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Sometimes when you pluck a string on a stringed instrument, the string turns into what looks like a nearly stationary loop with a semi-transparent middle.

Sometimes, the loop doesn't appear stationary, but you can see a wobbling "string" inside it.

Why is it that sometimes one happens and sometimes the other?

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When the instrument is plucked, it can go into one of the vibration modes below, depending exactly where and how it's plucked

enter image description here

from this webpage

https://www.earmaster.com/music-theory-online/ch03/chapter-3-2.html

The appearance then depends on which mode occurs.

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  • $\begingroup$ This is the first step towards an answer, but it ignores (1) the inharmonicity caused by the elastic stiffness of real strings and (2) the fact that the strings in a real stringed instrument can vibrate in 3D not 2D, and because of unsymmetrical real boundary conditions (which are not idealized rigid constraints) there will be pairs of vibration modes with similar (but not identical) frequencies and mode shapes. Note, part (2) is critical to understanding the playing technique of acoustic classical guitars. $\endgroup$
    – alephzero
    Jul 25 at 15:50
  • $\begingroup$ @alephzero Good point, maybe the OP will see this comment $\endgroup$ Jul 25 at 15:51

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