# Can Line Arrays Form Cylindrical Waves? A Line Array Theory Q & A

February 2005

What Is a Line Array?

A line array is a group of radiating elements arrayed in a straight line, closely spaced and operating with equal amplitude and in phase. Described by Olson in his 1957 classic text, Acoustical Engineering, line arrays are useful in applications where sound must be projected over long distances. This is because line arrays afford very directional vertical coverage and thus project sound effectively.

Fig. 1. Directional behavior of an eight meter long array of sixteen omni-directional sources

The MAPP plots of Figure 1 illustrate the directional characteristics of a line array composed of sixteen omni-directional sources uniformly spaced 0.5 meters apart. The array is highly directional to 500 Hz; above that, the directional characteristic begins to break down. Note the strong rear lobe at low frequencies; all conventional line arrays will exhibit this behavior because they are omnidirectional in this range. Note also the strong vertical lobes at 500 Hz. (The horizontal pattern of this system is independent of the vertical, and is omni-directional at all frequencies.)

Fig. 2. Directional behavior of an eight meter long array of thirty-two omni-directional sources

Figure 2 shows a line of thirty-two sources spaced 0.25 meters apart. Notice that this array maintains its directional characteristic to 1 kHz, where the strong vertical lobe appears. This illustrates the fact that directionality at high frequencies requires progressively more closely spaced elements.

How Do Line Arrays Work?

Line arrays achieve directivity through constructive and destructive interference. A simple thought experiment illustrates how this occurs.

Consider a speaker comprising a single twelve-inch cone radiator in an enclosure. We know from experience that this speaker’s directivity varies with frequency: at low frequencies, it is omni-directional; as the sound wavelength grows shorter, its directivity narrows; and above about 2 kHz, it becomes too beamy for most applications. This is why practical system designs employ crossovers and multiple elements to achieve more or less consistent directivity across the audio band.

Stacking two of these speakers one atop the other and driving both with the same signal results in a different radiation pattern. At points on-axis of the two there is constructive interference, and the sound pressure increases by 6 dB relative to a single unit. At other points off-axis, path length differences produce cancellation, resulting in a lower sound pressure level. In fact, if you drive both units with a sine wave, there will be points where the cancellation is complete (this is best demonstrated in an anechoic chamber). This is destructive interference, which is often referred to as combing.

A line array is a line of woofers carefully spaced so that constructive interference occurs on-axis of the array and destructive interference (combing) is aimed to the sides. While combing has traditionally been considered undesirable, line arrays use combing to work: without combing, there would be no directivity.

Can a Line Array Form Cylindrical Waves?

In a word, no.

The common misconception regarding line arrays is that they somehow magically enable sound waves to combine, forming a single "cylindrical wave" with special propagation characteristics. Under linear acoustic theory, however, this is impossible: the claim is not science but a marketing ploy.

Unlike shallow water waves, which are non-linear and can combine to form new waves, sound waves at the pressures common in sound reinforcement cannot join together: rather, they pass through one another linearly. Even at the high levels present in the throat of compression drivers, sound waves conform to linear theory and pass through one another transparently. Even at pressure levels of 130 dB nonlinear distortion is less than 1%.

The MAPP plot of Figure 3, which shows a cross-fired pair of Meyer MSL-4 loudspeakers, illustrates this point. At the area labeled A, in the crossfire region, there is significant destructive interference in the dark areas. At the area labeled B, however, the output of the corresponding MSL-4 is completely unaffected by the cross-firing unit. Though the waves interfere at A, the interference is local to that area in space, and they still pass through one another unaffected. In fact, you could turn off the cross-firing unit and hear virtually no change whatsoever at B.

Fig. 3. Cross-fired MSL-4 loudspeakers

This experiment is best done in an anechoic chamber or outdoors in an open field, away from reflecting surfaces. It’s also advisable to apply a low-cut filter to remove information below about 500 Hz, where the MSL-4 starts to lose directionality.

But don’t line arrays produce waves that only drop 3dB with every doubling of the distance from the array?

This simplistic marketing claim appears to be a misapplication of classical line array theory to practical systems. Classical line array mathematics assumes a line of infinitely small, perfectly omni-directional sources that is very large compared with the wavelength of the emitted energy. Obviously, practical systems cannot approach these conditions, and their behavior is far more complex than some audio company marketers suggest.

Modeling the behavior of a fifteen-inch woofer with Bessel functions (which describe a piston), Meyer Sound has written custom computer code to model line arrays with various numbers of loudspeakers at various spacings. This computation shows that it is theoretically possible to construct an audio line array that follows the theory at low frequencies, but it requires more than 1,000 fifteen-inch drivers, spaced twenty inches center-to-center, to do it!

A truncated continuous line array will produce waves that drop 3 dB per doubling of distance in the near field, but the extent of the near field depends on the frequency and the length of the array. Some would have us believe that, for a hybrid cone/wave guide system, the near field extends hundreds of meters at high frequencies. It can be shown mathematically that this is true for a line of 100 small omni-directional sources spaced an inch apart, but that is hardly a practical system for sound reinforcement and is not a model for the behavior of wave guides.

Nor does the purely theoretical computation reflect the reality of air absorption and its effects at high frequencies. The table below shows the attenuation at various distances from an array of 100 one-inch pistons spaced one inch apart, as modeled using a Bessel function. At 500 Hz and above, it also shows the total attenuation when air absorption is included using the calculation given in ANSI Standard S1.26-1995 (the conditions for this table are 20° C ambient temperature and 11% relative humidity). Note that, while at 16 kHz the array as modeled by the Bessel function is approaching 3 dB attenuation per doubling of distance, air absorption makes its actual behavior closer to 6 dB per distance doubling.

 2 meters 4 meters 8 meters 16 meters 32 meters 64 meters 128 meters 256 meters 125 Hz 0 5.5 11 17 23 29 35 41 250 Hz 0 5 11 17 23 29 35 41 500 Hz 0 2.3 7.2 13 19 25 31 37 w/air absorption 38 1 kHz 0 1.3 3.2 8.2 14 20 26 32 w/air absorption 15 21 28 35 2 kHz 0 3 5.2 7 12 18 24 30 w/air absorption 8 13 21 29 41 4 kHz 0 2.7 6.3 9 11 16 21 27 w/air absorption 3.1 7.1 11 14 23 35 59 8 kHz 0 2.8 5 8.6 11 13 18 24 w/air absorption 3.5 6 12 17 25 42 72 16 kHz 0 3.1 6.6 8.2 12 14 16 21 w/air absorption 4.1 8.6 12 20 33 49 88 3 dB per doubling 0 3 6 9 12 15 18 21 6 dB per doubling 0 6 12 18 24 30 36 42

Table 1. Attenuation in decibels for octave frequency bands at various distances
from a line array of 100 one-inch pistons spaced one inch apart

With a practical, real line array of sixteen cabinets (each using fifteen-inch low frequency cones), a slight "cylindrical wave" effect can be measured at about 350 Hz, where there is a 3 dB drop between two and four meters from the array. More than four meters from the array, however, the sound spreads spherically, losing 6 dB per distance doubling. This behavior can be confirmed with MAPP using the measured directionality of real loudspeakers.

At frequencies below 100 Hz, the drivers in a practical line array will be omni-directional but the array length will be small compared with the sound wavelength, so the system will not conform to line array theory. Above about 400 Hz the low-frequency cones become directional, again violating the theory’s assumptions. And at high frequencies, all practical systems use directional wave guides whose behavior cannot be described using line array theory.

In short, the geometry of real audio line arrays is far too complicated to be modeled accurately by antenna theory. They can only be accurately modeled by a computational code that uses a high-resolution measurement of the complex directionality of actual loudspeakers, such as MAPP.

That said, practical line array systems remain very useful tools, regardless of whether the continuous line array equation applies. They still achieve effective directional control, and skilled designers can make them behave very well in long-throw applications.

How Do Practical Line Array Systems Handle High Frequencies?

Figures 1 and 2 show that line array theory works best for low frequencies. As the sound wavelength decreases, more and more drivers, smaller in size and spaced more closely, are required to maintain directivity. This is why some line array systems cross over to eight-inch drivers for the midrange. Eventually, however, it becomes impractical to use, for example, hundreds of closely spaced one-inch cones.

Practical line array systems therefore act as line arrays only in the low and mid frequencies. For the high frequencies, some other method must be employed to attain directional characteristics that match those of the lows and mids. The most practical method for reinforcement systems is to use wave guides (horns) coupled to compression drivers.

Rather than using constructive and destructive interference, horns achieve directionality by reflecting sound into a specified coverage pattern. In a properly designed line array system, that pattern should closely match the low-frequency directional characteristic of the array: very narrow vertical coverage and wide horizontal coverage. (Narrow vertical coverage has the benefit that it minimizes multiple arrivals, which would harm intelligibility.) If this is achieved, then the wave guide elements can be integrated into the line array and, with proper equalization and crossovers, the beam from the high frequencies and the constructive interference of the low frequencies can be made to align so that the resulting arrayed system provides consistent coverage.

Can Line Array Loudspeakers Be Used Singly?

No, the cone drivers in a line array loudspeaker need the other cones in the array to create directionality. The cones in a single cabinet have the same directional characteristics as comparable cone drivers in other types of loudspeakers. In other words, each cabinet in a line array is not producing a "slice of a cylindrical wave." That is a marketing concept, not a scientific one.

Can You Curve a Line Array to Get Wider Coverage?

In practice, gently curving a line array (no more than five degrees of splay among cabinets) can aid in covering a broader area. Radically curving line arrays, however, introduces problems.

First, if the high-frequency section has the narrow vertical pattern that’s required to make a straight array work, curving the array can produce hot spots and areas of poor high-frequency coverage. Second, while the curvature can spread high frequencies over a larger area, it does nothing to the low frequencies, which remain directional because the curvature is trivial at long wavelengths.

Figure 4 illustrates these points. On the left is a series of MAPP plots for a curved array, and on the right are plots of a straight array. Both arrays are constructed of identical loudspeakers having a 12-inch cone low-frequency driver and a high-frequency horn with a 45-degree vertical pattern.

Notable in the left-hand plots is that, while the wider horn aids in spreading the high frequencies, it also introduces pronounced lobing due to interference. At 1 kHz and below, the array remains highly directional, following line array theory. In practice, this behavior would produce very uneven coverage, with the frequency response varying substantially across the coverage area and a large proportion of that area receiving almost no low-frequency energy.

The right-hand series of plots reveals that a loudspeaker with a moderately wide-coverage horn designed for curved arrays behaves poorly in a straight array. While the array is highly directional, pronounced vertical lobing occurs at 1 kHz and above. These strong side lobes divert energy from the intended coverage area and would excite the reverberant field excessively, reducing intelligibility.

Fig. 4. Directional characteristics of a curved (left) and straight (right) line array using a high-frequency horn with a 45-degree vertical pattern

Can You Combine Line Arrays With Other Types of Speakers?

Yes, since linear waves pass through one another regardless of whether they are created by a direct radiator or a wave guide, it is possible to combine line array systems with other types of loudspeakers as long as their phase response matches that of the line array speakers. There is nothing special about the sound waves that line arrays create. They are merely the output of low-frequency cones, spaced using line array theory, and high-frequency wave guides. Therefore, skilled designers with the proper tools can flexibly integrate other compatible types of loudspeakers to cover short-throw areas.

Fig. 5. A CQ-1 rigged under an M3D line array provides downfill coverage

How Do Line Arrays Behave in the Near and Far Field?

As we have seen, practical "line array" systems as used in high-power applications are actually a combination of "classical" line arrays for the low frequencies and highly directive wave guides for the high frequencies. Because of this hybrid nature, it is difficult to apply predictions from classical line array theory across the whole audio spectrum. Nonetheless, line array systems can be made to work reasonably well in both the far field and moderately close to the array.

Seen from the far field, the outputs of the individual sources in a line array combine constructively, and appear to operate as one source. Figure 6 illustrates this concept. The figure shows the far-field frequency response for line arrays of two, four and eight omni-directional radiators (a single-omni response is included for reference) spaced 0.4 meters apart. Notice that each doubling of the number of elements results in a uniform 6 dB level increase across the full frequency range of operation. The high frequency response is smooth, but reflects a natural roll off due to air absorption (20 degrees C and 50% relative humidity).

Fig. 6. Far-field frequency response for line arrays with various numbers of sources showing high-frequency loss due to air absorption and humidity

The near-field behavior of practical line arrays is more complex. Any given point in the near field is on axis of only one of the very directional high-frequency horns, yet "sees" the low-frequency energy from most of the cabinets in the array. For this reason, adding cabinets to the array boosts the near-field low-frequency energy, but the high frequencies remain the same.

This explains why line array systems need high-frequency boost equalization. In the far field, the equalization effectively compensates for air loss. In the near field, it compensates for the constructive addition of the low frequencies and the proximity to the directional high-frequency wave guide.

The Meyer 3D (M3D)

Figure 7 illustrates how a low-frequency line array and high-frequency waveguides can be integrated to form a well-behaved, consistent system. It shows the directional characteristics of a line array comprising sixteen Meyer 3D (M3D) Line Array Loudspeakers. By virtue of the M3D’s REM (ribbon emulation manifold) and constant-Q horn, the high frequency radiation pattern closely matches the low frequencies.

Note, also, the absence of any significant rear lobe at low frequencies. This illustrates the advantages of the M3D’s BroadbandQ low-frequency directional technology. There is virtually no vertical lobing at 500 Hz (as was seen in the omni array of Figure 1) because the 15-inch cone drivers and the high frequency horn are aligned in this region to work together and suppress off-axis energy.

Fig. 7. Directional behavior of an eight meter long array of sixteen Meyer 3D (M3D) Line Array Loudspeakers