One thing that has been on my mind lately, is the interaction between a speaker and the room that it’s in. Specifically, the real-world Qtc. I learned in engineering that a Q factor of 0.707 (or 1/sqrt(2)) is optimally damped, whereby the response to an impulse has no ringing, and the paths of overshoot and undershoot converge to produce an optimal curve with minimum settling time.
As a first approximation, it therefore makes sense for a speaker design to target a Q of 0.707.
However, this doesn’t take into account the real-world environment in which the speaker is used. For example, most ‘home’ speakers are used indoors, where the room forms another resonant cavity. Rather than a speaker model that assumes a “speaker box” that is floating in empty space (or in half-space because of a nearby boundary like a floor), we have a more complicated system with a box-within-a-box to deal with. This second box can be more or less resonant, when taking into account energy losses from things such as windows or doors. The speaker membrane balances both of these air volumes, which have vastly different spring constants. The spring constant of the speaker box is usually of the same order as its VAS parameter (equivalent air volume of its mechanical suspension), whereas the air suspension of the room will usually be much softer.
Using my pair of 10″ woofers in 40L boxes, playing in a 40-50m^3 room, as an example, we may end up with horn-like amplification at certain modal frequencies.
I conducted an experiment whereby I added 10 ohm resistors in series with my woofers, and discovered something very interesting. Despite increasing the Q of the speaker membrane, the in-room Q was measurably reduced. My method was very simple: identify and correct the most prominent room mode with a software based parametric equalizer, which offers a notch filter with adjustable centre frequency and adjustable Q. Add the series resistor and adjust the filter to suit.
But why would the in-room Q go down when it was supposed to go up?! Rather than thinking of the speaker as an isolated system that radiates energy into the room in a one-way process, notice that the speaker and box can act like a window for the room — an energy sink. This is a clue that it could be possible to optimise the speaker’s behaviour so that the room becomes far less resonant, at least from the location of the speaker.
With the added resistors, the notch in the parametric EQ had to be changed from about -10dB to -6dB, and the Q reduced from about 2.4 to 1.8. The method involved an iterative process of manually searching for the room mode by generating test frequencies with a MIDI keyboard, and listening for tell-tale signs of certain tones being much louder than others that were just a few semitones higher or lower, and making adjustments until the offending resonance seemed to disappear. It wasn’t just trial and error, though. We can call it dead reckoning. Once the centre frequency was established, it was easier to listen and check for anomalous “twin peaks” at nearby frequencies above and below, which could be attributed to the filter Q being too high, or, conversely, twin dips if the Q was too low. (Or a broad dip, if too many dB’s of attenuation were used, and so on.) With a centre frequency of 38~40Hz, I found that my method of live listening and adjustment was far quicker and more effective than attempting to use a microphone to convert the information to something visual.
OK, but what did I hear?
Boomy, loose bass? Nope. I would describe the bass as being warm and cuddly*. With tonal differences already accounted for, there had to be something else at play, which seemed to make the room disappear. I had some jazzy elevator music going, which had walking bass plucked on an acoustic bass. With the Q-reducing resistors and parametric EQ, it became a lot easier to imagine that big violin-family instrument being somewhere nearby. The speakers no-longer drew as much attention to themselves in the room. This led me think that harmonic distortion (or other forms of distortion) had gone down — and so, the rabbit hole of amplifier design…
Further work
The filter Q still seems high for my example system. What I’ve essentially done is I’ve reduced the settling time of the air vibrations in the room by anticipating the error (the overshoot and ringing) and providing a signal with “anti-ringing” to cancel it out. I was then able to reduce the settling time of the altered input signal around 40Hz by adding a series resistor. The main cost is reduced amplifier overhead. With a nominal speaker impedance of 8 ohms, it may be impractical to experiment with higher resistor values like 47 or 100 ohms — it would still work, but it would be wasteful. It seems far more elegant to design the amplifier differently in the first place.
One-dimensional model, piano analogy
Imagine that the system of air inside the speaker box and in the room together is like a long string under tension. The string is ‘driven’ near one end where the speaker motor is located, a bit like piano strings are hit by hammers near one end. The fixed ends of the string represent the solid walls surrounding the speaker membrane at a distance.
Ideally, a suitably sensitive amplifier should be able to influence the movement of the string, but also act like a soft damper that absorbs reflections from the room at an optimal rate. However, there seems to be a caveat, that sound creation and absorption is done by the same system, and I’m not even 100% sure what “current control” would look like with this analogy. Arguing for a high damping factor, and servo control in particular, is a bit like saying that those felt dampers inside a piano really should be made of steel, because steel is very hard and has a high mass. Obviously, a steel damper would be far more ‘authoritative’ in forcing a vibrating string to return to its idle resting position. Yeah, sort-of. As a musician, I know that that would be patently false, and that the dampers have to be soft, but also not too soft. Steel dampers would change a piano into a completely different instrument.
In fact, pianos have lots of different sized strings, and the mechanical parts are fine-tuned across the range, with some of the smallest strings forgoing dampers altogether. Kind-of like tweeters that are ‘padded’ with resistors in a crossover network. On the face of it to reduce sensitivity, even though the actual complaint always seems to be harshness, which is a more subtle phenomenon.
Final thoughts
One of these days, I’ll have to build a simulator to test and validate my thoughts on room modes. My intuition tells me that the strongest resonances imply that the system can be reduced to 1 dimension with low loss of accuracy, compared to a more elaborate 3D model. It’ll be interesting to see where it takes me.