To what degree does guitar construction affects the vibration of the strings? There's an old debate going on in the guitar community about how much does wood choice and body shape affect the sound of an electric guitar. No one denies that there's a difference acoustically (how the guitar sounds unplugged) because in this situation it's the wood and body shape that amplify the sound made by string vibrations, but when we're talking about the sound as it comes from the pickups, things get much more uncertain because there are a lot of variations even in supposedly identical guitar parts, and accounting for all of them for the purpose of doing an experiment where the only difference between two guitars is the wood is difficult (I certainly haven't heard of an experiment that was satisfying enough, but feel free to prove me wrong). I'm interested in a way to circumvent these practical difficulties using a theoretical explanation:
Since the pickup only sees the string's vibrations, the question basically becomes "does body shape and wood make enough difference in the way a string vibrates that it changes the sound in a noticeable way", and this sounds like something that may be possible to figure out mathematically, or at the very least should be much easier to test because there are less variables involved. My question is:
Is what I'm describing possible to calculate/test, and has this been done before?
 A: I do not know that it has been done before, but I have no doubt there is a difference. What is not clear is if it would be noticeable by human ear.
The difference is explained theoretically by the fact that the string will vibrate different with the supporting body. Only in the hypothetical scenario where the string is held by ideally unmovable holders would the string not be affected by them.
Since the pickups are just "translators" of the mechanical vibration profile, they will produce different electrical currents for different profiles of vibration of a string. This of course depends on its characteristics, since it may not by sensitive to all frequencies.
Therefore if we register with an oscilloscope the current produced from vibrating strings, and perform a Fourier analysis we could appreciate the differences. These would be present in the hypothetical case where the pickups would be sensitive to all frequencies equally, or at least some relevant ones.
Finally, the question if these differences are audible, well we would need to find difference in the audible range of frequencies, and the intensities in these frequencies would have to be such that they are apparent in the compound sound of a string. Also will depend on the person's hearing sensibility.
To design an experiment, I would propose choosing two guitars with exactly the same pickups (or use the same one on both) and record the output to perform a Fourier analysis on it to test for differences. 
To test for hearing differences, maybe play it to many people, possibly trying two different ways: say that you will play to different sounds to half of the persons, and say you will play the same sound to half of them. This should allow you to know when people are being prompted on their answers, because this is a known phenomenon and will occur so you better take into account.
Still, I sustain that there should be differences even if imperceptibles. 
