Why don't tuning forks produce overtones?
I want to know how a tuning fork can produce a pure tone.
I do not understand the process because I know, although not sure, the presence of air inside an instrument introduces the harmonics of the fundamental frequency (e.g. guitar or violin body).
Moreover, I have a theory which maybe you can test, because the tuning fork can be regarded as a rigid body, the oscillations of the fork alter the sound pressure nearby in a regular manner which results in a single frequency.
Overall, I need to understand the sound production in an instrument.
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The tuning fork does produce overtones. The amount of overtone depends on how the tuning fork is attacked. The modes of attack also depend on the pitch of the fork. I once had a very long tuning fork for a physics demo that was 80-100Hz. You could squeeze the ends together and slide your fingers off creating a smooth fundamental tone. If you struck it on one side you would create as many as three overtones that could be detected with a microphone and FFT software. Striking a fork on something hard is not good for it.
An interesting point is that the overtones are not integer multiples of the fundamental as with strings and air in pipes: these are ideal systems. The beam-bending equation that governs the fork's motion produces an interesting spectrum which depends on boundary conditions and other physical characteristics of the fork.
Hope I'm not beating a dead horse here.
Having worked in electronics for decades, I think of this question as "Why does a tuning fork produce a pure sin(e) wave with no harmonic overtones?" I don't think anything on our macroscopic scale produces an absolutely pure sin wave. There's always some distortion. It's a matter of degree.
I think you are quite confused about what overtones are and how they are produced.
For many mechanical oscillators (the air column in a flute, a vibrating string or free reed), there are modes of vibration satisfying the boundary conditions. A vibrating medium has inertial and elastic properties that combine in carrying the vibrational energy. This medium has significant boundaries which cannot support both movement or force. The boundary conditions usually support not just a simple fundamental vibration mode but also several higher modes of vibration: you can access them on some instruments as "harmonics" or "flageolet" by targetedly dampening of the fundamental.
For a string instrument like a piano, those higher modes can be almost but not quite proper harmonics of the fundamental frequency: they tend to be somewhat sharp, particularly for thicker strings and more compact instruments.
A tuning fork has torsional vibration modes around its tapped base. While it can in theory support higher modes than its fundamental mode, those are very much "inconvenienced" by the fork geometry, the location of the tap, and the thickness and curvature of the vibrating tines and thus tend to extinguish rather fast. They still make up some amount of the initial "ping" when striking the fork but fade out much faster than the fundamental and thus are not a significant part of the onsounding tone. The higher vibrational modes on a tuning fork also are not anywhere close to actual harmonics of the fundamental, so the fundamental cannot "feed" them in the manner it may happen with string instruments.
If you press it at a table you can more easily hear one of the overtones (an octave above just striking it). Good explanation on how tuning fork physics:
There is a nice paper by Rossing, "On the acoustics of tuning forks," aapt.scitation.org/doi/10.1119/1.17116 , which unfortunately is paywalled. The transverse momenta of the two tines cancel, which is why you can hold the stem in your hand without damping that mode. When the tines vibrate symmetrically at f, their center of mass vibrates longitudinally at 2f, but with a small amplitude. For a 1 mm vibration of the tines, Rossing found a 1 ?m longitudinal vibration of the stem.
In normal use, you touch the stem to a sounding board (guitar, violin, piano, ...), and it's the longitudinal vibration that causes the sounding board to vibrate. The reasons that the tiny vibration of the stem is so effective in transmitting sound are that (1) the sounding board is a dipole, whereas the tines act as a quadrupole; and (2) the sounding board has more surface area.
As others have noted, there will be other modes of vibration which will not be integer multiples of f.
It doesn't. Particularly when pressed against something that resonates (like a table-top) in order to add some volume. And the 'clang' sound, produced as it's struck, is a lot more complex. But it's a pretty clean waveform, as waveforms go.
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