Article originally published in Home Theater Magazine, December 1993.
Resonant Modes and Sound Cancellation
by Art Noxon
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There is another factor that limits the remaining options for speaker placement. The pressure zone is not a pinpoint-sized space; it spreads out. If the speaker is located near enough to the center of the pressure zone, the resonance can still be stimulated. A pressure zone effectively extends about one quarter of the distance between adjacent pressure zones and the speaker should not be located inside the effective pressure zone space. For all practical purposes, the speaker should be located 25 percent away from the end of the pipe to best avoid stimulating any of its first three harmonics. There is no location towards the middle of the pipe that suits a subwoofer position, as the pressure zones there are overlapping.
A listening room can be approximated as if composed of three intersecting pipes. These pipes would lay along the three-room axis — front to back, side to side, and floor to ceiling. This means that the subwoofer location for best, non-resonant playback will be about one-quarter of the ceiling height off the floor, one-quarter the width of the room off the side walls, and one-quarter the room length off the front or back wall. When discussing speaker location, it is only the dimensions to the center of the driver cone that count. The location of the edge of the box really doesn’t matter.
No computer program is needed to properly position the subwoofer in a room; a tape measure is your only investment. Note also that the currently popular “rule of thirds” placement formula is not consistent with the understanding of an a resonant speaker placement. This overpublicized “rule of thirds” would only be applicable if the subwoofer roll-off was set so that the speaker did not play the third harmonic.
The concepts of subwoofer placement have by now been well developed and now some practical applications can be considered. Two things need to be shown – the roll-off frequency of the subwoofer and the first resonance frequency of each pipe axis of the room. Typical roll off is set at 85 Hz.
The shortest dimension of a room is the floor-to-ceiling distance. If this dimension is eight feet, the first vertical resonance occurs at: 1130/2×8 = 70.6 Hz. The second at 141 Hz is well above roll-off and can be ignored as well as any higher partials. The vertical position range for a harmonic playback will be to locate the subwoofer anywhere in the middle half of the room, keeping it at least two feet away from either the floor or ceiling.
The next shortest distance in a room is the width, typically about 15 feet. The first resonance for this is 1130/2×15 = 37.7 Hz. The second is twice that at 75.4 Hz and the third is three times that or 113.1 Hz. The second harmonic is within the subwoofer range but not the third. The sub has to be placed more than 25 percent away from the wall because of the first harmonic, but not in the central one-eighth width of the room due to the second harmonic. The sub can be located anywhere between three-quarters and 6-3/4 feet from the side wall. Lastly, the length of a room night easily be 21 feet long. The first resonance for this would be 1130/2×21 = 26.9 Hz.. The second is 53.8 Hz and the third is 80.7 Hz. The fourth at 107.6 Hz md above are all well above the roll-off frequency and can be ignored. For the length of the room, the sub position should be one-quarter of the room length or five feet off either end wall.
So, a room 8 feet by 15 feet by 20 feet will have the smoothest bass if the piston of the subwoofer is located two to six feet off the floor, between 3-3/4 and 6-3/4 feet off the side walls, and five feet off the end wall. This is true as far as avoiding strong coupling of the speaker to the room modes, but there are more than modes to worry about as far as speaker smoothness is concerned.
Incidentally, these silent areas located between the pressure zones deserve a little attention as well. They are “cancel zones” because sound is canceled at these locations. Sound cancellation is being used a little more often these days, particularly with industrial noise control applications. Sound cancellation seems to possess a form of sci-fi lure for some people. The idea of beaming “anti-sound” waves to quiet freeway noise is one of the more popular of these energy-out-of-water type schemes. To the literal reader, words create reality. But to the engineer and scientist, reality exists independently from words. Just because someone can dream up a sentence that seems to make sense doesn’t mean that it physically does make sense.
Normally, sound cancellation applications remain limited to the control of sound in pipes. For example, if we take a closed pipe that contains a harmonic condition and drill a hole into the pipe, we will get varied results which depend on where the hole is located. For the first harmonic, with a pressure zone at either end and a cancel zone at the middle, we can drill a hole into the pressure zone at either end and kill the resonance. But, if we drill through the wall of the cancel zone, there is absolutely no change in the resonant condition. A hole in either pressure zone allows pressure energy to leak out. But there is no pressure energy in a cancel zone, so a hole that leaks pressure doesn’t affect anything.
This is not news — the ancients knew about it. The flute and clarinet-type instruments use this open/closed hole effect to select pipe resonances, heard by us as notes. Let’s consider what can happen if the closed pipe is engaged with its second harmonic. There are three pressure zones and two cancel zones. A hole could be drilled through the pipe wall at each cancel zone and not affect the existence of the resonance. Now we have made a closed pipe into an open pipe; and, if we blow air into one hole, it will come out the other hole. We have discovered a pathway to conduct air through a pipe filled with sound without having any of the sound leak out.
With industrial sound canceling, the tonal sounds of a blower that moves air in a closed duct can be canceled at an air outlet. One can use either this standing wave pipe process or a speaker/microphone/computer system to create this same sound-cancelling effect at the opening of the pipe. Although the sound at the opening can be canceled, the sound elsewhere in the pipe is very loud. If two forces are applied equal and opposite, there is no force imbalance, hence no movement. That doesn’t, however, mean there is no stress on the material. There is twice as much stress to the material than if only one force was applied.
So it is with sound. If two sounds are applied equal and opposite, there is no sound at some point, but that doesn’t mean there is no stress on the material. There is, in fact, twice as much stress in the material than if only one sound had been applied. If we move away from the point where there is no sound, we’ll find twice as much sound everywhere else. That’s the point. Sound cancellation doesn’t mean sound energy cancellation. The energy is still there. In fact, it has become twice as strong. Just because we can’t hear it at one location only means we will hear it twice as loud at another.