_directivity bandwidth of line arrays

8/12/2019 _Directivity Bandwidth of Line Arrays

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1. INTRODUCTION

Designing and installing loudspeaker linearrays involves compromises. We juggleacoustic power requirements, height, weight andcost constraints, inconvenient hang points anddifficult coverage angles. There is no setanswer. Often, we begin with an array that weknow works and tweak it for the next venue.

But why so some arrays seem to work betterthan others do and how far can we tweakthem?Why do some arrays seem to sound good in

every seat in the house and others a bit spotty?What factors contribute to good coverage andtonal balance?

The first things that come to mind are theobvious - good transducers, well-engineeredsystems, enough boxes in the array to providesufficient acoustic power, enough amplifiers toprovide plenty of headroom, etc. Oftenoverlooked, however, is whether an array hasenough boxes to provide smooth polar responseand sufficient low frequency directivity control two performance factors that can materiallyaffect overall coverage and tonal balance.

This Tech Note introduces the notion ofdirectivity bandwidth, defined as the frequencyrange across which a loudspeaker line arrayprovides coherent output within its intendedcoverage angle

1. Arrays with extended low

1 Directivity bandwidth should not be confused with frequency

response bandwidth. An array will have acoustical output

beyond where it can effectively control its directivity response.

frequency directivity control and smooth highfrequency polar response will have widedirectivity bandwidth. Those that cannot controlto very low frequency and/or have ragged highfrequency polar response will have narrowbandwidth.

The operative words in the definition arecoherent and within. Coherent is not usedhere in the mathematical sense; rather it refersto an arrays ability to act as a single source. Atsome high frequency, the elements of an arrayact separately, radiating sound as independent

sources. This point marks the upper limit of anarrays directivity bandwidth. Within means notwider than the intended coverage angle. Atsome low frequency, the wavelengths getsubstantially longer than the physicaldimensions of the array and overwhelm anarrays ability to control where the sound goes.This point marks the lower limit of an arraysdirectivity bandwidth.

Directivity bandwidth depends primarily on thesize and shape of the array. Large arrays withmoderate coverage angles (neither overlynarrow nor wide) tend to work best. Theyprovide extended directivity control in the lowend and smoother response in the high end.The net result is more uniform coverage andsuperior tonal balance.

This Tech Note describes how the number ofboxes and inter-box splay angles (shape) affectdirectivity bandwidth. We use two common linearray configurations to illustrate how thesegovern an arrays performance at both ends ofthe spectrum and show the expected

Directivity Bandwidth of

Loudspeaker Line Arrays

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performance on a grid. This grid provides thereader a graphical framework for visualizing theperformance trade-offs between array size,shape and directivity bandwidth.

2. ARRAY TYPES

Loudspeaker line arrays take various shapes.Certain attributes of venues dictate theseshapes including the number and geometry ofthe seating planes, the location of the array, anyheight and/or weight restrictions and so on. Twocommon array types are arc and progressivearrays

2. Arc line arrays employ a constant splay

angle between successive boxes. Figure 1shows a ten-box arc array with 5 inter-box splayangles. This provides a total coverage angle of45. One of the advantages of arc arrays is thatthey can achieve the maximum coverage anglefor any given number of boxes.

A progressive line array has an increasingsplay angle between successive boxes. Forinstance, an arithmetic progressive array hasinter-box angles of 1, 2, 3, 4 and so on.This yields a cumulative box-pointing angle of1, 3, 6 and 10 - an arithmetic expansion.Figure 2 shows a ten-box progressive array.The total coverage angle is 45, the same as thearc array in Figure 1. The difference is that thearc array achieves 45 in a linear fashion. Theprogressive array is flatter on top and curvesmore rapidly at the bottom.

Progressive arrays are useful in venues thatrequire asymmetrical polar response. Theirflatter shape at the top directs more sound to thetop of the coverage pattern than the bottom. Bycontrast, an arc array produces a completelysymmetrical polar pattern. Figure 3 shows theacoustical maps for both arrays at 4kHz.

3. MAXIMUM COVERAGE ANGLE

The maximum coverage angle achievable forall line array types is constrained by the draftangles of the boxes. An array built from boxeswith 5 draft angles (top and bottom) isconstrained to a maximum angle of 10 betweenadjacent boxes. Arc line arrays can provide themaximum coverage angle for any number ofboxes because they can use the full draft anglebetween each box. The maximum angle of anarc array aMaxis:

M 2 1

2M. S. Ureda, Analysis of Loudspeaker Line Arrays, J. Audio

Eng. Soc., Volume 52, Issue 5 pp. 467-495; May 2004.

where n is the number of boxes and is thedraft angle. So, for example, an arc array withfour boxes and a 5 draft angle will have amaximum coverage angle of 30. If you need alarger coverage angle, you will need to use more

boxes. Figure 4 shows a grid with number ofboxes along the top and coverage angle alongthe side. Number-angle combinations greaterthan 30 for four-box arrays and greater than 70for eight-box arrays are blocked out in redbecause these combinations are not achievable.Most of the cells, however, are blank indicatingnumber-angle combinations that areachievable

3. This shows the versatility of arc

arrays: Virtually any number of boxes canprovide almost any coverage angle.

Progressive arrays are not so versatile. Theyare straighter at the top to provide long throwand, therefore, do not utilize the full draft angles

of the boxes. This limits the maximumachievable coverage angle for a given numberof boxes. The maximum angle of a progressivearray pMaxis

M 1.

This (approximate) result for the maximumcoverage angle is based on an average anglebetween the boxes in the array. A progressivearray has virtually no splay angle (0) betweenboxes at the top and a maximum splay angle of2 at the bottom. The average of these two

extremes is. There are (n - 1) inter-box anglesin an array. Multiplying these together providesan estimate of the maximum total coverageangle achievable. Figure 5 shows number-anglecombinations for progressive arrays that areachievable. The combinations not achievableare colored red as before. Notice thatprogressive arrays are much more constrainedin achievable number-angle combinations thanarc arrays.

4. HIGH FREQUENCY DECOUPLING

An inescapable reality with loudspeaker

arrays is that at some high frequency, boxes inan array do not sum as nicely as we would like.This is physics. At low frequency, wherewavelengths are long, each box of an arrayradiates into a relatively wide angle. This allows

3 This number-angle grid is based on 5 draft angles (top and

bottom). Different draft angles would produce a different grid of

achievable and non-achievable arrays.

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the outputs of adjacent boxes to sum properly.As frequency increases, the wavelengths areshorter and the polar pattern narrows. At somehigh frequency, the angle of the radiated energyis smaller than the splay angle between boxes.This causes the array to behave as a set ofindependent acoustic sources. The array nolonger sums as a coherent whole. We call thisphenomenon decoupling.

It is quite easy to determine the frequencywhere decoupling begins. We know that thequarter-power angle of a line source

4is:

2.4 10

where 6dB is the quarter-power angle, f isfrequency and Ls is the length of the source

5.

We can rewrite this expression to solve for

frequency. Substituting the largest inter-boxsplay angle (LS)6 in the array for the quarter

power angle we get:

2.4 10

If a box has a source length Lsof meter anda largest inter-box splay angle of 10, thehighest usable frequency fhbefore decoupling isapproximately 5kHz. Figure 6 illustrates thisresult. It shows polar response curves at threedifferent frequencies of two line sources, m

long, splayed 10. The polar curves on the leftshow that at 4kHz the primary lobes of the twosources meet at approximately -3dB. The polarcurve on the right shows the result at 6.3kHzwhere the outputs of the two sources havedecoupled and the main lobes meet at -9dB.The transition frequency is 5kHz, shown in themiddle. Here, the main lobes of the two sourcesmeet at -6dB as expected.

Since we know the largest required inter-boxsplay angle for all number-angle combinations inthe grids used earlier, we can insert thefrequency at which decoupling begins. This

4M. S. Ureda, Analysis of Loudspeaker Line Arrays, J. Audio

Eng. Soc., Volume 52, Issue 5 pp. 467-495; May 2004.

5 The length of the source refers to the length of the acoustical

source. For a line array, it is approximately equal to the height ofa box.

6 An arc array has a constant inter-box angle and this, bydefinition, is the largest splay angle. The largest splay angle of a

progressive array is typically the angle between the bottom two

boxes.

frequency is the upper bound for uniformcoverage and smooth frequency response. Thisis shown in the following sections for arc andprogressive line arrays

7.

Decoupling of Arc Arrays

Figure 7 provides the closest ISO 1/3 octavecenter frequency at which decoupling occurs foreach number-angle combinations of arc arrays.To help visualize the patterns, the cells are colorcoded according to frequency as shown in thefollowing table.

>12.5kHz

12.5kHz

10kHz

8kHz

6.3kHz

5kHz

Referring to Figure 7, an eight-box arc arraywith a total coverage angle of 50 has a constantinter-box splay angle of 7. This yields a highfrequency limit of 6.3kHz before decoupling. Ifthe coverage angle were 30, the inter-box splayangle would be 4 and decoupling would notoccur until 10kHz. Smaller inter-box anglesextend smooth polar response to higherfrequencies.

Conversely, Figure 7 is useful for determininghow many boxes are required to provide smoothresponse up to a certain frequency for a given

required coverage angle. Lets say we need atotal coverage angle of 55. The first thing wemight notice is that it is not possible to achievethis angle with four boxes. An eight-box arraycould do it and would provide smooth responseup to 6.3kHz. Larger arrays of twelve andsixteen boxes would extend this to 10kHz and12.5kHz respectively. This illustrates a centralpoint of this Tech Note; larger arrays provideextended smooth polar response. This resultobtains because the inter-box splay angles aresmaller for large arrays.

Decoupling of Progressive Arrays

A similar result obtains for progressive arrays,shown in Figure 8. Notice that progressive

7All examples provided in this Tech Note are based on boxes with

5 draft angles and a source length (box height) of m.

Different draft angles and source lengths will produce differentresults. The reader can use the methodology provided in this

Tech Note to produce performance grids reflecting alternative

physical parameters.

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arrays require a larger number of boxes toachieve the same high frequency performanceas arc arrays. The eight-box 55 arc array inthe previous example provided smooth responseup to 6.3kHz. However, 55 is not obtainablewith eight boxes in a progressive configuration.Twelve boxes can achieve 55 but the highfrequency limit is only 5kHz. Achieving 6.3kHzperformance with a progressive array requiressixteen boxes twice as many as an arc array!

This is an important point as progressivearrays are so prevalent these days. They areused widely because their asymmetrical polarresponse accommodates so many venues.However, this asymmetry comes at a cost: Moreboxes are required to provide smooth response,especially toward the front of the house.Progressive arrays with a small number of boxesand large coverage angles do not work well athigh frequency. Asymmetrical response and

high frequency response are obtainable, butonly with large arrays. This illustrates one of thetrade-offs we discussed at the beginning of thisTech Note.

5. DIRECTIVITY RESPONSE

Now that we have explored performancetrade-offs at the high frequency end of thespectrum, let us turn our attention to the lowend. Here we can apply the physics ofloudspeaker horns to line arrays.

The mouth size and coverage angle of aloudspeaker horn determine the low frequency

to which it will maintain directivity control. Thislow frequency break point is the breakfrequency

8where

10

Here, is the horn angle (in degrees) and x isthe dimension across its mouth (in inches). Ahorn with a wider angle and/or larger mouth willprovide directivity control to a lower frequency.This is true for loudspeaker line arrays as well.Substituting the total coverage angle and

length L of the array for and x, we get anexpression for the break frequency of a linearray:

10

8C. Henricksen, M. S. Ureda, The Manta-Ray Horns, J. Audio

Eng. Soc., Volume 26, Issue 9 pp. 629-634; September 1978.

where L is in inches and is in degrees.Rewriting for Lexpressed in meters,

2.5 10

The break frequency for a loudspeaker linearray is largely independent of shape that is,an arc array and a progressive array of thesame total coverage angle and length will havethe same low frequency directivity control. Thisis because the wavelengths at low frequency arelong relative to the differences in the shape ofthe arrays.

Returning to an example used earlier, a ten-box array with a total coverage angle of 45 anda box height of m will have a low frequencybreak point of 100Hz:

2.5 10

4510.5 100Hz

Figure 9 shows the directivity response curvesof the arc and progressive ten-box arrays shownin Figures 2 and 3 respectively. Both have 45total coverage angles. It is clear that 100Hz isthe lowest frequency to which the arrays hold 45degrees

9. Figure 9 shows also that there is

virtually no difference in the directivity responsesof the arc and progressive configurations.

The inverse relationship between breakfrequency and angle-length product means thatnarrower angle arrays and shorter arrays willhave a higher break frequency - less lowfrequency control. Figure 10 shows thedirectivity response of a six-box, 30 array with abreak frequency of 300Hz. The response ofthe ten-box, 45 array is overlaid for comparison.

Let us now return to number-angle grids. Wehave just seen that array coverage angle andlength drive the break frequency and, of course,the number of boxes drives the length

10. The

number of boxes is therefore a proxy for lengthand we can insert the break frequency into thegrid for each number-angle combination. The

best way to see the impact of length and angleon break frequency is to color code the grid asbefore. Figure 11 is based on the following colorscheme:

9Notice that the directivity response narrows considerably in the

region just above the break frequency. This is called midrange

beaming. See Henricksen, (Footnote 8) for further discussion.

10This assumes a fixed box height. All examples in this Tech Note

use a box height of m.

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800Hz

The performance grid shows that large arraysand large angles drive the break frequencydown. This keeps low frequency energy fromspreading beyond the intended angle and ontopotentially unwanted surfaces such as reflectiveceilings or floors.

6. DIRECTIVITY BANDWIDTH

Our final step in characterizing the overalldirectivity bandwidth of line arrays is to overlaylow frequency performance and high frequencyrange on the number-angle grid. This provides

a graphical summary of the overall performanceof each number-angle combination and indicatesexpected directivity bandwidth.

Bandwidth of Arc Arrays

Figure 12 shows the directivity overlay for arcline arrays. It is the combination of Figures 7and 11. The purple area indicates arrays thatwill provide both extended low frequency controland a high frequency onset of decoupling, i.e.wide directivity bandwidth. The orange areaindicates poor low frequency control or severedecoupling resulting in relatively narrow

directivity bandwidth. Blue, green and yellowindicate in-between levels of directivitybandwidth.

Note that the orange areas indicate eitherpoor low frequency control or a relatively lowfrequency onset of decoupling. Arrays along thetop of the performance grid are low frequencylimited; that is, they do not control directivity to avery low frequency (400Hz to 800Hz). Arrayslong the bottom of the grid are high frequencylimited. They start decoupling at a relatively lowfrequency (between 5kHz and 6.3kHz).

The arrays in the green and blue cells have

directivity bandwidths from 100Hz to 400Hz inthe low end and 8kHz to 10kHz in the top end.This represents perhaps two octaves ofimproved directivity bandwidth over yellow andorange arrays. Arrays in the purple cells havethe widest bandwidth, from below 100Hz to12.5kHz and above.

Bandwidth of Progressive Arrays

Figure 13 shows the directivity overlay forprogressive arrays. It is the combination ofFigures 8 and 11. Here we see that the regionfor wide bandwidth progressive arrays isconsiderably smaller than for arc arrays. This is

a very important point and indicates the trade-offs we face in choosing line arrays.Progressive arrays are very useful for projectingsound energy asymmetrically but only relativelylarge arrays with moderate coverage angles willachieve extended directivity bandwidth. Asbefore, orange and yellow arrays along the topof the grid will have poor low frequencydirectivity control. Orange and yellow arraysalong the bottom will have an early onset ofdecoupling. Green and blue arrays will havedirectivity bandwidths from 100Hz to 400Hz inthe low end and 8kHz to 10kHz in the top end.

The purple region identifies arrays with thewidest directivity bandwidth; these are largearrays with moderate coverage angles.

7. INTERPOLATION AND DIRECT

CALCULATION

The number-angle grids illustrate performancecharacteristics from a macroscopic perspective.We purposefully selected a wide range ofnumber-angle combinations - from 5 to 80 incoverage angle and 4 to 24 boxes in size. Thishelps show broadly the directivity bandwidthcharacteristics. Arc arrays have a larger range

of wide bandwidth number-angle combinations.However, they always produce a symmetricalpolar response. Progressive arrays can provideasymmetrical response but at a cost; theygenerally need to be large to achieve widedirectivity bandwidth.

To illustrate these macro characteristics, thegrids used rather large increments in size (4boxes) and coverage angle (5). The reader caninterpolate from the grid the performance ofother, in-between, number-angle combinations.For example, the bandwidth of a 14-box arcarray with a total coverage angle of 52 can beinterpolated from the four cells bounded by 50 to55 degrees and 12 to 16 boxes. Referring toFigure 7, we see that a twelve-box arc arrayprovides smooth polar response up to 10kHz forboth 50 and 55 coverage angles. A sixteen-box array provides 12.5kHz and above (blankcell). If we choose a value of 16kHz (the nexthighest ISO 1/3 octave center frequency) for theblank cell and average the four values we get

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10 10 12.5 16

4 10 12

Now referring to Figure 11, we again averagethe values in the four cells bounded by 50 and55 degrees and 12 and 16 boxes. This providesan estimate of the low frequency break point:

83 63 76 57

4 70

Therefore, a 14-box arc array with a totalcoverage angle of 52 will have a directivitybandwidth of 70Hz 12kHz.

Alternatively, directivity bandwidth can bedetermined directly. We saw earlier that thelargest splay angle and the length of theacoustic source determine the onset ofdecoupling. The upper bound of directivitybandwidth for any line array type was given as:

2.4 10

The 14-box arc array will have a uniform inter-box splay angle of 4. Given a source length ofm (approximately the height of a single box),the high frequency onset of decoupling is

2.4 10

2.4 10

. 54 12

The low frequency break frequency for anyline array type is:

2.5 10

This expression is valid for any array shapebecause the break frequency depends only onsize (number of boxes) and total coverageangle. For the 14-box example array, we get:

2.5 10

5214.5 69

These results match the expected directivitybandwidth interpolated from the number-anglegrids. In summary, it should be clear thatdirectivity bandwidth depends only on the heightof the boxes, the number of boxes, the largestinter-box splay angle and the total coverageangle.

8. CLOSING

We began with a discussion regarding thetrade-offs one needs to evaluate in designingand installing loudspeaker line arrays. Often, weface constraints that severely limit our options.In these cases, the preceding discussion on

directivity bandwidth only informs ourunderstanding of the expected performancelimitations of our array(s). In other cases, weare less constrained. Here the reader isencouraged to actively assess the trade-offs andrelative merits among size, shape and directivitybandwidth with a goal of maximizing uniformcoverage and tonal balance.

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Figure 1: Ten-box arc array with 5 inter-box

s la an les. Total covera e an le is 45.

Figure 2: Ten-box arithmetic progressive

array. Total coverage angle is 45.

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Figure 3: Acoustical maps of 45, ten-box arc

array (on left) and progressive array (on right).

Number of boxes

Achievable number-angle

combinations.

lcoverage

angl

Fi ure 4: Performance rid for arc arra s.

Tot

Not achievable

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Number of boxes

Achievable number-angle

combinations.angle

Totalcoverage

Figure 5: Performance grid for progressive arrays.

Not achievable

Intersectionat -3dB

Intersectionat -6dB

Intersectionat -9dB

Figure 6: Polar response curves of two line

sources, .5m long and splayed 10

4kHz 5kHz 6.3kHz

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Number of boxes

angle

High frequency limit in

colored cells.

Totalcoverage

Figure 7: Performance grid for arc arrays.

Not achievable

Number of boxes

colored cells.

verage

angle

T

otalc

Not achievable


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Breakpoint

Figure 9: Directivity response of ten-box, 45 arc

(blue) and progressive (red) arrays.

Break oint

Figure 10: Directivity response of a ten-box, 45 arc

array and a six-box 30 arc array (blue).

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Figure 11: Break frequency gr id for arc and

progressive arrays.

4 8 12 16 20 24

5

10

Low

frequency

15

20

25

30

35

40

45

limitations

Wide

Moderate

directivity

bandwidth

50

55

60

65

70

75

High

frequency

limitations

bandwidth

Figure 12: Performance grid for arc arrays.

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4 8 12 16 20 24

5

1015

20

25

30

Low

frequencylimitations

Wide

directivit

Moderate

directivity

40

45

50

55

60

65

High

frequency

limitations

bandwidth

75

80


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_directivity bandwidth of line arrays

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