What kind of frequency effects does a guitar body have on sound?

Question

These are the questions I read through before asking this:

• Guitar string - feedback from the body
• Sound due to guitar
• Waves on a guitar string
• https://music.stackexchange.com/questions/5489/why-do-harmonics-happen
• How does an acoustic guitar amplify its sound?
• Why do harmonics occur when you pluck a string?

What I'm trying to understand (generally) is the effects of a guitar body on sound. In particular, tones being played either into the cavity or onto the face.

Here's my questions:

1. To what non-numerical degree are guitar bodies similar to formants? If one used a speaker to play pure tones into the cavity, would they find a meaningfully "bumpy" response? (in no way linear nor any other smooth function)

2. If #1 is "depends on the guitar," is this the reason for its shape and the craftsmanship involved with instrument making?

3. Does the non-linear response of the guitar body (and all physical bodies) create any overtones? If you took a surface transducer and played pure tones, would the non-linear response of the guitar body create any meaningful overtones?

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For further insight into what I'm trying to answer, here's my intuition for the three above:

1. yes there are "formants," if you tap on a guitar body, you'll hear a non-random sound. this is the frequency response.
2. if you made a box guitar, the response would be even worse. the shape of guitars is such that the distance between each successive "bounce" of sound is different, so no "bounce" can get too comfortable. a box would have the same bounces in each dimension
3. yes, non-linearity always introduces overtones, but given the tiny distance of movement this non-linearity is not particularly additive

Answer 1

1. yes, there are in principle discrete (eigen-)frequencies which are amplified by the guitar body, and because these are damped (due to sound radiating through the hole in the body), the resonance lines are no longer infinitely tall (which could theoretically cause a resonance catastrophe) and get broadened (supposedly what you mean by "bumps") so that they can be excited more easily by frequencies that are off the resonance center; this is the same principle that causes the formants of vocal articulation, but for a guitar one desires a more "neutral", less "bumpy" character of the spectrum (an "aaa" or "ooo" sound would make a guitar sound interesting, but would restrict its application in general music pieces)
2. a box guitar is possible, but there would be less (due to a phenomenon called degenerate eigen-values) and less fortunately distributed resonances, which would probably make the guitar sound too "regular", i.e. certain notes would almost be muted, while others would be quite loud; what is desirable is that the "bumpiness" of the guitar's response is as small as possible, such that every note sounds at about the same loudness when plucked with the same force; I suppose that is what craftsmanship for guitars (and probably all acoustic instruments) is mostly about
3. there is (or should be) no noticeable nonlinear response of a guitar body; otherwise the sound would suffer intermodulation (which is what you hear with a distortion effect device) that sounds rather ugly to ears not wanting to listen to rock music; since the guitar body is a linear system to very good approximation, there are also no harmonic or non-harmonic overtones added, that are not already in the excitation (plucked string); that being said, there can be nonlinearity under special circumstances: if you pull a string really hard, you notice the "zzzaaaooouung" sound, which I think is a nonlinear response of the string (not the body!); also if the string hits the fingerboard; but normally you try to avoid those effects when you are playing music, although they are relatively "mildly" nonlinear because they only concern the single string

Be careful with the term "nonlinearity". The guitar as a vibrating system is linear, but the frequency response is of course not a linear function of frequency (which is a completely different thing than the system being nonlinear).

I know of no nonlinear musical instruments outside of electronic music, but that may be a limitation of my knowledge. In electronic music, the non-linearity is most often applied only to single voices, because otherwise it will often sound noisy. Intermodulation is characterized by not only adding overtones (whole multiples of the base frequency), but rather fractional multiples and sum/difference frequencies. This is what makes nonlinearity sound noisy when multiple tones are played at the same time.