NewYork. the 107th Convention 1999 September AN AUDIO ENGINEERING SOCIETY PREPRINT

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1 A Generalzed Horn Desgn to Optmze Drectvty Control & Wavefront Curvature 5016 (F-6) Charles E. Hughes Peavey Electroncs Merdan, MS, USA Presented at the 107th Conventon 1999 September NewYork ^UDIO Ths preprnt has been reproduced from the author's advance manuscrpt, wthout edtng, correctons or consderaton by the Revew Board. The AES takes no responsblty for the contents. Addtonal preprnts may be obtaned by sendng request and remttance to the Audo Engneerng Socety, 60 East 42nd St., New York, New York , USA. All rghts reserved. Reproducton of ths preprnt, or any porton thereof, s not permtted wthout drect permsson from the Journal of the Audo Engneerng Socety. AN AUDIO ENGINEERING SOCIETY PREPRINT

2 A Generalzed Horn Desgn to Optmze Drectvty Control & Wavefront Curvature Charles E. Hughes Peavey Electroncs Merdan, MS A new horn desgn s presented Ths approach yelds good loadng characterstcs and reduced harmonc dstorton, The new hornspolar patterns are that of a constant drectvty type horn. The novel feature of ths new horn s thatts apparent apcesfor the horzontal and vertcal planes are n the same physcal locaton regardless of coverage angle for the horzontal or vertcal plane. 0. Introducton Hems have been used n loudspeaker systems for decades. Ther use can be attrbuted to two man factors: 1) ncreased output for a gven nput and 2) drectvty control. The latter has been seen to be the more domnant reason n recent years. A good degree of loadng of the loudspeaker drver must stll be accomplshed for the horn to be useful. However, wth hgher power handlng and ncreased effcency avalable from typcal compresson drvers today, horn desgners can concentrate more on the drectvty response of a horn. In recent years Geddes I and Putland2have put forth works on hems that have the propertes of propagatng a oneparameter acoustc wave. The moton of such a wave can be descrbed by a sngle patal coordnate. These types of hems have been referred to as wavegudes. A good dfferentaton between a horn and awavegude can be thought of as a hem beng prmarly concerned wth the optmal loadng of ts drver, whle a wavegude s prmarly concerned wth ts drectonal characterstcs 3. There are a several terms that wll be used later. Ther defnton s as follows. r m radus of connectng arc Yt =- dameter of throat entrance rh _- radus of horzontal plane connectng arc rv -=radus of vertcal plane connectng arc tg - horzontal coverage half - angle 0 v --- vertcalcoverage half - angle 1. Constant Drectv_, In 1975 Keele4 outlned a concept whereby a horn could be constructed of essentally separate, but joned, sectons. The frst secton of ths type of horn has a cross sectonal area that expands exponentally. The second secton has a cross sectonal areathat expands n a manner smlar to that of a smple cone. Ths combned horn had two very sought arer trats. It presented good loadng to the drver to whch t was attached allowng for an ncrease n the effcency ofthe hem. It also mantaned good drectvty control over a wde range of frequences.

3 Up untl ths tme the exponental horn was a manstay of the ndustry. It loaded the drver to whch t was attached very well. Ths gave t good effcency to reasonably low frequences. Its prmary drawback was that as frequency ncreased, the drectvty, or coverage pattern, of the horn would start to narrow. Ths was very undesrable because as lsteners became more off-axs to the horn they would not receve as much hgh frequency nformaton. Keele's horn gave good loadng and a more consstent beamwdth or drectvty, hence the term Constant Drectvty. In 1977 Henrcksen & Ureda5 ntroduced what they called the Manta-Ray horn. It was named for ts shape. Ths horn had good loadng as well as good drectvty control. However, t dd suffer n some areas. One n partcular was that t had rather severe astgmatsm n the curvature of ts wave front. Put smply, the radus of curvature for the wavefront s dfferent for the horzontal and vertcal planes. Accordngly, the shape of the wavefront from such a devce can, at best, be ellpsodal. Whle ths doesn't seem to he cause for concern when the horn s used by tself, when more than one horn s employed n an array t can be problematc. Ths astgmatsm s common to a number of horn desgns wthn the ndustry today. In addton to the astgmatsm, these types of horns suffered n other areas as well. The slowly expandng exponental secton can tend to cause dstorton as hgh sound pressure levels are reached. The dscontnutes n some of the horns, where the sectons are connect, can cause reflectons back down the horn that can be problematc. These dscontnutes also cause dffracton, whch ntroduce another set of dstorton products. Walls that are parallel, or nearly parallel, n the throat secton of some of these horns can lead to unwanted resonance condtons. 2. Solvng the Problem of Astgmatsm Most conventonal constant drectvty type horns have at least one tem n common. That s the man wavegude secton of the horn s concal n nature. That s to say that t has _redomnately straght walls. Such ahorn falls nto the category of admttng and propagatng a one-parameter wave. Ths s probably the reason for the constant coverage of such a devce. As such, the man body of ths new wavegude wll also be comprsed of straght walls. A soluton was sought for the astgmatc wavefront that s typcal for conventonal constant drectvty horns. Snce ths problem s caused by ahorn havng dfferent ponts from whch the wavefront appears to orgnate n the horzontal and vertcal planes, one must smply make the pont the same for both coverage planes. Ths pont was located n the center of the throat entrance to the wavegude as shown n fgure 1. The challenge now was how to jon the requred throat dameter to the straght walls of the requred desgn coverage angles. A crcular arc tums out to perform ths functon qute ncely. Ths arc s from a pont on the permeter of the throat to the straght wall of the wavegude. The radus of the arc s such that one end of the arc s tangent to the straght wal at one pont whle the other end of the arc s perpendcular to the plane of the throat at the permeter of the throat entrance. Ths can be seen n fgure 2. The new wavegude can be thought of as beng constructed n dstnctly separate, but joned sectons. The frst secton s the throat secton that s comprsed of the connectng arc descrbed above. The last secton s comprsed of straght walls n order to obtan the desred drectonal propertes. Snce dfferent coverage angles are typcally requred for the horzontal and vertcal planes, the throat secton becomes subdvded nto two sectons. The frst of these sectons wll be where, n both the horzontal and vertcal coverage planes; the connectng arcs defne the wall shapes. The second of these sectons wll be where only one of the coverage planes, ether horzontal or vertcal has a connectng arc to defne ts wall shape. The other coverage plane n ths secton has ts wall shape defned by a straght wall at the desgn coverage angle. A method had to be mplemented to gofrom the crcular throat entry to the rectangular cross sectonal shape of the man body of the wavegude. The throat secton can be made to change from crcular to rectangular by allowng ellptcally shaped fllets to develop n the comers. These ellptc comer fllets begn as a crcle at the throat; the horzontal radus equals the vertcal radus. As the length down the horn ncreases the comers rad become smaller so that they attan ther fnal desred value at the pont where that secton termnates nto the straght wall secton. The fnal value of the comer fllet s arbtrary and can be chosen based on aesthetcs or manufacturng constrants. Snce the horzontal coverage angle s dfferent from the vertcal coverage angle, the horzontal radus of the comer

4 fllet wll change at a dfferent rate than the vertcal radus of the corner fllet. Ths s what gves rse to the ellptcal nature of the fllets. An ntal prototype was constructed at ths pont to test the performance. The results were extremely promsng. Ths prototype exhbted good ampltude response, good loadng characterstcs and very good drectonal characterstcs. 3. Development of the Desgn Equatons To analyze the new desgn, equatons descrbng the cross sectonal area expanson wthn the throat secton had to be developed. A relatonshp between known desgn parameters and the connectng arc, r, were also needed. To smplfy matters we wll only concern ourselves wth the axsymmetrc, or crcular, case for the tme beng. Referrng to the fgure 3. a2(r+y') 2 Ir+y,' -_- +r-- _ - --_ l+tan20 Equaton (1) gves us the relatonshp between the radus of the connectng arc, r, the throat dameter, y,, and the desgn coverage angle, O. To derve an expresson for the heght, or dameter, wthn the throat secton, let h he the ncremental heght dfference from the throat entry dameter, yt, and the wavegude boundary. y= y, +2h (r-h) 2 =r 2-x 2 h=r_x_r2_x 2 y = y, + 2(r- r2x/-_-x2 ) (2) Equaton (2) gves us an expresson for the heght wthn the throat secton of the wavegude. We may now proceed to the dervaton for the cross sectonal area expanson. Ths reduces to S = Ax 2 + B_r 2 - x 2 + C

5 Equaton (3), whle not n the classcal form of a quadratc equaton, does posses the trats of havng an x_,an x and a constant term. As such, ths new wavegude desgn has been dubbed a Quadratc Throat wavegude. For the non-axsymmetre cases the expansons do not lend themselves to ths level of algebrac reducton. Hence ther expanson s gven by - From equaton (1) rh=_( I_0 H -1) rv=_-(l_tan 20 v -1) Equaton (4) s the expanson for the rectangular case that was used for the development of the wavegudes presented n the paper. 4. Analyss of the New Wavegude Desgn As was stated n Secton 2 the loadng that ths new wavegude presented to ts drvers seemed to be good. However, a more quanttatve measure was desred. For some tme now ths author has been usng a method of analyzng the loadng of a horn that was brought to hs attenton by Gunness6. Ths method compares the nstantaneous flare rate of an arbtrary shape horn to that of a classcal exponental horns flare rate. The nstantaneous flare rate s determned for any pont along the length of the horn. Ths Instantaneous fccan be graphed aganst the horn length. A spreadsheet was desgned wth the approprate equatons to perform ths analyss. Fgure 4 shows how ths method compares an exponental horn, a concal horn and an axsymmetrc Quadratc Throat wavegude. For ths comparson the throat entry and horn length are the same for all three. The mouth ext s the same for the exponental and concal horns. The mouth of the QT wavegude s slghtly smaller than the other horns. It s smaller by the exact dmenson of the throat entry. Ths s due to the geometry of ths type of horn. As would be expected, snce ts flare rate s the reference, the exponental horn has a constant value off. The concal horn can be seen to have an ntal value atthe throat entry. Ths value steadly decreases untl t reaches ts mnmum value at the mouth. Ths s ndcatve of the gradual decrease n the acoustc resstance of concal horns as frequency decreases. The QT wavegude presents a very dfferent loadng characterstc. It has an ntal value of 0 Hz atthe throat. Ths can be attrbuted to the fact that at the throat entry the walls have no flare; they are normal to the plane of the throat entry. The value of fosteadly ncreases untl t reaches ts maxmum value at the pont where the throat secton jons the straght wall secton. From ths pont t steadly decreases n the exact manner that the concal horn does. Ths gves us a very good ndcaton as to the loadng propertes of the QT wavegude. Whle t s not qute as good as an exponental horn, t s almost so. The development of the proper wavefront shape s of paramount mportance for the drectvty response to be as ntended. The requrement that must be met for proper 2 development s for the wavefront to reman normal to the wavegude boundary at all ponts along the boundary. Ths means the wavegude must effectvely transform the planar wavefront, presented to the throat entry bythe drvng unt, to a sphercal wavefront. Fgure 5 llustrates how ths s accomplshed. The wavefront s shown at regularly spaced ntervals to depct ts transformaton and development. The sphercal wavefront s acheved at the pont where the wavegude transtons from ts throat secton to ts straght wall secton. From ths pont forward, the sphercal wavefront progresses toward the mouth of the wavegude. The radus of curvature of the wavefront n and beyond the straght wall secton s equal to ts dstance from the center of the throat entry.

6 It was observed that for a number of dfferent throat entry dameters and desgn coverage angles, the sze of the aperture between the two sectons of the wavegude was near ts optmum sze accordng to Keele's pror workn'7. Ths s not unexpected gven the geometry of the desgn of the wavegude boundares. Ths can be llustrated by examnng a 60 x 40 wavegude wth a maxmum nstantaneous f, of 500 Hz. For an optmzed aperture over the frequency range of foto 10f,Keele's kcamvalue s.403. The kcamvalue for the QT wavegude descrbed above s Performance evaluaton of the new desgn Two exstng horns were redesgned usng the QT wavegude technque. The new wavegudes had the same coverage angles and walls n the outer secton as the exstng horns. The only major dfference beng that of the ntal throat secton. The frst exstng horn s that ora.875" entrance nto an exponental throat secton. Ths throat secton jons to a straght wall secton. It s a conventonal, radal, constant drectvty horn desgn. The second exstng horn s that of a 2.0" entrance nto a straght wall secton. Ths secton s mantaned untl the last _A of the horn length. At ths pont secondary flanges are added to mnmze beamwdth narrowng pror to the horn losng ts drectvty control n the lower frequency regon. It s a conventonal straght wall horn. (It should be noted that the wavegude desgned from ths second exstng horn was done wth a 1.6" throat entry and not the orgnal 2.0" entry. Ths new wavegude was beng developed for a specfc product and ths smaller entry was more approprate. The exstng 2.0" entry horn has an adapter secton that allows the same 1.6" ext drver to mount on t as well as the new wavegude. Ths smaller entrance may account for some of the ncreased output of the new wavegude as well as ts mproved drectvty response n the very hgh frequency regon. It otherwse should have no effect on the comparsons made for the purposes of the study.) The new wavegudes were extensvely compared to the exstng horn desgns. Ampltude response, mpedance, harmonc dstorton at dfferent power levels and drectvty response measurements were made on the exstng horns and the new wavegudes. The same drve unts were used for each comparable horn so as to mnmze any measurement errors. Graphs of these measurements for the.875" throat entry devces are shown n fgures 6-9. Graphs of these measurements forthe larger throat entry devces are shown n fgures All of the measurements n these graphs are shown wth 1/3 octave smoothng. Fgure 6 A s the ampltude response, B s the phase response of the two devces. The two devces are comparable n ths area. Fgure 7 shows the horzontal and vertcal beamwdth measurements. The horzontal beamwdth of the two s almost dentcal. However, the vertcal beamwdth of the QT wavegude s much closer to ts ntended coverage angle than the conventonal desgn. Fgures 8 & 9 present 2nd& 3raharmonc dstorton, respectvely. It s nd clearly obvous that the QT wavegude has lower dstorton. The 2 harmonc dstorton s 3.5 to 4 db lower whle 3raharmonc dstorton s reduced n excess of 9 db over the conventonal horn. Fgure 10 shows the ampltude response of the two large throat entry devces. The QT wavegude has an ncrease n the on-axs output above 5 khz. At least part of ths s attrbutable to the horzontal drectvty n the same frequency regon, shown n fgure 11. The conventonal horn's energy s beng spread over a wder beamwdth above 5 khz. The 2ndharmonc dstorton graphs n fgure 12 shows no real dfference n the two devces. The QT wavegude actually has an average of 0.16 db less to 0.4 db more dstorton than the conventonal horn. Fgure 13 reveals an overall decrease n the 3rdharmonc dstorton of the QT wavegude. Snce, for the larger entry devces, the throat entry s larger for the conventonal desgn, one would expect that ts dstorton would be lower. However, due to the fact that ts dstorton s margnally lower (2ha)to slghtly hgher (3rd) t s assumed that, had the throat entry szes been equal, the same amount of reducton n dstorton would have been realzed as that of the.875" entry devces.

7 6. Conclusons It can be surmsed that the Quadratc Throat wavegude has a large reducton of 3rdharmonc dstorton, whle havng a sgnfcant, yet smaller, reducton of 2"dharmonc dstorton when compared to a concal horn or a conventonal constant drectvty horn comprsed of an exponental secton followed by a concal secton. Due to the desgn of the throat secton, ths new wavegude wll admt and propagate a one-parameter wave when t s drven at ts throat entry by a plane wave. The astgmatsm typcally found n conventonal horns s elmnated n ths new type of wavegude. Ths has a defnte advantage when multple horns are employed n an array as the apparent apex of the wavefront s n the same place for any gven orentaton of the wavegude. Ths feature makes spatal algnment of the ndvdual elements n the array much easer. Once placed n the array these wavegudes may be rotated, ptched or yawed as needed wthout affectng the spatal orentaton of the wavefront, as t s truly sphercal. Wth conventonal horns, a change n orentaton results n a change n the orentaton of the wavefront curvature, as t s not sphercal. 7. Acknowledgments The author would lke to thank hs boss, Tm Tardo, and hs employer, Peavey Electroncs, for allowng hm the tme and resourcesto pursue and develop ths new desgn concept. I would also lke to thank JorRsch, Ed Heath and John Murray for ther help n the revew of ths manuscrpt. Fnally, I wsh to thank my wfe, Beth, wthout whose seemngly unendng patence and understandng, ths work would have not been possble. 8. References 1. Earl R. Geddes, "Acoustc Wavegude Theory", presented at the 83rdAES Conventon, Oct Gavn R. Putland, "Every One-Parameter Acoustc Feld Obeys Webster's Horn Equaton", Journal of the AES, Vol. 41, No. 6, pp (June 1993) 3. Davd Gunness, emal correspondence, Jan. 6, D.B. Keele, Jr., "What's So Sacred About Exponental Horns", presented atthe 51stAES Conventon, May Clfford A. Henrcksen & Mark S. Ureda, "The Manta Ray Horns", LoudspeakersVol.2/In Anthology,pp Davd Gunness, "Instantaneous Flare Rate", notes from Synergetc Audo Concepts - Loudspeakers Workshop, June D.B. Keele, Jr., "Optmum Horn Mouth Sze", presented at the 46thAES Conventon, Sept. 1973

8 Throat Entry Throat Entry / / "_, actng Arc StraghtWalls at StraghtWalls at _ Desgn Coverage _ Desgn Coverage /// _n /,\ j', Angl_ gles Fgure1 Fgure2 r F,_ r-... Fgure3: 0 Geometry of Quadratc 0 Y Throat wavegude Yt _ constructon. _x _J

9 Instantaneous fo 1, I-... Exponental _' 400 _ Quadratc '_ [ Throat u. 200, Horn Length (x) Fgure 4 / \ // \ \ '\ Fgure 5: Wav&ront development _ nsdewavegude. Notethat,/ '\ the wavefront s always / '_ normalto thewavegude / _ boundary. \\ \\

10 Ampltude Response _ 120 _ I _ T ' I r : ' : ' ' ' 90_! t + l I ',! m ' 8o._ ] _. r 1O0 1,000 10,000 Ampltude Response 6090 '"_--'... I I '_ : 30 = -90 -_ _' 100 1,000 10,000 )... Conventonal horn#1 I Fgure 6 - A & B g 1 [ QTwave u!de# r

11 I Horzontal Beamwdth 1000 : --_ - :. -6dBAngles I " Z J 100 1,000 10,000! Vertcal Beamwdth -6dBAngles (degrees) C- 100 = - _ '.... 1C loo ooo lo,ooo _q & B... Conventonalhorn#1_ QT waveg ude #1

12 2nd Harmonc - tw _ ,, _- 1,000 10,000 2nd Harmonc - 10W lo0 9O 80 [, rn 'o 70 [ 60L I I -- ', [ J 1,000 10,000 2nd Harmonc - 20W 11o --_ '... l... k I I _ ' ' _ oo _ _'-_-"--'_... --:2 '_... --_... _2_ ,000 10,000 [ _:--' :.Convenonal horn_l l

13 I 3rd Harmonc - IW 9o, 1,_:7 16oI... -.I --2 _. ' :_, _,I m L,_!., o.,o'''-- 1,000 10,000 3rd Harmonc - 10W 9o _..,, so!....,r'-_:-:c...,... j, _ t_; l 60_[ I '! + 40[ [ I I! 1,000 10,000 3rd Harmonc - 20W 100T a 80 - _._o_.o-- 70 _'''''' 90I I _;' ;5_ ' : m 1,000 10,000 Fre que ncy - ;- ; _/- ]... Conventonal horn#1 I[ Fgure 9- A, B & C [[-. -QT wavegude #1 z_ - '_!!

14 j... r _ AmpltudeResponse 120,._ !-. ' l... : J L k ; : 100 J go... ; ] ; r ; 1O0 1,000 10,000 Ampltude Response _ !... I '--_ _o 100 1,000 10,000 Fgure lo - A & B I otwaveu,de,2... Conventonal horn #2 j! I

15 _6dB l... Z Horzontal Beamwdth Angles (degrees) _ 100 t_ * ; _ ,00O 10,000 Vertcal Beamwdth 1000 '6dBAngles _. _... _.: - _, _ (degrees) 100, 100 1,O00 10, Conventonal horn #2 Fgure 11- A & B r QT waveg ude #2

16 2nd Harmonc - IW 100_ -! I _. 90 t - 4- I- - I!, -! 8o{..,--= 7o._ _!. -...,... L-.._ 60',, - -!! '. [. L_..-- J. _,, L ' ' 5o J... 1,ooo lo,ooo 2nd Harmonc - low f 11o!-, _ r r loo ', L _o -! _o -... J - _ ' 10,000 1,000 2nd Harmonc - 40W _- t _.:[- _ ,.o70 50 _ 10,000 1,000 L... - _ ll OTw've.u'"e# /

17 I I 3rd Harmonc - IW 100, ' I ' - f = : L -o 60, ', [ I r" 50 4o I 1,000 10,000 loo, rd Harmonc - IOW 9oL E L' _... I u_ 70,o[ _o[ I I I 1,000 10,000 l 3rd Harmonc - 40W J 106 ' ' I 00::2.2-'-'-'-'_ I _ 7o!LLL 50 / : 40 [ / ; 1,000, O0 --. ::.. 6onventonal hon#2 Fgure 13- A, B & C -- QTwavegude02