Question

When you pluck a guitar string, initially the vibration is chaotic and complex, but the components of the vibration that aren't eigenmodes die out over time due destructive interference. This ostensibly explains why the harmonics quickly become dominant.

The problem with this model is that there infinitely more non-eigenmodes than eigenmodes. (If you formalize the space of all possible modes, the subspace of eigenmodes will have measure 0.) So the naive mathematical extrapolation suggests that if the non-eigenmodes cancel themselves out, only an infinitesimal part of the original energy remains. Obviously this is not the case, given that the initial energy from a pluck is finite (actually pretty small), and we can only hear vibrations above some threshold of energy.

So why do the harmonics constitute a significant fraction of the total energy? Does it have to do with the discreteness of the universe (particles having nonzero size and mass)? Or is it that vibrations that are close-to-but-not-exactly eigenmodes also die out slowly?

Answer 1

When a guitar string is plucked, the vibration initially chaotic and complex. However, over time, the non-eigenmodes of the vibration die out due to destructive interference, allowing the harmonics to become dominant. While there are more non-eigenmodes than eigenmodes, the non-eigenmodes do not completely cancel out due to vibrations close to, but not exactly, the eigenmodes.

Step-by-step explanation:

When you pluck a guitar string, the vibration initially is chaotic and complex, but over time, the components of the vibration that aren't eigenmodes die out due to destructive interference. This phenomenon explains why the harmonics quickly become dominant. While there are indeed infinitely more non-eigenmodes than eigenmodes, the non-eigenmodes do not cancel themselves out completely. This is because vibrations that are close to, but not exactly, eigenmodes also die out slowly. The harmonics constitute a significant fraction of the total energy because they correspond to frequencies that are close to the eigenmodes of the string, and these frequencies have a higher probability of being excited and sustained.