Cutting Corners

If anyone ever thinks to compile a list of the 100 seminal audio papers that should be found in every tech-aware audiophile's filing cabinet, Harry Olson's "Direct Radiator Loudspeaker Enclosures" deserves to feature in it. Originally presented at the second Audio Engineering Society Convention, in October 1950, it was published in Audio Engineering in 1951. In 1969—in a rare and certain acknowledgement of its classic status—the AES republished it in its Journal (footnote 1).

Olson's paper was all about the shapes of loudspeaker cabinets and the influence they have on frequency response. Olson worked in the frequency domain because this was easiest for him to measure, but the effects he was investigating in fact have their origins in the time domain. Generally, they are referred to as diffraction effects. Developments elsewhere, in digital audio—specifically, concerns about the energy smear of brick-wall digital filters—ought to have brought speaker diffraction back into the headlines, but it remains oddly disregarded.

Diffraction, to quote a dictionary definition, is "a deviation in the direction of a wave at the edge of an obstacle in its path," the obstacle in this case being the loudspeaker cabinet and the wave a soundwave generated by the speaker's drive-unit(s). Although you can think of diffraction in terms of waves "bending" around an object, it is much better for what follows to view it in terms of secondary radiation. When a soundwave traveling at grazing incidence to a speaker baffle reaches that baffle's edge, there is a more or less sudden change—depending on the form of the edge—in the radiation impedance the wave experiences. All such impedance transitions give rise to a division of the wave energy between transmitted and reflected components, the reflected part being what concerns us here. In effect, the edge of the baffle re-radiates a proportion of the incident wave energy, which, because it is delayed and undergoes a 180 degrees phase shift, interferes with the sound radiated directly from the speaker drive-unit(s) in a manner that is frequency-dependent.

And how. Olson's work showed that, if you were foolish enough to use a circular front baffle and place your drive-unit smack in the middle of it, you could expect response ripples of up to ±5dB, even if you were lucky enough to have a drive-unit with an inherently flat response. Whereas if you placed your blameless driver flush with the surface of a spherical cabinet, the response ripples would virtually disappear. (To appreciate just how strong the secondary radiation from the baffle edges is in the first instance, consider that if you wanted to imitate those response ripples using a second drive-unit with its output suitably delayed, its operating level would be only 5.7dB below that of the main driver.)

Few loudspeaker designers are so misguided as to use the former baffle shape—or, unfortunately, so enlightened as to use the latter. (Of course, there are exceptions.) Instead, the default moving-coil loudspeaker cabinet form is, as it always has been, the rectangular box. As Olson showed, this introduces significant frequency-response ripples too, although it doesn't perform as badly as the circular baffle, particularly if, as is usual, the driver is not mounted dead-center. This is not because the edges of a rectangular baffle radiate less energy—why should they, when the transition in radiation impedance is just as sudden?—but because the distance between the driver and baffle edge is no longer the same in all directions, so the secondary radiation is more spread out in time. This tames the response ripples, although whether it reduces the subjective impact of the edge reflections' time-domain cluttering is another matter.

To validate a means of suppressing the rectangular cabinet's remaining response ripples, Olson included among the 12 cabinet forms he tested one with 45 degrees chamfers at three of the four front baffle edges. By making the edge transition more gradual, these chamfers suppressed the secondary radiation sufficiently to almost match the spherical cabinet's smooth response. This explains why some modern loudspeakers use chamfers or bevels around the periphery of their front baffles. But a great many others don't, while some designers are even rash enough to toss in further edges in the form of constructional or aesthetic features.

Does this widespread apparent indifference to diffraction effects mean that they are, within reason, irrelevant? Is Olson's work more a historical curiosity than an informer of modern design practice? These are questions I began asking myself 20 years ago, when, for a while, I took to experimenting with review speakers by Blu-Tacking lengths of 2"-radius quadrant molding around their baffle edges, so as to perform quick A/B, with/without comparisons for myself and interested visitors. Adding or removing the molding certainly made a difference to the sound (an experiment the more hands-on reader might care to repeat), but back then, I had no measurement equipment with which to quantify the effect. Meanwhile, the majority of speaker manufacturers—not wanting to upset their bean counters by adding to production costs—resolutely soldiered on with easy-to-build cabinet wraps with 90 degree edges.

On the one hand, nothing much has changed in the two decades since. The box cabinet still rules, and edge chamfers or bevels of worthwhile dimensions are still the exceptions. The difference is that I now have the necessary measurement tools to hand—which explains, in a tangential way, how I recently came to build a 3m (10') square chipboard baffle in my listening room. This would allow me to do what I'd wanted to do all those years ago: quantify the effect of baffle-edge secondary radiation and begin to get a better handle on its practical significance.

The concept was simple. First, choose as a subject a rectangular box loudspeaker with 90 degree baffle edges, preferably one that is many years out of production, so that the manufacturer can't complain of being picked on. Second, build a baffle large enough that—with the speaker poked through a closely dimensioned hole in the middle of it—no baffle edge re-radiation can occur within the measurement window (about 6 milliseconds in my listening room, dictated by the 3m wall-floor spacing). Third, use DRA Labs' MLSSA system to measure the speaker's frequency response with and without the baffle extension—ie, with and without the effects of secondary radiation—and see what the differences are.

Time Scale
Before we get on to the results of that experiment, though, let's begin by considering the time scales involved here and the frequencies we expect to be affected. The speaker I chose to use was Acoustic Energy's AE100, a compact two-way I reviewed many years ago which was somehow never collected afterward. The AE100 has a front baffle measuring 295mm high by 180mm wide, on which the 25mm dome tweeter and 100mm bass-midrange driver are mounted on the long axis centerline, with their centers respectively 67mm from the top edge and 80mm from the lower edge. Knowing these dimensions and the speed of sound (344m/s), we can build a simplistic picture of the time-domain behavior of the edge reflections simply by counting the number of baffle edges that a wavefront intersects as it progressively expands from each driver.

Assuming that both drive-units behave as point sources, the outcome is shown in fig.1, which plots time along the horizontal axis and the number of "active" baffle edges along the vertical axis. The red trace is for the bass-mid unit, the blue trace for the tweeter. You might be surprised to see the plots peaking at six active edges when a rectangular baffle has only four edges in total, but this is because what is actually being counted here are intersections between the wavefront and baffle extremities. This means that each edge can potentially count as either one or two.

Footnote 1: H.F. Olson, "Direct Radiating Loudspeaker Enclosures," JAES (January 1969), Vol.17 No.1, pp.22-29.
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