What happens when you pluck a guitar string?

[olivermsun]

My feeling is that stiffness and frequency dependent loss terms would deform the shape as time progressed (i.e. Simple triangle wave wouldn't remain a triangle wave for very long), which would mean a Fourier analysis of the initial shape would yield much less information about the frequency output of the system overall.

You're absolutely right about stiffness and damping in a real string (especially one fitted on an instrument). You might model this with a frequency-dependent time decay (e.g., a complex impedence for wave transmission down the string), in which case the Fourier decomposition would be quite helpful. For weak damping, this should give the frequency shift due to damping as well. On the other hand, the true behavior really is nonlinear, so this approach wouldn't work perfectly as you've already pointed out.

Then again, the complex impedence might also be a useful way to allow partial transmission through the bridge...

 

[chingel]

You are correct, however that's not really a good way of looking at it. It's true that the spatial 'modes' (i.e. the dividing the string into even parts and making sinusoids) affect the output, but it's not nearly as direct as it would seem. Like you said, thanks to Fourier analysis, we can indeed take the shape of the string at any time and turn it into a sum of sinusoidal terms based on some frequency omega, which is directly related to dividing the length of the string into even parts.

Thank you for your reply. I still don't understand why does a string divide itself nodes (or parts), why doesn't it just oscillate at the fundamental, why does it also start oscillating at the partials? I also didn't understand why the harmonics last longer and the other frequencies die out quickly, if the other frequencies are even produced when plucking a string which I am not sure about either? There is supposedly some sort of an impedance for non-harmonic frequencies, but how it works I didn't understand. It had something to do with the nodes not fitting on the strings length, but then the videos were posted showing moving nodes on a plucked string.

 

[mikeph]

However, the 1-d wave equation has traveling wave solutions along the characteristics x+ct, x-ct, (c.f. d'Alembert's formula) so that the time dependence is related to the space dependence through the wave speed c.

A good way of showing there are no traveling waves on a guitar string,
f(x - ct) = 0, f(x + ct) = 0 at the nodes, say the nodes are x = 0,1, (wlog)
x = 0; f(-ct) = 0 and f(+ct) = 0, for all t
so f = 0.

 

[Pythagorean]

why does a string divide itself nodes (or parts), why doesn't it just oscillate at the fundamental, why does it also start oscillating at the partials?

The standing wave (with the nodes) is not the whole picture. When you pluck the string, you actually send out traveling waves. The disturbance goes out from your pluck, encounters both bounded ends, and travels back, hits the opposite side, etc, etc, superimposing on each other in such a way that gives rise to the pattern of standing wave harmonics due to the two fixed boundaries. As you change the tension on the string (with the keys) you change the speed of the wave, as you fret different strings, you change the length (shortening/lengthening the path of the traveling waves).

I also didn't understand why the harmonics last longer and the other frequencies die out quickly, if the other frequencies are even produced when plucking a string which I am not sure about either?

The string isn't massless, the tension isn't uniform, thermodynamics applies, etc, etc. So yeah, there's impurities in the harmonics, but they die out rather quickly because they don't "propel" each other like harmonics do (dosey do!) They're not synchronized in time like harmonics are with the fundamental (and the fundamental is the strongest force present, the "dominant" tone.)

 

[chingel]

I don't understand, are there nodes or not, is there a traveling wave or not? Reading this thread the answers seem to me very conflicting. Just looking at the last two posts, one says there are no traveling waves, the other says there are, which is correct?

The standing wave (with the nodes) is not the whole picture. When you pluck the string, you actually send out traveling waves. The disturbance goes out from your pluck, encounters both bounded ends, and travels back, hits the opposite side, etc, etc, superimposing on each other in such a way that gives rise to the pattern of standing wave harmonics due to the two fixed boundaries. As you change the tension on the string (with the keys) you change the speed of the wave, as you fret different strings, you change the length (shortening/lengthening the path of the traveling waves).



The string isn't massless, the tension isn't uniform, thermodynamics applies, etc, etc. So yeah, there's impurities in the harmonics, but they die out rather quickly because they don't "propel" each other like harmonics do (dosey do!) They're not synchronized in time like harmonics are with the fundamental (and the fundamental is the strongest force present, the "dominant" tone.)

Could you please elaborate on how they superimpose to give rise to the wave harmonics? I am having difficulty understanding. Also how do the harmonics propel each other? If the string divides itself into two parts/nodes, one is going up, the other is going down, is that correct? Doesn't that mean that one is necessarily not synchronized with the fundamental, but rather working against it? How does the synchronization with the fundamental work?

 

[Pythagorean]

I don't understand, are there nodes or not, is there a traveling wave or not? Reading this thread the answers seem to me very conflicting. Just looking at the last two posts, one says there are no traveling waves, the other says there are, which is correct?

Initially, they are traveling waves. As the waves meet each other and superimpose, they produce a standing wave.

Also how do the harmonics propel each other? If the string divides itself into two parts/nodes, one is going up, the other is going down, is that correct?

yes

Doesn't that mean that one is necessarily not synchronized with the fundamental, but rather working against it? How does the synchronization with the fundamental work?

You could call it "anti-synchronized" if you like, (just like anti-parallel vectors are still parallel lines). The work "against" the opposing bend sling shots it to the other side with more force than if the string would have been flat, due to tension.

Anyway, they're synchronized in that they reach their maxima at the same time (albeit in opposite directions).

 

[mikeph]

Please correct me if I'm wrong, this is a hunch,

If you pluck the string to create an initial shape as a V shape, for example, there will only be traveling waves if you use your finger.

If you have a V-shaped mould that you push against the string, and suddenly remove it, there will be no traveling waves, even at the start. The traveling waves only arise because it takes a finite time for the signal to move from the place where the finger was to the edges. If the entire string is supported and then instantaneously let free, there are no traveling waves.

I think that's they key.

 

[olivermsun]

A good way of showing there are no traveling waves on a guitar string,
f(x - ct) = 0, f(x + ct) = 0 at the nodes, say the nodes are x = 0,1, (wlog)
x = 0; f(-ct) = 0 and f(+ct) = 0, for all t
so f = 0.

This is incorrect. The wave reflects from the (clamped) boundary with a phase reversal. This allows the boundary condition to be satisfied for all time while conserving energy and momentum.

 

[olivermsun]

I don't understand, are there nodes or not, is there a traveling wave or not? Reading this thread the answers seem to me very conflicting. Just looking at the last two posts, one says there are no traveling waves, the other says there are, which is correct?

You should look at the videos linked earlier or the paper which discusses waves on a string and decide for yourself which one is correct.

Initially, they are traveling waves. As the waves meet each other and superimpose, they produce a standing wave.

Hint: for general initial conditions, a standing wave will not be created.

 

[sophiecentaur]

Remember that it takes time for a standing wave to establish itself. The wave has to travel the full length of the string and be reflected before a standing wave is formed.

Also, (@chingel) there is no conflict in first saying traveling waves and then saying standing waves. A standing wave is only the resultant interference pattern of a set of traveling waves.

 

[chingel]

I found a bunch of high speed videos showing a plucked string

http://physics.doane.edu/physicsvideolibrary/string/StringDirectory.html

Clicking on the links under the headline Videos will open them, some strings are plucked towards the end, some in the middle, they use different strings and weights etc.

It seems like initially there are traveling waves, but they seem to turn into standing waves.

Also, (@chingel) there is no conflict in first saying traveling waves and then saying standing waves. A standing wave is only the resultant interference pattern of a set of traveling waves.

How does that work? How do the interference patterns of traveling waves create standing waves?

 

[olivermsun]

I found a bunch of high speed videos showing a plucked string

http://physics.doane.edu/physicsvideolibrary/string/StringDirectory.html

Clicking on the links under the headline Videos will open them, some strings are plucked towards the end, some in the middle, they use different strings and weights etc.

It seems like initially there are traveling waves, but they seem to turn into standing waves.

Just looking at a couple of the examples where the strings are plucked near one end, I don't see any that turn into standing waves.

 

[sophiecentaur]

I found a bunch of high speed videos showing a plucked string

http://physics.doane.edu/physicsvideolibrary/string/StringDirectory.html

It seems like initially there are traveling waves, but they seem to turn into standing waves.
How does that work? How do the interference patterns of traveling waves create standing waves?

That's what standing waves are. They are the sum of traveling waves moving in different directions. You only get a standing wave resonance on a string (etc) if the length between the ends causes the constructive and destructive interference to coincide for all reflections from both ends. If you wiggle one end of a string (heavy rope is better) with the other end tied (a node), you launch a traveling along it. Choose the right frequency of wiggling and you will get a resonance with one. two, three or more antinodes. Your hand is more or less at a node ( the movement need only be small, compared with the maximum swings at the antinodes) and you only need to be putting in a relatively small amount of power to maintain a large amplitude of resonance.

 

[mikeph]

This is incorrect. The wave reflects from the (clamped) boundary with a phase reversal. This allows the boundary condition to be satisfied for all time while conserving energy and momentum.

But from the mathematics, it's not a d'Alembert solution...?

 

[sophiecentaur]

Just looking at a couple of the examples where the strings are plucked near one end, I don't see any that turn into standing waves.

That's because they are happening slowly and, unlike a real guitar string, you don't see multiple images at once. I don't understand - are you actually disagreeing that standing waves are due to multiple traveling waves? You'll be disagreeing about Young's Slits and Lasers too, perhaps - it's all the same phenomenon.
Just 'cos you can't see something doesn't mean it's not there.

 

[thegreenlaser]

But from the mathematics, it's not a d'Alembert solution...?

I think the issue with your post was your first claim that f(x+ct)=0 and f(x-ct)=0 at the nodes. Correct me if I'm wrong, but I think the only thing you can say is f(x+ct)+f(x-ct)=0, since d'Alembert's solution is
$$u(x,t)=\frac{1}{2} \left( f(x+ct)+f(x-ct) \right)$$
assuming initial velocity is zero. So if you supply the boundary condition u(0,t)=0 for all t, you end up with f(ct)+f(-ct)=0, which does not imply that f=0. If it were true that f=0, then the solution for the displacement of the string would always be zero if no initial velocity were involved, wouldn't it?

Please correct me if I'm wrong, this is a hunch,

If you pluck the string to create an initial shape as a V shape, for example, there will only be traveling waves if you use your finger.

If you have a V-shaped mould that you push against the string, and suddenly remove it, there will be no traveling waves, even at the start. The traveling waves only arise because it takes a finite time for the signal to move from the place where the finger was to the edges. If the entire string is supported and then instantaneously let free, there are no traveling waves.

I think that's they key.

If my understanding is correct, you would still end up with traveling waves. If you have some sort of symmetry in your starting position/velocity (e.g. a triangle whose peak is exactly in the centre) I'm pretty sure you would end up with a standing wave in the ideal case. However, if you were to, say, offset the triangle, even if you do set it off in the perfect manner like you've described, it will create a traveling wave. I'm guessing that that's explained by an imbalance of tension that could probably be shown with some trig, but I do know that such a starting position will cause a horizontally moving wave, and that can be shown with the solution to the wave equation.

 

[sophiecentaur]

I really don't understand this 'obsession' with traveling waves and how it gets in the way of understanding the final condition of standing waves. If the energy distribution on the string is not uniform (if it's plucked off centre, for instance) then energy will flow, once it's been let go (no?). So the energy will be carried by waves. Before many cycles have elapsed, can it be possible to identify the full interference pattern of these waves (i.e. the modes)? Any form of resonance has a build up or decay time associated with it so I can't see that there is a paradox / clash. It takes time to define a frequency precisely and there will be a bandwidth associated with the build up of all these oscillations. You can't just discuss the Frequency domain of the oscillations unless you specify a time over which the analysis is done. Before things have settled down I think the analysis has to be done in the time domain and would involve the equations of motion of each elemental part of the string. Too damn hard I think. So you just have to think in general terms of traveling waves on the string until a suitable time has elapsed, after which the problem becomes more straightforward and the natural modes are the result.

I'm not sure that the excellent slo-mo movies actually help, either, if people have already made up their minds about what they are showing.

 

[olivermsun]

But from the mathematics, it's not a d'Alembert solution...?

It is. I think thegreenlaser explains it quite well in a post above.

I don't understand - are you actually disagreeing that standing waves are due to multiple traveling waves?

I'm disagreeing that constructive and destructive interference between traveling waves is sufficient to create a standing wave pattern. The standing wave is a very special case where the spatial dependence is completely fixed in time, that is, the shape of the string is not time dependent. For example, the maxima and nodes have to stay in exactly the same places on the string for all time. Since there is no time dependence, the standing wave pattern must exist from t = 0 onward -- there is no "set up time" for the resonance to begin.

A pair of (equal) pure sinusoidal waves traveling in opposite directions along the string will automatically create a standing wave. On the other hand, arbitrary combinations of standing wave modes will not result in standing waves. To convince yourself of this, you can try plotting the first two standing modes on the string, where the second mode oscillates at twice the frequency of the fundamental, and their sum.

Just 'cos you can't see something doesn't mean it's not there.

True, but you can also do the math and get the same result. In fact, I'm surprised it has generated this much controversy without people just breaking out the pencil and paper and trying it themselves.

 

[chingel]

Just looking at a couple of the examples where the strings are plucked near one end, I don't see any that turn into standing waves.

In the first video for example, doesn't the kink lose it's kinkiness and just a general pattern of up and down oscillation emerge? Or is that still a traveling wave, only that the kink is smoother and less kinky?

 

[sophiecentaur]

@oliversum
Any reflection will produce a standing wave but you need multiple reflections to get a resonance. Think about transmission lines. You need a matched load to avoid a standing wave, independent of the source impedance.

With no losses, the plucked string would be expected to return, every so often, to the same peaky shape as when released but it doesn't happen in practice, because the high overtones don't survive.

 

[olivermsun]

In the first video for example, doesn't the kink lose it's kinkiness and just a general pattern of up and down oscillation emerge? Or is that still a traveling wave, only that the kink is smoother and less kinky?

The kink slowly becomes a bump but continues to travel back and forth between the ends of the string.

A standing wave doesn't travel, it stands still.

 

[chingel]

The kink slowly becomes a bump but continues to travel back and forth between the ends of the string.

A standing wave doesn't travel, it stands still.

There is still a moving wave near the end of the video, but it seems to me there is some general up and down moving as well happening when the bump is on the other side. Doesn't that mean that there are standing waves evolving? Isn't it the case that the bump is just an illusion created by the sum of the standing waves of the harmonics?

 

[sophiecentaur]

What's in a name? Travelling / standing. The way you would recognise a 'standing wave' would not be obvious if there were multiple waves. What would you expect to see? Certainly not a simple set of nodes and antinodes. But the maths tells you that you will only get combinations of the normal modes. They are the only ones that can exist, I think, so you need to think more layerally and decide what you would look for. Would it be obvious?

 

[chingel]

I think that if the vibrating string is actually the result of a sum of standing waves, the big kink (traveling wave) makes it's roundtrip at the speed of the fundamental, but if the string is actually divided into nodes, the edges (every part actually I think but the edges are easier to discern when the big kink is on the other side) would wobble up an down before the big kink arrives, because the frequency of the nodes is several times higher. I am not too sure but I think that is what I see happening in the high speed videos towards the end.

I am probably sounding stubborn, but I still don't understand why/how does the string divide itself into nodes, why does it vibrate at multiple frequencies simultaneously, if it does so at all?

 

[sophiecentaur]

I think you need to ignore what the movies appear to tell you because they are not ideal. The high frequencies are lost too soon for you to see the pattern - and what pattern would you actually expect to see?
You have to believe the sums, which tell you the only possible solutions to the wave equation. (That could be your problem?)
All the possible modes can exist at once (with respective levels according to the start shape) because superposition is 'allowed'. Intuition can be a problem with this sort of thing an you may need to FIGHT it. ;-)

 

[chingel]

I think you need to ignore what the movies appear to tell you because they are not ideal. The high frequencies are lost too soon for you to see the pattern - and what pattern would you actually expect to see?
You have to believe the sums, which tell you the only possible solutions to the wave equation. (That could be your problem?)
All the possible modes can exist at once (with respective levels according to the start shape) because superposition is 'allowed'. Intuition can be a problem with this sort of thing an you may need to FIGHT it. ;-)

But the overtones are there for as long as the string is vibrating, otherwise wouldn't it's timbre change to a sine wave? I think there should be some vibrating nodes distinguishable from the string if there are nodes, because they vibrate multiple times faster than the big bump is moving.

Do you mean the sums that make up the shape of the bump on the string? But that gives me again the question that the sums making up the shape aren't the same sums making up the pressure waves.

Why does a string decide, when it is pulled to the side, alright, I'll divide myself at this point and make the part on the other side move up while I'm going down and the opposite. What makes it behave like that?

 

[sophiecentaur]

The timber of the attack is very different from that of the sustain.
Each piece only "knows" to follow the net force on it and it has a certain mass. The boundary conditions tell you what each bit will do. Why do you demand that the problem must be solved in 'your way'? You could do a simulation, I suppose, which would look at each element in turn. But, as an analytical solution exists, why not use it?

 

[chingel]

The timber of the attack is very different from that of the sustain.
Each piece only "knows" to follow the net force on it and it has a certain mass. The boundary conditions tell you what each bit will do. Why do you demand that the problem must be solved in 'your way'? You could do a simulation, I suppose, which would look at each element in turn. But, as an analytical solution exists, why not use it?

How and what forces apply to make nodes happen? How do the boundary conditions tell the nodes what to do?

I am just trying to understand why/how does a string divide itself into nodes.

 

[sophiecentaur]

Assuming that you accept that multiple modes can exist (superposition) then you can consider each overtones separately.
Every time a wave travels towards an end, a wave will be reflected and it will be in antiphase. One half wavelength away from the end, the incident and reflected waves will again be in antiphase. This is a node and part of an interference pattern, on ANY string and for ANY wave. If the string happens to be an integer number of half waves long the interference patterns for both ends coincide and you get resonance.

 

[chingel]

Do I understand correctly that if I only consider one overtone, when the wave is half the wavelength away from the end, the string is straight for that overtone?

How do the interference patterns give me the resonance? Are there waves on both ends at the same time and they need to be in sync? What would happen if the string isn't a multiple of half waves long, how would they interfere? Why do the overtones appear in the first place?

 

[sophiecentaur]

How could the string be straight all the time. Do you actually have a proper picture in your mind of what a standing wave looks like over time?

 

[sophiecentaur]

How do the interference patterns give me the resonance? Are there waves on both ends at the same time and they need to be in sync? What would happen if the string isn't a multiple of half waves long, how would they interfere? Why do the overtones appear in the first place?

You are still having problems, aren't you?
You need to appreciate that this all takes time to establish. Enough time for waves of the highest frequency to have traveled back and forth and to have formed an interference pattern.
A stable interference pattern cannot be established for any waves but those allowed modes _ the overtones and fundamental.
When you get down to it, you have a system that can only vibrate at certain frequencies so it can't oscillate any other way if it is left to itself.
Imagine a mass on a spring. There is only one frequency for it to vibrate at once you let it go. Now try adding another mass, hanging on the bottom on another spring. More complex but you could still only get vibrations at a limited set of natural frequencies. The masses will go up and down and together and apart. You could choose any setup to start with and you'd only get those normal modes of vibration. (Violent disturbance so they collide is not allowed, of course) I don't think you could argue with it so far.
A taught string is just another system that can oscillate in certain modes. Once it's been let go it cannot vibrate in any other way but in a combination of these modes. Before you try to argue against the why's and wherefores of traveling waves, nodes and standing waves you just HAVE TO accept the above.
This means there can only be certain waves on the string. These have to have lengths which actually fit into the space between the ends of the string.

I think it's time for you to do more thinking, reading round and not to keep coming back more questions. You need to cut the apron strings.
:-)