The Science of Subwoofing Page 2

What about bass "speed"? Some pundits have branded this term technically inaccurate, because, as they point out, the steepness of the transient's leading edge is determined by its upper harmonics, which may after all not be reproduced by the woofer but rather by the midbass or midrange drive-units. Yet, I think that we have all observed slow, sluggish, or woolly bass response. So the real question is: Does a woofer have to be fast? An excellent figure of merit for woofer speed is its acceleration factor, defined as the ratio of the woofer's magnetic force to its moving mass. This is nothing more than a restatement of Newton's First Law of Motion: Force equals mass times acceleration. To increase speed you either reduce the woofer's moving mass (which consists of the voice-coil and cone), or you increase the magnetic force. The latter can be accomplished by increasing the magnetic flux in the voice-coil gap by using a larger magnet.

Clearly, if a woofer is crossed-over in the midrange, say around 2kHz, as it might be in a two-way system, then its speed does indeed matter if it is to integrate with the upper-range drivers. And just as clearly, if the woofer's output can be confined to the range below 100Hz, then its speed or rise time is no longer critical. But this is hard to do. Too often, shallow crossovers are used so that there is significant contribution from the woofer in the midrange. In which case, the woofer had better be sufficiently fast to blend in with the rest of the system.

Ultimately, audiophiles are polarized into one of two camps. One camp tends to prefer bass quantity at the expense of quality, while the other side almost cheerfully seeks out quality at the expense of tonal balance and/or dynamic range. The situation has come about because so very few speakers out there successfully negotiate both sides of the bass equation. And the ones that do are only affordable by a select few whose lifestyles can soak up $10,000/pair price tags. Few of us can have our cake and eat it too.

What is A Subwoofer?
This is not as simpleminded a question as it first appears. In the mind's eye, a subwoofer is a hippopotamus-sized box with a substantial foot-long or larger driver. Getting a bit more technical, a reasonable definition would be a speaker for the reproduction of the range from 20 to 100Hz. But even this description merely defines the tip of the iceberg and hardly reveals the submerged mass of performance criteria necessary for high-fidelity performance in the subwoofer range. Faithful reproduction of lowest octaves presents the most severe speaker-design challenge, and is the area most often compromised in mass-market loudspeakers.

Nature is pretty stingy in the bottom end. Sidestepping the issue of bass horns, the larger the direct radiator the more partial Mother Nature is toward it. Another way to say this is that the acoustic radiation efficiency of a direct radiator is proportional to its pistonic cross-sectional area. It's a question of having a larger sail to flap into the air: the larger piston provides a better impedance match with the air load and hence a more efficient transfer of energy. A 15" woofer is better in this respect than a 12" woofer, and a 24" woofer is even better. Of course, there's a practical limit to woofer size. It's difficult to make large cones sufficiently stiff to prevent serious buckling from the accelerating forces applied at the apex. And large cones also mean huge moving masses that are very difficult to accelerate quickly.

So why aren't large woofers used routinely in commercial loudspeakers? Cost considerations aside, large woofers are a speaker designer's worst nightmare—simply because they require large cabinets. And that is a serious fly in the ointment; almost nobody likes large boxes. For the same reason they're more efficient in energizing ambient air, large woofers are more efficient in pressurizing the air inside the speaker cabinet. Typically, the cabinet-air stiffness drives the speaker's resonant frequency well above the woofer's free-air resonance. The result is a loss of bass extension and a boomy response characterized by a Q well over one.

OK, large woofers and small boxes don't mix well, but how large does the cabinet have to be to minimize the stiffness of the trapped air? Well, let's look at a commercial example using an 18" woofer with a compliant suspension. The subwoofer in Dave Wilson's WAMM system consists of an expensive 18-incher housed in a coffin-sized affair probably large enough to comfortably accommodate even Andre the Giant. Think of the difficulty—read expense—of making such a large box sufficiently stiff to avoid major panel flexure.

(The brave home constructor does have a bit more imaginative flexibility available to him. A closet could be used as a large enclosure. If you've got a basement, floor-mounting the woofer would work very nicely indeed. And finally, by wall-mounting the woofer you could take advantage of the biggest infinite baffle of them all: the great outdoors.)

I have focused on bass efficiency because, to sound loud, bass frequencies must be reproduced at very high SPLs. The ear is insensitive to bass frequencies, the hearing threshold curve rising at about 18dB/octave from 50Hz to about 20Hz. At 20Hz, an 85dB SPL will be just barely audible in an average room.

How loudly a subwoofer will be required to reproduce the deep bass is a crucial design consideration. Louis Fielder and Eric Benjamin have examined this and many other subwoofer performance aspects in a recent milestone paper, titled "Subwoofer Performance For Accurate Reproduction Of Music" (footnote 1). For the first time, all of the pertinent performance aspects of subwoofers were carefully researched, and some new information was brought to bear on the subject. I will lean heavily on this paper in the following discussion.

As Fielder and Benjamin note, how loudly a subwoofer must reproduce the deep bass depends on the desired low-frequency cutoff and the amplitude of the bass information present on the recording. The authors investigated the LF-cutoff requirements by analyzing a sample of CDs for minimum audible frequency at two benchmark SPLs. The discs were selected on the basis of either previous knowledge or by recommendation that they had substantial low-frequency content. Each CD was played back through a player with a –3dB cutoff at 3Hz, and the output was analyzed by a spectrum analyzer. Discs that were found to contain significant bass content below 30Hz made up a final sample for further analysis. These 13 CDs were analyzed in detail for minimum audible frequencies based on the threshold of hearing and maximum peak CD outputs equivalent to SPLs of 110 and 120dB. The results are shown in Table 1.

Table 1: CD Possessing Low-Bass Information, Minimum Audible Frequency (Hz)

120dB/110dB Composer Selection Label/Catalog Number
10/12.5 Tchaikovsky 1812 Overture Telarc/CD-80041
16.5/16.5 Dupre Symphony in g Telarc/CD-80136
15/17.5 Grofe Grand Canyon Suite Telarc/CD-80086
18/18 Hindemith Organ Sonata 1 Argo/417-159-2
18.5/18.5 Jongen Symphony Concertante Telarc/CD-80096
12.5/22 Flim & the BB's Big Notes DMP/CD-454
16.5/22 R. Strauss Also Sprach Zarathustra Telarc/CD-80106
22/22 J.S. Bach Kyrie, Gott helliger Geist Telarc/CD-80097
24/24 Saint-Saens Symphony No.3 Telarc/CD-80051
25/25 Williams Star Wars Theme Telarc/CD-80094
19/25 J.S. Bach Toccata & Fugue in d Telarc/CD-80088
29/29 Billy Cobham Warning GRP/GRP-D-9528
29/29 Various Country soundtrack Wyndham Hill/DIDX-141

What this means is that if you want to feel the 10Hz component of the Telarc cannon, you'd better be able to reproduce that track at 120dB peak SPL. If, on the other hand, you care little for Telarc's sonic spectaculars, pipe organs, synthesizers, or sound effects (not necessarily in that order), then civilized SPLs of around 100dB will do just fine. In fact, Fielder and Benjamin offer the following general conclusions: First, recordings with audible bass below 30Hz are relatively rare. Second, these very low frequencies are generated by pipe organs, synthesizers, or special effects and environmental noises. Other instruments, such as bass guitar, bass viol, tympani, or bass drum, produce relatively little output below about 40Hz, although they may have very high levels at or above that frequency.



Footnote 1: Originally presented at the 83rd convention of the Audio Engineering Society, October 1987 (preprint 2537); since published in the Journal of the AES, June 1988.

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