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Author Topic: What is the shape of a vibrating string?  (Read 12210 times)

Offline Helicalred

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What is the shape of a vibrating string?
« on: 09/07/2011 09:14:05 »
Soon after being plucked or struck, a string vibrating at its fundamental frequency adopts a characteristic shape. Does this shape have a name? Is it a vesica Pisces, some sort of catenary, a tropskein or something else?


 

Offline Mr. Data

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What is the shape of a vibrating string?
« Reply #1 on: 09/07/2011 11:06:44 »
Physics would call it a standing wave, with the ends making the nodes. So it's shape is really wavelike.
 

Offline Helicalred

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What is the shape of a vibrating string?
« Reply #2 on: 10/07/2011 02:37:30 »
Thanks Mr Data.

"Standing wave" describes one of its properties, but I was seeking a name for the curve.

BTW, I spelt troposkein incorrectly in my original post.

- Helicalred
 

Offline RD

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What is the shape of a vibrating string?
« Reply #3 on: 10/07/2011 02:49:32 »
"Standing wave" describes one of its properties, but I was seeking a name for the curve.

sinusoidal ... http://en.wikipedia.org/wiki/Standing_waves#Mathematical_description
« Last Edit: 10/07/2011 02:51:41 by RD »
 

Offline techmind

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What is the shape of a vibrating string?
« Reply #4 on: 19/07/2011 23:03:21 »
Soon after being plucked or struck, a string vibrating at its fundamental frequency adopts a characteristic shape. Does this shape have a name? Is it a vesica Pisces, some sort of catenary, a tropskein or something else?
It'll be a half-sinusoid.
 

Offline Soul Surfer

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What is the shape of a vibrating string?
« Reply #5 on: 20/07/2011 00:22:22 »
Think a little bit about what is going on.  you pick or hit a straight tensioned string at some point along its length. The initial shape of the string just before it is released is a sort of triangular wave.  This then vibrates at the fundamental frequency and harmonics of the string dependant on its mass and tension.  It is not sinusoidal.

Think a little bit more.  The fixed points at the ends of the string are like mirrors for the vibration and the high harmonics on the string are amplitude modulated travelling backwards and forwards reflected from these mirrors so tin the first vibrations the shape of the string oscillates between the initial triangle and the triangle the other way round  that is assuming that the string was not plucked in the middle when the reverse pattern is the same as the first pattern. 

As one might expect the higher harmonics loose their energy more quickly than the fundamental note so, as the note fades the waves become more sinusoidal with the final vibrations being sinusoidal. 

You can watch all this happen if you take a reasonably large elastic band an stretch it with a relatively low tension between two solid points so that it vibrates at only a few cycles per second and pluck it some distance away from the end.
 

Offline Geezer

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What is the shape of a vibrating string?
« Reply #6 on: 20/07/2011 06:26:22 »
It's actually quite easy to make a guitar string vibrate with a dominant second harmonic. You just have to discourage any initial oscillation around the midpoint. You can also force the third and fourth harmonics to dominate by a similar technique.
 

Offline Bored chemist

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What is the shape of a vibrating string?
« Reply #7 on: 20/07/2011 06:58:48 »
I thought a bit about what was going on.
The OP asked about the shape of a string vibrating at its fundamental frequency, that is, without any harmonics.

I thought about it some more.
If the whole string is free from harmonics then each part of it must be exhibiting simple harmonic motion.
What I'm not sure of is what the distribution of amplitudes is along the string, but sinusoidal looks reasonable to me.
 

Offline Soul Surfer

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What is the shape of a vibrating string?
« Reply #8 on: 20/07/2011 09:23:44 »
Think again, the harmonics of the fundamental are sine waves with zeros in their waveform at the fixed points at each end of the string they also have zeros at positions along the string.  When you pluck or strike the string the harmonics that are exited would be those which have an amplitude at the position where the string is plucked and the ones that have zero amplitude at this position will be discouraged (although non linear processes will excite them later as the string vibrates) so if you pluck the string in the middle second fourth and other even harmonics are suppressed and first third fifth and odd harmonics are strongest.

This is why musical instruments sound very different depending on where the string is plucked.   The nearer the end of the string that you excite the more high harmonics you generate. Most instruments chose around the third or fifth harmonic peak because that creates the most pleasing sounds.
 

Offline Geezer

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What is the shape of a vibrating string?
« Reply #9 on: 21/07/2011 04:55:36 »
so if you pluck the string in the middle second fourth and other even harmonics are suppressed and first third fifth and odd harmonics are strongest.


I think that is true SS.

You can also "force" the even harmonics by temporarily creating a null at the midpoint of a string. Thereafter, the fundamental seems to remain the second harmonic of the string! Presumably this has to do with the energy transfer within the string.
 

Offline Bored chemist

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What is the shape of a vibrating string?
« Reply #10 on: 21/07/2011 18:43:08 »
It doesn't seem to matter how often I think about it, the fundamental still isn't the same as the harmonics.
If you add various amounts of higher frequencies you will get a different shape.
That would mean that the answer to the question would be "it depends" which is not helpful.
 

Offline JP

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What is the shape of a vibrating string?
« Reply #11 on: 21/07/2011 20:01:00 »
The fundamental by definition is an order zero harmonic, so it's a half period of a sine wave for a string with both ends clamped down. 
http://en.wikipedia.org/wiki/Fundamental_frequency

It's really hard to excite only the fundamental frequency, though, so generally when you pluck a string you generate many higher order harmonics as well.  Because your final shape is the addition of a bunch of harmonics, it doesn't have to look like any one of them.  (Those familiar with Fourier series will know well that adding up a bunch of sine waves of different periods can yield a lot of different shaped functions.) http://en.wikipedia.org/wiki/Fourier_series
 

Offline Geezer

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What is the shape of a vibrating string?
« Reply #12 on: 21/07/2011 21:25:05 »
BTW, the lateral oscillation of a string that is producing only its fundamental = first-harmonic = order-zero-harmonic is sinusoidal, but does that translate to a sinusoidal shape along the length of the string? As SoulSurfer points out, it does not start out that way.

Are we being conned into thinking the shape is sinusoidal because of the relationship between the wavelength and the string length?
 

Offline JP

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What is the shape of a vibrating string?
« Reply #13 on: 21/07/2011 21:34:53 »
The question asked what the fundamental looks like.  It looks like a sine wave in this case.  It is, by definition, the lowest order harmonic. 

A more general solution doesn't have to look like a sine wave, but he fundamental does.  A more general solution does have to be expressible as the sum of harmonics, however.  But if you add a lot of them together, the result doesn't have to look like a sine wave at all. 

It is, in fact, very hard to pluck a string to only get the fundamental.  If only the fundamental sounded when a string were plucked, the tone would sound very pure and very boring.  Part of the richness of the notes coming from string instruments is due to the fact that you get a lot of overtones in addition to the fundamental.
« Last Edit: 21/07/2011 21:37:13 by JP »
 

Offline Geezer

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What is the shape of a vibrating string?
« Reply #14 on: 21/07/2011 21:54:01 »
Isn't the question was about the shape of the string? The fundamental sound pressure is obviously sinusoidal, but the fundamental is produced by lateral displacement of the string in time. That oscillating displacement must be sinusoidal, but I'm not sure that means the shape of the string is necessarily sinusoidal.

A taught string could possibly be modelled as a series of masses with springs that act to return each mass to the rest state. The resonant frequency of each mass/spring combination would be at the fundamental frequency. If the maximum amplitudes of the excursions of all the masses were traced, would that necessarily produce a sine wave?

It certainly does not start out sinusoidal. As SS says, it's initially triangular. 
 

Offline Geezer

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What is the shape of a vibrating string?
« Reply #15 on: 21/07/2011 22:11:57 »
Here we go. I think this says the shape is sinusoidal at the fundamental, although it suggests that may not be the case at greater amplitudes.

http://en.wikipedia.org/wiki/Vibrating_string
 

Offline JP

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What is the shape of a vibrating string?
« Reply #16 on: 21/07/2011 22:43:16 »
That's exactly what I was about to post.  Musical instruments tend to be well approximated by the wave equation, though, since the strings don't vibrate with huge amplitude. 

The harmonics are important because they form a basis, which is a technical term for saying that any shape of a plucked string (Soul Surfer's triangle, for example), can be formed by adding up harmonics in the proper amounts.  Since harmonics are easy to write out, this means even the complex behavior of a triangle shaped string can be modeled easily by expressing it as the sum of much better-behaved harmonics. 

The fundamental frequency is just a technical term for frequency of the lowest order harmonic (i.e. the one with the longest wavelength).  A plucked guitar string generally doesn't vibrate in only the lowest order mode unless you very carefully pluck it in a half-sinusoid shape.

I believe that in reality, the different modes die out at different rates in time as well, so that even if you begin with a triangle, you might end up with something looking more like the fundamental mode as the higher order ones decay faster. 
 

Offline Helicalred

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What is the shape of a vibrating string?
« Reply #17 on: 30/07/2011 16:16:08 »
Thanks for your efforts people. I think I'll go for half-sinusoidal although I'm not entirely convinced. What prompted me to ask the question was that having read a number of texts that discussed vibrating strings, none of the authors actually named the shape although one did say that it had a sinusoidal appearance. So if the shape is not exactly sinusoidal then apparently it doesn't have a name. I guess for practical purposes, sinusoidal is close enough. I'm not going to lose any sleep over it. 
 

Offline Helicalred

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What is the shape of a vibrating string?
« Reply #18 on: 19/08/2011 06:08:14 »
When I asked the question I was unaware that there had been much debate about this in the 18th century which is summarised in this article: newbielink:http://lit.gfax.ch//ScientificBackgroundToRameau%27sPrinciplesOfHarmony.pdf [nonactive]
After some more searching I came across this splendid animation - newbielink:http://www.falstad.com/loadedstring/ [nonactive]- that can clearly show that the shape of a plucked string is not half a sine wave. Also, I realise now that my question was flawed - a plucked string will not vibrate at just its fundamental frequency; there will always be harmonics. Thus the shape can't be named because, as Bored Chemist noted, "it depends". 
 

Offline Geezer

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What is the shape of a vibrating string?
« Reply #19 on: 20/08/2011 02:27:10 »
Nice simulation! I would think the "shape" approximates to a half-sine as the harmonics decay, so at some point it does become "more-or-less" sinusoidal.

But here's another question: How do the individual "loads" in the simulation move? Do they move with simple harmonic motion, or is it not as simple as that?
 

Offline wolfekeeper

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What is the shape of a vibrating string?
« Reply #20 on: 20/08/2011 15:44:23 »
"Standing wave" describes one of its properties, but I was seeking a name for the curve.
I think that that actually is the name of any of the curves you get when you vibrate a string; I'm pretty sure there's no other name for them.

Note that standing waves are not necessarily sinusoidal, although they are made of sinusoids, but then literally every curve can be made of sinusoids!
 

Offline wolfekeeper

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What is the shape of a vibrating string?
« Reply #21 on: 20/08/2011 15:48:38 »
And it's not quite correct to say it's A sine wave for the fundamental. It's actually two added together:

sin(x-wt) + sin(x+wt)
 

Offline Geezer

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What is the shape of a vibrating string?
« Reply #22 on: 21/08/2011 08:34:49 »
And it's not quite correct to say it's A sine wave for the fundamental. It's actually two added together:

sin(x-wt) + sin(x+wt)

I'm probably thick, but I don't see why it necessarily follows that the shape of the string is sinusoidal at the fundamental (although I suspect it is).

The thing that is sinusoidal is the orthogonal displacement of any element of the string (assuming, of course, that the oscillation of the string never decays) and it is that displacement, or really those displacements, that produce the note.
 

Offline wolfekeeper

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What is the shape of a vibrating string?
« Reply #23 on: 21/08/2011 14:32:21 »
Well, you can change it to a cos and plug in the identity:

0.5*(cos(a+b)+cos(a-b)) = cos(a)cos(b)

To get:

y = 2 cos(x)cos(wt)

which is presumably what you're asking.
« Last Edit: 21/08/2011 14:34:20 by wolfekeeper »
 

Offline JP

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What is the shape of a vibrating string?
« Reply #24 on: 22/08/2011 03:02:02 »
And it's not quite correct to say it's A sine wave for the fundamental. It's actually two added together:

sin(x-wt) + sin(x+wt)

As you showed in the above post, you can rearrange this to be a sinusoid in time multiplied by a sinusoid in position.  At any time, it's a position sinusoid with an amplitude that also oscillates in time.  But both answers are right, and the above equation tells the story that it's the sum of a wave traveling left to right plus the reflected one traveling right to left.
 

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What is the shape of a vibrating string?
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