About two weeks ago there was a mock test in Korea, and a physics question asked if a plucked guitar (it was actually a gayageum, a traditional instrument, but I'll just call it a guitar for convenience) string creates a standing wave.
I've learned in school that this is true, and the answer was true as well. But today my physics teacher said that this is actually false. Because a standing wave is caused by two identical waves traveling in opposite directions, a guitar string cannot create a standing wave. So a plucked guitar string only makes a vibration, not a standing wave.
But this is also mentioned in school textbooks. On the page explaining standing waves, there's a picture of a vibrating string and the caption says, "A string tied at both ends makes a standing wave, causing resonance."
I am confused. Does plucking a guitar string make a standing wave on the string? Or is this just a vibration?
Answer
Yes, plucking a guitar string does create standing waves, but...
No, plucking a guitar string does not create a standing wave, as the sum of standing waves is in general not a standing wave (thanks for Ben Crowell for pointing this out), since a standing wave must have a stationary spatial dependence and a well-defined frequency:
$$ y(x,t) \propto \sin(2\pi x/\lambda)\cos(\omega t).$$
The initial perturbation is not sinusoidal, but instead contains a plethora of frequencies, of which only remain, after a transient, the resonant ones - which correspond to some of the possible standing waves. It's the sum of those that compose the vibration you'll observe.
The counter-propagating waves, if you want to model each of the standing waves this way, you get from the reflections at the cord's ends.
For more details see this answer and, especially, the answers to the question Why do harmonics occur when you pluck a string?.
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