Alan Carruth wrote: ↑
Sun Jan 21, 2018 5:40 pm
In fact, there are some things about the way the guitar works that differentiate it from other similar instruments, and I'm pretty sure that at least one of them has to do with the air in the box.
Some years back one of my students rigged up a small accelerometer for me. I stuck it on the bridge of a guitar, hung it up with the strings damped, and used a 'sweep' signal from my computer to drive the bridge from about 50-1000 Hz. I recorded the output of the accelerometer on the computer, and ran it through a spectrum analysis program. The result was a chart with clear peaks and dips that showed how active tat spot on the top was at every frequency. I then used Chladni patterns to find all the resonances of the top and back that I could, to see how they corresponded with the peaks and dips on the chart. As you'd expect, all the lower top resonances that were active at the bridge showed up as peaks. The back resonance tended to show up as dips, since the back gets all of it's energy from the top. I was able to account for all of the peaks pretty well, except for one very close to A=440; fifth fret on the high E string. There was no 'wood' resonance near that pitch, and the closest 'air' resonance in the literature was at about 350Hz. Other tests showed that the peak was really there in the played sound, and the sound was coming out of the hole. A microphone inserted into the hole showed that the resonance was, in fact, an 'air' mode; in fact, the one that was supposed to be at 350 Hz. This is sometimes called a 'lengthwise bathtub' mode, as the air 'sloshes' the length of the body. The sound level is high at either end of the box, and there's a 'null' in the middle (you can think of this as analogous to a string that you pluck while touching it in the middle, at the 12th fret. A pickup in the middle would hear no sound). In fact, there was
a resonance like this at 351 Hz and another at 438Hz !
One of the rules about this stuff is that you won't see the same resonance shape at two different frequencies unless there's another resonance coupling in and influencing it. In fact, the two 'air' resonances, at 351Hz and 436Hz, had slightly different shapes: the pressure nulls were in a somewhat different place at the two frequencies. I thought this might have to do with the sound hole and the waist.
I put together a simplified test using a cardboard mailing tube the length of the guitar body. I made up a 'waist' out of foam coffee cups that could be inserted in the right place, a little above the center of the length. I drove the air through a tube hooked into a loudspeaker at one end, and put a microphone in the other end to measure the sound level along the length of the tube. I then looked at the mode under four different conditions. One was the closed tube. Anther was the closed tube with the 'waist' inserted. The third was the tube with no waist, but a sound hole in the 'right' position, just above the waist. Finally, I looked at the mode with both the waist and the hole.
The plain tube had the mode near 350 Hz with the null exactly in the middle. Adding the waist changed the pitch a little, and moved the null upward, due to the speeding up of the air flow through the restriction of the waist. The hole by itself moved the null downward, and dropped the pitch a little. There is some sound produced at the hole, and that moves air in and out, which, apparently, increases the effective mass of the air in the upper end of the tube. Finally, with both the waist and the hole I got both modes. I wrote up a short paper on this, and submitted it to the Catgut Acoustical Society journal.
A few weeks later the subject of 'air' modes came up on a violin maker's list, and I mentioned this. I got an e-mail from a physics professor in England expressing some doubt, so I sent him a copy of the paper. A few weeks later I got an e-mail back from him saying that he had checked my result using a ceramic drain pipe and a roll of lead for the waist, and had not seen what I saw. "Check your apparatus". I withdrew the paper, and got back to work.
The long and short of it is that the split into two modes only happens when the walls of the box can vibrate. With rigid walls the hole and waist produce a single 'compromise' mode, but when the walls can move you see two, but only
if there is a pronounced waist above the center of the length of the box, AND a sound hole just above that. 'Dreadnought' guitars without a well defined waist don't show this, and have a different characteristic timbre. Similarly, when the sound hole is moved up into the corner of the box to drop the 'main air' pitch the effect goes away.
This was not, of course, something that was planned when the shape of the guitar was first devised. Rather, they made the shape, and found that it had a 'characteristic' timbre that was pleasant. This is one of the reasons we keep making them this way: if you don't they don't sound as much like guitars.
A few years after that a group in Spain published a finite element model of the guitar that included both the wood and the internal air. This 'added' air resonance, which had been missed in tests in which the box was embedded in sand to immobilize the walls, showed up clearly. It took a pretty complicated model to catch it.
Making such a model that would behave exactly like a specific guitar would be very difficult. All the very small inconsistencies in the wood properties from point to point would need to be modeled, and you'd need t measure them first. This would pretty much require chopping up the wood into little pieces and measuring them all, and then, of course, you won't have the guitar any more. We can learn a lot from these models using 'average' values of the materials, but there are limits.
Given the normal sort of variation that you get in wood, I strongly doubt that exact 'tonal copies' can be made. Tight control, which IMO includes things like 'free' plate tuning, can get you 'arbitrarily close' though.