Can someone explain how plucking a guitar string produces multiple harmonics?

[BigMacnFries]

When a musical string (guitar etc) is struck a standing wave the length of the string is set up with a frequency X. Also many harmonics are set up with frequencies 2X, 3X, 4X...
From my physics textbooks I can understand how by shaking a piece of string (with one end fixed) at the right frequency you could set up a standing wave of X, or by shaking it 3 times faster you could set up a standing wave of 3X.
I don't understand how by plucking a guitar string once you get all of the possible harmonics, common sense says to me you should only get X but obviously this is not true. Can somebody please explain or point me to a web page, Thanks.

 

[Galileo]

If you pluck a string you are kind of dumping an entire spectrum of frequencies on it. It will only pick out its resonance frequencies X,2X,3X etc. Those are the only ways a standing wave can be produced.
If you would only get the fundamental harmonic, the note would sound like a sine wave. It is the distribution of the intensity over the various harmonics that gives an instrument its distinctive 'color' (or sound).

 

[Pythagorean]

I think it's actually 1/x, where x increases by integers, i.e. 1/1, 1/2, 1/3, 1/4... and so on, which is the harmonic series, (i think)

Either way, the physics is easy to understand conceptually. When you strike a string, it resonates in (possibly) an infinite amount of ways, along the harmonic series. But only the first couple terms of the harmonic series are strong enough to be recognized by the average ear.

Those are:

1/1 - the fundamental, the whole string ringing in an arc back and forth (open string, no fretting)
1/2 - the octave, the string also vibrates in two separate halves, the arcs going opposite ways, like a sine wave symbol. The deffinition of an octave is twice the frequency, which is what half a string length generates harmonically. (12th fret on fret board)
1/3 - The string also vibrates in three equal parts. (7th fret)
1/4 - The tring also vibrates in four equal parts. (5th fret).
1/5 - the string also vibrates in five equal parts... I believe the fret point is about three and a half frets...

Past that, harmonics start getting more and more faint.

You can find the harmonics on the 5th, 7th, and 12th fret on a guitar if you lightly dampen the particular fret with your finger (without pushing all the way down like you normally would) and let off right after you pick the string.

The three notes generated on 5, 7, and 12 (1/2, 1/3, and 1/4th of the string) are also the three most common notes in pop music, generally known as the roots of the I, IV, and V chords, and also surely the fundamental structure western music in general

 

[madeinmsia]

i have a related question. what happens when u pluck two guitar strings together, but one of the string is fretted. let's assume the fretted string is just one octave higher.

so for example, the 5th string is open and the 3rd string is fretted on the 2nd fret. u pluck these two strings together with the same amplitude. what's going to happen?

i tried to understand these by looking at the standing wave equation sin(n*pi*x/L)*cos(n*pi*c*t/L). but the problem is even though both strings are playing the same note, one double the other's frequency, they both have different lengths! so when i plot the sin-cos graph, they both have nodes at different points now.

this is really weird. i thought each wave of the two string will reinforce each other since they are playing the same note.