Sound decay in a rectangular room with specular and diffuse reflecting surfaces

Transcription

1 Soun ecay n a recangular room wh pecular an ffue reflecng urface Nkolay Kanev Anreyev Acouc Inue, Mocow, Rua Summary Soun ecay n room uually efne by aborpon an caerng propere of urface In ome cae oun ecay calculaon a raher mple ak Bu n mo problem we eal wh nonunformly aborpon rbuon on he urface, nonffue reflecon, nonergoc room an oher obacle prevenng he exac analycal oluon Smulaon proceure can help u n praccal cae when we nvegae a ceran room here are ome fferen moel for calculaon of ffue oun fel, wherea pecular reflecon can be aken no accoun n a mple way In he preen paper we propoe analycal moel for oun ecay calculaon n a recangular room In h moel oun fel n he room eparae no wo componen Fr componen ecrbng pecular reflece oun fel gven by a um of pecular reflecon Secon componen ecrbng ffuely reflece oun fel a oluon of a fferenal equaon Whou ffue reflecon oun ecay n he room wh nonunformly aborpon rbuon alway nonexponenal If he ffue reflecon coeffcen uffcen hen oun ecay exponenal an cloe o Sabne law Separaon of oun fel no ecular reflece an ffuely reflece componen perm o efne wha of hem omnan Conon uner whch oun ecay manly ecrbe by ffue reflecon can be emae from he moel a well PACS no 4355Br, 4355D Inroucon he reverberaon me generally recognze a he mo mporan acoucal arbue for any kn of room I characerze a oun ecay rae on conon ha a ecay curve aume o be cloe o exponen Many formula have been propoe for precng he reverberaon me [-3] A neceary conon of exence for he reverberaon me ha he oun fel n he encloure ffue an oun ecay cloe o exponen For ha he encloure mu be uffcenly ranomzng [4] Ranomzaon of he oun fel can be prove by he encloure hape or by he roughne of urface In he encloure wh hee propere any nal oun fel become ffue or homogeneou wh me Bu h conon no uffcen for he encloure wh a hgh nonunform rbuon of aborpon Sgnfcan evaon from a pure exponenal ecay law ake place n rongly (c) European Acouc Aocaon chaoc room wh aborpon localze n ceran regon of he room [5] An exreme cae a nonranomzng encloure wh nonunform rbuon of aborpon whou any caerng obacle an urface Boh propere can reul n grea evaon of oun ecay from he exponenal law For example [6] oun energy n a recangular room wh an aborbng celng proporonal o / akng no accoun caerng propere of encloure wall perm o acheve more ranomze oun fel In orer o ecrbe caerng by he urface a ffue reflecon coeffcen uually nrouce I convenen o eparae oun energy no wo componen [4,6] Fr of hem efne by pecular reflece oun energy Secon one efne by caere energy h approach evelope for cae when boh componen are efne by ffue fel an ecay n accorance wh exponenal law [7] Bu n nonranomzng encloure nonexponenal ecay of pecular energy have o be aken no coneraon In h work we coner a nonranomzng encloure by he example of a recangular paralleleppe wh pecular an ffue reflecng urface Fr of all we fn a ecay law for (c) European Acouc Aocaon, ISBN: , ISSN:

2 FORUM ACUSICUM Kanev: Soun ecay n a recangular room 7 June - July, Aalborg wh pecular an ffue reflecng urface energy of pecular reflecon an hen nvegae he nfluence of oun caerng by urface on ecay of oal oun energy Specular reflecng wall Le coner a recangular paralleleppe encloure wh menon L, D, H he begnnng of he coornae yem place n he encloure corner an he axe are rece along he encloure ege a hown n Fgure a Sx wall of he encloure are numerae n he followng way: a number of he wall lyng n he plane z H ; a number of he wall lyng n he plane z ; number of he wall lyng n he plane x L, x, y D, y are 3, 4, 5, 6 repecvely Aborbng propere of he encloure wall are characerze by pecular reflecon coeffcen, where he wall number We aume ha he wall are mooh an o no ffue oun So he aborpon coeffcen of he wall are equal o Inal oun fel n he encloure ffue In accorance wh Kuruff [3] mean ha a any pon n he encloure oun wave are ncen from all recon wh equal neny an ranom phae So a he momen he oun fel a uperpoon of ncoheren plane wave wh equal amplue unformly rbue on he reconal angle an Noe ha he plane wave can be replace by oun ray or oun parcle [4,5] bu he followng compuaon are correc for hem a well Soun energy n he encloure efne a um neny of all wave (or oun parcle) an gven by E A (,, ) co, () 4 where A (,, ) he amplue of he wave propagang n recon eermne by he angle an In orer o nvegae he oun ecay proce we apply he mage room meho [3] he encloure connuouly mrrore a whole a wall Image of he orgnal encloure fll whole pace whou leavng uncovere regon an whou any overlap he paern of he mage encloure n he plane xz hown n Fgure b an connue n a mlar manner n he y -recon perpencular o he rawng plane Inal oun fel mrrore n enre pace a well Fnally, we oban ffue oun fel n all pace wh he ame propere a nal oun fel n he orgnal encloure ha We uppoe ha an nal conon for he mrrore oun fel A (,, ), () In he propoe moel we nee no longer coner reflecon of oun from he wall Each reflecon ubue by he nerecon of he plane conng of he wall mage by he wave ravelng owar he orgnal encloure he ance beween he neghbour mage plane perpencular o he ax x ( y or z ) equal L ( D or H ) Afer any nerecon he wave loe a fracon of neny correponng o he wall aborpon coeffcen Soun energy n he encloure a he momen a) b) Fgure A recangular encloure (a) an mrrore encloure n he plane xz (b) 936 (c) European Acouc Aocaon, ISBN: , ISSN: -3767

3 FORUM ACUSICUM Kanev: Soun ecay n a recangular room 7 June - July, Aalborg wh pecular an ffue reflecng urface efne by he wave pang ance c owar he encloure, where c pee of oun In orer o fn neny reucon of he wave ue o wall aborpon we have o coun he nerecon wh he mage plane For he ake of mplcy we calculae oun energy a he pon (,,), e a he encloure corner he wave wh he reconal angle an ravel ance c n along he ax z n me Along he axe x an y ravel ance c co co an c co n repecvely he number of reflecon n from he wall wh number n me equal, ( c n,, ) n, (3) H 3,4 ( c n,, ) co co, (4) L 5,6 ( c n,, ) co n (5) D Ineny reucon of he wave ue o aborpon on he wall equal 6 n (,, ),, ) exp n (,, ) ln 6 oal oun energy can be foun from equaon by negraon by he angle an 6 n (,, )ln co E e (6) 4 Equaon 6 efne he energy ecay law n he encloure If all wall are aboluely reflecng an her reflecon coeffcen are equal o hen oun energy n he encloure reman conan Le u coner he encloure wh one aborbng wall Suppoe ha an Equaon 6 gve H E ~, c ln H (7) c ln Soun ecay nverely proporonal o me Equaon 7 conce wh Kuruff reul [3] for he mlar encloure Suppoe ha wo nonparallel wall can aborb oun wave If he reflecon coeffcen of he wall are equal, 3 an han from equaon 6 we can fn E 8HL ~, (8) ln ln ( c) 3 for H c ln an L c ln 3 Soun ecay nverely proporonal o quare me Soun energy ecay faer hen n he encloure wh one aborbng wall In general cae all wall can aborb oun wave For, 6 we can fn from Equaon 6 for H ( c ln ), L c ln 3 ) an D c ln 5 ) E ( 4 le l e lh he ( 6 h, (9) c c where ln 5 6, l ln 3 4, D L c h ln H We can ee from equaon 9 ha oun ecay no exponenal uner any aborpon rbuon on he wall Phycal ene of l, an h ha hey characerze exponenal ecay rae of oun propagang along he axe x, y an z repecvely hey epen on boh he ance beween parallel wall an he aborpon coeffcen of hee wall he um n equaon 9 can be eparae no hree umman each of hem ecrbe oun energy ecay along one of he ax Soun propagang along he ax wh maxmal exponenal ecay rae aborbe faer hen along wo oher axe Mnmum exponenal ecay rae eermne oun ecay a If l, h hen he lowe oun ecay along he ax x an he oun energy foun from equaon 9 a l e E () h Comparng equaon 7, 8 an we can ee ha he oun ecay become faer wh ncreang of number of aborbng wall A an example le u coner a recangular encloure wh maxmal menon H an oher menon D 7H, L 5H an nrouce menonle me c H for hree fferen rbuon of he oun aborpon on wall In fr cae all aborpon concenrae on one wall wh he number he aborpon coeffcen of h wall equal 7 In econ cae he aborpon rbue on wo nonparallel wall wh he number an 5 (c) European Acouc Aocaon, ISBN: , ISSN:

4 FORUM ACUSICUM Kanev: Soun ecay n a recangular room 7 June - July, Aalborg wh pecular an ffue reflecng urface her aborpon coeffcen 9 In hr cae he aborpon rbue on hree nonparallel wall whch aborpon coeffcen are equal 6 In all cae he mean value of he aborpon coeffcen are he ame an equal 8 Clacal Sabne law of oun energy ecay gven by cs E( ) exp, () 4V where V he encloure volume an S he area of all wall Decay curve for here aborpon rbuon on he wall calculae by equaon 6 an Sabne ecay curve calculae by equaon are hown n Fgure A we can ee he ecay curve for nonunform aborpon ffer rongly from he exponenal Sabne law Wherea he ecay curve for unform aborpon cloe o he exponen bu oe no conce wh a 3 Scaerng wall In orer o ake no accoun caerng propere of he wall we nrouce a ffue reflecon coeffcen whch efne a he rao of caere oun energy o ncen oun energy I connece wh he aborpon coeffcen an he pecular reflecon coeffcen by () Le u eparae oun energy n he encloure no wo componen Fr componen pecular energy E (), whch eermne by equaon 6 Secon componen ffue energy E (), whch forme by caere energy he nal oun fel efne only by pecular energy, ffue energy equal zero A every reflecon he fracon of ncen energy ranform no ffue energy from pecular energy Aborpon of ffue energy ecrbe by ornary exponenal law, whch correc for ffue oun fel Suppoe ha ffue energy ecay n accorance wh he Sabne law gven by equaon If afer any reflecon from he wall wh he number he wave wh reconal angle an ha energy E (,, ) efne by prevou pecular reflecon, hen before h reflecon pecular energy equal E (,, ) So caere energy urng h reflecon equal E (,, ) A he me nerval from o he number of reflecon from each wall equal n, where n n Accorng o equaon 3-5 n oe no epen on me Increae n ffue energy ue o he wave wh reconal angle an efne by all reflecon n he me nerval (, ) E (,, ) n (, ) E (,, (3) ) oal ncreae of ffue energy gven by negraon by all angle E E ( ), B e n (, )ln n (, ) co (4) In accorance wh he Sabne law ffue energy ecay a e, where 4V cs Reucon of ffue energy a he me nerval equal E E (5) From equaon 4 an 5 we can fn he fferenoal equaon for ffue energy E E, ) e B(, ) (6) wh he nal conon E ( ), (7) 4 Fgure Soun energy ecay n he recangular encloure wh menon H : D : L=:7:5 wh one aborbng wall (), wo aborbng wall (), hree aborbng wall (3) n comparon wh he Sabne law (4) (c) European Acouc Aocaon, ISBN: , ISSN: -3767

5 FORUM ACUSICUM Kanev: Soun ecay n a recangular room 7 June - July, Aalborg wh pecular an ffue reflecng urface where, ) n (, ) ln (, ) (, ) co 4 B n Soluon of equaon 6 an 7 gven by,, ) e E e B(, ) (8), ) oal oun energy n he encloure eermne by a um of equaon 6 an 8 E E E (9) In orer o fn ffue energy a we rewre equaon 9 for pecular energy n he followng way E E l ( l) E e h ( ) e lh E ( h) e l h () an nrouce value characerzng caerng propere of he wall c 3 4 l, L 3 4 c h H c 5 6, D 5 6 From equaon 8 one can fn a l ( ) ( l) E E ( ) e l ( ) E ( ) e h ( h) E ( ) e h () Noe ha equaon no correc f value cloe o l, or h We can ee from equaon ha ffue energy rongly epen on pecular energy In orer o ffue energy preval over pecular energy requre gnfcan value of l,, h an mall aborpon of ffue fel l,, h In he encloure wh menon H : D : L : 7 : 5 conere above uppoe ha only one wall ( ) can aborb oun an aborpon an reflecon coeffcen are equal 7, 3, Four wall caer oun whou aborpon ( , ) he wall wh he number oe no aborb an caer oun, e Fgure 3 how reul of calculaon of pecular energy E () efne by equaon 6, ffue energy E () efne by equaon 8 an oal energy E () efne by equaon 9 for four value of he ffue reflecon coeffcen, 5,, Wh ncreang of caerng on he wall ffue energy ncreae n relaon o pecular energy an he ecay rae of oal energy ncreae a well If he ffue reflecon coeffcen grea enough ( ) ffue energy omnae an he oun ecay law en o he Sabne law When caerng on he wall mall ( 5 ) oal energy efne only by pecular energy an oun caerng equvalen o oun aborpon We ee n Fgure 3 pecular energy an ffue energy are approxmaely equal a Le u emae value of he ffue reflecon coeffcen whch prove he mlar ecay law for boh energe, e E ~ E a n he conere encloure wh one aborbng wall an four caerng wall perpencular o aborbng one he malle exponenal ecay rae of pecular energy componen gven by equaon l cln 3 4 L Becaue of 3 4 he exponenal ecay rae of pecular energy equal l c L n cae of mall caerng he exponenal ecay rae of ffue energy characerze by 4V cs In cae of 3 H ~ L ~ D we can emae V ~ L an S ~ 6L So ~ L c Specular an ffue energe are approxmaely equal f her exponenal ecay rae are approxmaly equal a well l ~ Subung here emaon for l an we can fn he followng conon ~ (3) Equaon (3) allow emang he ffue reflecon coeffcen eenal for gnfcan oun fel ffuon n he encloure For example n a room wh an aborbng celng ffue energy gnfcan n comparon wh pecular energy f he ffue reflecon coeffcen of wall equal or greaer hen - (c) European Acouc Aocaon, ISBN: , ISSN:

6 FORUM ACUSICUM Kanev: Soun ecay n a recangular room 7 June - July, Aalborg wh pecular an ffue reflecng urface E ( ), B E ( ), B E ( ), B E ( ), B Fgure 3 Decay of pecular energy, ffue energy an oal energy n he recangular encloure wh one aborbng wall an four caerng wall for fferen value of he ffue reflecon coeffcen E () ( ), E () ( ), E () ( ), Sabne law ( ) 4 Concluon An analycal moel of oun ecay n a recangular encloure propoe In he moel energy of peculary reflece oun an energy of oun caere by encloure wall are calculae eparaely Decay of pecular energy nonexponenal uner any aborpon rbuon akng no accoun he caerng propere of wall perm o acheve an exponenal ecay law I hown ha uner nonunform aborpon rbuon ffue energy gnfcan f he ffue reflecon coeffcen of he wall cloe o he average aborpon coeffcen Applcaon of he reverberaon me correc only for exponenal ecay Wherea oun ecay n a recangular encloure wh pecular reflecng wall no exponenal So eem ha reverberaon me formula ung only aborpon coeffcen are unjufe n calculaon for recangular encloure an pobly for he encloure wh a poor ranomzng rengh Reference [] WC Sabne: Collece Paper on Acouc Pennula, Lo Alo, CA, 99 [] H Arau-Puchae: An mprove reverberaon formula Acuca 65 (988) 63 8 [3] H Kuruff: Room Acouc Elever Apple Scence, Lonon, 99 [4] WBJoyce: Exac effec of urface roughne on he reverberaon me of a unformly aborbng phercal encloure J Acou Soc Am 64 (978) [5] F Moreange, O Legran, D Sornee: Role of he aborpon rbuon an generalzaon exponenal reverberaon law n chaoc room J Acou Soc Am 94 (993) 54 6 [6] RN Mle: Soun fel n a recangular encloure wh ffuely reflece bounare J Soun Vb 9 (984) 3 6 [7] Hanyu: A heorecal framework for quanavely characerzng oun fel ffuon bae on caerng coeffcen an aborpon coeffcen of wall J Acou Soc Am 8 () (c) European Acouc Aocaon, ISBN: , ISSN: -3767