What happens when you pluck a guitar string?

[sophiecentaur]

How can nodes move back and forth? The concept is meaningless by the definition of the word node.

If a mass on a spring will only oscillate at one frequency, why should not a string only oscillate its overtones and fundamental frequency? The only waves that can sustain are those which can meet the boundary conditions. That's basic theory isn't it?

 

[rcgldr]

If a mass on a spring will only oscillate at one frequency ,,,

Couldn't a high speed low mass impact result in a high frequency vibration in the spring (due to momentum of the spring itself)? Also a string vibrates side to side, what would a spring and mass system due if the spring were "plucked" from the side? For example, I'm thinking of a long stretched out "slinky" like spring fixed at both ends and "plucked" or oscillated to induce moving waves into the spring.

How can nodes move back and forth?

Bad terminology on my part, I meant waves moving back and forth, and it was easiest to see this at the "nodes" of those waves. Again, although the high speed video shows these, I don't know how quickly these moving waves would dissipate in the case of a real guitar string.

 

[sophiecentaur]

The mass on spring model assumes a massless spring, naturally. Only the natural resonant frequency can occur.

 

[sophiecentaur]

This strikes me as being in common with QM. In an ideal string system, energy can only be taken in for certain frequencies. When you pluck the string (at least when you just let it go from some displaced position) you are just introducing potential energy and there is no frequency specified. After that, the only modes to be excited are the permitted ones. Is that not good enough?

 

[chingel]

This strikes me as being in common with QM. In an ideal string system, energy can only be taken in for certain frequencies. When you pluck the string (at least when you just let it go from some displaced position) you are just introducing potential energy and there is no frequency specified. After that, the only modes to be excited are the permitted ones. Is that not good enough?

What about the video showing the moving wave on the string?

If I put a mass on a spring and pull it, the strings pulls it up too far up, then it goes down too far etc. The mass on a spring doesn't produce harmonics, does it? Why does the string start producing overtones? Why does a string divide itself into nodes, or does it at all?

 

[mikeph]

What about the video showing the moving wave on the string?

If I put a mass on a spring and pull it, the strings pulls it up too far up, then it goes down too far etc. The mass on a spring doesn't produce harmonics, does it? Why does the string start producing overtones? Why does a string divide itself into nodes, or does it at all?

1) There is no moving wave, it's been said before, the appearance of movement is an illusion generated by the shutter speed of the camera sampling a standing wave at different points in its oscillation

2) The mass of a spring is a totally different system

3) The string doesn't produce overtones, you GIVE the string overtones when you pluck it and leave it to oscillate from an initial displacement resembling a triangle. The triangular wave has a load of harmonic components built into it which decay slowly, because they are all standing waves.

 

[I like Serena]

As I've understood it, waves travel in the string with something like the speed of sound.
At the end of the string the wave reflects and travels back again, and so forth.

If a multiple of the wave length matches the length of the string, it amplifies itself (it resonates).
If it doesn't match an interference pattern is created, which in effect means that the wave cancels itself out.
Some frequencies will dampen out quicker than others.

This effect would be very strong in a single linear string of a homogeneous material.
In a 2D material like the sound box made of non-homogeneous material, this effect would be almost non-existent.Btw, I find it hard to believe that the moving wave in the video is an illusion caused by the shutter speed.
If the wave was properly standing still, regardless of the shutter speed, we should see nice nodes and anti-nodes.

 

[rcgldr]

1) There is no moving wave, it's been said before, the appearance of movement is an illusion generated by the shutter speed of the camera sampling a standing wave at different points in its oscillation.

The article at that site mentions that in the second video, a strobe light was used instead of a moving shutter to eliminate that issue. No shutter was used at all, just the strobe light putting images onto film moving at high speed. The duration of each strobe pulse is short enough that the speed of the film isn't an issue (no significant blurring of the image).

 

[sophiecentaur]

The only difference, in principle, between mass and spring and vibrating string is the number of possible resonances.
Not having seen the movie, I can only comment that the results of temporal subsampling can often be misleading.
If waves "cancel themselves out" over the whole length of the string then there is no energy in them and so they don't exist. What happens in the first period of oscillation of a wave has no real meaning in terms of frequency as the time for frequency analysis is too short for a valid answer. I can only reiterate the fact that the only energy, after the system has settled down, must be in oscillations of 'possible' frequencies. You cannot discuss the concept of frequency in a time interval which is as short as the initial 'attack' time.

 

[rcgldr]

Not having seen the movie

Link to the youtube video:



Another video:

 

[sophiecentaur]

Thanks for that.
No menion of sampling rate, so my reservations still hold. Also. First clip seems to show the string resting on a surface (?). I can't make it out. There seems to be a jangling sound well after the release of the string. Any contact would totally upset the situation. Is the string a string or a long coiled spring? It looks very fat.
The second clip is better, maybe, but only shows the overtones - which I would have expected.
Some basic resonance theory: a resonance takes many cycles to establish itself, just as it takes time to decay. Potential energy when the string is released is shared with KE as parts of the string start to move. This input energy has to couple with something and can go in two ways. Some of the impulse will transfer straight to the sound board via the bridge, giving a non-tuned, percussive attack sound. For a light, yielding sound board, a lot of energy can go this way. The rest of the energy is absorbed into the string resonances which then decay. Why the energy only goes into the overtones might be explained in terms of matching impedances. A non resonant wave will have a much higher impedance (someone may correct me and tell us it's a low impedance but it still represents a bad mismatch) and energy just can't transfer well. The natural string modes present a 'good match' and can extract energy .
I think that this is yet another example of having to look at a phenomenon in terms that may not be intuitive if you want to understand better. If the unfamiliar explanation works then go with it. There is no need to lose sleep over difficulty with the intuitive explanation. Let's face it, they had to bring in QM ideas before atomic theory could progress: totally non-intuitive.

 

[olivermsun]

For those interested, Googling on 'plucked string' turned up this nice exposition by Robert Johns in the March 1977 Physics Teacher: http://homepages.ius.edu/kforinas/ClassRefs/sound/strings/PlluckedstringTPT.pdf" .

 

[sophiecentaur]

That link is a seriously useful piece of work with a definite practical approach. The effect of 'bowing' a string is particularly interesting.

 

[Pythagorean]

Oh - you just opened another can of worms!
Yes; overtones from all musical instruments do not coincide exactly with harmonics of the fundamental. Nice to listen to but not nice to analyse.

It's not just the instruments! Musical theory frequencies differ from mathematical frequencies by a "syntonic comma".

 

[Pythagorean]

As a sidenote, this is why we need tuning procedures like equal temperament, to nudge that error around and make the scale cyclically consistent.

 

[sophiecentaur]

That only works for certain instruments. You don't get a horn with an even tempered scale.
But it's a subjective thing in the end.

 

[Pythagorean]

I imagine modeling a real vibrating brass instrument would be a much more complicated undertaking than a string, in deed!

 

[chingel]

I read the link too, but I still have questions. What causes the harmonics? Does the string actually divide itself into nodes? If the harmonics are sharp, does that mean that the string tries to divide itself for example into two nodes, but since it doesn't bend perfectly at the middle section due to stiffness, the nodes are slightly shorter than theoretical and therefore sharp. Is this a correct understanding?

Does the observation that a plucked string has sharp triangular kinks mean that it also creates pressure waves that are sharp and contain harmonics? Why does the string's shape matter, as long as it is moving back and forth at a consistent speed? If a sharp kink is consistently moving at me and then away, shouldn't it also create a consistent pressure increase and decrease? I mean that for example a loudspeaker can have triangular, conical or all sorts of shapes, what matters is how it moves back and forth and what pressure waves it creates.

 

[sophiecentaur]

If you pluck a string half way along then there is not much chance that a node will form at the mid point - because it has already been displaced. So you might expect a lot of fundamental and some odd harmonics but only very low level even harmonics.

If you held the string in an already sine shaped former then let it go you could ensure a pretty clean fundamental ( or any other overtone that the former was shaped to).

As you say, practical, rather than ideal strings will not behave ideally. It's part of what makes the sound of musical instruments so appealing.

 

[mikeph]

Something I am unsure about. Does energy ever get transferred between frequencies?

eg. I pluck a string in a specific way to only excite two frequencies:
frequency #1 has amplitude 2, and frequency = 2.12934*fundamental frequency
frequency #2 has amplitude 1, and the fundamental frequency.

We know the higher amplitude component of the wave will be damped much faster, but it begins with much more energy, so is there any mechanism to transfer that energy to other frequency components? Neglecting 2nd order effects of the nodes themselves being displaced (neck, bridge, etc.) and then re-oscillating the string.

 

[sophiecentaur]

Something I am unsure about. Does energy ever get transferred between frequencies?

eg. I pluck a string in a specific way to only excite two frequencies:
frequency #1 has amplitude 2, and frequency = 2.12934*fundamental frequency
frequency #2 has amplitude 1, and the fundamental frequency.

We know the higher amplitude component of the wave will be damped much faster, but it begins with much more energy, so is there any mechanism to transfer that energy to other frequency components? Neglecting 2nd order effects of the nodes themselves being displaced (neck, bridge, etc.) and then re-oscillating the string.

I think the problem is that you are associating the plucking of the string with actual frequencies. All you are doing when plucking the string is to displace it, physically (no frequencies or wavelengths involved yet). You then let it go with potential energy that gradually transfers and shares with kinetic energy. The way that this sharing is achieved is up to the system. The oscillations that will take place are then a function of the system. This is very like the time domain (or impulse) response of a electrical filter. What you see there is the response of the system to an infinitely short burst of energy, to which you can't give a meaningful description of spectral content. For a resonant circuit, this will be a decaying sine wave of a frequency given by the LC combination in the circuit.

You could, however, discuss what happens if you try to excite a string with a continuous wave of a single frequency (say with a vibrator, loosely coupled to the string). The amplitude at which it will resonate is a maximum at the string's fundamental, of whichever overtone your tone is at. This, as we have discussed, is because the waves progressive waves on the string happen to interfere consistently along its length, producing nodes and antinodes. Slightly off frequency, there will also be some response and the response will depend on the damping factor (or Q) of the resonator. If you remove the excitation, there is no way that the frequency can suddenly shift because that would violate all sorts of boundary conditions***. All that will happen is that the natural losses in the system will cause the tone to dissipate - once you have removed the off-frequency excitation (which forces some some pattern on the waves on the string) the wavelength of the forced oscillation will not correspond to the length of the string so you will expect to have waves moving from end to end and back again as they gradually dissipate. The rate of decay should be similar to the decay of a natural resonation, I think, because the resistive mechanism will be the same.
*** To get any frequency shift, you need a non-linearity in the system. All the above (and the rest of the thread, mainly) assumes an ideal, linear system.

 

[I like Serena]

This raises an interesting problem.

Suppose you have 2 sound sources that excite the air with a certain frequency.
The second sound source shifted in phase as to oppose the first sound source.
We put them close enough together so that the interference pattern will cancel out the sound almost completely.

Where does the energy go?

 

[sophiecentaur]

It goes somewhere else, other than the specific place where there was cancellation. You just manufactured a Node so there will be an Antinode, somewhere else.

 

[mikeph]

I think the problem is that you are associating the plucking of the string with actual frequencies. All you are doing when plucking the string is to displace it, physically (no frequencies or wavelengths involved yet)

But the initial displacement due to plucking is a function, and can be as the initial conditions when solving the wave equation. And nothing stops us from expressing this function as a Fourier series, right? Then each term in the Fourier series is a part of the overall superposition that you're allowed in the (linear) wave equation, so each frequency should evolve independently? I don't see how they can't, mathematically.

 

[sophiecentaur]

But the initial displacement due to plucking is a function, and can be as the initial conditions when solving the wave equation. And nothing stops us from expressing this function as a Fourier series, right? Then each term in the Fourier series is a part of the overall superposition that you're allowed in the (linear) wave equation, so each frequency should evolve independently? I don't see how they can't, mathematically.

A Fourier analysis of the Shape of the string is not a frequency analysis nor a description of the initial waves on the string. The string is static at the instant of release. You are assuming that you have injected a particular set of waves onto the string (Static and Dynamic conditions), which is an entirely different situation. It is not valid to proceed any further with that argument.
If you were to excite the string in the way that you are implying then it would be reasonable to suggest that all those waves would carry on sloshing about on the string until they decayed due to friction. A different situation entirely, though.

 

[I like Serena]

It goes somewhere else, other than the specific place where there was cancellation. You just manufactured a Node so there will be an Antinode, somewhere else.

So how does that work with noise canceling techniques applied in factories?
Where are the antinodes?
Are they in frequencies that are higher than the human ear can hear or what?

 

[olivermsun]

Something I am unsure about. Does energy ever get transferred between frequencies?
…We know the higher amplitude component of the wave will be damped much faster, but it begins with much more energy, so is there any mechanism to transfer that energy to other frequency components? Neglecting 2nd order effects .

Not in the linearized system, almost by definition. The damping is nonlinear, of course, and will produce a frequency shift.

But the initial displacement due to plucking is a function, and can be as the initial conditions when solving the wave equation. And nothing stops us from expressing this function as a Fourier series, right?

That's my understanding as well.

A Fourier analysis of the Shape of the string is not a frequency analysis nor a description of the initial waves on the string. The string is static at the instant of release. You are assuming that you have injected a particular set of waves onto the string (Static and Dynamic conditions), which is an entirely different situation.

First of all, there's a dispersion relation for waves traveling along the string. In fact, it's a very simple one since the phase speed c is fixed for all waves. That means that the frequency content is completely determined by the wavenumber content, that is, by a Fourier analysis of the spatial shape.

Secondly, the linear wave behavior is for a deflected point y(x) to return directly toward the unperturbed state, y(x) = 0. At the maximum deflection, v(x) = 0. Therefore, the initial conditions f(x,0) = f0(x), v(x,t) = v0(x) = 0 are exactly the right ones for "injecting" the usual pair of wave disturbances f(x+ct)/2, f(x-ct)/2 which propagate in the + and - directions.

 

[olivermsun]

This raises an interesting problem.

Suppose you have 2 sound sources that excite the air with a certain frequency.
The second sound source shifted in phase as to oppose the first sound source.
We put them close enough together so that the interference pattern will cancel out the sound almost completely.

Where does the energy go?

Just imagine a pair of point sources in space which are close but not exactly in the same place. Both radiate symmetrically radially. You set them up so that the phases cancel at some location of interest. What's going to happen elsewhere?

 

[sophiecentaur]

First of all, there's a dispersion relation for waves traveling along the string. In fact, it's a very simple one since the phase speed c is fixed for all waves. That means that the frequency content is completely determined by the wavenumber content, that is, by a Fourier analysis of the spatial shape.

1. Are you absolutely rock solid sure about that?
If you are, then:
2. If you are using the term "wave number", then that assumes you are only dealing with overtones, I think, and not any other waves - which means that you can only get overtones.

 

[olivermsun]

1. Are you absolutely rock solid sure about that?

Pretty sure, yes.

If you are, then:
2. If you are using the term "wave number", then that assumes you are only dealing with overtones, I think, and not any other waves - which means that you can only get overtones.

Wavenumber is usually defined as 2π/wavelength, where the term itself doesn't imply integer multiples or overtones.

In this case, however, the boundary conditions (the clamped ends of the string at x = 0, L such that y(0, t) = y(L, t) = 0) along with the dispersion relation do make them overtones, yes.