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I've answered Ron off list: he e-mailed me, but in the interest of spreading the wealth I'll re-post the reply here...
The lowest resonance on the guitar that can radiate any sound is the 'main air' part of the 'bass reflex' couple. Usually that's between F and A on the low E string, with G being very common. However, on most guitars, the 'neck' resonance is lower than that in pitch.
It's not really just a 'neck' resonance, of course: it's actually the whole assembly vibrating like a xylophone bar. There are two stationary node lines; one across the lower bout at about the widest part, and one near the nut or first fret. You can hear it if you hold the guitar up by pinching the neck up around the nut and tapping on the top of the headstock in back. The head is moving more than any other part of the guitar in this mode, but it's not moving enough to put out much sound, since it's so small relative to the wave length, which will be more than ten feet. On the other hand, this resonance can be a pretty good 'flywheel', since it tends to have low losses, and can thus store a bit of energy.
There is a way in which the 'neck' mode can be made more effective, though: you can couple it to a larger surface. If you think about the guitar as bending 'upward', with the headstock and tail block moving 'up' in the direction of the top, you'll see that the top is being compressed along its length. Since the top on most guitars is domed a bit, this causes it to puff up, and suck a little air in through the soundhole. Similarly, as the neck moves 'back' the top gets stretched and flattens out a bit. This air movement is just what the 'main air' resonance is doing; moving air in and out of the soundhole. Thus, if you can get the 'neck' resonance to be somewhere near as high in pitch as the 'main air' mode the two will work together. What happens in this case is that the two things 'push' on each other, and, as is usual, the coupled system has two modes that are somewhat higher and lower in pitch than either part would have on it's own.
In the spectrum of the guitar the result will be a double peak around the air mode/neck mode pitch. The actual peak height will be lower, and the response will be spread out over a broader frequency range. The lower level means that you're less likely to have the 'wolf' on the low G: a note that lacks sustain because the guitar is so effective at extracting string energy at that pitch. Instead there will be several notes that have a particularly 'full' and 'strong' response. In my experience the whole tone of the guitar becomes 'darker' and 'richer'. The 'flywheel effect' of the neck mode energy storage makes the attack of the low notes a bit less sudden, and helps them 'build' over a short time, although the overtones usually take over in the 'attack' department.
It's hard to tune the neck mode. Adding in CF rods helps, but less than you'd hope. Keeping the neck weight down, by using cedro instead of mahogany, for example, helps. Weight on the headstock is particularly influential: wood buttons on Schallers cut the weight of two whole machines, for example. Tapering the neck thickness, so that it's heavier at the body end, also seems to work well. Short scale guitars tend to have higher 'neck modes', for obvious reasons: in one case I had to add almost 20 grams to the headstock of a short scale guitar to get the neck mode down! Because there is only about 5 hz between semitones at that low range the tuning has to be pretty exact to get it to work really well.
I'll add for the list:
It's hard to say how much of an effect neck motion has in absorbing sound from the strings. Those animations really exagerate the scale of motion; otherwise it would not be visible at all. Certainly the bridge moves more, as it must. Certainly stiffneing the neck can't hurt in that respect, but IMO the greater acoustic effect comes through the couple with the top and air, if it's happening.
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