Published in September 2005

Loudspeaker Directivity
By Richard Honeycutt, PhD

What it’s good for, how you get it.

Figure 1. Small sanctuary having custom-built speaker system with 180° horizontal coverage above 800Hz.

Most audio professionals know that some rooms absolutely require directional speaker systems, while others will let you get by with an omni system. The choice finally comes down to one of speaker location and room architecture, but in general, the equations show that maximum acoustic gain before feedback depends directly on speaker Q. In this case, Q refers to the ability of the speaker system to radiate sound in one particular direction, as opposed to spraying it all over the place. Specifically, Q is defined as “the ratio of sound pressure squared, at some fixed distance and specified direction, to the mean squared sound pressure at the same distance averaged over all directions from the transducer.”1 Q is related to the horizontal and vertical coverage angles of an ideal loudspeaker as follows:

Although Q is useful in calculating acoustic gain, a knowledge of the effective horizontal and vertical coverage angles at various frequencies often is more helpful in the system design process.
The key words in that last sentence are at various frequencies. Directivity of any speaker or system always varies with frequency. We’ll try to help you understand something about that variation here, and give you an idea of what is and what is not possible in the way of directional performance, because you are otherwise at the mercy of manufacturers’ advertising departments.

Huygens’ Principle

Any acoustic radiator with constant phase over its surface, such as a woofer cone operating at low frequencies or an electrostatic panel, radiates omnidirectionally at low frequencies, becoming more directional at higher frequencies. The reason for the directionality can be appreciated if we remember Huygens’ Principle, which is that any radiating surface can be represented accurately as if it were a set of individual small surfaces, each radiating omnidirectionally. Thus, if we name one such small subdivision of a cone “surface A” and another, “surface B” (Figure 2), we can identify the effects of travel time from each of these two surfaces to any given listening point.

Figure 2. How directional effects are calculated.

 Figure 3. Cone driver. Figure 4. Directivity of a piston radiator.

If a listening point is equidistant from both surfaces, sound radiations from the two will add. If at some frequency, the travel time for the wave from surface A is half a period longer than that from surface B, the two waves will arrive out of phase, and cancel. Through the use of calculus, it is possible to determine the combined response to the radiation from all of the small surfaces of a cone or other radiator, for any listening position. These combined responses can be plotted to give a picture of the directional radiation pattern of the cone at one specific frequency (Figure 4).2
The graphs in Figure 4 are labeled according to the value of ka, which is the product of cone radius and wavenumber. (Wavenumber is given by

where f is frequency and c is the speed of sound.) With a little algebra, we find that ka is just the ratio of circumference over wavelength. Recognizing that the wavelength of a 1kHz sound is about a foot, and remembering that wavelength is inversely proportional to frequency, we see that, for a 15-inch speaker, the circumference is a bit less than 4 feet, so the ka=1 graph applies to a 15-inch speaker reproducing about 250Hz. For a 12-inch speaker, the corresponding frequency would be about 333Hz.
Notice that significant directionality does not occur until the frequency is higher than the one for which ka=1. It is also generally true that cones behave pretty much as true pistons, radiating in phase over their whole surfaces, for frequencies where ka<1. Real cones do not radiate in phase over their entire surfaces at frequencies much above this value, though, so the curves in Figure 2 for ka>1 generally are not valid in the real world.

 Figure 5. Klipsch MCM Grand all-horn speaker system.

Thus, at higher frequencies, the directional pattern of a real cone is different from the theoretical pattern for a piston of the same size. In general, the effects of cone breakup (different parts of the cone radiating out of phase with each other) cause a speaker to radiate over a wider solid angle at higher frequencies than would a true piston. It is also generally true that the directional pattern of any cone speaker narrows as the frequency increases. This effect is known as beaming.

Two Methods
Where it is desired to radiate directionally in order to control the spread of sound, two methods are generally used: horns and arrays. Well-designed horns make smaller diaphragms act as if they were larger, producing a spherical wavefront whose radius is controlled largely by the horn profile and mouth area. Arrays use interference effects among a group of smaller radiators to create the same sort of wavefront. In either case, the directivity is only controllable for frequencies whose wavelengths are equal to or less than the circumference of the device.
In the case of a line array, the length of the line must be about 0.6 wavelengths or longer for directional control to result. Neither of these devices provides a magic way of achieving low-frequency directivity: In fact, a single speaker of the same dimensions would do as well at the lowest frequencies. At higher frequencies, though, the directivity of a horn or array is much better controlled than that of a single driver, and more amenable to design manipulation.

Unbaffled Speaker
One other class of radiator can be used to achieve directivity, and this one does work at low frequencies, even without requiring huge radiators. The simplest member of this class is an unbaffled speaker, also called a doublet radiator. Because the speaker radiates equally, but in opposite phase, from both sides of the cone, all radiation is cancelled in the plane perpendicular to the axis of the cone. Thus, the directivity takes the form of a figure-8. If appropriate acoustic resistance is placed on the back side of the cone, other patterns, such as hypercardioid and cardioid (like microphone patterns) are available. The cost associated with this technique is much greater cone excursion for a given amount of low-frequency output. This places high demands on speaker mechanical and thermal design.
An acoustic horn is a device in which the sound is channeled through a gradually expanding cross-sectional area from the driver end (throat) to the radiating end (mouth). At very low frequencies, the horn essentially is acoustically transparent, and the driver acts as if the horn were not present. Then, above a certain cutoff frequency, the horn begins to act as an acoustic transformer, converting the low acoustic impedance (low-pressure, high-volume-velocity) mouth termination into a high acoustic impedance (high-pressure, low-volume-velocity) at the throat. This transformation greatly increases the efficiency of the radiating system, often by 12dB or more, and reduces excursion, and thus excursion-related distortions (the most common kind).
In between the mouth and the throat, the manner in which the cross-sectional area increases as the sound travels down the horn affects both the low-end cutoff frequency and the directional radiation.

Figure 6. Directional coverage of horns.

Directional radiation by horns can be discussed in three frequency regions (see Figure 6).3 In the lowest region, radiation changes from omnidirectional to directional exactly as it would for a cone the size of the horn mouth. In the middle region, the horn directs sound as you would expect by examining its walls: fairly constant directivity for flat walls and increasing directivity with frequency for curved flares. At the highest frequencies, the driver diameter controls directivity.
There are two non-obvious features to be aware of here, though. The first is that, just above the mouth control region, the directivity may narrow for about half an octave or so; this is called “waist-banding.” The second is that, if the horn walls are not straight in profile, the horn will beam at higher frequencies. Essentially, for about the first half-wavelength of travel, a sound wave conforms to the surface of the horn walls. Then, even if the walls diverge, as in an exponential horn, the sound waves no longer follow the walls. Thus, horns with very slow expansions at the throat will be much more directional at high frequencies than at low frequencies.

The so-called radial horn exploits this fact by maintaining a constant conical flare in the horizontal plane, with an extreme deviation from conical in the vertical plane. Thus, the horizontal directivity is almost constant with frequency, but the horn beams vary significantly in the vertical plane. Such horns make use of the fact that, in many applications, a 10° or 20° vertical radiation is adequate, but a 90° or wider horizontal directivity is needed.
The top region of horn directivity is that produced by the directional characteristics of the throat, generally as defined by the opening where the compression driver is mounted. Thus a horn with a 25mm driver can have a wider high-frequency radiation pattern than will a horn with a 50mm driver. Transition to this region takes place somewhat above the frequency where the driver circumference equals the wavelength: about 5kHz for a 25mm driver or 2.5kHz for a 50mm driver. (Of course, the frequency at which the transition actually occurs is determined by the wall angle of the horn as well as by the driver opening.)

 Figure 7. A Radia Pro 1.9 ribbon line array.

One way to increase the high-frequency coverage angle is by the use of “bullets” in front of the phase plug of the driver. Simulations and measurements of horn operation indicate that these devices introduce frequency-response irregularities.4
The bottom line is that a “90°x40°” horn only has the 90°x40° radiation pattern within the middle region already discussed. A fair approximation of the lowest frequency at which a horn can provide directional control in a given plane (horizontal or vertical) is the frequency at which the horn dimension is about 0.6 wavelengths. Thus, the common 4"x10" horn used in low-priced systems, if mounted with the long dimension horizontal, would provide directional control above frequencies that can be approximated this way:
Vertical dimension equals 0.6 wavelength when wavelength is 4"/0.6=6.67".

This corresponds to a frequency of

Horizontal dimension is 2.5 times as great, so the frequency is 2.5 times lower, or 902Hz. Most of these horns have a true exponential flare (curved walls), so they beam at high frequencies, the directionality dropping to perhaps 45°x20° at 10kHz. Constant-directivity (CD) horns have much flatter sidewalls, and thus do not beam nearly as badly in the 1000-10kHz range. However, the throat design of a CD horn usually involves a narrow tunnel terminating in a diffraction slot.
Often this slot is of such dimensions as to cause significant beaming at the highest audio frequencies. For good horizontal coverage (90° to 120°) at the highest audio frequencies, a slotloaded compression tweeter is hard to beat. Often the slot is about ½-inch wide, avoiding beaming at any frequency below about 16kHz.
Where multiple horns are arrayed to cover the same frequency range, interference effects will cause response irregularities (called comb filtering because of the shape of the resulting frequency response curve) in the overlap zones. The closer together the horn drivers are located, the higher the frequency at which these irregularities will take place and, thus, the less objectionable they will be. Many designers deliberately underlap the horn patterns in areas where both horns will cover the audience, reasoning that slight droops in the level (meaning lower direct-to-reverberant sound ratio for the audience) are less objectionable than the response irregularities resulting from interference effects.

Near, Far Fields
The behavior of the sound field radiated by any source depends on the listening distance. Generally, listening distances can be separated into the near field and the far field. The near field extends from the surface of the radiator to the greater of these distances:

where L is the longest dimension of the source, and the distance, L, and l are all measured in the same units. In the near field, the sound level from the source decreases only slowly with distance. In the far field, extending from the near-field transition distance to infinity, the sound level from the source decreases at a rate of 6dB for each doubling of distance if the source is outdoors. Inside, room effects cause a slower decrease, somewhere between 3dB and 6dB per doubling of distance. These effects are important for measurement, in which the microphone should be in the far field for all frequencies of concern, for accurate results. They are also important in the evaluation and application of array speakers.

Line Arrays
Every few years, someone in the audio world rediscovers a principle that was investigated and usually patented in or before the 1940s, proclaims it to be a great breakthrough, and may even patent it again. One such principle is that of the line array. A true line array is a continuous strip radiator that can be oriented horizontally or vertically, and that produces directional control through interference effects among separate portions of the array. Only two true arrays are presently available to this author’s knowledge: the electrostatic strip radiator and the electromagnetic strip (“ribbon”) radiators sold by several companies. Numerous manufacturers produce stacks of boxes that are meant to act more or less as line arrays, but these are not true line arrays.
A true line array radiates in phase all along its length. A close approximation to a true line array can be achieved if individual radiators are fed in phase, and the centers of adjacent units are within ¼ wavelength of each other at the highest frequencies. Arrays that do not meet this requirement will exhibit irregular frequency response.

Figure 8. Directional radiation of a line source as a function of length.

Figure 8 5 shows the directional characteristics of a true line array. Notice that, in addition to the major lobe, at lengths greater than one wavelength, minor lobes also appear. Listeners at these angles will experience extremely irregular frequency response. However, also note that the levels of the minor lobes are reduced greatly when compared to the major lobes. The approximate equation6 for the angular width of vertical coverage for a vertical line array is
q = 2 sin-1(l/L). (The angle will be expressed in radians.)
Also note that the directivity is not constant with frequency, but becomes narrower as frequency increases. In some applications, this effect is not a problem. As an example, when a line array is mounted with its center in the plane of the listeners’ ears, a very narrow vertical directivity is permissible, and all listeners can hear pretty much the full frequency range.
An improvement on the true line array would involve segmenting the array so the central portion of the radiator handles the full frequency range, with segments farther from the center being fed low-pass-filtered signals with progressively lower cutoff frequencies. Thus, the effective length of the array, as measured in wavelengths, could be kept essentially the same, resulting in a more uniform directivity. This approach is called “frequency shading,” and it has been demonstrated that the Bessel filter is the best topology to use in frequency-shaded arrays. Philips owns a patent on the Bessel array.
Another variation on array design is the curved array, which provides less high-frequency beaming.

 Figure 9. Amina DML panel with a graphic printed on it.

To allow the sound produced by a line array to be electronically controlled, the array elements or drivers can be fed through separate amplifiers, with different delays applied to the different elements. Such “steerable arrays” are available from most of the manufacturers of touring sound systems.

Near-Field-to-Far-Field Transition
Another seldom discussed aspect of line-array application is the effect of the near-field-to-far-field transition. To the extent that a line array radiates sound cylindrically, sound level will decrease at a rate of 3dB per doubling of distance as the listener moves away from the array, assuming anechoic or outdoor conditions. Indoors, the rate is even slower. If the array extends from floor to ceiling in a room, it will act as a true cylindrical radiator at all frequencies. Shorter arrays will act more or less as cylindrical radiators at frequencies where l<L. Thus, if we examine the vertical directivity of an array two meters long, it will be essentially omnidirectional below about 70Hz, with sound pressure decreasing at 6dB per doubling of distance (outdoors) at listening distances in the far field
(> about 8m at 70Hz for this array).
The array will have about a 40° vertical pattern at 200Hz, with sound pressure decreasing at 3dB per doubling of distance in the far field, which is > about 4m at 345Hz. At 1kHz, the pattern will be about 10°, with sound pressure decreasing at 3dB per doubling of distance beyond about 3m. Theoretically, at 10kHz, the 3dB rate does not occur until the listening distance is about 30m.7 Notice the trend. At high frequencies, the listener has to be farther and farther away from the array before the 3dB rule takes effect. At closer distances, the rate changes from 0dB (per doubling of distance) at very close distances to 3dB at the far-field transition distance, which is frequency dependent.

 A sanctuary with a custom speaker consisting of two coaxial full-range horns in a single cabinet designed to match the sanctuary woodwork. The horns provide just the angular coverage required by the venue.

In applications in which the array is to be equalized for a predetermined frequency response, it becomes necessary to identify the reference listening distance at which a flat response is desired, as the response flatness will vary over several dB for different listening distances. This effect is mitigated significantly when line arrays are used indoors because room reflections tend to average out the anomalies to some extent.

And Finally, DMLs
Some years ago, a new variety of loudspeaker was introduced, one that actually had not been examined prior to the 1940s! It is the distributed mode loudspeaker, or DML. This loudspeaker consists of one or more transducers mounted to a carefully designed panel. In operation, the transducer imparts a bending wave to the panel, which is then radiated as sound as it travels to the perimeter of the panel, is reflected back, and so forth until it is damped out. By appropriate choice of panel materials, the duration of “reverberation” of the sound in the panel is carefully controlled, so pretty good impulse response is maintained. However, the radiation is phase incoherent, so interference effects essentially are cancelled out, and the panel can produce an overall flat response. Usually, the panels are used with no back enclosure, and in such application, they radiate in a 360° pattern in free space.
When mounted near a wall, they produce an almost 180° (half-spherical) pattern. Further, their incoherent radiation frees them from deleterious interference effects (comb filtering) when more than one panel is used to cover a single area. And they are less prone to excite room modes than are phase-coherent speakers.

 Church sanctuary with two unobtrusively-positioned ribbon line arrays (look at right and left edges of the image).

Finally, because of the fairly large area that is usual for DML panels, the radiation is distributed such that listeners in what would normally be the near field are not exposed to as high acoustic levels as one would expect. Thus, for example, a DML panel can serve double duty as a boardroom whiteboard, with little chance of feedback when the presenter with a wireless mic stands near the board.
Certainly, the DML is no cure-all. It has no place in applications requiring high directivity to produce good intelligibility. It cannot produce concert sound levels. Its transient response is noticeably somewhat less than highest fidelity. But its unique combination of characteristics does qualify it for some applications for which more common speakers are less than ideal.

In Closing
We’ve presented the effects of sound radiation and loading here that are thought to be of greatest importance to practicing sound professionals. Minimal mathematics has been used.

References
1 Peterson, APG; and Gross, Ervin E., Jr.: Handbook of Noise Measurement, Seventh Edition, General Radio, Inc., 1974.
2 Holland, Keith R.: “Principles of Sound Radiation,” in Loudspeaker and Headphone Handbook, Third Edition, John Borwick, editor, Focal Press, Oxford, 2001, p. 14.
3 After Holland, op. cit., p. 35.
4 Morita, Shigeru, et al: “Acoustic Radiation of a Horn Loudspeaker by the Finite Element Method—A Consideration of the Acoustic Characteristics of Horns,” in Loudspeakers, Vol. 2, published by the Audio Engineering Society, 1984, pp. 161-168.
5 Olson, Harry F., Acoustical Engineering, Professional Audio Journals, 1991, p. 36.
6 Kinsler, Lawrence, et al, Fundamentals of Acoustics, 3rd Ed., John Wiley and Sons, New York, 1982.
7 See Beranek Leo L.: Acoustics, McGraw-Hill, New York, 1954, p. 100; Kinsler, op. cit., p. 188; and AES2-1984 (r1997): AES Recomended Practice—Specification of Loudspeaker Components Used in Professional Audio and Sound Reinforcement, Audio Engineering Society, 1984, Appendix A, p. 10. Note that these references do not exactly agree as to the extent of the near field!

Richard Honeycutt, BS Physics; PhD, Electroacoustics, is a freelance audio/electroacoustical engineer and writer. His work includes writing for numerous audio publications, and assisting consultants and contractors with audio system design.

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