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Old May 19th 2015, 05:13 PM   #1
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Resonance and conservation of energy

I am currently stuck on a exam question which says,

A standing wave is produced by small transverse oscillations of one end of a taught string stretched between an oscillator and a fixed point. The frequency of oscillation is adjusted to be resonant at the third harmonic. The amplitude of the string's vibration is very much greater than that of the oscillator. Does this violate the principle of conservation of energy?

I am pretty sure that it does not violate it however, for it to have a greater amplitude it must gain some energy and I am unsure where this extra energy would have come from. Would it be because the wavelength decreases or I am thinking about the problem in the wrong way? Any help would be appreciated.
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Old May 19th 2015, 05:19 PM   #2
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For any oscillation does the energy depend on the amplitude or some other property?

Hint what did Plank have to say on the subject.
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Old May 19th 2015, 05:27 PM   #3
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Well Planck said that energy depended upon frequency, which depends on speed and wavelength. So would the speed also increase as well as the wavelength, so overall the energy would still be the same?
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Old May 20th 2015, 08:11 AM   #4
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In the normal or large scale world of classical physics energy conservation is never violated.
You should have the confidence to know that if any proposal appears to contradict it there is either a mistake or a misunderstanding, so well done for asking.
At the very small quantum scale conservation may be bent for a very short timescale, but evens out over the longer term.

The energy of an oscillator or wave is proportional to the square of the amplitude and also proportional to the frequency of the oscillation.

Oscillators donít have a speed as such. Wave motion couples the motion of a series of oscillators in and organised way and allows the transmission of energy from one oscillator to the next.
The speed of this transmission is called the wave speed and the energy of the wave is independent of this speed

Think about the electromagnetic spectrum.
All EM waves travel at the same velocity, radio, infra-red, visible, ultra-violet, X-rays etc.
But these different EM waves have different frequencies and energies.

Back to your question.

The oscillator is continually inputting energy to the string at all frequencies.
At most frequencies this is lost through dissipative processes, and if measured the string would be found to have increased in temperature.

What your question is telling you is that when the oscillator is operating at a whole multiple (3x) of the elastic resonant frequency of the stretched string the amplitude is much larger than that of the oscillator so less energy is being lost than is input so energy is being stored in the oscillation of the string.

You are correct, this is characteristic of a resonant system.

Some energy however continues to be lost through dissipation. If this were not so then the energy input would continue to build up and the amplitude continue to increase until the string broke.

So energy conservation is satisfied.

Incidentally can you think why the effect was noticed on the third harmonic, not the second (or fourth) ?
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Old May 20th 2015, 10:07 AM   #5
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Thanks for your response. I am not really sure what is special about the third harmonic, compared to the fourth or second harmonics. Does it have to do with the third not being a whole number of wavelengths? That's the only thing I can think of.
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Old Jun 8th 2015, 11:49 PM   #6
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There is nothing special about it being the third harmonic. Any "allowed" frequency will cause resonance. Since the string is clamped only at one end, the respective resonant lengths will be given by L = 1/2 , 3/2, 5/2 lambda etc, and the respective frequencies at which resonance occurs will be f, 3f, 5f etc. The extra energy required is provided by the oscillator, which itself draws energy from some other source.

Studiot has explained this beautifully.

However, I don't quite agree with the statement "The oscillator is continually inputting energy to the string at all frequencies." in this case, as it could be driven by a motor with a fixed rpm. However there are cases where multiple frequencies could be generated and only the resonant frequency is prominent while the others lead to dissipation like for example when blowing into a flute.
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Old Dec 21st 2015, 08:23 AM   #7
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Reonance and Energy

HI, jUST CANNOT CONTINUE WITH THIS WITHOUT butting in.There is much confusion here due mainly to innacurate thinking.
With resdpect - the sentence below "The energy of an oscillator or wave is proportional to the square of the amplitude and also proportional to the frequency of the oscillation" is strictly incorrect. That word "amplitude" lacks a clear meaning. Is it being applied to the instantaneous amplitude or to the general (peak) amplitude overall?
The instananeous enrgy is proportional to the instaneneous amplitude and, to obtain the overall energy of the wave, this must be inteegrated over a full cycle of the oscillation ? This is the basis of the RMS value?

The harmonic content of an audio waveform has a drastic effect on the sound as perceived by the ear. Post WWll all domestc radio receivers in this country were designed to be enriched with 2nd (and even) harmonics bcause they had a soothing effect as preferred by European listeners. By contrast American designs featured 3rd (and odd) harmonics because their harsher sound effect wqs preferred "over there".

En passant the flute is renowned for the purity of its sound - practically free of harmonic content.

Ken Green

Originally Posted by studiot View Post
In the normal or large scale world of classical physics energy conservation is never violated.
You should have the confidence to know that if any proposal appears to contradict it there is either a mistake or a misunderstanding, so well done for asking.
At the very small quantum scale conservation may be bent for a very short timescale, but evens out over the longer term.

The energy of an oscillator or wave is proportional to the square of the amplitude and also proportional to the frequency of the oscillation.

Oscillators donít have a speed as such. Wave motion couples the motion of a series of oscillators in and organised way and allows the transmission of energy from one oscillator to the next.
The speed of this transmission is called the wave speed and the energy of the wave is independent of this speed

Think about the electromagnetic spectrum.
All EM waves travel at the same velocity, radio, infra-red, visible, ultra-violet, X-rays etc.
But these different EM waves have different frequencies and energies.

Back to your question.

The oscillator is continually inputting energy to the string at all frequencies.
At most frequencies this is lost through dissipative processes, and if measured the string would be found to have increased in temperature.

What your question is telling you is that when the oscillator is operating at a whole multiple (3x) of the elastic resonant frequency of the stretched string the amplitude is much larger than that of the oscillator so less energy is being lost than is input so energy is being stored in the oscillation of the string.

You are correct, this is characteristic of a resonant system.

Some energy however continues to be lost through dissipation. If this were not so then the energy input would continue to build up and the amplitude continue to increase until the string broke.

So energy conservation is satisfied.

Incidentally can you think why the effect was noticed on the third harmonic, not the second (or fourth) ?
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Old Dec 21st 2015, 03:29 PM   #8
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More on Resonance

You mention a "standing wave"? Do you understand that such a phenomenon is not truly a wave?

Your mechanical oscillator is vibrating one end of a string so what is the immediate effect?

If your answer is that a travelling wave passes down the string - then half marks ... what eventually happens to that wave?

What dissipates the energy of that wave
OR
where does that energy end up?

Let us hear your answer.

Ken Green
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