# Harmonics of a vibrating string.

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How can I determine which harmonics will be present and in what amplitudes when a string is plucked at a given point along the string. For example let’s say I pluck the string in the exact middle I want to find out what will be amplitude of the fundamental and the subsequent harmonics?

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there are more things to a pluck than just its position. but even if you were to idealize it as an impulse, i am not aware that the analitical solution exists. youd probably be able to simulate it with a numerical package like matlab.

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on second thought, a good approximation would probably be:

position in range [0,1]

amp = maxamp * sin(position*Pi*i)/(i!)

but thats mostly guessing im affraid.

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Wouldn’t each harmonic have a different maxamp? Like inversely proportional to its frequency.

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The frequency depends by the tension applied to the string; more tension means higher frequencies...like in the guitar!

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Quote:
 Original post by blizzard999The frequency depends by the tension applied to the string; more tension means higher frequencies...like in the guitar!

Actually frequency depends on tension, length, and mass per meter. And that just gives you the fundamental frequency. All subsequent harmonics are multiples of the fundamental.

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Quote:
 Original post by GrainWouldn’t each harmonic have a different maxamp? Like inversely proportional to its frequency.

notice the /(i!) part. its the bit im least sure about, but it means dividing by i*(i-1)*...*1, which takes into account higher harmonics having a lower amplitude.

since 1/(i!)is 1 at most, and so is the sinus, maxamp is simply a scaling value setting the maximum amplitude.

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Look into solutions of the wave equation; this is a well known problem. If you're interested in amplitudes and harmonics then you will probably want a Fourier series solution. If I remember correctly the problem is only as hard as finding the Fourier series of the impulse (pluck).

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it may be better to use numberic integration, the only analitical equations of waves that I know of involve only the steady state solutions. what I mean is if you have a string with tension T what are it's resonance frequency and how many humps you got in the wave etc. I don't know of a dynamic solution, but for sure the wave equation must be satisfied, I'd look into the post refering to that, I'm just not that familiar with solving it, it probably got an anelitical solution for 1 dimensional string subject to impulse force, it seems simple enough, I dont know tho.

Tim

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Quote:
 Original post by Eelcoon second thought, a good approximation would probably be:position in range [0,1]amp = maxamp * sin(position*Pi*i)/(i!)but thats mostly guessing im affraid.

Ok I tested it out and this is definitely not the solution. I get an exponential peak in my wave form at the place where I apply the impulse, While I logically should get a peak there, its no way it should be that big.

Also on further consideration of the problem, Its not just the amplitudes I want but the phases as well.

@jonnyfish & timw: Thanks, I'll look in to wave equations and Fourier series and report back here.

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