# Producing harmonics of standing wave on a string at non-coprime divisions

I am currently researching the properties of producing harmonics on a vibrating string fixed at its ends (e.g. violin). Lightly touching a node at division of the string a/b will sound harmonic b iff a and b are coprime. So touching the node at 1/3 or 2/3 the string will produce the 3rd harmonic, but touching the string at 2/6 the string will not produce the 6th harmonic as 2/6 --> 1/3 --> 3rd harmonic. Even though the 6th harmonic does also have a node at 2/6 (1/3) the string, only touching at 1/6 and 5/6 will sound the 6th harmonic.

I'm wondering what the physical reasoning for this was? I always felt like it was because the simpler modes of vibration require less energy, so the string prefers them if one is possible when touching a given node. Maybe someone has a more refined answer?

The question can actually be reduced to the following: "Why is the first harmonic the strongest?" I went into SE archives and found the following statements:

...and the amplitude of each harmonic in frequency space is proportional to its amplitude in the initial configuration in wavenumber space. Now the initial configuration for plucked instruments happens to be roughly what I depicted in the first figure in the Music.SE answer: something between a triangle and a sawtooth, and as we know both have a monotonically descreasing1 sequence of Fourier coefficients ($$\mathcal{O}(\frac{1}{n^2})$$ for triangle, $$\mathcal{O}(\frac{1}{n})$$ for sawtooth), so the fundamental does tend to have the strongest amplitude on the string in the beginning. [1]

leftaroundabout actually goes into some deeper explanation further linked in the post, so I advise you go there.

My small summary would be that with usual initial conditions, i.e. how you usually first strike the string, the first mode is has stronger than the others. For example, for a sine wave, the frequency is purely 1st mode, and as exemplified above, most other periodic shapes have their fundamental modes more prominent, although there can be some exceptions (also provided at [1])

(I also suspect that higher modes get damped faster too, but I couldn't find any sources on this, so it's only a speculation. Anybody that who can prove/refute this is more than welcome.)

Some relevant questions on SE:

[1] 1st answer at, "Does the Fundamental Frequency in a Vibrating String NOT Necessarily Have the Strongest Amplitude?", phys.se

[2] "Is it possible for a harmonic to be louder than the fundamental frequency?", phys.se

[3] "WHY do harmonics happen?", music.se

• Your speculation about higher harmonics being damped more than lower harmonics make sense to me in the context of fingers on a violin string. The finger has a certain width and the higher frequency harmonics would have shorter wavelengths so the finger would be impinging on a greater fraction of the loop with a higher harmonic. – M. Enns Jun 4 at 2:18
• @acarturk This makes sense to me. Let me see if I understand: introducing a node at 1/3 does not exclude the 6th harmonic (it continues to sound as an overtone), it is simply that we hear the 3rd as the loudest (sort of a new "fundamental") because it has the mode with the largest amplitude with a node at 1/3 given the string's starting amplitude conditions. – Thomas Nicholson Jun 4 at 8:10
• @ThomasNicholson Yes that's the main idea. – acarturk Jun 4 at 9:16