Naked Science Forum
Non Life Sciences => Physics, Astronomy & Cosmology => Topic started by: Helicalred on 09/07/2011 09:14:05
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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?
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Physics would call it a standing wave, with the ends making the nodes. So it's shape is really wavelike.
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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
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"Standing wave" describes one of its properties, but I was seeking a name for the curve.
sinusoidal (http://en.wikipedia.org/wiki/Sinusoidal) ... http://en.wikipedia.org/wiki/Standing_waves#Mathematical_description
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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
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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?
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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.
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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.
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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
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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.
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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.
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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: http://lit.gfax.ch//ScientificBackgroundToRameau%27sPrinciplesOfHarmony.pdf (http://lit.gfax.ch//ScientificBackgroundToRameau%27sPrinciplesOfHarmony.pdf)
After some more searching I came across this splendid animation -http://www.falstad.com/loadedstring/ (http://www.falstad.com/loadedstring/)- 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".
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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?
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"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!
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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)
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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.
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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.
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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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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.
Geezer, it's a sinusoid because it's defined that way. If you solve for the physics of a string, the displacement of the string is a wave equation. If you solve the wave equation subject to the conditions that the ends of the string are fixed, you can of course get a bunch of funky shapes that aren't sinusoids. But these shapes aren't modes, either. The modes are defined to be sinusoids, and the fundamental mode is the lowest frequency sinusoid allowed by physics (aside from an unmoving string, which is technically a sinusoid of infinitely long wavelength :p ).
The reason why we bother defining modes at all has to do with the way the physics can be described mathematically. No matter what shape the vibrating string takes, it should be the sum of a bunch of modes. To specify the more complex shape, you only need to specify the coefficients of the modes, which is a set of numbers. It's often easier to deal with a set of numbers than to try to find an equation for an arbitrary wave. This is especially true if you're representing waves in a computer, where you can figure out how many modes actually are important, and truncate your set of numbers there, making a very efficient way of storing that wave's complex shape.
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Geezer, it's a sinusoid because it's defined that way. If you solve for the physics of a string, the displacement of the string is a wave equation. If you solve the wave equation subject to the conditions that the ends of the string are fixed, you can of course get a bunch of funky shapes that aren't sinusoids. But these shapes aren't modes, either. The modes are defined to be sinusoids, and the fundamental mode is the lowest frequency sinusoid allowed by physics (aside from an unmoving string, which is technically a sinusoid of infinitely long wavelength :p ).
The reason why we bother defining modes at all has to do with the way the physics can be described mathematically. No matter what shape the vibrating string takes, it should be the sum of a bunch of modes. To specify the more complex shape, you only need to specify the coefficients of the modes, which is a set of numbers. It's often easier to deal with a set of numbers than to try to find an equation for an arbitrary wave. This is especially true if you're representing waves in a computer, where you can figure out how many modes actually are important, and truncate your set of numbers there, making a very efficient way of storing that wave's complex shape.
Pardon me JP, but you have obviously mistaken me for someone who might have a clue about the math involved [:D]
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Geezer, it's a sinusoid because it's defined that way.
No, that's not quite right I think. Sinusoids aren't defined in terms of this, nor vice versa.
It's pretty much an accident of physics.
I suppose it's because there's two modes of energy storage here: potential energy (energy stored in the tension) and kinetic energy (energy of motion), and the energy oscillates between the two in a similar way that distances change along the x and y axes as you spin around a circle.
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Geezer, it's a sinusoid because it's defined that way.
No, that's not quite right I think. Sinusoids aren't defined in terms of this, nor vice versa.
It's pretty much an accident of physics.
I suppose it's because there's two modes of energy storage here: potential energy (energy stored in the tension) and kinetic energy (energy of motion), and the energy oscillates between the two in a similar way that distances change along the x and y axes as you spin around a circle.
It's not defined to be a sinusoid, but it is defined in terms of a certain mathematical construction. For a vibrating string, fixed at both ends, this definition requires it to be a sinusoid.
If you go through the derivation of waves on a string, you get the wave equation: http://en.wikipedia.org/wiki/Vibrating_string#Derivation
Modes are defined here: http://en.wikipedia.org/wiki/Normal_mode: "A normal mode is a mode of a linear field among a chosen set of orthogonal modes." That's fancy mathematical language which places requirements on what a mode is. This, along with the underlying physics (the wave equation + both ends of the string fixed), require each mode to be a sinusoid in the case of a vibrating string.
More accurately, it's a sinusoid in position along the string, and it's amplitude also oscillates sinusoidally in time.
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No. Nature doesn't use maths. Humans use maths to predict what nature does. Maths is just a collection of patterns that humans have noticed about the world and about patterns in patterns. And we give these patterns symbols and we write them down, and we call that maths.
But nothing, ever, in nature is due to our human definitions.
The fact that the fundamental must be a sine wave is because of the relationship between the energy modes and the x/y dimensions of something spinning around a circle. It's the same pattern. That's why it's sinusoidal. There's probably other ways to see it as well.
Your argument that you just have to solve the wave equation doesn't really solve anything, since ANY waveform can travel obeying the wave equation. Square, triangular, anything. You need to show why sinusoids are so common.
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No. Nature doesn't use maths. Humans use maths to predict what nature does. Maths is just a collection of patterns that humans have noticed about the world and about patterns in patterns. And we give these patterns symbols and we write them down, and we call that maths.
But nothing, ever, in nature is due to our human definitions.
I completely agree, and I wasn't arguing otherwise. But modes are a human definition. What we do in physics is to look at the string and try to model it by mathematics. Of all the waves that can be described on the string by our model, some have a special property, and we call those modes. That property is that the waves form a complete, normal set. That's math-speak to say that we can mathematically describe, with our model, any wave that exists on the string by adding up a bunch of these modes, and also that any one mode cannot be written in terms of any other modes.
The real power is what I've mentioned twice above: that you can mathematically describe a wave on a string in terms of just a list of the coefficients of these modes, i.e. a list of numbers rather than a very complicated equation.
The fact that the fundamental must be a sine wave is because of the relationship between the energy modes and the x/y dimensions of something spinning around a circle. It's the same pattern. That's why it's sinusoidal. There's probably other ways to see it as well.
I'm not sure what you're getting at here, but the fundamental mode is a sine wave because all modes are sine waves for the string. There aren't really other ways of getting at this, since modes are a mathematical definition within our model, and you only get them by enforcing completeness and normality.
Your argument that you just have to solve the wave equation doesn't really solve anything, since ANY waveform can travel obeying the wave equation. Square, triangular, anything. You need to show why sinusoids are so common.
Actually it does solve everything. Modes are a particular wave within our mathematical model, so it makes sense we have to solve the equation on which we base our model. As I mentioned, these more complex waves don't meet the requirements of what we call modes, but within our model, we can mathematically express them as a sum of modes.
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No, sorry. Point of fact you can analyse vibrations using many different bases, including various orthogonal square waves. You're essentially implicitly using fourier analysis, but you absolutely don't have to.
In fact, for example, in image processing, sine waves are less commonly used since sine waves are mathematically cumbersome.
What you're saying actually doesn't explain it at all; it's more of an 'when you mathematically do this, it works' argument, which is 'it just happens to be true' rather than a 'it must be a sine wave because...' argument which is what the original question was.
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Oh dear! It seemed like an innocent enough question, but now I'm not so sure [:D] It will take me a considerable amount of time to try to catch up here.
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No, sorry. Point of fact you can analyse vibrations using many different bases, including various orthogonal square waves. You're essentially implicitly using fourier analysis, but you absolutely don't have to.
You have a good point. I was misled by talking about modes, rather than harmonics. We're actually talking about the harmonics here. By definition, harmonics are solutions to the Helmholtz equation. What that physically means for this string is that they're waves whose vibration in time is of a single frequency (in the Fourier sense). Because of the physics of having both ends of the string tied down, these waves bounce back and forth and form standing waves.
These harmonics are also members of a complete, normal basis, but you're right--they certainly aren't the only possible basis. They are the only one that is made up of harmonics.
I can see why the argument that they're special because they're harmonics or standing waves makes sense, and I agree those are special properties. But I still think their use is primarily mathematical. If you pluck a string, you don't get just one mode. You get a traveling wave of a complex shape (probably a triangle). The use of modes in this case is to describe the wave in terms of it's harmonic solutions.
The harmonics might also be useful because as the wave dies out, the higher order modes, which vibrate faster, will probably tend to die out sooner, due to air resistance. As long as the original wave had a lot of the fundamental present, the wave will probably look a lot like the fundamental as the higher frequency harmonics die out.
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Er, well, what I was getting at is, in the case where there is only the first harmonic, you can model the string as a large number of masses, each moving with simple harmonic motion orthogonal to the "center line" of the string. If there is no damping, there never will be any other harmonics and each mass will oscillate indefinitely.
The envelope formed by the amplitudes of the oscillations of the masses determines the shape of the string. (Now for the tricky bit.)
Why does that insist that the envelope is sinusoidal? I'm pretty sure it is, but it's not obvious to me why it has to be. If it turned out to be triangular, as long as all the masses oscillated with the same frequency and phase, there would still only be a single harmonic.
The answer, I think, is because the tension in the string varies sinusoidally due to the sinusoidal orthogonal displacement.
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The answer, I think, is because the tension in the string varies sinusoidally due to the sinusoidal orthogonal displacement.
Yeah, essentially. Of course, your initial displacement has to be sinusoidal for this to work. If you pluck it as a triangle shape you don't get a nicely behaved sine wave. I believe that if you let the vibration go on for a while, the higher frequency harmonics that make up that triangle will be damped, so you will end up with something that looks like the sine wave, since that's the most resistant to air resistance.
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If you pluck it as a triangle shape you don't get a nicely behaved sine wave. I believe that if you let the vibration go on for a while, the higher frequency harmonics that make up that triangle will be damped, so you will end up with something that looks like the sine wave, since that's the most resistant to air resistance.
Yes. I think so. Isn't there also less energy in the higher harmonics to start with?
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Higher order harmonics tend to be important if there's a lot of fine structure in the wave. (Which makes sense from the Fourier standpoint.) If you pluck it as a fairly smooth wave to begin with, there won't be a lot of energy in the higher order harmonics from the start.
Even if you somehow put a lot of energy into the high order harmonics, they're higher temporal frequency, so they oscillate faster than the low order ones, so I would expect them to lose that energy to air resistance (and sound waves) quickly.