Mark Hatcher wrote:
"All things kept equal, increasing the body depth will lower the body resonance of the guitar. "
In theory, yes, but in practice it's less of an effect than you'd think.
Some years ago Fred Dickens, a researcher at Bell Labs who made (very nice) classical guitars on the side, did an experiment to check this out. He built a guitar with sides that were about 6" deep, played it and measured all the resonances. Then he removed the back, cut the sides down by an inch, reassembled it, and checked it all out again. He kept doing this until the sides got so shallow the top and back braces were about to hit each other. Through the whole process he found that the 'main air' resonant frequency rose by only 7%; not much more than a semitone.
We usually think of the 'main air' resonance as a 'Helmholtz' mode; the sort of thing you hear when you blow across the mouth of a wine bottle. It acts pretty much like one: the air pressure changes in phase everywhere in the box and the pressure pumps air in and out of the soundhole, just as a Helmholtz resonator is supposed to do. However, there's one big difference: in a real Helmholtz resonator, the walls are 'rigid', where the walls of a guitar, an, in particular, the top, are pretty flexible. What we've really got here is a 'bass reflex enclosure', at least in the low range, and you need to look at both the Helmholtz resonance, and the 'loudspeaker' resonance of the top, to understand what's going on.
You can see the 'real' Helmholtz resonance of a guitar by immobilizing the sides; say by burying the thing in sand but leaving the soundhole open. When you do that, you'll get a Helmholtz mode that's probably pitched up around 125 Hz, or even higher. Similarly, you could cut away the back of the guitar, leaving only a narrow rim to stiffen the bottom edge, but eliminating the enclosure and it's effect on the top, and you'd see the 'main top' loudspeaker mode at around 160-180 Hz, most likely (Thomas Rossing has done this experiment, and I'm pulling the numbers up from memory: they may not be exactly right, but I think they're close enough for now).
When you get the whole box together the top and the air have to work together. Air flow in and out of the hole changes the pressure in the box, and that pressure change pushes on the top. By the same token, as the op moves in and out, it causes the pressure in the box to change, and that pushes air in and out of the hole. In physical terms, the top and air resonant modes are 'tightly coupled', and will thus affect each other.
In a situation like this, you end up with two resonant modes in which both the top and air are moving. In one case, most of the energy in the system is in the 'air' part of the motion, and we call that the 'main air' mode. The other resonance has more of the energy in the 'top', so that becomes the 'main top' mode.
One way to think about how these mode frequencies compare with the independent Helmholtz and 'loudspeaker' mode pitches is by picturing how things are moving at the two frequencies. At the 'main air' pitch, air is flowing into the soundhole as the top is moving 'out' of the box. The air has to push the mass of the top, and that shifts the pitch of the 'main air' mode downward relative to the pitch of the 'pure' Helmholtz mode. At the 'main top' frequency, the air is moving into the box as the top is pulling outward. In effect, the pressure change in the box is a 'spring' that adds to the stiffness of the top, and shifts the pitch of the 'main top' mode upward relative to what the 'free air' loudspeaker mode would have been. The amount of the frequency shift of each mode depends partly on the relative masses of the top and the air, but also on how tightly coupled the two are.
When Fred cut down the height of the sides on his guitar, the 'real Helmholtz' mode frequency rose. However, with the shallower box there was more pressure change inside for a given amount of top motion, so the coupling was stronger. The added strength of the coupling shifted the 'air' pitch down more relative to the (higher) real Helmholtz pitch, and it ended up practically the same as it was with the deeper box.
What Fred didn't tell me, but I saw later in a similar situation when I cut a guitar down, was that the 'main top' mode pitch can actually shift _upward_ when you cut the box down. This seems counter intuitive: you didn't do anything to the top but the pitch changed, whereas the thing you expected to change, the air pitch, pretty well stayed put. (I'll note here that the real situation is even a little more complicated than this due to the coupling of the top and back through the sides, as Trevor Gore has shown, but we can stay with this for now).
What I've seen in comparing similar guitars with different box depths is that the 'main air' peak in the output spectrum tends to be lower, but possibly broader in frequency, with a deeper box. Again, for a given amount of top motion, there's less pressure change in the box. This tends to give a 'rounder' bass sound, and less 'punch' on the attack.
Sorry about the long post...
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