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I believe the purpose of a tuning fork is to produce a single pure frequency of vibration. How do two coupled vibrating prongs isolate a single frequency? Is it possible to produce the same effect using only 1 prong? Can a single prong not generate a pure frequency? Does the addition of more prongs produce a "more pure" frequency?

The two prong system only supports a single standing wave mode, why is that?

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Nice question, though I wish there were more formulae or at least images to support the answers... –  Tobias Kienzler Jan 22 '13 at 7:05
    
@Tobias, Agreed I was hoping for something a little more in depth. –  acadien Jan 22 '13 at 12:41
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FYI someone asked essentially the same question in response to this one, and got a very nice answer: physics.stackexchange.com/questions/51847 –  BlueRaja - Danny Pflughoeft Jan 22 '13 at 23:39
    
Yeah I saw, I was surprised it wasn't closed as a duplicate. –  acadien Jan 23 '13 at 0:00
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6 Answers

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I am by no means an expert in tuning fork design, but here are some physical considerations:

  • Different designs may have different "purities," but don't take this too far. It is certainly possible to tune to something not a pure tone; after all, orchestras usually tune to instruments, not tuning forks.
  • Whatever mode(s) you want to excite, you don't want to damp with your hand. Imagine a single bar. If you struck it in free space, a good deal of the power would go into the lowest frequency mode, which would involve motion at both ends. However, clamping a resonator at an antinode is the best way to damp it - all the energy would go into your hand. A fork, on the other hand, has a natural bending mode that will not couple very well to a clamp in the middle.
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I was about to post almost exactly the same thing. In slightly simpler terms, the vibrations of the two prongs cancel out at the point where they're joined together, so that you can hold it by the handle without letting any energy get transferred to your hand, so that it will continue vibrating for longer. –  Nathaniel Jan 22 '13 at 2:48
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"after all, orchestras usually tune to instruments, not tuning forks." - They tune to instruments because some instruments (particularly, oboes) cannot be easily tuned, so everyone else has to tune with respect to them. It has nothing to do with whether or not tuning forks are good enough. –  BlueRaja - Danny Pflughoeft Jan 22 '13 at 9:31
    
@BlueRaja-DannyPflughoeft Of course, I don't mean to imply tuning to instruments is better, just that it is easy enough that we don't need to worry about having a pure sine wave generator. –  Chris White Jan 22 '13 at 9:45
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Personally, I find it easier to tune a guitar to a tone that has a few harmonics than one that doesn't. The reason is that you can hear the guitar's harmonics beating agains the tuning tone's harmonics, as well as the fundamentals beating against one another, which manifests itself as a kind of roughness in the sound when they're not quite in tune. Pianos are unusual in that their harmonics are "stretched" (the first harmonic is a bit more than an octave above the fundamental, and so on), so for tuning a piano, having something close to a pure tone is much more important. –  Nathaniel Jan 22 '13 at 14:58
    
@Nathaniel: all freely-vibrating–string instruments have their harmonics stretched to some degree, not just pianos. It's only most obvious in small pianos, because the strings of these have a particularly big thickness-to-length ratio. –  leftaroundabout Jan 22 '13 at 18:32
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The reason is that to work properly the tuning fork has to have a balanced motion. It is normally used held in the hand. If you just had one prong, the energy of the oscillation would very quickly be transferred from the handle to the skin of the hand, and would be lost. The result would be that the oscillation would die away very quickly. If you have a tuning fork with two prongs of equal size, they can oscillate with motion equal and opposite to each other - balanced in other words. Because the motion of one prong balances out the motion of the other, there is no motion of the handle. Because there is no mechanical energy going into the handle, no energy can be lost into the hand, so the oscillation lasts a long time.

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Resonance amplifies and sustains the tone much longer than with just one prong.

Think about this: if you've ever sung in the shower or in the car, have you noticed that some pitches sound abnormally louder than others? That's because the dimensions of the shower are just right such that those notes are amplified via resonance. For example the shower width might be an integer multiple of a certain pitch's wavelength, so the wave bounces back and fourth, riding itself and getting bigger. Like when you push a kid on the swing at just the right times so she gets higher and higher.

The 2 prongs on the fork resonates the sound, just like your shower walls. Each prong forces the other prong to vibrate at the same rate, thus sustaining the sound longer. If there were only 1 prong, the sound would be much quieter and it would die off much quicker. Try it with a butter knife.

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Can you elaborate on that? –  Tobias Kienzler Jan 22 '13 at 9:59
    
If that were the case, would the frequency of tuning fork not depend on density of air? –  qarma Jan 22 '13 at 13:35
    
@qarma Who says it doesn't? –  Tobias Kienzler Jan 22 '13 at 13:42
    
@qarma no. The frequency always depends on the density, humidity, and temperature. –  chharvey Jan 23 '13 at 1:51
    
I would prefer a tuning fork that produces set tone regardless where in the world I happen to be. That is with as little influence of surrounding air as possible. Same goes for expansion coeff. of metal. –  qarma Jan 23 '13 at 8:44
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Q. How do two coupled vibrating prongs isolate a single frequency?

howstuffworks.com has an article on How Tuning Forks Work

The way a tuning fork's vibrations interact with the surrounding air is what causes sound to form. When a tuning fork's tines are moving away from one another, it pushes surrounding air molecules together, forming small, high-pressure areas known as compressions. When the tines snap back toward each other, they suck surrounding air molecules apart, forming small, low-pressure areas known as rarefactions. The result is a steady collection of rarefactions and compressions that, together, form a sound wave.

The faster a tuning fork's frequency, the higher the pitch of the note it plays. For instance, for a tuning fork to mimic the top key on a piano, it needs to vibrate at 4,000 Hz. To mimic the lowest key, on the other hand, it would only need to vibrate at 28 Hz.

Two prongs on a tuning fork oscillate such that they both move together, then they both move apart. These compressions and rarefactions of air between and behind the prongs is what creates the stronger compression waves in the air and hence louder sound of this primary mode of vibration.

In contrast, when you pluck a string, the fundamental frequency is produced by the vibration of the whole string, but the string is also vibrating in halves, thirds, fourths, fifths, etc. This causes overtones making the frequency not as pure, but rather harmonic.

via wikipedia:

String Harmonics

Same thing in woodwind and brass instruments when you blow air through a tube, or vibrate a reed playing air through a tube, or strike a bell, whose shape is set up to accentuate different harmonics. The relative loudness of the different harmonic overtones gives each instrument its own timbre.

A tuning fork is designed such that the harmonic overtones are quiet compared to its fundamental pitch. I found this great YouTube video showing a tuning fork model which shows the different modes the fork vibrates in, and models the strength of each mode of vibration.

The video also shows the constraints of holding the tuning fork on the end, which eliminates the rigid body modes (which were already quiet to begin with) but also dampens some of the other harmonic modes, creating and even more pure tone with very low amplitude harmonics. Daniel A. Russell at The Pennsylvania State University has a page showing animations of these vibrational modes.

Holding the tuning fork at the end does little to dampen the mode of vibration which creates the primary frequency. If you also hold the end of the fork against a hard surface, the small up and down movement will cause resonance in the surface, amplifying the primary frequency even more.

Tuning fork primary modes of vibration

Q. Is it possible to produce the same effect using only 1 prong? Can a single prong not generate a pure frequency? Does the addition of more prongs produce a "more pure" frequency?

One prong wouldn't have the additional compression effect of two prongs moving closer together, creating a louder primary frequency. But more importantly, the second loudest mode of a tuning fork (the "clang" mode, the high-pitched sound you hear when it is first struck) is dampened because you strike the fork at a modal point about 1/4 the length of the prongs from its vibrating end.

Additional prongs don't create more dampening effects, yet they also create more vibrating modes, so the sound is less "pure".

Related question: Why don't tuning forks have three prongs?

Edit: Reference paper, with formulae and data on vibration mode frequencies, etc.

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How does this answer the question? –  Peter Shor Jan 22 '13 at 16:36
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@PeterShor you have sufficiently shamed me into expanding my answer. :) –  ghoppe Jan 22 '13 at 19:11
    
Great youtube video you recommended. I'm especially surprised the torquing modes have such low frequencies. –  Chris White Jan 25 '13 at 16:53
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Two-prong system surely supports more than one mode, consider:

  • squeezing the prongs together (mode that you want)
  • twisting the both prongs relative to stem
  • twisting each prong relative to its base
  • wobble/barrel of both prongs
  • sound wave travelling in the metal from one edge to another
  • etc...

If you are a designer of the tuning fork you want one mode to dominate, that is you want all other modes to dissipate quickly.

In fact holding the tuning fork in your hand already helps to dampen some of the modes.

Furthermore, the stem is sometimes put on a table or similar amplify the sound.

My guess is, tuning fork is made of particular metal to achieve stability, the cross-section of the prong is considered to get rid of some secondary modes, the losses in are optimised to ensure narrow enough frequency as well as being able to hear the tuning fork, relative ease of manufacturing and perhaps a dozen more considerations that only musicians could think of.

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All but 1 primary mode would be damped out, from other comments it seems the vibration resulting from the prongs vibrating in anti-phase is the only one that is not substantially damped. –  acadien Jan 22 '13 at 16:33
    
and that's by design. a random two-pronged object would not necessarily have that porperty. a random shaped object usable as a crude tuning fork (e.g. a tincan) certainly does not have that property. –  qarma Jan 22 '13 at 16:38
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If there were only one prong (imagine holding a metal rod in your hand), then the oscillation energy of the prong would quickly be dissipated by its contact with your hand. On the other hand, a fork with two prongs oscillates in such a way that the point of contact with your hand does not move much due to the oscillation of the fork. This causes the oscillations to be safe from damping due to contact with your hand, so they continue for a longer period of time.

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"a fork with two prongs oscillates in such a way that the point of contact with your hand does not move much due to the oscillation of the fork" - Er, why? If we removed one of the prongs, would that still be true? What if we added a third prong? Do we have any equations to explain this? –  BlueRaja - Danny Pflughoeft Jan 22 '13 at 9:34
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@BlueRaja-DannyPflughoeft Regarding removing a prong: The fundamental mode (two prongs vibrating symmetrically) would no longer exist. By conservation of momentum, if one rod is moving to the right, and there isn't another rod moving left at the same time, then something (i.e. your hand) would have to push back to vibrate the prong that is there. And unless your hand can vibrate several hundred times a second, you won't be able to keep it vibrating. –  Chris White Jan 22 '13 at 9:40
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For three equal tongs, there would be more low-energy overtones, resulting in a richer (less pure) sound. –  Chris White Jan 22 '13 at 9:41
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