Figure: The first 7 harmonics of standing waves on a vibrating string

Image 1 Caption: An image illustrating the first 7 normal modes (i.e., which are called harmonics in music jargon) of standing waves on a vibrating string with fixed endpoints.

The standing waves are not traveling waves. Each point on the string oscillates between the upper and lower bounds shown in the image. The absolute value of the upper and lower bounds at a point is the amplitude of the oscillation at that point.

A wavelength is one complete up-and-down cycle of any of the standing wave curves.

Further explication:

1. A vibrating string is a 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 image because the waves are standing waves.

2. 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.

3. From the second definition of antinode, we note that an antinode equals half a wavelength or two antinodes equal one full wavelength.

   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 as aforementioned above.

   This definition of wavelength is the useful one for analysis of wave phenomenon.

   

4. Image 2 Caption: Note, the universal symbol for wavelength is the small Greek lambda = λ.

5. 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 explicate this case below.

6. The allowed resonances are determined by (among other things) the boundary conditions---which in the case of a vibrating string are ideally exactly fixed endpoints as in the image. Less ideally, the endpoints vibrate a little.

7. Below we give some mathematical details:

8. Clearly, the number of antinodes n for a standing wave on an (ideal) vibrating string must be an integer. Since an antinode length is half a wavelength (i.e., λ/2), we have the 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)  

   Note, n = 1, implies λ = 2L and so there is a standing wave that is only half a wavelength since the string length is L = λ/2.

   Now we do 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). This formula is proven, e.g., IAL 6: The Relationship Between Wavelength and Frequency. Using this formula, we find the resonance frequency formula for a vibrating string to be

     f_n = v/λ = n[v/(2L)]  .  

9. For musical instruments (like string instruments) where the vibration of the instrument produces sound waves with the same frequencies as whatever is vibrating in the instrument, 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).

10. Now when you strike a vibrating string, you typically simultaneously excite standing waves with multiple superimposed resonance frequencies, and other frequency waves are strongly suppressed by canceling pulls.

    The above description generalizes for musical instruments. Whenever you strike a vibrator in them, you excite multiple resonances: i.e., harmonics. The weighting of harmonics in intensity determines much of the timbre of the musical instrument which makes its sound pleasing.

    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!

    

11. Image 3 Caption: The image shows a schematic diagram of guitar and guitar string with superimposed harmonics. The first harmonic is blue (1 antinode), the second harmonic is green (2 antinodes), and the third harmonic is is brown (3 antinodes). The net string motion (which is what is actually observed) is purple and is somewhat irregular.

12. 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). However, for an adjustable online frequency generator, see:
    Online Tone Generator for any sound frequency. Turn the sound intensity slider on the left way down: e.g., ≤ 3 % for high frequency; ≤ 50 % for low frequency.

    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.

13. Optionally, see the unauthoritative video Standing Waves Part I: Demonstration: James Dann | 4:37. Yours truly thinks the demonstrator is mistaken at 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: For a stretched string with one fixed end and the other end being a vertical driver, does the driver frequency set the number of antinodes which can be non-integer with the driver amplitude just being part of the wave form with a ghost node beyond the driver?

    Answer (on one occasion): Yes, your physical intuition is absolutely correct. The mechanical driver sets a boundary condition that forces the physical string to form what looks like a fractional or non-integer number of antinodes, with the driver amplitude simply fitting into a larger "virtual" standing wave pattern that projects a ghost node beyond the physical boundary.

    It took a bit of reformulating the question so that Google AI knew what yours truly was actually asking and to get it to agree (flatteringly) with yours truly's understanding (which yours truly believes to be correct).