At what frequency does a string vibrate? When a string with fixed ends vibrates (e.g. plucking a guitar string) Fourier Theorem says that the vibration can be expressed as a sum of its normal modes, which are sinusoidal vibrations with frequencies that are all integer multiples of the fundamental frequency.
My question is really simple: since the resulting vibration is a sum of a large number of simple vibrations, with what frequency is the string really vibrating (since, after all, it is a vibration)? Are all the points in the string vibrating with the same frequency but with amplitude modulated in space as happens with the normal modes? 
I'm not asking which pitch do we perceive (the fundamental frequency), but at what frequency is the string vibrating.
 A: We perceive the fundamental, the lowest frequency.  Even though there are other harmonics present in the vibrating string when no damping is present the shape will move across the string and come back to its initial configuration at the fundamental frequency.  The other harmonics are perceived as tone.  For example plucking a string closer to one of the supports (a.k.a. boundaries) will create more high end harmonics and in guitar speak that sound "twangy" or "hot", "bright".  Plucking near the middle accentuates the fundamental and sounds "warm", or "smooth".  But, all are perceived as the same note.  Experienced musicians are trained to hear some of the harmonics, once you are trained to hear them you can't un-hear them and music is never the same.  The octave is pretty easy to hear but each note contains (at the very least) a major triad in just tuning.  So every note you play creates a soft major chord.  Very strange from a music theory point of view.   
A: The content of Fourier Theorem is that every periodic function of period T can be represented as a series of (in principle infinite) harmonics, i.e. harmonic motions of frequencies of the kind:
$$
\nu_i = i \cdot \nu_1 ~~~~~ i = 1, 2,3 \dots 
$$
each with its own phase.
Therefore, although made by many different frequencies, the signal which is the synthesis of all its harmonics, as a function of time is a periodic function of frequency $\nu_1$. An example of time variation for a signal made by five harmonics is the following (I have obtained it using the applet in the page http://www.falstad.com/fourier/; it is a good applet and I recommend to play a little with it to get a first understanding how different harmonics do combine;  in the abscissa there is time ):
 
The plot contains almost 3 periods of the fundamental mode. From the plot is equally clear that the superposition of normal modes is a periodic function, but definitely it is not a harmonic oscillation.
