uile eeen n 1, uggeing h he pproximion h e hve ue for he rio ( ( / i no eque Figure 2 ho he me reul oine in he me ui room, u ih α3 The onluion re imilr T3( T3( ering oeffiien Figure 2: me figure 1 ih α ering oeffiien Ry ring ine Eyring Moel eq (2 Ry ring ine Eyring Moel eq (2 Figure 3: Revererion ime T3 ompue y four moel in n irregulr room (733m³, 56m² n α1 The ering oeffiien i he me for ll urfe An explnion of he ifferene oerve eeen he ry ring moel n (2 lo ering vlue oul e h lrge prllel urfe ree imge oure in ell efine ireion, hih oviouly onri he umpion h le u o he peulr onriuion (18 To verify hi, e ompue T3 vlue in n irregulr room ih pproximely he me volume n urfe hn our ui room The reul illure in figure 3 lerly ho h he inree of T3 for h lmo ippere (ompre ih figure 1 6 Conluion The oui riive rnfer equion i poile pproh o moel he influene of he ering oeffiien on he revererion ime Hoeve muh reerh ork mu ill e one o fin n eque formulion of he moel, priulrly for: - he prmeer μ in (2, - he onriuion of lrge prllel urfe hih houl e iole, - he onriuion of urfe ih ifferen ering properie Referene Proeeing of he Aoui 212 Nne Conferene [1] TJ Cox, P D Anonio, Aoui Aorer n Diffue 2 n e, Tylor & Frni, Lonon n Ne York (29 [2] MR Hogon, Eviene of iffue urfe refleion in room, J Aou o Am 89, (1991 [3] YW Lm, A omprion of hree iffue refleion moelling meho ue in room oui ompuer moel, J Aou o Am 1, (1996 [] H Kuruff, Room Aoui, h e, Tylor & Frni, Lonon n Ne York (2 [5] T Hnyu, A heoreil frmeork for quniively hrerizing oun fiel iffuion e on ering oeffiien n orpion oeffiien of ll, J Aou o Am 128(3, (21 [6] H Ong, JH Rinel, Aoui hrerii of urn ree in relion o ering ue y uiling fe, Applie Aoui 68, (27 [7] D umr-pvlovi, M Miji, An inigh ino he influene of geomeril feure of room on heir oui repone e on free ph lengh iriuion, A Aui unie ih Aui 93, (27 [8] JM Nvrro, F Joen, J Eolno, JJ Lopez, A heoreil pproh o room oui imulion e on riive rnfer moel, A Aui unie ih Aui 96, (21 [9] Y Jing, EW Lren, N Xing, One-imenionl rnpor equion moel for oun energy propgion in long pe: Theory, J Aou o Am 127(, (21 [1]A Billon, JJ Emreh, Numeril eviene of mixing in room uing he free ph emporl iriuion, J Aou o Am 13(3, (211 [11]JJ Emreh, Bro perum iffuion moel for room oui ry-ring lgorihm, J Aou o Am 17(, (2 22