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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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    Just to be clear, at least in the US, the word we use is a "tuning fork." A pitchfork is a large farming instrument used to move around bales of hay, like this one. – user1118321 Jun 30 '18 at 23:09
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    I thought they were primarily for threatening Frankenstein's monster. – luser droog Jul 1 '18 at 4:35
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    Love the term 'pitchfork'. Sums its use up succinctly! Real pitchforks give more of a dull thud. More suited to tuning bass drums? – Tim Jul 1 '18 at 9:28
  • Related discussion on Physics: physics.stackexchange.com/questions/51838/… (and links therein). – dmckee --- ex-moderator kitten Jul 1 '18 at 17:29
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    A perfect hemispherical shell vibrates without overtones. Maybe also that it was the only such shape, but I don't remember. – Arthur Jul 2 '18 at 7:48
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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.

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    "the overtones are not integer multiples of the fundamental as with strings and air in pipes," - they are not integer multiples for strings and air in pipes either, though they are near enough that for "Physics 101" courses to ignore the reasons why not. See en.wikipedia.org/wiki/Inharmonicity for strings. For pipes, the details are probably at postgrad level in most university physics courses. – user19146 Jun 30 '18 at 19:16
  • There is a second issue: how well does any particular harmonic transmit energy into the air, i.e. can you hear it even if it exists. For the fundamental frequency of the fork, the base of the fork moves up and down as the prongs move closer and further apart, and that up-and-down motion can transmit the vibration into a table-top, etc, which can move a relatively large amount of air to make a sound. For the higher modes, this might not occur. – user19146 Jun 30 '18 at 19:23
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    @alephzero, not sure about what you are comparing with your first comment. The measured string to the model or one model to another, or deviations from ideal due to movement of the bridge on an acoustic etc. For an electric guitar the spectrum is very close to the ideal based on FFT of measurements. Of course a steel string is really a very thin beam and the stiffness aspect of the system comes into play. Perhaps on an upright bass or the very low or very high strings of a piano would start to display these features. – ggcg Jun 30 '18 at 19:59
  • While you are correct about the audible nature of the harmonics in theory I can say from personal experience that the two overtones about the fundamental of the 80Hz fork are definitely audible, they can be discerned by the ear. To answer the OP it's enough to state the fact that overtones exist. But there is definitely a wealth of physics to learn beyond the simple models of these instruments. I never really bought the model of standing waves in a pipe and was revealed to learn the real physics of it later. The string is more believable (but still ideal) ;-). – ggcg Jun 30 '18 at 20:03
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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.

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  • From this, should I infer that the overtones( or harmonics in some cases) are produced strictly by the vibrating body (the string or the fork itself) and are not a product of some acoustical process that the air goes through inside the body of the instrument? – Oguz Jun 30 '18 at 21:08
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    @RecepOğuzAraz correct. Well, for wind instruments, the vibrating body is the air going through the body, but for solid-mechanic instruments like guitars or pitchforks, the vibrations have little to do with air. You would get almost exactly the same harmonics in a vacuum (inaudible, but still there). – leftaroundabout Jun 30 '18 at 22:55
  • @leftaroundabout thanks a lot! The harmonics in vacuum argument really helps. – Oguz Jul 7 '18 at 12:51
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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:

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There is a nice paper by Rossing, "On the acoustics of tuning forks," https://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 μm 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.

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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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  • When you place the fork against the top of a guitar or other resonator to amplify it you will only significantly amplify those vibration that the resonator will respond to. So you have filtered out the overtones of the fork. You are correct in that the "clang" is more complex, all impulsive excitation are. But the fork does have an overtone sequence. – ggcg Jun 30 '18 at 20:42
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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.

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