Wavefront Sculpture Technology

Transcription

1 Auio Engineering Society Convention Paer Presente at the th Convention 00 Setember New York, NY, USA This convention aer has been rerouce from the author's avance manuscrit, without eiting, corrections, or consieration by the Review Boar. The AES takes no resonsibility for the contents. Aitional aers may be obtaine by sening request an remittance to Auio Engineering Society, 60 East n Street, New York, New York , USA; also see All rights reserve. Rerouction of this aer, or any ortion thereof, is not ermitte without irect ermission from the Journal of the Auio Engineering Society. Wavefront Sculture Technology MARCEL URBAN, CRISTIAN EIL, PAUL BAUMAN L-ACOUSTICS Gometz-La-Ville, 900 France ABSTRACT We introuce Fresnel s ieas in otics to the fiel of acoustics. Fresnel analysis rovies an effective, intuitive aroach to the unerstaning of comlex interference henomena an thus oens the roa to establishing the criteria for the effective couling of soun sources an for the coverage of a given auience geometry in soun reinforcement alications. The erive criteria form the basis of what is terme Wavefront Sculture Technology. 0 INTRODUCTION This aer is a continuation of the rerint resente at the 9 n AES Convention in 99 []. Revisiting the conclusions of this article, which were base on etaile mathematical analysis an numerical methos, we now resent a more qualitative aroach base on Fresnel analysis that enables a better unerstaning of the hysical henomena involve in arraying iscrete soun sources. From this analysis, we establish criteria that efine how an array of iscrete soun sources can be assemble to create a continuous line source. Consiering a flat array, these criteria turn out to be the same as those which were originally eveloe in []. We also consier a variable curvature line source an efine other criteria require to rouce a wave fiel that is free of estructive interference over a reefine coverage region for the array, as well as a wave fiel intensity that ecreases as the inverse of the istance over the auience area. These collective criteria are terme Wavefront Sculture Technology (WST) Criteria. MULTIPLE SOUND SOURCE RADIATION - A REVIEW The nee for more soun ower to cover large auience areas in soun reinforcement alications imlies the use of more an more soun sources. A common ractice is to configure many louseakers in arrays or clusters in orer to achieve the require Wavefront Sculture Technology an WST are traemarks of L- ACOUSTICS soun ressure level (SPL). While an SPL olar lot can characterize a single louseaker, an array of multile louseakers is not so simle. Tyically, traezoial horn-loae louseakers are assemble in fan-shae arrays accoring to the angles etermine by the nominal horizontal an vertical coverage angles of each enclosure in an attemt to reuce overlaing zones that cause estructive interference. owever, since the irectivity of the iniviual louseakers varies with frequency, the soun waves raiate by the arraye louseakers o not coule coherently, resulting in interference that changes with both frequency an listener osition. Consiering early line array systems (column seakers), aart from narrowing of the vertical irectivity, another roblem is the aearance of seconary lobes outsie the main beamwith whose SPL can be as high as the on-axis level. This can be imrove with various taering or shaing schemes, for examle, Bessel weighting. The main rawback is a reuce SPL an, for the case of Bessel weighting, it was shown that the otimum number of sources was five []. This is far from being enough for oen-air erformances. In [] we avocate the solution of a line source array to rouce a wave front that is as continuous as ossible. Consiering first a flat, continuous an isohasic (constant hase) line source, we emonstrate that the soun fiel exhibits two satially istinct regions: the near fiel an the far fiel. In the near fiel, wave

2 fronts roagate with 3 B attenuation er oubling of istance (cylinrical wave roagation) whereas in the far fiel there is 6 B attenuation er oubling of istance (sherical wave roagation). It is to be note that usual concets of irectivity, olar iagrams an lobes only make sense in the far fiel (this is eveloe in aenix ). Consiering next a line source with iscontinuities, we also escribe a rogressively chaotic behavior of the soun fiel as these iscontinuities become rogressively larger. This was confirme in 997 [3] when Smith, working on an array of 3 louseakers, iscovere that 7 B SPL variations over foot was a common feature in the near fiel. Smith trie raise cosine weighting aroaches in orer to iminish this chaotic SPL an was somewhat successful, but it is not ossible to have, at the same time, raise cosine weighting for the near fiel an Bessel weighting for the far fiel. In [] we showe that a way to minimize these effects is to buil a quasi-continuous wave front. The location of the borer between the near fiel an the far fiel is a key arameter that escribes the wave fiel. Let us call B the istance from the array to this borer. We will make the aroximation that if F is the frequency in kz then λ=/(3f) where λ is the wavelength in metres. Consiering a flat, continuous line source of height that is raiating a flat isohasic wavefront, we emonstrate in [] that a reasonable average of the ifferent ossible exressions for B obtaine using either geometric, numerical or Fresnel calculations is: B = 3 F ( 3F) We also emonstrate in [] that a line array of sources, each of them raiating a flat isohase wave front, will rouce seconary lobes not greater than B with resect to the main lobe in the far fiel an SPL variations not greater than ± 3 B within the near fiel region, rovie that: Either the sum of the flat, iniviual raiating areas covers more than 80% of the vertical frame of the array, i.e., the target raiating area Or the sacing between the iniviual soun sources is smaller than /(6F), i.e., λ/. These two requirements form the basis of WST Criteria which, in turn, efine conitions for the effective couling of multile soun sources. In the following sections, we will erive these results using the Fresnel aroach along with further results that are useful for line source acoustical reictions. Figure islays a cut view of the raiate soun fiel. The SPL is significant only in the otte zone (ABCD + cone beyon BC). A more etaile escrition is eferre to Section. where B an are in meters, F is in kz. There are three things to note about this formula: ) The root factor inicates that there is no near fiel for frequencies lower than /(3). ence a m high array will raiate immeiately in the far fiel moe for frequencies less than 80 z. ) For frequencies above /(3) the near fiel extension is almost linear with frequency. 3) The eenence on the imension of the array is not linear but quaratic. All of this inicates that the near fiel can exten quite far away. For examle, a 5. m high flat line source array will have a near fiel extening as far as 88 meters at kz. Figure : Raiate SPL of a line source AD of height. In the near fiel, the SPL ecreases as 3 B er oubling of istance, whereas in the far fiel, the SPL ecreases as 6 B er oubling of istance. It shoul be note that ifferent authors have come u with various exressions for the borer istance: B = 3 Smith [3] B = /π Rathe [] B = maximum of (, λ/6) Beranek [5] Most of these exressions omit the frequency eenency an are incorrect concerning the size eenence. Figure illustrates the variation of borer istance an far fiel ivergence angle with frequency for a flat line source array of height = 5. m. Figure : Reresentation of the variation of borer istance an far fiel ivergence angle with frequency for a flat line source array of height 5. metres. AES T CONVENTION, NEW YORK, NY, USA, 00 SEPTEMBER

3 TE FRESNEL APPROAC FOR A CONTINUOUS LINE SOURCE The fact that light is a wave imlies interference henomena when an isohasic an extene light source is looke at from a given oint of view. These interference atterns are not easy to reict but Fresnel, in 89, escribe a way to semi-quantitatively icture these atterns. Fresnel's iea was to artition the main light source into fictitious zones mae u of elementary light sources. The zones are classifie accoring to their arrival time ifferences to the observer in such a way that the first zone aears in hase to the observer (within a fraction of the wave length λ). The next zone consists of elementary sources that are in-hase at the observer osition, but are collectively in hase oosition with resect to the first zone, an so on. A more recise analysis shows that the fraction of wave length is λ/ for a -imensional source an λ/.7 for a -imensional source (lease see aenix for further etails). We will aly Fresnel s concets to the soun fiel of extene sources. Let us consier first a erfectly flat, continuous an isohasic line source. To etermine how this continuous wave front will erform with resect to a given listener osition, we raw sheres centere on the listener osition whose raii are incremente by stes of λ/ (see figures 3 an ). The first raius equals the tangential istance that searates the line source an the listener. Basically two cases can be observe:. A ominant zone aears: The outer zones are alternatively in-hase an out-of-hase. Their size is aroximately equal an they cancel each other out. We can then consier only the largest, ominant zone an neglect all others. We assume that this ominant zone is reresentative of the SPL raiate by the line source. This is illustrate in Figure 3 where it is seen that for an observer facing the line source the soun intensity corresons roughly to the soun raiate by the first zone. Figure : The observer O, is no longer facing the line source. The corresoning Fresnel zones are shown on the left art (front view). There is no ominant zone an iniviual zones cancel each other off-axis. Moving the observation oint to a few locations aroun the line source an reeating the exercise, we can get a goo qualitative icture of the soun fiel raiate by the line source at a given frequency. Note that the Fresnel reresentations of figures 3 an are at a single frequency. The effects of changing frequency an the onaxis listener osition are shown in Figure 5.. No ominant zone aears in the attern an almost no soun is raiate to the observer osition. Referring to figure, this illustrates the case for an off-axis observer. Figure 5: The effect of changing frequency an listener osition. As the frequency is ecrease, the size of the Fresnel zone grows so that a larger ortion of the line source is locate within the first ominant zone. Conversely, as the frequency increases, a reuce ortion of the line source is locate insie the first ominant zone. If the frequency is hel constant an the listener osition is closer to the array, less of the line source is locate within the first ominant zone ue to the increase curvature. As we move further away, the entire line source falls within the first ominant zone. Figure 3: Observer facing the line source. On the right art (sie view), circles are rawn centere on the observer O, with raii increasing by stes of λ/. The attern of intersections on the source AB is shown on the left art (front view). These efine the Fresnel zones. 3 EFFECTS OF DISCONTINUITIES ON LINE SOURCE ARRAYS In the real worl, a line source array results from the vertical assembly of searate louseaker enclosures. The raiating transucers o not touch each other because of the enclosure wall thicknesses. Assuming that each transucer originally raiates a flat wave front, the line source array is no longer continuous. In this section, our goal is to analyze the ifferences versus a AES T CONVENTION, NEW YORK, NY, USA, 00 SEPTEMBER 3

4 continuous line source in orer to efine accetable limits for a line source array. Let us consier a collection of flat isohasic line sources of height D, with their centers sace by. To unerstan the soun fiel raiate by this array, we relace the real array by the coherent sum of two virtual sources as islaye in figure 6. The real array is equivalent to the sum of a continuous line source an a isrution gri which is in hase oosition with this erfect continuous source. Figure 6: The left art shows a real array consisting of sources of size D sace aart by. The right art shows two virtual sources consiere as a erturbation an a continuous ieal source. Their sum is equivalent to the real array. 3. Angular SPL of the Disrution Gri The ressure magnitue rouce by the isrution gri is roortional to the thickness of the walls of the louseaker enclosures. Figure 7 illustrates how to reict the effect of the isrution gri in articular irections at a given frequency. The comlex aition of the virtual soun sources of the gri creates an interference attern that cannot be neglecte, unless by reucing their size. Figure 7: When the observer osition is very far, Fresnel circles are transforme into segments. The left figure shows that when observing at the angle θ i, half the sources are in hase oosition with the other half thus roucing a null ressure. On the right, it is seen that as we move further off-axis, all sources are in hase thus roucing a strong ressure. Let us erform Fresnel analysis for an observer at infinity. In this case, circles crossing the gri become straight lines. Now let us consier the interference attern as a function of olar angle. In the forwar irection (θ = 0), all sources aear in hase. At θ i, half the sources are in hase an the other half are in hase oosition, thus they cancel each other an the resulting SPL is very small. At θ eak, all sources are back in hase an rouce an SPL as strong as in the forwar irection. Therefore, the iscontinuities in a line source generate seconary lobes outsie the beamwith whose effects are roortional to the size of the iscontinuities. This is the first reason why it is esirable to attemt to aroximate a continuous line source as closely as ossible. From this qualitative aroach, we unerstan that seconary lobes aear in the soun fiel ue to the gri effect. The angles where the seconary eak an the seconary i arise are given by: sin( θ eak) = λ sin( θ i) = λ / If the first notch aears at θ i > π/, it will not be etrimental to the raiate soun fiel. This translates to: sin( θ i ) F 6 As before, F is in kz an is in meters. Alternatively, exressing in terms of wavelength: λ In other wors, the maximum searation or between iniviual soun sources must be less than λ/ at the highest frequency of the oerating banwith in orer for the iniviual soun sources to roerly coule without introucing strong offaxis lobes. As an examle, if = 0.5m, notches will not aear in the soun fiel rovie that F < 300 z. In the next section, we inten to quantify the isrution ue to the walls of enclosures an to establish limits on the sacing between raiating transucers. 3. The Active Raiating Factor () Now we have to o some math to etermine the suerosition of ressure. The ressure elivere by the ieal continuous source in the far fiel is: continuous The ressure of the isrution gri is: sin k sin θ k sin θ sin(( N + ) k sin θ ) ( ) isrut D sin( k sin θ ) Where D is the active raiating height of an iniviual soun element as shown in Figure 6. In the forwar irection, i.e: θ = 0, we have: isrut real continuous (θ = 0) ( θ = 0) ( N + )( D) = continuous + Since = N we have: isrut ( N + )( D) AES T CONVENTION, NEW YORK, NY, USA, 00 SEPTEMBER

5 real (θ = 0) ( N + ) D At the seconary eak we have: sin( θ eak) = λ π N λ k sin( θ eak ) = = Nπ λ = real continuous sin( Nπ) ( θ eak) = = 0 Nπ ( θ eak ) = isrut( θ eak ) ( 0) = ( N + )( D ) isrut We now have to efine an accetable ratio for the height of a seconary lobe with resect to the main on-axis lobe. Base on the attern of a erfect line source that rouces seconary lobes in the far fiel not higher than 3.5 B of the main lobe, it seems otimal to secify in our case a 3.5 B ratio. Therefore we require: real( θ = θeak) ( θ = 0) real. [ ] ( N + )( D) ( N + ) D. D D N +.73 We efine the Active Raiating Factor () as: thus, D = 0.8 ( + ).73 ( N + ) Note: Frequency eenency oes not show u in the final formula for. This is because we have assume that the angle θ eak was between 0 an π/. owever, it shoul be note that if the frequency is low enough there will be no seconary eak an this is the only way that frequency eenency can enter into this calculation. 3.3 The First WST Criteria an Linear Arrays Assuming that the line array consists of a collection of iniviual flat isohasic sources, we have just reefine the two criteria require in orer to assimilate this assembly into the equivalent of a continuous line source as erive in []. These two conitions are terme Wavefront Sculture Technology (WST) criteria: The sum of the iniviual flat raiating areas is greater than 80% of the array frame (target raiating area) or The frequency range of the oerating banwith is limite to F</(6 ), i.e., the istance between sources is less than λ/. Note: further WST criteria will be erive in the following sections. For a slot whose with is small comare to its height D, is D/. For the case of touching circular soun sources, the average is π/ =75%. It is therefore imossible to satisfy the first criterion an for circular istons the only way to avoi the seconary lobes is to secify that the maximum oerating frequency be less than /(6D). In other wors, the iameter of a circular iston has to be smaller than /(6F). While this is ossible for frequencies lower than a few kz it becomes imossible at higher frequencies. For examle, at 6 kz we woul nee touching istons with iameters of a few millimeters. From this examle, we unerstan that there is a challenge as to how to fulfil the first criterion at higher frequencies. One solution might consist of arraying rectangular horns in such a way that their eges touch each other. owever, an imortant consieration is that such evices o not raiate a flat isohasic wave front. Then, the next question to be answere becomes: how flat oes the wave front have to be in orer for the sources to coule correctly? Let us consier a collection of vertically arraye horns, searate only by their eges. The raiate wave front exhibits riles of magnitue s, as shown in figure 8. The most critical case occurs at high frequencies where the wavelength is becoming small, e.g., cm at 6 kz. Accoring to Fresnel, when staning in the far fiel the raiate wavefront curvature (s), shoul not be greater than half the wavelength, i.e., cm at 6 kz. Unfortunately, the conitions are much more restrictive in the near fiel when consiering high frequencies. Therefore, when N is large, we fin that has to be larger than 8% if the seconary eak is to be 3.5 B below the main forwar eak. This confirms what was originally obtaine in []. Note that a seconary eak of only 0B below the main forwar eak is obtaine when is equal to 76%. is thus a factor which has to be carefully looke at. When N is large, a ractical formula relating to the attenuation of the seconary sie lobes in ecibels, Atten(B), is: = + 0 Atten(B) 0 Figure 8: This illustrates that vertically arraying conventional horns will not rouce a flat wave front. AES T CONVENTION, NEW YORK, NY, USA, 00 SEPTEMBER 5

6 Figure 9 islays the calculate SPL versus istance for a line array of 30 horns, 0.5 m high, each of them roucing a curve wave front of 0.3 m (rile s = 0 mm). Comarison with a flat line source shows chaotic behavior of the line array, starting at 8 kz an increasing with frequency. Aart from severe fluctuations in the SPL at higher frequencies, there is also a B loss at 6 kz from 0-00 metres. It is therefore necessary to reuce wave front curvature by half (s < 5 mm), in orer to create an as goo as erfect line source u to 6 kz. In effect, this will shift the sielobe attern observe in Figure 0 an the on-axis behaviour observe in Figure 9 from 8 kz to 6 kz. We conclue by stating that the eviation from a flat wave front shoul be less than λ/ at the highest oerating frequency (corresoning to 5 mm at 6 kz). For this reason, a waveguie has been secifically eveloe in orer to generate a flat, isohasic wavefront at the exit of the evice. The atente DOSC waveguie is incororate in several commercially-available soun reinforcement systems that are esigne to erform in accorance with WST criteria. For the frequency range of.3-6 kz, the soun ressure of a circular iston (i.e., the outut of a comression river) is asse through the waveguie where all ossible acoustic ath lengths are ientical in length. This rouces a wave front that is flat an isohasic (constant hase) at the rectangular aerture of the oening (see figure ). This geometric transformation from circular to rectangular creates a wavefront that is sufficiently flat to satisfy the limits of accetable curvature erive above an exeriments have shown that wavefront curvature is less than mm at 6 kz. When multile DOSC waveguies are vertically arraye, this allows for satisfaction of the 80% criterion rovie that the angle between ajacent enclosures is less than 5 egrees (see section 6. for further etails). Figure 9: SPL vs istance for a vertical array of 30 horns (total height =.5 m, wavefront curvature s=0 mm) calculate at,, 8 an 6 kz. White ots: line array, black ots: continuous line source. Another comarison islaye in figure 0 illustrates the cross section of the beam with which exhibits strong seconary eaks in the near fiel (0 m) at frequencies higher than 8 kz. Figure 0: SPL along a vertical ath, 0 m away from the vertical array of 30 horns (total height =.5 m, wavefront curvature s=0 mm) calculate at,, 8 an 6 kz. White ots: line array, black ots: continuous line source. Figure : WST Criteria Illustrate. On to we see the central ortion of the DOSC waveguie that geometrically sets all ossible soun ath lengths to be ientical from the circular entrance to the rectangular exit of the evice, thus roucing a flat, isohasic source for the high frequency section. The bottom figure shows a stack of 5 such evices (incluing the outer housing) which rouces a vertical, flat soun source satisfying WST criteria. AES T CONVENTION, NEW YORK, NY, USA, 00 SEPTEMBER 6

7 SOUND FIELD RADIATED BY A FLAT LINE SOURCE ARRAY. Raiation as a Function of Distance Consiering a flat line source array, we want to unerstan why there is a near fiel (cylinrical wave roagation) an a far fiel (sherical wave roagation) an to erive an exression for the borer istance. We will use Fresnel analysis to locate the borer between the two regions (see [] for analytical calculations). When h > I = I I nearfiel flat nearfiel flat h 3F Let us consier a flat line source array of N iscrete elements, oerating at a given frequency. The observer is moving along the main axis of the raiation, as shown in figure. As the observer moves away from the line source, the number of sources in the ominant zone, N eff, increases until it reaches the maximum number of available sources (h = ). Moving beyon this istance, the number of sources no longer varies. I = I farfiel flat We verify that as long as N eff < N max, the SPL ecreases as /, efining the cylinrical wave roagation region. It is simle to extract the exression for borer istance B. I nearfiel flat ( B flat B flat = 3 ) = F farfiel flat ( B flat where B an are in meters, F is in kz. The formula erive in [] for F>>/3, is 3/ F, therefore Fresnel analysis reicts that the borer istance is 50% closer. When oes a near fiel exist? With Fresnel we unerstoo that as the istance of the listener ecreases, the number of sources in the first zone ecreases too, excet for when λ/ > / because then the entire array is always in the first zone. Therefore, with Fresnel analysis, we have erive the same result as foun in [], i.e., there is no near fiel when F</(3) I ) Figure : The first Fresnel zone height is h. This height grows as istance increases until h =. At greater istances, no more increase of the raiate ower is execte. Each source raiates a soun fiel as eicte in figure. We will lace ourselves in the far fiel of each source (where sherical roagation alies). It will be etaile in section 6. that the conition that we are locate in the far fiel of each element imlies certain restrictions on the tilt angles between ajacent elements. The total ressure magnitue is thus roortional to: eff N eff while the SPL is roortional to the square of eff. We nee to know how N eff (or h) varies with the listener istance. From figure, we calculate: N eff = h = λ ( + λ/ ) For λ<<, we have two simlifie formulations for the SPL eening on the size of h: There is, however, the basic fact that even a continuous source islays riles in the SPL of the near fiel, but with magnitue less than ± 3B about the average. This is the secon reason for assigning ourselves the goal of roucing a wave front as close as ossible to a continuous soun source, i.e, in orer to reuce riles in the nearfiel resonse. (recall that the first reason was in orer to reuce sielobe levels in the far fiel). To illustrate this, the array stuie in [3] consists of 3 ome tweeters with iameters of 5 mm. The is 80 mm. The secon criterion for arrayability is that the frequency be less than /(6) = /(6*0.08) = kz. The first criterion is that for frequencies higher than kz, the shoul be greater than 80%. ere the is less than 5/80=30%. an we can conclue that above kz, this array will exhibit severe roblems in the near fiel. In figure 3, we comare the SPL of a continuous soun source.76 m high, with the array of 3 tweeters an calculate the SPL as a function of istance at an 8 kz. We see that below kz the continuous an iscrete arrays are similar while for higher frequencies, the iscrete array shows unaccetable SPL riles over very small istances. We also lot the -3 B (near fiel) an the -6 B (far fiel) lines reicte from our analysis. When h < : AES T CONVENTION, NEW YORK, NY, USA, 00 SEPTEMBER 7

8 Figure : A) The first Fresnel zone is larger than the height of the array an the entire array is in hase. B) As we turn aroun the array there may be an angle where half the array is in hase oosition to the other half. Figure 3: SPL as a function of istance. Emty circles: 3 tweeters totaling a height of.76m. Full circles: a continuous array of the same height. The 3B an the 6B er oubling of istance lines are shown on the same figure to inicate the borer between the near an far fiels. At Position A: we are staning in the far fiel region an the entire source is hear in hase so that we have maximum SPL. At Position B: we have cancellation since half the sources are inhase while the rest are out of hase. This occurs at the angle θ i (see Figure 5).. Vertical Pattern in the Far Fiel With the hel of Fresnel analysis, we now investigate the vertical irectivity in the far fiel for a flat line array. The horizontal irectivity is equivalent to that raiate by a single element. We saw reviously in figures 3 an, that being off-axis raically changes the Fresnel zone attern on the source. We want to show what haens with some simle examles. The ressure in the far fiel for a flat line array of height is known analytically: k sin θ sin ressure( θ) ksin θ The first i in ressure is given by: k sin θ = π sin θi = π = k 3F Let us see how Fresnel analysis allows us to unerstan why an where there will be ressure cancellations in the far fiel. At a given frequency, we rotate aroun the source at a fixe istance, as shown in figure. Figure 5: Defining the quantities use to etermine θ i. We want IJ to be equal to λ/, thus: IJ = sinθi sin θ i = λ = 3F This is exactly the same result as the analytical formula. Remarks: We see here that the Fresnel aroach oes not give the exact functional behavior of the SPL. Instea it gives us, in a simle way, the characteristic oints. We unerstan hysically why there will be an angle where no SPL is rouce an we can calculate that angle, but we cannot erive the sinx/x behavior. In more comlex situations Fresnel will tell us the gross features of the soun fiel - if it comes to a oint where we nee more etaile information we will have to use numerical analysis, but the characteristic features are more easily unerstoo with Fresnel analysis. When staying on-axis, the soun fiel is cylinrical u to B. We see now that moving just a bit off the main axis can cause the SPL to change tremenously. If there are several listeners at ifferent ositions an they are aligne on the main axis, then a flat array is fine. Most of the time, however, the auience is more off-axis than on-axis. AES T CONVENTION, NEW YORK, NY, USA, 00 SEPTEMBER 8

9 .3 Vertical Pattern in the Near Fiel As state above in Section an [], contrary to the far fiel, the SPL in the near fiel is not amenable to close form exressions. This is unfortunate since the near fiel can exten very far, esecially at higher frequencies. owever, using Fresnel analysis, (as escribe in Section ) we can escribe the vertical attern of the soun fiel in both the near an far fiels. At this oint, we woul like to escribe the SPL in the near fiel (otte region of figure ) in greater etail. The SPL as calculate along A'D' of figure is shown in figure 6 (black oints). For this examle we have = m, the istance AA' is = 9 m an the frequency is F = kz. Figure 6 shows that the SPL is nearly constant between A'D' until it ros to 6 B at the ege of the array. We see also that as we go beyon the ege of the array, the SPL has ecrease by more than B. The size of the first Fresnel zone is a very characteristic imension an for this examle its value is.5 m. It is reicte that the SPL will fall to 6 B at the ege of the array an over half the first Fresnel zone istance. This is seen to be in excellent agreement with the results shown in figure 6. In figure 6 we also lot the SPL corresoning to the ressure in the far fiel (emty circles): sin( k sin θ) farfiel = k sin θ This clearly illustrates that a olar lot or an angular formula that is vali in the far fiel is totally wrong in the near fiel. For further etails, lease see Aenix. 5 SOUND FIELD RADIATED BY A CURVED LINE SOURCE ARRAY 5. Raiation as a Function of Distance Consiering now a convex line source array of constant curvature, Fresnel analysis can be use to fin the borer between the near fiel an the far fiel at a given frequency. It will be shown in the following that this borer istance is always further away for a convex line source than for a flat line source, eening on the raius of curvature. This surrising result raises a new question as to how the soun fiel behaves in the near fiel with resect to the far fiel. This question will be answere analytically in the following section where it will be seen that in some cases, the transition between near an far fiels is asymtotic so that the ifference in the soun fiel behaviour is less ronounce than as is for a flat soun source. owever, at this oint of the iscussion we are aware that an extene soun source an, more secifically, a curve line source array cannot be assimilate to a oint source that raiates a sherical wavefront. Attemts to reresent the extene soun source with a oint source moel necessarily imlies comromise results which turn out to be unaccetable when the soun source becomes large with resect to the consiere wavelength. With reference to figure 7, when the observer is at osition A, the flat line source (black vertical line) is not yet entirely in the first Fresnel zone, thus at A the fiel is cylinrical. Moving to osition B, the flat line source is entirely in the first Fresnel zone an consequently we have reache the borer between the near an far fiel for this kin of source. Curving the array in a convex shae, we realize that the listener osition in the near fiel is extene rovie that the curve source is not entirely inclue in the first zone. Figure 6: Full circles: SPL along A'D' for a line source ( = m) at a istance of 9 m, for F = kz. Emty circles: the SPL calculate using the analytical exression for the farfiel irectivity for the same line source. This is rawn along A'D', each oint on A'D' efining the angle θ as shown in Figure. Figure 7: At osition A, the fiel of the flat line source is cylinrical. At osition B we have reache the borer between near an far fiels for the flat line source. By curving the array, we can lace the listener farther away than B because the entire curve line source is not yet in the first Fresnel zone. Thus the far fiel of a curve array begins farther away than the corresoning one for a flat array. The amount of increase eens uon R, the raius of curvature of the array. aving R very large imlies a borer line slightly larger than for a flat source, as is execte, since the flat source is just a articular case of a curve array with R very large. Conversely, with a reuce raius of curvature, the near fiel can exten very far away from the array (otentially infinite). owever, as seen below in Section 5., the traeoff is that there is a reuction in the on-axis SPL in comarison with a flat line array. AES T CONVENTION, NEW YORK, NY, USA, 00 SEPTEMBER 9

10 5. Vertical Pattern of the Raiate Soun Fiel Although we o not have an analytical formula for the far fiel attern of a curve line source, with Fresnel analysis we can see that a curve array rojects a uniform soun fiel excet near the eges. We lace ourselves at infinity an instea of the Fresnel circles we raw straight lines. Figure 8 shows a curve array with constant curvature R. Figure 8: The left art shows a curve array AB an the first Fresnel zone in black for a listener osition locate at infinity. The right figure shows what haens for an observer far away an listening to the array at angle θ ege. The size of the first Fresnel zone (in black) is half of what it use to be. We get the same number of effective sources until we reach angle θ ege where the number ros by a factor of. This corresons to a 6 B reuction in SPL an therefore efines the vertical irectivity of the curve array. A curve array has a uniform SPL that is efine by the angles of its eges. A straight array is non-uniform but on-axis, rojects a higher SPL. Therefore, the uniform vertical angular SPL of a curve array has a rice. We show in the following, that the SPL of a curve line source is, on average, 3 B less than the on-axis SPL of a flat line source. Figure 9: Comarison of flat an curve line source arrays of the same height. The SPL is calculate on a vertical line 0 m away from the sources. It is aarent that the curve source resents less variation in its vertical SPL attern but with reuce on-axis SPL. 6 We have investigate two generic tyes of extene soun sources: the flat line source an the constant curvature line source. In an effort to aat the shae of a line source to a secific auience geometry, we will now look at variable curvature line sources. In oing this, our intent is to focus more energy at the most remote listeners ositions, while istributing the energy better at closer locations (see figure 0). In figure 9, we comare curve versus flat line source arrays. The height of both sources is 3 m an the raius of curvature is 5 m for the curve line source. For a frequency of kz, borer for the flat source is situate at 7 m. We calculate the SPL along a vertical line, 0 m away from the array. The curve source is still roucing a cylinrical fiel at 0 m an it is seen in figure 9 that the vertical attern of the SPL for the curve source is clearly less chaotic than that of the flat source. owever, comaring the average SPLs between ±.5 m, the flat line array shows a 3 B avantage over the curve array. Figure 0: Comarison between flat an variable curvature arrays. The SPL istribution of the variable curvature array is aate to suit the auience geometry. 6. Raiation as a Function of Distance Using Fresnel analysis, we will consier the size of the ominant zone at various locations an istances from the array in orer to etermine how to otimize the shae of the line source to match the auience geometry requirements. Consiering this aroach, we are aware that the ressure magnitue of the array at one AES T CONVENTION, NEW YORK, NY, USA, 00 SEPTEMBER 0

11 location is roortional to the size of the ominant zone looke at from this osition. We have seen in revious sections that the size is larger for a flat line source an gets smaller as the raius of curvature ecreases. To formalize this, let s calculate the size of the ominant zone with resect to the number of effective soun sources inclue in the first zone. Figure islays the metho use to calculate the size of the zone inclue in the ga [, +λ/], being the tangential istance from the listener to the array. As in section, the array consists of N iscrete elements, each of them raiating a flat isohasic wave front an oerating at a given frequency. These elements are articulate with angular stes to form an array of variable curvature. Figure : Nomenclature for the calculation of the size of the first Fresnel zone for a curve array. As in section., the total ressure magnitue is roortional to: eff N eff while the SPL is roortional to the square of eff. λ h ( + ) = + ( R + R cosθ) The height of the first Fresnel zone is h (see figure ) such that: where h = R sin θ We will use the small angle aroximation for θ to get: from which we fin that: λ λ θ + = R + Rθ + R θ + = λ θ λ R θ + + R The quantity θ / is smaller than an smaller than /R, thus: λ + λ R θ + R The active height of the first Fresnel zone is: h Rθ an the soun intensity from this zone is: ( R ) θ I We now make the aroximation that the closest listener is at a istance that is larger than the wavelength of interest. I curve 3 F α + where is a constant an is the istance to the listener. Recall that α is the tilt angle between ajacent raiating elements. This angle varies along the array an the raius of curvature R at a given oint on the array is just /α. The exression for a flat line array is obtaine by setting α = 0. It is to be note that this exression is vali rovie that the curve line source array is not entirely inclue within the ominant Fresnel zone. This is the case for : α > 3 F Three major results can be erive by comaring the above exression for I curve with the exressions reviously erive for a flat line source, i.e,: I nearfiel flat I farfiel flat 3F For α = 0, the curve array is flat. The two exressions for I flat an I curve converge an both exressions emonstrate near fiel behaviour with cylinrical soun fiel roagation. For α = constant, the transition between the near fiel an far fiel is smooth. At short istances where α <, the near fiel goes from cylinrical to sherical. At long istances where α >, the far fiel tyically becomes sherical with an asymtotic limit for I curve, that is: Conversely, when α < ( )/ (3 F ), I curve This exression for I curve tyically alies at lower frequencies. AES T CONVENTION, NEW YORK, NY, USA, 00 SEPTEMBER

12 3 α F I curve that the elements are raiating flat wavefronts an are angle with resect to each other. The interesting art comes when a constant value for α is secifie. This can be achieve by aating the angular ste α searating two ajacent soun sources to the istance of their focus target on the auience. See figure for an illustration of this. Setting α = K = constant throughout the entire auience rofile, the exression for I curve becomes: I aate 3F K + This exression shows a / soun ressure level eenence, thus a 3 B attenuation er oubling of the istance. Although exhibiting cylinrical behaviour for the soun fiel, it shoul be note that the structure of the soun fiel has cylinrical effects (/ eenence) on the auience only, while the roagation in a fixe irection (through the air), is still somehow in between cylinrical an sherical moes (accoring to revious consierations). For this reason, we will term the aate soun fiel raiate by a variable curvature line source having α constant as a seuocylinrical soun fiel. In aition, the metho consisting of aating the soun fiel to the auience geometry is terme Wavefront Sculture. As a matter of fact, shaing the line source in such a way so that α = constant corresons to an aitional Wavefront Sculture Technology criterion. Figure 3: Two sources searate by istance an tilte by angle α with resect to each other. The SPL is shown as a otte region. Let us efine φ as the far fiel coverage angle of a single element at frequency F. Using a small angle aroximation: λ φ = The istance an angle that searate two ajacent elements are an α, resectively. In figure 3, the otte zones reresent the soun fiel of each source an the blank zone AC corresons to a zone with oor SPL. We aim at reucing the blank zone an shoul clearly avoi allowing oint C to reach the auience Figure : Illustration of wavefront sculture where a variable curvature line array is esigne so that α=constant over the auience geometry. 6. Limits On the Angular Incrementation of a Curve Line Source In section 3, we investigate the effects of iscontinuities on line source arrays. Another consieration for a variable curvature line source array is the amount of angular searation that is allowe between two iscrete sources before lobing occurs. As shown in figure 3, each source iniviually raiates a near fiel over a istance that eens on its size an the frequency of interest. The SPL is mainly focusse in the otte regions (recall figure ) an the zone AC is a small SPL region. This efines a maximum searation angle between two iscrete elements, base on the nee to roject a soun fiel with no iscontinuities on the auience. Figure 3 is also interesting because it illustrates that the soun fiel can be ba in the air above the auience but that it will be fine on the auience. Note: even if there is no hysical ga between the fronts of raiating elements, this region AC will still exist ue to the fact Using the small angle aroximation, the istance AC is given by: AC = φ α Rewriting φ in terms of frequency an secifying that AC is smaller than, we have: α < 3 F where α is in raians, F in kz an is in meters. We require that α is greater than zero, thus: < 3F The worst case is F=6 kz an = min, the minimum istance where a listener will be locate. This corresons to: max = min Substituting max into the above exression for α we get the following exression for the maximum tilt angle α max : = max α max min max ( ) max AES T CONVENTION, NEW YORK, NY, USA, 00 SEPTEMBER

13 From this exression it is seen that there is a traeoff between the maximum element size an the maximum allowable inter-element angles, i.e., if we want to increase the angles between sources then we have to reuce the element size. As an examle, let us consier min to be 0 meters. Since the iameter of a 5'' low frequency comonent is tyically 0.0 m, this imlies a minimum of 0. m when we allow for the aitional thickness of the louseaker enclosure walls. When we take this minimum value of 0. m we fin for α max the conition that: α max 5.7 =. 6 Since must remain between 0.8 an, therefore α max will be between.5 an 3., which reresents the maximum allowable angle between enclosures. What about the intensity? I = 3 F α + Assuming a of 0. m, min of 0 m an a frequency of 6 kz, we fin that α/ is of orer. Therefore the intensity will be roughly a factor of smaller (-3 B) than the on-axis intensity of a straight array (α=0). 7 CONCLUSION Our technique to unerstan an characterize the soun fiel raiate by linear arrays is the Fresnel aroach in otics, as alie to acoustics. Fresnel analysis oes not rovie recise numerical results but gives a semi-quantitative, intuitive unerstaning. More recise results can come later using numerical analysis techniques, but only when one knows or can reict the answers in a semi-quantitative way. For a curve array: The same criteria as for the flat array lus enclosure tilt angles shoul vary in inverse roortion to the listener istance. the vertical size of each enclosure an the relative tilt angles between ajacent enclosures shoul conform within the limits establishe in section REFERENCES [] C. eil, M. Urban, Soun Fiels Raiate by Multile Soun Source Arrays, rerint #369, resente at the 9 n AES Convention, Vienna, March -7, 99 [] D.B. Keele, Effective Performance of Bessel Arrays, J. Auio Eng. Soc. Vol 38, ages 73-78, October 990 [3] D.L. Smith, Discrete-Element Line Arrays. Their Moeling an Otimization, J. Auio Eng. Soc. Vol. 5, No, November 997 [] E. Rathe, Notes on Two Common Problems of Soun Proagation, J. Soun Vibration, Vol. 0, ages 7-79, 969 [5] L. L. Beranek ACOUSTICS Publishe by the American Institute of Physics, Inc. New York. Thir rinting 990. APPENDIX It is only in the far fiel region that irectivity, olar lots an seconary lobes make sense. In the near fiel, these concets cannot be use as they are greatly misleaing. This is ue to the fact that the line source cannot be reresente as a oint source in the near fiel since a olar iagram makes the assumtion that the energy flow is raial. For examle, in orer to raw a olar lot we woul measure the SPL along a circle like the one icture on the right art of figure A-. This woul result in the olar lot as shown on the left art of the figure an we woul wrongly conclue that a large fraction of energy is sent to the floor an to the ceiling. This is incorrect since in the near fiel the energy flow is only forwar (erenicular to the line source array). We tackle the roblem of efining when an assembly of iscrete sources can be consiere equivalent to a continuous source an why a continuous source is esirable. We unerstoo why a continuous line source exhibits two ifferent regimes: when close to the source the SPL varies as / (cylinrical wave roagation) an far away the SPL varies as / (sherical wave roagation). We foun that the osition of the borer is roortional to the frequency an to the square of the height of the array an also that for low enough frequencies there is no near fiel. Following this, by stuying the roerties of curve arrays using Fresnel analysis we etermine conitions concerning the tilt angles between louseaker enclosures an the size of these enclosures require in orer to rovie a uniform cylinrical SPL over a given auience. Summarizing the Wavefront Sculture Technology criteria for arrayability: For a flat array: either the sum of the iniviual flat raiating areas covers more than 80% of the vertical frame of the array, i.e., the target raiating area or the sacing between soun sources is smaller than /(6F), i.e., less than λ/ at the highest oerating frequency the eviation from a flat wavefront shoul be less than λ/ at the highest oerating frequency. Figure A-: The rawing on the left, islays a olar iagram where the flow of energy is suose to emanate from O, along OA for instance. Using such a olar iagram in the near fiel (right) woul inicate an incorrect flow of energy. APPENDIX For Fresnel analysis we raw circles with λ/ increments in their raii. This may aear somewhat surrising since half a wavelength leas to a hase oosition. One ege of the zone is in hase oosition to the other ege of the zone an consequently we woul exect a small resultant SPL. Qualitatively we can emonstrate why Fresnel chose that value an why the SPL is not small but, on the contrary, reaches its maximum level. Consier the first zone to be ivie into small but finite ieces as shown in figure A-. AES T CONVENTION, NEW YORK, NY, USA, 00 SEPTEMBER 3

14 in the figure A-3. ere we can see that the maximum SPL is not reache for λ/ but for λ/.7. The reason for this is that Fresnel consiere -imensional sources whereas we are consiering only -imensional sources. Since we are intereste in the qualitative reictions of the metho we use λ/ as a reference value. It is easier to remember an figure A-3 shows that the SPL ifference between λ/ an λ/.7 is only 0.5b. Figure A-: The left art shows the first Fresnel zone broken into 5 ieces equal in soun ressure. The right art islays the Argan iagram of the comlex amlitues associate with these ieces, an their resultant sum. Finally, we note that, on average, the light intensity from the ominant zone is roughly 6 B higher than the light intensity from the comlete source. SPL{( r)} / SPL first zone OA is the resultant SPL from the first Fresnel zone an it is larger than OB which incororates art of the secon zone. In orer to see this more rigorously, we calculate the SPL of a continuous line source whose height is variable. The observation oint is meters away, on the main axis (see figure A-). r = λ/ r / λ Figure A-3: The normalize SPL of the segment ( r), efine in figure A3-, islaye as a function of the increase in the raius of the circle. Figure A-: From the observation oint we raw a circle of raius. This circle is tangent to the line AB. Drawing a circle whose raius is larger (+ r) efines a segment of height on AB. We normalize the SPL ue to ( r) by the SPL arising ue to the first Fresnel zone at the same istance, while assuming the entire zone to be at the center (this is equivalent to neglecting the λ/ variation from the center to the ege of the zone). This is icture AES T CONVENTION, NEW YORK, NY, USA, 00 SEPTEMBER