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/q/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
up vote 52 down vote accepted

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
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    @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. – user10851 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
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    @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

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. – user10851 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. – user10851 Jan 22 '13 at 9:41

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. – user10851 Jan 25 '13 at 16:53

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.

  • 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. – Dima Tisnek Jan 22 '13 at 16:38

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.

  • 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? – Dima Tisnek 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. – Dima Tisnek Jan 23 '13 at 8:44

First, a general observation: oscillators made of solids of simple shapes are not all that great at acoustically generating pure tones. The simplest way to acoustically generate a fairly pure tone is to use a flute organ pipe! Such pipes are voiced specifically to suppress all harmonics, and in steady state produce perhaps the purest of tones that could be generated using devices having a fairly simple geometry.

The general requirement fulfilled by a tuning fork is to have a solid vibrating device that you can hold in your hand, with a reasonably long decay time constant and a reasonably stable frequency, and not requiring a supply of pressurized air to work (i.e. not an organ pipe).

In order for the contact with the squishy tissues in your hand not to act as a damper, the handle should not vibrate. Note that the momentum is always conserved: if something moves one way, there must be something else moving the opposite way for the total momentum to be zero - else the handle will be moving.

The closest we can come to this with something that is a single piece of metal - and thus easy to manufacture - is to have two prongs attached to a handle - called a tuning fork. The handle still moves longitudinally somewhat, since as the forks deflect sideways their centers of mass follow an arc, and thus the handle has to move back-and-forth along its length to conserve the momentum. Fortunately, this motion is 2-3 orders of magnitude smaller than the motion of the prongs - think microns in a typical tuning fork. It can be coupled to your ear by conduction through the bone: strike the fork and then push the handle on the skull behind your ear.

The longitudinal motion of the handle can be harnessed by coupling it to a sounding box (a resonator) tuned to the fundamental frequency. We get a tuning forks standing on top of a box that is open on one side. The sounding box is its own quasi-monopole radiator and thus fairly efficient. It is an additional element, though, and somewhat unwieldy to hold. It should also be noted that the stem vibrates with higher amplitude at the 2nd harmonic than at the fundamental. The 2nd harmonic in the handle can be suppressed by an order of magnitude by bending the prongs of the fork inwards, setting the distance at their ends to ~1/3rd of the distance at the base.

A single cantilevered beam would require a handle with comparatively large inertia so that the motion of the beam would not move the entire device much. The same problem would be faced by systems of odd number of beams vibrating in plane.

Alas, in terms of acoustics, a freestanding tuning fork is somewhat inadequate, as the two prongs form a quadrupole radiator whose radiation efficiency scales with the 6th power of the frequency. Thus low frequency tuning forks are very quiet. Also, the clang mode - the second mode and about 6 times the fundamental - thus radiates 6^6 or about 50,000 times stronger. It's very audible! Why is it a quadrupole? As the prongs move, they create a region of lower pressure on their one side, and higher pressure on their other side. Since there are 4 regions in total, with the inner regions quite separated - it's a quadrupole. A linear quadrupole, in fact.

One solution to getting rid of the deficiencies of a quadrupole and to suppress the clang mode is to convert the tuning fork to a monopole. This is done by acoustically coupling the prongs to a resonator tuned to the fundamental frequency. In practical terms: take a length of pipe, and cut a longitudinal slot in it. The sections of the pipe on the sides of the slot are the prongs of the fork, and the remaining unslotted length of the pipe is the acoustic resonator. These are called by various names, such as tone chimes or choirchimes. The resonator can be either open or closed. A closed-end quarter-wavelength resonator suppresses the sound generated between the prongs, converting the outside of the prongs to a monopole. An open half-wavelength resonator carries the sound from between the prongs to its other end, shifting it 180 degrees in phase, and the whole chime becomes a pair of in-phase monopoles: the outsides of the prongs are one monopole, and the open end of the resonator is another monopole.

Another solution would be to have a free vibrating beam (suspended at the nodes), coupled to a resonator. That's how xylophones and marimbas are made. The stronger the coupling, the more the other modes are suppressed. The strongest coupling would be achieved by having resonators at each of the antinodes, on both sides of the beam. Since a free vibrating beam has 3 antinodes, there would be 6 resonators: 3 half-wave and 3 quarter-wave, to produce 3 monopole in-phase sound sources. It's obvious that this would be unwieldy and expensive. Marimbas and xylophones make do with just one half-wave resonator.

Yet another solution would be to orient the prongs like the sides of a regular polygon: several such prongs in proximity would approximate an acoustic dipole - now the problem becomes how to excite them all initially at the same phase and amplitude. With two in-plane prongs, striking just one of them excites both the symmetric- and anti-symmetric modes, but the anti-symmetric modes are stronger and decay slower. The symmetric modes are damped by the hand holding the handle!

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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