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Hitting a string of a violin or a guitar will cause that string to vibrate, but after short time the amplitude of the vibration will decay, consequently the produced sound will die out.

I suppose this decay happens because of friction with air. If this is true then how longer will the string keep vibrating in a vacuumed room? Any way to estimate this? Are there other effects that cause the damping?

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Apart form friction, there is a loss of energy due to, say, potential interaction with air. In absence of air, the damping will be due to inelastic deformations inside the system containing the string. –  Vladimir Kalitvianski Oct 26 '11 at 19:14
@VladimirKalitvianski Sorry, Could you elaborate please. –  Revo Oct 26 '11 at 19:23
For sound propagation one needs pressure oscillations that are mainly elastic, potential. For example, a membrane pushes air and compresses it. As to inelastic deformations in a guitar, they always exist in classical systems so the energy stored in the string oscillations is gradually transformed into heat. –  Vladimir Kalitvianski Oct 26 '11 at 19:45
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3 Answers

The instrument is designed to make sound. The loss of energy is not due to friction, but to emitting sound. In a vacuum, a suspended plucked guitar would ring for minutes, not seconds.

EDIT: More detail

The loss of energy to sound is a direct mode-coupling, and it takes energy away regardless of internal friction. But you can estimate the degree to which internal friction is important by comparing the ringing time of materials with negligible internal friction for sound propagation--- crystalline materials like metals--- vs. complex polymers like wood or plastic, where the propagation of sound leads to losses because the restoring forces are partially entropic.

For a steel body guitar with steel strings, there is no plausible avenue for sound modes to decay fast in the resonator, because the steel is a crystal material. To get an estimate for internal losses in wood, compare the resonating time for a note/chord steel body guitar and a wood guitar in air. The wood guitar frictional losses are estimated by the decay time of the tone in a wood guitar vs. steel body.

Here is a steel body demonstration: http://www.youtube.com/watch?v=tVx62GpWKOE

I don't hear noticably less decay in the metal, so I assume that internal losses in wood are small compared to radiated sound energy.

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While I think what you say is most likely true, I can't construe have any good arguments to justify it. We know there is significant energy dissipated in air because we can hear it, but we don't have a good basis to say the thermal energy is considerably less, or less at all. I think it is less, I just can't show it. –  AlanSE Oct 27 '11 at 0:03
@Zassounotsukushi: I am sure you are right. My thinking was as follows: to generate heat, you either need internal slippage of domain walls or polymer junctions, or rubbing of parts, like a violin string against the bridge. The rubbing effects are at nodes, so they are asymptotically negligible-- the loss goes to zero with the amplitude, and I will ignore this. For a metal body guitar violin/guitar with metal strings, there will be no internal slippage, and the answer is correct. For wood and catgut, the answer is probably different. I will modify the answer to reflect this. –  Ron Maimon Oct 27 '11 at 2:04
@Zassounotsukushi: Of course, I missed what is probably the dominant decay method for sound waves--- the flow of heat from hotter to colder regions in the adiabatic compression. This type of damping is different in wood and in metal because of the different thermal conductivities, but metal is a better heat conductor, and so would have quicker attenuation. I will have to reconsider this answer. –  Ron Maimon Oct 27 '11 at 4:12
@Zassounotsukushi: I didn't miss it---it is the entropic part of the restoring force that determines the temperature gradients in the sound wave, I just had a mental lapse. –  Ron Maimon Oct 27 '11 at 5:12
This attanuation is negligible, sound is too fast compared to thermal flow. But nevertheless I voted down, because You rechurned. I recommend to read the entire thread before answering. The classical physics demonstration is a monochord on a solid block of hardwood, this shows the damping of all but sound transfer to body and air. –  Georg Oct 27 '11 at 9:30
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Friction doesn't just come from air, it comes from two sources.

  1. Drag in the air, which makes heat as well as the sound
  2. Friction in the string itself

For the vibration to work in the first place, the string must be stretchable. As it is stretched, the tension increases. When it vibrates in standing waves it oscillates between a high tension, no velocity state to a low tension, high velocity state.

Although the system looks different, we can treat this fairly similarly to a normal dampened harmonic oscillator system. You can say the string starts stretched to some length, $l_o$ and tension $F_o$. It is not uncommon to treat this system with a drag force proportional to "velocity", although I would suggest a superficial definition of velocity in this case, which is the rate of contraction or elongation of the string over time.

$$F = F_0 - k (l-l_o) - c \frac{dl}{dt}$$

This equation, however, is not a complete differential equation. This is because I'm using it in an analogous form to a mass on a spring, and what to use in place of the mass is non-obvious. I won't go into that because I'm not sure how much detail is wanted.

Basically, the energy is still dissipated as heat in the string. The heat is stored there unless it radiates out. The string will eventually stop oscillating, although it will last longer than if it were in air. Obviously, no music is produced unless you consider the vibrations in the structures of the violin and whatever else it's touching music

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Lot of wrong statements. In a string, there is no friction. The friction to air is negligible, as You will see when experimenting with a monochord. The damping is nearly solely by transfer of vibration to the body of the instrument and from there to air as sound. Forgetting this main purpose of string instruments is ridiculous. –  Georg Oct 26 '11 at 19:16
@Georg It is correct that this answer is written for a vibrating string, with nothing specific for it being a violin. This would most accurately reflect a plucked instrument. I am likewise concerned that someone reading your comment would pick up incorrect physical notions. No friction in a string? Yes, yes a string should have friction when oscillating! –  AlanSE Oct 26 '11 at 19:50
@Georg Consider: If the primary source of elasticity is from the instrument case then the selection of the case, not the string would determine the pitch. The amount of elasticity (which yes, is a combination of the string and case) determines the vibration frequency (in addition to the mode of vibration). We know from experience that the string type determines the note, that being the entire principle of a guitar. The source of elasticity is probably also the source of energy dissipation in the absence of air. –  AlanSE Oct 26 '11 at 23:46
Dear Zas, that "primary source of elasticity" is Your idea, not mine. Please note that string instruments are made to generate sound. This sound energy is created by pluck or bow, then transferred by that "saddle" or "bridge" to the corpus, which acts as a membrane to radiate air waves. I hope this rather common facts will illuminate You. –  Georg Oct 27 '11 at 9:25
@Zassounotsukushi: I understood. Georg is right. The sound wave is produced by the instrument body itself--- reflections don't make a wave intensity overall louder. The body vibrates and produces nearly all of the sound due to vibrations transferred through the place where the strings are tied to the wood. This is the whole purpose of the wood, to act as a resonator. The mode-coupling of the string to the body is very strong. It would be impossible to keep the body from shaking. –  Ron Maimon Oct 27 '11 at 14:17
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Well, as far as my intuition goes air friction should not cause much of damping in the vibrating string of violin, though it surely plays a small role in damping. For speeds of the oscillator not so large (do not cause turbulence) certainly a friction force exist in viscous medium, like air, which is proportional to velocity as $ f = -bv $ where $b$ is constant of proportionality which depends on medium and object.

Major damping as I believe should be due to heating of the string. Other source of damping will be transfer of energy to instrument (as Georg pointed).

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Most of the energy makes sound. –  Ron Maimon Oct 26 '11 at 22:05
@Zassounotsukushi: link may lead you to some more serious sources.. –  orion Oct 27 '11 at 2:53
I am serious, and the source you provide does not give an answer for the non-sonic damping (which I believe is zero for all intents and purposes). -1, dude. –  Ron Maimon Oct 27 '11 at 3:27
Sorry, Ron If I have hurt you. I actually appreciated your view, but just came around some research, which of course directly may not provide answer, may lead to answer. I posted the comment below my post, hence the "seriousness" term was not for you, instead it was with respect to my answer. But anyway, my mistake as it turned that way, which was not my intent, friend. –  orion Oct 27 '11 at 13:20
@mehulpht: I wasn't hurt--- I was annoyed. It was clear you didn't bother to read your own link. Where is the data which answers the question? You want to know what is the percent of damping due to sound production, and the percent due to friction. But the source is all about measuring the frequency and damping of various modes in air, to reproduce the attack and sustain of a plucked violin. Giving a dense irrelevant reference wastes people's time, so -1. –  Ron Maimon Oct 27 '11 at 14:08
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