PRESSURE RESPONSE OF INE ARRAYS Audio Engining Socit Convntion Pap 5649 Psntd at th 113th Convntion Octob 5-8 os Angls, CA. USA This convntion pap has bn poducd fom th autho's advanc manuscipt, without diting, coctions, o considation b th Rviw Boad. Th AES taks no sponsibilit fo th contnts. Additional paps ma b obtaind b snding qust and mittanc to Audio Engining Socit, 6 East 4 nd Stt, Nw Yok, Nw Yok 1165-5, USA; also s www.as.og. All ights svd. Rpoduction of this pap, o an potion thof, is not pmittd without dict pmission fom th Jounal of th Audio Engining Socit. Pssu Rspons of in Soucs Mak S. Uda JB Pofssional, Nothidg, CA USA ABSTRACT Th on-axis pssu spons of a vtical lin souc is known to dcas at 3dB p doubling of distanc in th na fild and at 6dB in th fa fild. Th psnt pap shows that th convntional mathmatics usd to achiv this sult undstats th distanc at which th -3dB to -6dB tansition occus. An xamination of th pssu fild of a lin souc vals that th na fild xtnds to a gat distanc at positions latall displacd fom th cntlin, nomal to th souc. Th pap intoducs th ndpoint convntion fo th pssu spons and compas th on-axis spons of staight and hbid lin soucs. 1. INTRODUCTION Vtical lin aas of loudspaks continu to gow in populait. This is du in pat bcaus loudspak manufactus hav cntl undtakn dvlopmnt of sstms dsignd spcificall to optimiz th pfomanc of lin aas. Vtical lin aas povid naow vtical pola spons, high dictivit indx, and smooth hoizontal covag compad to hoizontal aas of loudspaks. Th dictional pfomanc of lin aas is wll documntd [1,,3]. Th litatu show that th pola spons naows both with incasing lngth and fqunc. Ths lationships a basd on analss of idalizd lin soucs a. In gnal, th nw gnation of lin aa loudspak sstms povids an impovd appoximation to a continuous lin souc than thi pdcssos. A viw of th litatu vals that th pola spons of a lin souc is gnall divd und fa fild conditions. This qualification simplifis th mathmatics and povids closd-fom solutions fo a lin souc s dictivit function and quat-pow angl. Fo pactical puposs, th sults obtaind a usful. This is bcaus fa fild conditions a mt fo man nominal-sizd lin aas ov a boad fqunc ang. Howv, whn v long aas a usd in clos poximit and/o at v high fqunc, it is impotant to consid at what distanc fa fild conditions a obtaind. Th fa fild is gnall accptd as th gion in which th pssu spons dcass at 6dB p doubling of distanc. Th point at which this occus is commonl (and confusingl b ) fd to as th citical distanc. Up to this point, that is, btwn th souc and th citical distanc, th on-axis pssu spons undulats and gnall dcass at 3dB p doubling of distanc. a This pap uss th convntion that a lin souc is an idalizd, continuous acoustic adiato whil a lin aa is an aa of loudspaks aangd to appoximat a lin souc. oudspaks and nclosus hav inhnt fqunc spons, dictional chaactistics, and diffaction and shadowing ffcts, all which modulat th thotical spons of a pu lin souc. b Th citical distanc is also dfind in th contxt of achitctual acoustics as th point at which th dict sound and th vbant sound a at th sam lvl. S Davis, Sound Sstm Engining, Howad W. Sams and Co., Indianapolis, IN., 1975. AES 113 TH CONVENTION, OS ANGEES, CA, USA, OCT 5-8 1
PRESSURE RESPONSE OF INE ARRAYS l Na fild Citical distanc Midpoint Path 4 Fa fild 6 in souc Figu 1: Gomtic constuction fo calculating th on-axis pssu spons of a lin souc (midpoint convntion). 8 1 1 1 1. 1 3 1. 1 4 Distanc (mts) Figu : On-axis pssu spons (fom th midpoint) of a unifom lin souc (4m, 8kHz). Bond ths gnalizations, th litatu offing a dtaild analsis of a lin souc s on-axis pssu spons is somwhat obscu. Txts gnall povid th on-axis spons of piston soucs but not of lin soucs. Som stimat th distanc to th fa fild of linlik o pistonlik soucs [4], but do not povid th on-axis spons p s. Rcntl, a fw paps [5,6] hav dscibd with mo spcificit how th on-axis pssu spons vais with souc lngth and fqunc. In addition, th show how th on-axis pssu spons of a continuous lin souc is affctd b gaps and oth discontinuitis. A common convntion among th litatu is that th onaxis pssu spons, whth fo a piston o lin souc, is takn along a path nomal to th souc bginning at its midpoint. Th psnt pap shows that this convntion potntiall undstats th distanc to th fa fild. Futhmo, man of th gnalizations gading th bhavio of th pssu spons a uniqu to th on-axis path fom th midpoint. Finall, man lin aas usd in pactic toda a not staight lin aas at all. Th a oftn cuvd slightl, gnall at th low nd. This pap concluds b diving th pssu spons of sval hbid lin soucs and compas thm to staight-lin soucs.. ON AXIS RESPONSE OF A UNIFORM INE SOURCE - CONVENTIONA METHOD W will bgin with a bif viw of th on-axis spons of a lin souc. It is obtaind at an abita distanc b summing th pssu adiatd along th souc. Figu 1 shows th gomtic constuction usd to solv fo th pssu along a path nomal to th souc, bginning at its midpoint. Rfing to Figu 1, is th total lngth of th souc and is th distanc fom an adiating lmnt l of th souc to an point along th path. Assuming unifom amplitud and phas along th lin souc c, th pssu at is givn b wh p mid ( ) / / '(, + jk'(, dl '(, and p mid fs to th pssu along a path bginning at th midpoint of th lin souc. Th pssu spons is th logaithmic atio of th magnitud of th pssu squad at ov th magnitud of th pssu squad at som fnc distanc, i.. p( ) Pssu spons( ) log. p( ) l f Th on-axis pssu spons of 4-mt lin souc at 8kHz is shown in Figu. Th pssu spons gnall dcass at a at of 3dB p doubling of distanc out to appoximatl 1 mts. It xhibits undulations acoss this gion, th magnituds of which incas as th distanc c Th gnal fom of th pssu spons is p( ) / / A( '( j( k'( + φ ( ) dl wh A( and ( a th amplitud and phas distibutions along th souc. In th psnt pap onl unifom lin soucs a valuatd. Unifom lin soucs st A( 1 and (. AES 113 TH CONVENTION, OS ANGEES, CA, USA, OCT 5-8
PRESSURE RESPONSE OF INE ARRAYS m p 5Hz 4m khz 4 8m 4 8kHz 6 6 8 1 1 1 1.1 3 1.1 4 Distanc (mts) Figu 3: On-axis spons (fom th midpoint) of a, 4, and 8-mt long unifom lin souc at 8kHz. Th 4 and 8-mt spons cuvs a offst b 1dB and db spctivl. 8 1 1 1 1. 1 3 1. 1 4 ( ) Figu 4: On-axis spons (fom th midpoint) of a 4-mt unifom lin souc at 5Hz, khz, and 8kHz. Th khz and 8kHz spons cuvs a offst b 1dB and db spctivl. appoachs 1 mts. Bond 1 mts, th pssu amplitud no long undulats and dcass at 6dB p doubling of distanc. Th point at which this tansition occus is fd to as th citical distanc. Th gion btwn th souc and th citical distanc is fd to as th na fild and th gion bond is th fa fild. and p nd ( ) / / jknd nd dl Th citical distanc vais with souc lngth and fqunc. Figu 3 shows th on-axis spons of th diffnt lngth lin soucs at 8kHz. As lngth incass, th citical distanc incass. Figu 4 shows th on-axis spons of a 4-mt long aa at 5Hz, khz, and 8kHz. It shows that th citical distanc also incass with fqunc. Figus, 3, and 4 summaiz th common wisdom gading th on-axis pssu spons of lin aas. It holds that th spons dcass at 3dB in th na fild and at 6dB in th fa fild. Ths gnalizations, howv, significantl undstat th complxit of a lin souc s adiatd pssu fild. Th nxt sctions of this pap dscib som of ths complxitis. 3. MIDPOINT VERSUS ENDPOINT Th convntional appoach to dtmining th pssu spons of a lin souc is to tak a path nomal to th souc bginning at its midpoint. This ilds a sult, howv, that is uniqu to this path. Choosing th midpoint as a stating point minimizs th appant aptu of th souc. If th oigin fo th path is chosn at a point latall displacd fom th midpoint, th lin souc appas long in on diction than th oth. This appant lngth is maximizd at th ndpoint of th souc. If th pssu is summd along a path nomal to th souc bginning at th ndpoint of th souc, th citical distanc movs futh out. Figu 5 shows a modifid gomtic constuction fo calculating th pssu spons. Th pssu, summd along a path nomal to th ndpoint of a unifom lin souc is nd + l +. This modifid xpssion fo captus th full aptu of l in souc x path Figu 5: Gomtic constuction to calculat th pssu spons along a path oiginating at th ndpoint. AES 113 TH CONVENTION, OS ANGEES, CA, USA, OCT 5-8 3
PRESSURE RESPONSE OF INE ARRAYS Midpoint / (Midpoint) 4 Endpoint 4 3/ 6 6 8 1 1 1 1. 1 3 1. 1 4 8 1 1 1 1 x. 1 3 1. 1 4 Figu 6: Compaison of midpoint and ndpoint pssu spons of a 4-m lin souc at 8kHz. Figu 7: Pssu spons along paths nomal to th lin souc at vaious points of oigin. (4-mt, 8kHz) th lin souc. Figu 6 compas th on-axis spons of a 4-mt long lin souc at 8khz using th midpoint and ndpoint convntions. Th last pak of th midpoint na fild spons occus at appoximatl 1 mts. Th last pak of th ndpoint spons occus at appoximatl 4 mts. Not that ths spons cuvs a not offst b som abita amount as in Figus 3 and 4. Th ndpoint spons is at a low lvl than th midpoint spons in th na fild. In th fa fild, th cuvs convg and ild th monotonicall 6dB p doubling of distanc fall-off at dscibd abov. If th oigin of th path is movd latall bond th ndpoint, th distanc to th last pak occus at gat and gat distancs. This indicats that th distanc to th fa fild continus to incas as on chooss a path futh off th midpoint axis. Howv, th pssu lvls fall off quit damaticall onc th ndpoint is bchd. This is shown in Figu 7. Th fist two cuvs labld and / a th midpoint and ndpoint pssu sponss shown in Figu 6. Th nxt two cuvs a on-axis pssu spons cuvs bginning at points on lngth and th-halvs lngths latall displacd fom th midpoint. In this xampl, th lngth of th souc is 4 mts. Th spons labld is th on-axis pssu spons, nomal to th souc, bginning at a point 4 mts fom th midpoint of th souc. Th spons labld /3 stats at 6 mts. Th cuvs in Figu 7 a slics of th pssu fild, nomal to th souc, at incasing latal distancs fom th midpoint. Th illustat two impotant points. Fist, as th oigin of th path is movd latall off th midpoint, th magnitud of th pssu dcass. It is onl 6dB at th ndpoint, but appoximatl 4dB at and 3/. Scond, th pssu spons cuvs and 3/ undulat in a fashion that is nal flat fom th souc to th fa fild. Th and 3/ spons cuvs a so low in lvl lativ to th midpoint spons that th will hav littl impact on th ovall pssu fild. Howv, th ndpoint cuv is a matial fatu of th pssu fild and should b considd b sound sstm dsigns whn using vtical lin aas. 4. PRESSURE FIED OF A VERTICA INE SOURCE Anoth wa to obsv th pssu spons of a lin souc is to comput its nti pssu fild. This is obtaind b witing th xpssion fo th adiatd pssu in tms of Catsian coodinats. Th pssu at an point (x,) is wh p x, / jkx, / x, dl x, x + ( Figu 8 shows th pssu fild of a 4-mt lin souc at 8kHz. Th souc is locatd in th upp lft con (not shown) facing to th ight. Th x-axis in Figu 8 d is coincidnt with a path, nomal to th souc, stating at its midpoint. Onl th low quadant of th pssu fild is shown. Th quadant dictl abov would b a mio imag smmtical about th x-axis. Figu 8 shows th contxt in which th slics of Figu 7 a obtaind. Th pssu undulations incas in numb and dcas in magnitud as th slic movs latall fom th midpoint. Also, th undulations continu out to a gat distanc at lag latal displacmnt. d Th pssu fild is plottd on a log-log gid. This is to accommodat th lag distancs of intst but maks th main lob of th pssu fild appa wid than it would in oth fomats. AES 113 TH CONVENTION, OS ANGEES, CA, USA, OCT 5-8 4
PRESSURE RESPONSE OF INE ARRAYS 1 1 1 1 1 x 1 θ R φ A Ac souc 1 (,) path Figu 8: Pssu fild of a 4-m lin souc at 8kHz. Th souc is locatd in th upp lft con (not shown) facing to th ight. Th x- axis in th figu is coincidnt with a path, nomal to th souc, stating at its midpoint. Figu 9: Gomtic constuction of an ac souc. As w concludd in th pvious sction, th pssu undulations at lag latal displacmnts ma b immatial. Howv, th undulations along th ndpoint path a onl -6 to -1 db fom th midpoint spons and will hav a matial influnc on th sound fild. This mans, conta to convntional wisdom, that th pssu in th fa fild dos not monotonicall dcas vwh past th midpoint citical distanc. 5. DISTANCE TO THE FAR FIED - ENDPOINT CONVENTION Sval paps and txts on lin aas off fomula fo stimating th distanc to th fa fild [7,8,9]. ik th onaxis spons, ths a basd on a midpoint convntion. Th pvious sction shows that this undstimats th xtnt of th na fild. In this sction, w will wit th fomula consistnt with th ndpoint convntion. Th distanc to th fa fild of a lin souc can b stimatd b dtmining at what distanc th popagation distanc fom th nds of a lin souc a within on-quat wavlngth. That is, fing back to Figu 5, wh λ ' 4 Solving fo th citical distanc c w gt c λ λ 8 As a pactical matt, th scond tm of this xpssion can b doppd with v littl impact on th stimat of th citical distanc. Also, it is oftn usful to xpss c in tms of fqunc ath than wavlngth. This lavs c.6 f wh is in mts and f is in Hz. 6. ON-AXIS PRESSURE RESPONSE OF A CURVED SOURCE As mntiond ali, man lin aas of loudspaks usd in pactic a actuall cuvd. This is bcaus pu staightlin aas of nominal lngth poduc an xtml naow vtical pola spons oftn too naow to ach audincs bnath and slightl in font of th aa. A slightl cuvd aa povids supio covag in this aa. On tp of souc that povids a wid covag angl is an ac souc. It is compisd of adiating lmnts aangd along a sgmnt of a cicl. At all fquncis, it povids a wid dictivit spons than a staight-lin aa of th sam lngth. At high fqunc, it povids a pola pattn cosponding to th includd angl of th ac. Th on-axis pssu spons of a cuvd souc can b xpssd in a fom simila to th ali xpssions fo staight-lin soucs. Figu 9 shows th gomtic constuction of an ac souc with adius R and total includd angl. Th ac is facing downwad, that is, is masud downwad fom th hoizontal. Th pssu, using th ndpoint convntion, of a unifom ac souc at distanc is This shows that th distanc to th fa fild is pimail a function of th squa of th souc lngth. Using an xampl fom abov, th citical distanc fo a 4-mt lin souc at 8kHz is 74 mts. Rfing to Figu 6, th ndpoint on-axis spons cuv shows that this is a good stimat of th point wh th fa fild gion bgins. wh θ jka pa( ) dφ A AES 113 TH CONVENTION, OS ANGEES, CA, USA, OCT 5-8 5
PRESSURE RESPONSE OF INE ARRAYS 4 6 4 6 1 m (15 ) m (3 ) 4m (6 ) 8 1 1 1 1. 1 3 1. 1 4 x Figu 1: On-axis pssu spons of an ac souc and a staight lin souc at khz. Th lin souc has a lngth of 4 mts. Th ac souc is 3dg and a adius of 8 mts poviding an quivalnt total lngth of 4m. A [( + R( 1 cosφ )) R (sinθ sinφ) Th path is chosn to bgin at th bottom of th ac bcaus all adiating lmnts d hav cla lin-of-sight to an summation point at distanc. A path bginning at th top of th ac would poduc nonsnsical sults fo lag and small as would intsct th souc. Also, th oigin (,) is chosn to b coincidnt with th lading dg of th ac at th top. This facilitats compaing th on-axis sponss of staight lin and ac soucs. Figu 1 povids a compaison of th pssu spons of quivalnt-lngth staight lin and ac soucs. Though th ac souc is cuvd onl 3 dgs, its on-axis spons is smooth in th na fild than th on poducd b th lin souc. ik a lin souc, th pssu spons of an ac souc changs with ac lngth and fqunc. Figu 11 shows th on-axis spons of th ac soucs of vaious lngths. Th diffnt lngths a povidd b a constant adius (4m) and diffnt includd angls. Th citical distanc incass with ac lngth as xpctd. Figu 1 shows how th pssu spons changs with fqunc. As with lin soucs, th citical distanc incass with fqunc. Not that th tansition fom th na fild to th fa fild is gnall smooth fo an ac than fo a lin souc at all lngths and fqunc. 7. ON-AXIS RESPONSE OF A J-SOURCE A J-souc [1] a hbid lin souc compisd of lin souc and an cuvd (ac) souc. In sound infocmnt applications, th staight sgmnt is locatd abov th cuvd sgmnt and is intndd to povid th long thow componnt of th pola spons. Th cuvd potion is intndd to povid covag in th aa blow and in font of th aa. + ] 1 8 1 1 1 1. 1 3 1. 1 4 Figu 11: On-axis pssu spons of th ac soucs (θ 15, 3 and 6 ) at 8kHz wh R4m. Ths cospond to ac lngths of 1,, and 4 mts. Togth, th sgmnts povid an asmmtic pola spons in th vtical plan. Th on-axis pssu spons of a J-souc is obtaind b combining th pssu spons functions of th lin and cuvd soucs psntd abov. Th gomtic constuction is shown in Figu 13 wh is th lngth of th staight sgmnt and R and spcif th cuvd sgmnt. Basd on this gomt, th pssu adiatd fom a unifom J-souc is 4 6 8 5Hz khz 8kHz 1 1 1 1. 1 3 1. 1 4 x Figu 1: On-axis pssu spons of th ac soucs at vaious fquncis wh θ 6 and R 4m. Th khz and 8kHz cuvs a offst b 1 and db spctivl. AES 113 TH CONVENTION, OS ANGEES, CA, USA, OCT 5-8 6
PRESSURE RESPONSE OF INE ARRAYS J souc in θ R φ (,) J Figu 13: Gomtic Constuction of th Pssu Fild of a J-Souc path 4 6 8 J-souc 1 1 1 1. 1 3 1. 1 4 Figu 14: On-axis pssu spons of a J- souc wh m, Rm, θ6 and fkhz. wh p ( ) J jk dl + R θ + R sinθ + jka A ( dφ wavlngth of intst. Th total numb of lmntal lngths M is M and th incmntal angl btwn th lmnts is and A R ( + R(1 cosφ)) + (sinθ sinφ) Figu 14 shows th on-axis pssu spons of quivalntlngth staight lin and J-soucs. Th staight sgmnt of th J-souc dominats th spons, poducing undulations in th na fild v simila to thos of th staight lin souc. Howv, th on-axis aptu of th J-souc is foshotnd lativ to th qual-lngth lin souc so th distanc to th fa fild is maginall shot. 8. ON-AXIS RESPONSE OF A SPIRA SOURCE ik th J-souc, a spial souc [11] povids an asmmtic pola spons in th vtical plan. Howv, unlik th J-souc, it is a continuous cuv ath than two distinct sgmnts. Th cuvatu incass with distanc along th cuv. This sults in an upp potion that is lagl - but not pfctl - staight, and a low potion that is cuvd downwad. Spial soucs povid pola spons cuvs that a makabl constant with fqunc. A spial souc is full dfind b its lngth and tminal angl. Th pssu spons is dtmind b summing th adiation fom ach lmntal lngth along th souc. is chosn to b a small faction of th shotst Ω M ( M + 1) ψ. Th spial can thn b xpssd in paamtic fom as and x( s) s η ( s) + sin s η [ 1 η ( η + 1 ) ψ ] cos [ 1 ( η + 1 ) ψ ] η. Th coss-sction of a spial souc is shown is Figu 15. Its on-axis pssu spons is p ( ) S M σ jks S Fo instanc, to obtain th dictivit function up to 4kHz, should b lss than ¼ wavlngth, i.. appoximatl. mts. AES 113 TH CONVENTION, OS ANGEES, CA, USA, OCT 5-8 7
PRESSURE RESPONSE OF INE ARRAYS Spial souc in (x(s), (s)) S 4 Spial Ω path 6 1 1 1 1 ξ. 1 3 1. 1 4 wh Figu 15: Gomtic constuction of a spial souc, S ( x( σ )) + ( ( M ) ( σ )). Figu 16 compas th on-axis spons cuvs of quivalnt-lngth staight-lin and spial soucs. Ths cuvs show that th spial souc has ducd undulations in th na fild and a smooth tansition fom th na fild to th fa fild. Coupld with its favoabl pola spons chaactistics, a spial souc povids an xcllnt gomt upon which to bas loudspak aas fo sound infocmnt applications. 9. SUMMARY This pap shows that th convntional mthod fo calculating th citical distanc of lin soucs undstats th xtnt of th undulations in th na fild. Th convntional mthod taks a path nomal to th souc bginning at its midpoint and povids a limitd pictu of th adiatd pssu fild. Choosing th ndpoint as th oigin of th path taks th full aptu of th lin souc into account and shows that th pssu undulations chaactistic of th na fild continu wll past th midpoint-divd citical distanc. This sult is impotant to sound sstm dsigns bcaus th will not achiv a monotonicall dcasing pssu spons vwh past th midpoint citical distanc. Th ndpoint convntion is applid to th analsis of hbid lin soucs tps including ac, J, and spial. Ths gnall povid a smooth tansition fom th na fild to th fa fild than staight lin soucs. 1. REFERENCES 1 Wolff, I., and Malt,., Jou. Acous. Soc. Am., Vol., No., p. 1, 193. Th sminal wok on lin aas. Figu 16: Pssu spons compaison of a 45 tminal angl, 4 mt long spial souc and lin souc of th sam lngth at khz. Authos div dictivit functions fo lin soucs and cuvd soucs. S appndix of th pap fo dtails Olson, H. F., Elmnts of Acoustical Engining, 1st d., D. Van Nostand Compan, Inc., Nw Yok, 194. Autho shows th pssu spons of lin soucs (p. 4) and cuvd soucs (p.5). 3 Bank,.. Acoustics, fist dition, McGaw-Hill, 1954. Autho shows pola spons (in db) of lin soucs (p. 96) and cuvd soucs (p. 16). 4 Kinsl and F, Fundamntals of Acoustics, thid dition, John Wil and Sons, 198. 5 Hil, C. Sound Filds Radiatd b Multipl Sound Souc Aas, Psntd at th 9nd Convntion of th Audio Engining Socit, 199 Mach, Vinna, ppint 369. Autho discusss th spons of lin aas in th na fild (Fsnl gion) and th fa fild (Faunhof gion). 6 Uda, Mak, in Aas: Tho and Applications, Psntd at th 11th Convntion of th Audio Engining Socit, 1 Ma, Amstdam, ppint 534. S pag 8 fo analsis of gaps in continuous lin soucs. 7 Smith, D.., Disct-Elmnt in Aas Thi Modling and Optimization, J. Audio Eng. Soc., Vol. 45, No. 11, 1997.. 8 Kinsl and F, Fundamntals of Acoustics, thid dition, John Wil and Sons, 198. Pag 188 shows a simila divation using ½ wavlngth as th cition 9 Uda, Mak, in Aas: Tho and Applications, Psntd at th 11th Convntion of th Audio Engining Socit, 1 Ma, Amstdam, ppint 534. S pag 6 fo fa fild quation. 1 Uda, Mak, J and Spial in Aas, Psntd at th 111 th Convntion of th Audio Engining Socit, 1 Dcmb, Nw Yok, Nw Yok, ppint 5485. 11 IBID. AES 113 TH CONVENTION, OS ANGEES, CA, USA, OCT 5-8 8