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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?

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marked as duplicate by Sebastian Riese, John Rennie, Kyle Kanos, ACuriousMind, Chris White Mar 18 at 1:08

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

Just to make this clear: one can oscillate a string at any frequency. It's not "quantized". The standing waves are just the eigenfrequencies that it carries out on its own and that have the least damping. When you "pluck" a string, you excite all kinds of modes and frequencies, not just the first mode or the first few modes. However, the higher modes decay quicker and they leave the fundamental ringing longer. Under stroboscopic lighting one can see that plucked strings are not harmonic:… – CuriousOne Mar 16 at 13:48
@CuriousOne Actually the "stroboscopic" lighting is specifically designed to reproduce an artifact of the CMOS "rolling shutter" in most digital cameras by illuminating different parts of the strings at different times. I suspect that if you illuminated a vibrating guitar string with an ordinary strobe light whose frequency matched the fundamental or the harmonics for the string, you would see sinusoidal displacements. – rob Mar 16 at 18:10
@rob: Really? To the best of my knowledge an ordinary strobe light is called a stroboscope and stroboscopic lighting refers to lighting by ordinary strobe light – slebetman Mar 17 at 3:21
Search for Victor Wooten Amazing Grace. Enjoy a little musical physics in action. – Bill N Mar 17 at 3:35
@slebetman I mean specifically in the video linked by CuriousOne, which is using a second-generation clever trick. Here's a link to the middle of the video where you can see that different parts of the guitar neck are illuminated at different times to duplicate the "rolling shutter" illusion. An ordinary strobe light would illuminate the entire neck at once. – rob Mar 17 at 3:43
up vote 23 down vote accepted

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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(and this string instrument technique is called flageolet.) – Sebastian Riese Mar 16 at 13:51
BTW, you can do this at other node locations. E.g. placing your finger at a point 1/3 from the end will suppress a different set of harmonics. It gets harder and harder, but I've gotten up to the 11th harmonic. The pitch of that should be near F# (starting from C), but far enough away to be discernibly different. I was studying the lydian dominant scale at the time. It turns out that the fourth octave above the fundamental contains all the pitches (not exactly, due to the "comma") of the lydian scale, plus a B-flat. So it contains all the pitches of the lydian dominant scale as well. – garyp Mar 16 at 16:38
@SebastianRiese: I've always heard this technique referred to as "playing a harmonic" by guitar players. – BlueRaja - Danny Pflughoeft Mar 16 at 17:45
You can also briefly touch a string that's already playing to silence the fundamental and any harmonics that don't already have a node at that point, which is a cool-sounding technique on electric guitar. – hobbs Mar 16 at 21:53

You always make harmonics when you pluck a string. You can change the spectrum of the harmonics --- that is, their relative intensities --- by changing where on the guitar string you pluck.

The usual place to strum a guitar is near the sound hole, about a third of the way from the bridge. If you strum near the center of the string (12th/octave fret) you don't excite the first/octave harmonic, which has a node there, nor any harmonics in the half of the spectrum with a node at the center. I think that strumming near the center of the strings makes the sound of the string become noticeably more "o"-shaped. If you strum very close to the bridge, on the other hand, you excite lots of very high harmonics, and the sound becomes more "eee"-shaped.

You can excite only harmonics by lightly touching the string near one of their nodes. Here's the method. Pluck one of the strings like usual with the string open. While the string is vibrating, slowly bring your finger down onto the string at the 12th/octave fret. After you touch the string, but before you make contact between the string and the fret, you'll hear the pitch jump by an octave. You've just killed the fundamental frequency, and all the harmonics with antinodes at the center, without much changing the harmonics with nodes at the center. (Once you've figured out how much pressure to apply, there's no need to let the string sound open first; as another commenter points out this is called "flagolet".)

You can excite other harmonics as well at different frets. The 7th fret (which divides the string into thirds) raises the tone by an octave plus a fifth; the 5th fret (which divides the string into quarters) raises the tone by two octaves. My experience is that I can isolate higher harmonics on expensive guitars than I can on cheap guitars.

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