in September 2005
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.
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
Figure 2. How directional effects
Figure 3. Cone driver.
Figure 4. Directivity of a
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.
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.
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
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
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
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
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
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.
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
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
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.
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.
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.
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.
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.