September 2012

On Loudspeaker Directivity Part 1
Article By Jeff Poth

  I've talked a lot about directivity in these pages, particularly in my OSWG Building Oblate Spheroid Waveguides and Heil AMT1 And The Heil Horn articles. Directivity is a very significant topic, poorly understood by many people, so a more focused review of the topic is warranted. Fortunately, with the right sort of documentation and examples, directivity is a lot easier to understand. Let's start the easy way- with a definition.

Directivity: The gross behavior of an acoustic source in terms of three dimensional space.

That is a little more simplified than some definitions, but captures the key point- that the way the sound changes with changes in measurement (or listening) position relative to the acoustic source is largely defined by its directivity. To give a couple basic examples, a small coverage pattern horn (say, 40x60 degrees) would be considered high directivity and a dome tweeter would be considered low directivity. The light equivalent of extremely high directivity is a laser beam, where light exists at high intensity along a single, extremely narrow axis (and very little light anywhere else). The low directivity light equivalent would be a light bulb with no shade, spraying light in all directions equally and without encumbrance.

There are a number of ways to display directivity, I'm going to use a very simple display method. Each chart will be a frequency response simulation for each of many microphone positions, all equidistant from the acoustic source (speaker) and along the same horizontal plane. Vertical directivity has a role to play, but we'll focus on horizontal, which is by most accounts the far more important component. Below is a diagram- if a circle is centered on the acoustic source (the speaker) each measurement position is the same distance, but a certain number of degrees off axis, with 0 degrees being immediately in front of the loudspeaker, and 90 degrees being directly to the side of the loudspeaker, and the same distance away from it as the 0 degrees location.

Keep in mind that hemispherical or spherical wavefronts are just a 3D representation of the circle shown in the 2D diagram - they're expanding 3D circles, either half a circle or full, in which the pressure is constant at any point on the surface. A beach ball, expanding out from the acoustic source is a good approximation- the pressure on the surface from the air inside is equal at every point. The hemispherical version is simply the beach ball, cut in half. A speaker's baffle constrains radiation into a hemisphere until such time as the baffle size becomes small relative to the wavelength being reproduced. This phenomenon is known as baffle step. This detour will have to wait for another article, as it's quite an important piece with a number of solutions/approaches associated with it.

Omni
A perfect omnidirectional loudspeaker distributes sound equally in every direction, at every frequency. In a circle around a perfect omnidirectional loudspeaker, the frequency response will be the same at every position- not only in terms of response linearity, but also in amplitude. In other words, you could move around the loudspeaker and hear the same response at any position (absent any room reflections). It would have an identical tonal balance and amplitude (loudness). In the case of a true perfect omnidirectional loudspeaker, the source would be as small as possible, a single point in space radiating sound equally in all directions (sphere).

As you can see, the idealized case has a single response curve. This is because in an ideal omnidirectional loudspeaker, the frequency response is unchanging in the measurement condition described. An ideal omnidirectional speaker disperses sound equally in every direction and because of this, maximizes room reflections, though the energy exciting the room is equal in all frequencies. While some philosophies of loudspeaker design believe in minimizing the influence of the room, the manner of room influence is important too. A good omnidirectional design ensures that the reflections of the room are at spectrally balanced, and thus do not have a large difference between the tonal balance of the direct sound and the reflected. Duevel makes a number of omnidirectional speakers, using acoustic lenses.

Another user of omnidirectional loudspeakers is Dr. Sigfried Linkwitz. Dr. Linkwitz is one of the pioneers of audio, and anyone wanting to begin getting higher level understanding of loudspeaker and audio technology would be well served to spend a significant number of hours perusing his site. The Pluto uses an upfiring mid-woofer and a small format mid-tweeter with no baffle.

The smaller the baffle, the fewer obstacles there is to sound radiating to the sides or rearwards. In this case, there's no baffle to speak of, and thus the driver behaves omni-directionally up until frequencies where the driver diameter becomes significant (very high treble).

Two-Way 6.5 / 1" Dome
Speaking of frequencies where the driver diameter becomes significant, we'll come to the shortcomings of a "normal" hi-fi loudspeaker. Usually these have something along the lines of a 6.5" cone mid-woofer and a 1" dome tweeter, with a crossover frequency somewhere between 1.5 kHz and 2.5 kHz. Most designers stay away from 1500 Hz as it puts a lot of demand on output on the 1" driver and can lead to audible distortion. We'll discuss a 2 kHz crossover example, commonly chosen as the best balance between power handling in the 1" driver and directionality in the 6.5".

And therein lies the crux of the issue- a mid-woofer big enough to provide decent bass will also have a high enough diameter to have off-axis cancellation. This is known as "beaming", where the distance between driver edges is significant enough to have phase cancellations. This means that a driver, depending primarily on its diameter, will become increasingly directional with increasing frequency. A 15" for example will tend to be -6dB @ 45 degrees, around 1 kHz. A smaller driver diameter will become directional higher in frequency, and for a 6.5" two-way, the loudspeaker becomes directional before the 2 kHz crossover. This creates a dip in the frequency response off-axis, from the lower frequencies where the speaker behaves largely like an omnidirectional speaker, to becoming directional more like a horn loudspeaker. The same thing happens higher in frequency with the dome tweeter, again, beginning where the driver diameter begins to be large enough to begin beaming.

This means that not only will the tonal response change as you move off-axis, but also that the spectrum of the room reflections will always be different than that of the direct, on-axis response. This means that if you want the overall in-room frequency response to be "flat", you have to have an on-axis peak of a significant size. Likewise, if you want the on-axis response to be flat, you have to have a room response dip of significant size, leading to a perceived dip in this region. Not only will the loudspeaker be inconsistent with respect to listening position, but there's no way to get a predictable perceived response. This is a major shortcoming. Typically flat on-axis response is chosen over a more balanced power response and the associated peak. Humans notice dips less than peaks, but it's still an inconsistency that reduces the quality of the sound, even listening directly on-axis with a flat axial frequency response.

To deal with a larger mid-woofer crossing to a small tweeter and the associated power/room response challenge, loudspeaker designs have often gone to more drivers operating in their "omni" range- tapering in diameter based upon the bandwidth they're intended to cover. Depending upon baffle arrangement and number of drivers, this method can range from quite inconsistent like the example above, to relatively close to omnidirectional, relying upon many drivers, close together, each covering a limited bandwidth. The B&W Nautilus loudspeaker is one example of such a design, and would only suffer the inconsistency between room/power response and on-axis response in the highest frequencies. Such a design is necessarily extremely complex and expensive.

You'll note the similarity in the driver mountings between the Nautilus design and the Pluto from Linkwitz Lab. Key to achieving omnidirectional sound is the lack of a baffle -- a baffle tends to direct frequencies forwards, where it's size is meaningful relative to wavelength. Normal tweeters with 4" faceplates should not be used to execute such a design, as it will direct some frequencies into a hemispherical (forward only) pattern more than a spherical (omnidirectional) pattern based upon that acoustic barrier. Many loudspeakers use a pyramidal design, where a conventional box is used but the size of the baffles tapers to track the driver diameter.

Constant Directivity
In simplest terms, Constant Directivity means that as the listener moves off-axis, the frequency response remains unchanged. This is a rather inclusive term, as a perfect omnidirectional loudspeaker as above would meet the criteria. Omnidirectional would qualify as "Constant wide directivity". A speaker with sufficiently large, well designed, waveguides/horns would be "Constant narrow directivity", if you consider 90 degrees or so narrow. The key component of constant directivity is that the relative level of the frequencies, or frequency response, is unchanged as you move off-axis. Below is an example of what an ideal "90 Degree Constant Directivity" loudspeaker would look like- one that falls off more rapidly in level as you move more than 45 degrees off-axis. This has equal response amplitude within the coverage pattern, a 90 degree hemisphere. Beyond this 90 degree pattern (45 degrees on either side of the forward axis), the response drops off rapidly but the response linearity stays the same.

As with the ideal case of the omnidirectional speaker, the tonal balance is unchanged at any position of the listener. This ensures that the reflected energy of the room will only minimally differ (based upon room acoustics variations) from the direct response, and thus a flat on-axis response will mean that the perceived response will be flat. This is different than the 6.5" two-way example above, where the response will have dips in the perceived response at the top end of the 6.5"s range, and also at the top frequencies for the 1" tweeter.

To achieve this response, however, would require an unmanageably large loudspeaker. The horns or waveguides would have to be so large as to only work on a massive scale, like a 15 foot by 20 foot loudspeaker front surface area. As this is unrealistic, most designers choose to attempt to control directivity only within a limited range. Due to the practicalities of horn design, designs like my OSWG setup are pretty typical- a large mid-woofer, run up until where it becomes directional, and crossed over somewhere around 1 kHz to a constant-directivity horn/compression driver combo.

While the response is simplified for the sake of explanation, you can see that the responses are equal as you move along the equidistant arc, until you reach 45 degrees off-axis, after which it falls off rapidly, but while retaining the same frequency response in the horn range. Also notable is a small narrowing of the pattern above 10 kHz, but that's a very minor effect. This design ensures that you have the same reflected energy balance as that of the on-axis response, but also limits the off-axis energy. In other words, room reflections are spectrally balanced above 800 Hz or so, but also limited in amplitude by the directionality applied by the horn. This reduces the overall influence of the room, and allows for lower diaphragm deflection for a given SPL, as it's only maintaining full axial SPL over a 90 degree cone, a small area relative to full space (as in an omni loudspeaker). Such speakers excel in max SPL, low distortion, dynamics, low room coloration, and superb imaging made possible by cross-axial placement.

Some commercial examples are some speakers offered by:

www.gedlee.com (all speakers)

www.audiokinesis.com (all audiokinesis brand speakers)

www.pispeakers.com (4pi loudspeaker)

There are many DIY examples, particularly popular is the "Econowave" design, as described by our now passed-on compatriot Zilch. As he was fond of saying, "More data, less wank." RIP amigo.

Enough already!

That is it for now. In part two we'll delve into the behavior of dipoles. They represent a much more complex case than either constant directivity horn/waveguide loudspeakers, omnis, or "conventional" loudspeakers. I'll also talk a little bit about bipoles and the "Offset Bipole" arrangement favored by Duke LeJune at AudioKinesis, an interesting arrangement.

Correction of Omission in Distributed Bass:

Many of my articles tread a fine line between scientific and accessible. As such, I try to give reasonable, but not inclusive, references. In the article describing distributed bass, I missed one key fellow -- Dr. Floyd Toole, of Harman International. He’s done quite a bit of work on the topic and while I’m familiar with his work, I missed including him. Here are some of his white papers, including one on multiple subwoofer locations.

Click here for Part 2.