For an oscillating string that is clamped at both ends (I am thinking of a guitar string specifically) there will be a standing wave with specific nodes and anti-nodes at defined $x$ positions.
I understand and can work through the maths to obtain the fact that the frequency is quantised and is inversely dependent on $L$, the length of the string, and $n$, some integer.
If I pluck a guitar string, this oscillates at the fundamental frequency, $n=1$. If I change to a different fret, I am changing $L$ and this is changing the frequency. Is it possible to get to higher modes ($n=2$, $n=3$ etc)? I don't understand how by plucking a string you could get to 1st or 2nd overtones. Are you just stuck in the $n=1$ mode? Or would the string needed to be oscillated (plucked) faster and faster to reach these modes?
Answer
When you pluck the string you excite many many overtones, not just the fundamental. You can observe this by suppressing the fundamental. Pluck the string while holding a finger lightly at the center of the string. That point is an antinode for the fundamental and all odd harmonics, but a node for the even harmonics. Putting your finger at that point damps the odd harmonics (especially the fundamental), but has little effect on the even harmonics. (There's a node at that point.) You may have to experiment a little to find exactly the right spot and pressure. Guitar players do this all the time to get a different sound out of the instrument.
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