First 6 harmonics of standing waves on a vibrating string

Image 1 Caption: An animation dynamically illustrating the first 6 normal modes of standing waves on a vibrating string with fixed endpoints.

Features:

A vibrating string is transverse wave case since the oscillation of the medium (i.e., the string) is perpendicular to the propagation direction when the waves propagate---which they do NOT do in the animation because the waves are standing waves.
The nodes are the unmoving points and the antinodes are the points where the wave amplitude is maximum or in another meaning the regions between the nodes.
From the second definition of antinode, we note that an antinode is 1/2 wavelength or a wavelength λ is the length of 2 antinodes.
The above statement is because a wavelength is by definition a full spatial cycle of the wave phenomenon. Usually a full spatial cycle can be characterized as a full up-and-down cycle.
This definition of wavelength is the useful one for analysis of wave phenomenon.
Image 2 Caption: Note, the universal symbol for wavelength is the small Greek lambda = λ.
How are the standing waves produced?
If we drive a vibrating string with a vibrating machine at exactly a resonance frequency f_n (which we explain below), we get standing waves with exactly n antinodes. What if we drive the vibrating string off all resonance frequencies? We get small traveling waves that varied in time. For off resonance frequencies, the pulls on the vibrating string strongly cancel instead of adding up.
Now when you strike a vibrating string, you simultaneously excite standing waves with multiple superimposed resonance frequencies, but other frequency waves are strongly suppressed by canceling pulls.
The allowed resonances are determined by, among other things) the boundary conditions---which in the case of a vibrating string are usually fixed endpoints as in the animation.
Clearly, the number of antinodes n for a standing wave on a vibrating string must be an integer. Since an antinode length is half a wavelength (i.e., λ/2), we have the of number of antinodes formula
```
       L
  n = --- , where L is the length of the vibrating string.
      λ/2 
```
Thus, the wavelength formula is
```
        L
  λ = ----- = 2L/n  , 
      (n/2)  
```
where the first version makes sense since there are 2 antinodes per standing wave. Note, n = 1, implies λ = 2L.
Now we have a general formula for phase velocity (the velocity at which traveling waves propagate):
```
 phase velocity formula fλ=v  , 
```
where v is the phase velocity and f is frequency). Using this formula, we find the resonance frequency formula for a vibrating string to be
```
  f_n = v/λ = n[v/(2L)]  .  
```
For musical instruments (like string instruments) where the vibration of the instrument produces sound waves, the resonance frequencies are called harmonics---the longest wavelength (which is the lowest frequency) harmonic is the 1st harmonic or the fundamental and the others are higher order harmonics: i.e., 2nd harmonic, 3rd harmonic, etc. In music, one often uses the alternative terms fundamental tone (or 1st partial), 2nd tone (or 2nd partial), 3rd tone (or 3rd partial), etc. All, the "tones" except the fundamental tone are overtones, but the word overtone is also used for non-integer multiples of the fundamental (see Wikipedia: Overtone: Musical usage term).
The weighting of harmonics in intensity determines much of the timbre of the musical instrument which makes its sound pleasing.
Pure frequencies are, in fact, rather boring. See Wikipedia: Audio frequencies: Sound file table (scroll down) and Wikipedia: Concert A (standard music reference frequency) (or Wikipedia: A440 (pitch standard): Modern practices). But since the buttons do NOT always work for on the Wikipedia sites, see:
To hear Concert A 440 Hz with overtones for a violin (much more pleasant than the pure frequency Concert A), see File:A440 violin.mid, but it's probably just a synthesizer violin.
How does one pronounce timbre? See Forvo: timbre---but you'll be no wiser after you've heard all the pronunciations---anyway as lumberjacks say Timber!
See video Standing Waves Part I: Demonstration: James Dann | 4:37. Yours truly thinks James Dann is mistaken 0:25 when he suggests that a random frequency gives "unsynchronized" waves at least for an ideal version of his demonstration, although he may be correct for his less-than-ideal setup.
To clarify the situation quasi-authoritatively: Google AI question: If you drive standing waves off resonance are the oscillations just small standing waves matching the boundary conditions?: Answer on one occasion:

Images:

Credit/Permission: © User:Adjwilley, 2013 / CC BY-SA 3.0.
Image link: Wikimedia Commons: File:Standing waves on a string.gif.
Credit/Permission: Derrick Coetzee (AKA User:Dcoetzee), User:F l a n k e r, 2006 / Public domain.
Image link: Wikipedia: File:Lambda_uc_lc.svg.

local link: standing_waves.html

Waves file

standing_waves.html