What happens when you pluck a guitar string?

[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.

 

[sophiecentaur]

So one would only expect resonances corresponding to overtones from a plucked string?
(Which was my original thought about resonant systems in general).

 

[thegreenlaser]

I'm actually working on a project related to this right now, so I'm hoping I can help here.

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?

No, it doesn't actually divide itself into nodes like that. Ideally, it will form into a sum of those different waves. You won't actually observe most, if any, of the nodes present, though in a way they're there, because they're contributing to the sum. If you haven't taken any Fourier analysis, at least read a basic intro to Fourier series, because that's essentially what this is. You can take nicely spaced frequencies like the ones that show up on a guitar string, multiply them by various amounts and add them together and end up with a really messy-looking signal.

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.

The initial shape of the string does indeed affect the frequency output, but you can't just find the Fourier series of the initial shape to find the Fourier series of the output. i.e. A triangle-shaped string does not guarantee a triangle-shaped output. For example, if you start an ideal string as a triangular wave and let it go, at any given time, the string will be in some sort of triangular wave shape. You can watch the triangular wave move back and forth. However, if you sample the displacement at one point (say, a third of the way along the string), you'll actually find that the output is more like a rectangular wave, and the proportions of that rectangular wave will actually change depending on where you sample along the string. Closer to the edges, there's more emphasis on high frequencies, which is why the bridge pickup on an electric guitar sounds brighter than the neck pickup.

There is no moving wave. How can the wave possibly move when there are always two nodes either end?

When you hold a string ready to be plucked, you have a piecewise continuous function (displacement vs distance from bridge). That function can be expressed using a Fourier series (imagining it is periodic beyond either node). Each frequency has an amplitude.

When you let go, the amplitudes will evolve. Within milliseconds (depending on the length of the string and the speed of sound within the material) most of the frequencies will decay, leaving the harmonics. They decay far slower, creating the sound.

Try picking up a guitar and plucking it at an extreme point. You can hear a "noise" before you hear the note, and the note is tinny. This is because the amplitudes of the non-resonant sinusoids making up the initial displacement are high, so a lot of energy is dissipated instantly. And then, a lot more energy is put into the 5th,6th,... harmonics which have a higher frequency, making it sound tinny.

If you pluck the string precisely in the centre, you don't hear that noise, and you hear a much purer sound. That's because more of the energy in the pluck is put into the fundamental frequency and not "wasted" standing waves (not harmonics) nor higher harmonics (in fact the 2nd, 4th, etc. harmonics will contain no energy).

I'm not so sure that this is completely accurate. In any linear string model, the only possible solutions will be linear combinations of frequencies, all of which are multiples of the fundamental frequency. I think what you're describing actually comes from frequency-dependent loss: A guitar string will dissipate high frequencies faster than low frequencies. When you first pluck a string, you're getting a much more even spread of frequencies, but they're still all multiples of the fundamental frequency of the string. Very quickly, though, the higher frequencies die off and the ones closer to the fundamental start to dominate, making the tone sound more pure. The reason that plucking in the middle rather than on the edges sounds 'purer' is similar: there's more emphasis on the frequencies close to the fundamental frequency. Again, you don't actually need to have non-fundamental frequencies in order to explain this behavior. The more emphasis you have on high frequencies far away from the fundamental, the more tinny and messy your sound is going to be. In a real string, there will obviously be some non-linearities, but the frequency-dependent loss is mostly what contributes to that initial distorted 'twang.' Non-linearities tend to mostly come into play when you pluck a string really hard or something.

 

[mikeph]

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.

I really don't understand why not.

Any wave at maximum displacement has zero velocity at the instant of release. The shape of the string when I pluck it is at maximum displacement, so the fact that it's static is not a problem.

 

[chingel]

No, it doesn't actually divide itself into nodes like that. Ideally, it will form into a sum of those different waves. You won't actually observe most, if any, of the nodes present, though in a way they're there, because they're contributing to the sum. If you haven't taken any Fourier analysis, at least read a basic intro to Fourier series, because that's essentially what this is. You can take nicely spaced frequencies like the ones that show up on a guitar string, multiply them by various amounts and add them together and end up with a really messy-looking signal.

The initial shape of the string does indeed affect the frequency output, but you can't just find the Fourier series of the initial shape to find the Fourier series of the output. i.e. A triangle-shaped string does not guarantee a triangle-shaped output. For example, if you start an ideal string as a triangular wave and let it go, at any given time, the string will be in some sort of triangular wave shape. You can watch the triangular wave move back and forth. However, if you sample the displacement at one point (say, a third of the way along the string), you'll actually find that the output is more like a rectangular wave, and the proportions of that rectangular wave will actually change depending on where you sample along the string. Closer to the edges, there's more emphasis on high frequencies, which is why the bridge pickup on an electric guitar sounds brighter than the neck pickup.

Do I understand it correctly, if I say that the string does divide itself into two parts, three parts, four parts etc, and these parts vibrate and the sum of these movements of the string (since they are vibrating at the same time and in some places add, in some places interfere) creates the overall triangular kinky wave seen on the string, that moves back and forth on the string and was discussed in the pdf linked beforehand?

Does the shape of displacement on a particular point on the string correspond to the shape of sound waves created?

 

[sophiecentaur]

I really don't understand why not.

Any wave at maximum displacement has zero velocity at the instant of release. The shape of the string when I pluck it is at maximum displacement, so the fact that it's static is not a problem.

I see what you are saying but why should one assume that the whole length of the string would be stationary at the same time? For instance, the different overtones are not at rest at the same time or place - the phases of any two could (would?) not be the same - hence one would be moving whilst the other is stationary. The string being stationary would require all the overtones to be to be at a suitable relative phase to each other to achieve a net KE of zero. I find that hard to believe - despite the justification given earlier. I appreciate that what I say would apply to the majority of cases but that the condition of zero KE need only be met for one particular situation.

 

[olivermsun]

...why should one assume that the whole length of the string would be stationary at the same time?...The string being stationary would require all the overtones to be to be at a suitable relative phase to each other to achieve a net KE of zero.

At the moment of initial release, the string is not moving and hence KE = 0. All the energy in the system resides in PE (the stretch of the string). Immediately after t = 0, the displacement at every point on the string decreases toward equilibrium, so that PE decreases and KE increases. The phases of all Fourier modes are necessarily such that the initial condition is fulfilled.

 

[sophiecentaur]

That makes sense. You actually impress those initial conditions so that defines the relative levels and phases of the normal modes. Fair enough.

 

[thegreenlaser]

Do I understand it correctly, if I say that the string does divide itself into two parts, three parts, four parts etc, and these parts vibrate and the sum of these movements of the string (since they are vibrating at the same time and in some places add, in some places interfere) creates the overall triangular kinky wave seen on the string, that moves back and forth on the string and was discussed in the pdf linked beforehand?

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. So if f(x,t) is the displacement of the string at position x at time t, then we can say, for any t:
$$f(x,t)=\sum_{k=0}^{\infty} A_k(t) \cdot \sin (k\omega_0 x)$$
Where A_k are some functions of time, which we can find with Fourier analysis. In fact, the solution to the simple 1D wave equation comes exactly in a form like this. However, notice that the sin term is a function of x, and not t. This means that any time-variance has to come from the A_k(t) term. It is these terms that will determine the frequency output at any point x, not the sine terms, yet it's the sine terms that generate the shape profile. From what I've said here, there's not even a guarantee that the A_k(t) oscillate at predictable frequencies, yet the sine terms have predictable frequencies. It so happens that they do in most cases, but just looking at the shape of the string doesn't guarantee that. It's a much more finicky relationship. So yes, you can always break the string shape down into a sum of sinusoid-shaped strings, but this isn't as closely related to the frequency output as it might seem. What determines the frequency output is the way in which these 'sinusoidal strings' oscillate back and forth, which, while related, is a different story.

In response to your question about the displacement of the string at one point being the output: that is actually a fairly reasonable assumption, especially in the case of an electric guitar. In the case of acoustic instruments like an acoustic guitar or piano, such an assumption can't be made as easily, but even then, just summing the displacements of a couple points on the string can be a pretty good approximation. In that case, actually, a Fourier analysis of the shape can be useful because you can look at the sinusoids in the spatial domain to see where each frequency component is strongest (peaks of the spatial sinusoid) and weakest (nodes of the spatial sinusoid). Analysis like that explains why if you listen at the edge of a string, it sounds more 'tinny' than if you listen at the middle. However, this is only helping you find information about how those frequencies change as you move along the string, and not about the frequencies themselves.

At the moment of initial release, the string is not moving and hence KE = 0. All the energy in the system resides in PE (the stretch of the string). Immediately after t = 0, the displacement at every point on the string decreases toward equilibrium, so that PE decreases and KE increases. The phases of all Fourier modes are necessarily such that the initial condition is fulfilled.

(I'm kind of responding to the conversation you've been having with sophiecentaur rather than this particular post) The statement that a Fourier analysis of the string shape does not correspond to an analysis of the frequency output is true, as I've explained above. When you turn the shape at time t into a Fourier series, you're finding one particular set of A_k values. You can make arguments based on energy analysis and analysis of how the string should respond to draw conclusions about the time-dependence of the A_k(t), however, just from turning the shape profile at anyone time into a Fourier series, you cannot glean any information about the actual frequency output. It's true that the initial shape/velocity will uniquely determine the output, but that's less to do with the spatial frequency characteristic of the shape and more to do with the fact that those are boundary conditions for the PDE model being used. You must consider other factors in order to get information about the frequency output, as it's not a direct relationship. Particularly if you add non-linearities to the model, Fourier analysis of the shape can become almost meaningless. In that case, you can start with a triangular shape, which can easily be expressed as a sum of sine terms, and yet you might find that the frequency content of the output shifts over time, and the string may go in and out of tune. There would be no reason to believe that just from the shape profile, and yet if you pluck a guitar string really hard, you can hear this frequency distortion.

 

[olivermsun]

...if f(x,t) is the displacement of the string at position x at time t, then we can say, for any t:
$$f(x,t)=\sum_{k=0}^{\infty} A_k(t) \cdot \sin (k\omega_0 x)$$
Where A_k are some functions of time, which we can find with Fourier analysis. In fact, the solution to the simple 1D wave equation comes exactly in a form like this. However, notice that the sin term is a function of x, and not t. This means that any time-variance has to come from the A_k(t) term. It is these terms that will determine the frequency output at any point x, not the sine terms, yet it's the sine terms that generate the shape profile.

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. Put another way, the signal detected at a fixed point over time should be exactly the 'incoming' spatial wave shape(s). A similar principle applies for 'analogue' recording media such as LPs and magnetic tapes.

In response to your question about the displacement of the string at one point being the output: that is actually a fairly reasonable assumption, especially in the case of an electric guitar. In the case of acoustic instruments like an acoustic guitar or piano, such an assumption can't be made as easily, but even then, just summing the displacements of a couple points on the string can be a pretty good approximation.

For an acoustic instrument, I would think that most of the 'output' is actually whatever goes through the bridge to the soundboard(s) of the instrument.

You must consider other factors in order to get information about the frequency output, as it's not a direct relationship. Particularly if you add non-linearities to the model, Fourier analysis of the shape can become almost meaningless.

Most of the preceding discussion has been assuming a linear model of the string, but I'd argue that most real-world stringed instruments have behavior which is at least recognizably close to linear -- otherwise, they would be terribly hard to play.

 

[thegreenlaser]

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. Put another way, the signal detected at a fixed point over time should be exactly the 'incoming' spatial wave shape(s). A similar principle applies for 'analogue' recording media such as LPs and magnetic tapes.

I agree, in the case of a perfect string. Does a similar relationship still hold in a linear model of a string with stiffness and loss terms? I'll admit, I don't know enough about PDE's to know whether it would or not. 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 would have to know how the Fourier transform of the shape changes with time, and my gut feeling is that in the case of a more complicated PDE model, you wouldn't be able to predict that with very much ease at all. That would be in contrast with the ideal string case, where the manner in which the string moves is indeed quite easy to predict based on the initial position.

For an acoustic instrument, I would think that most of the 'output' is actually whatever goes through the bridge to the soundboard(s) of the instrument.

Point taken. Though I think it's still reasonable to talk about the displacement of the string at a single point as being the 'output.' It's crude, but it's not so different from the real sound that it's useless.

Most of the preceding discussion has been assuming a linear model of the string, but I'd argue that most real-world stringed instruments have behavior which is at least recognizably close to linear -- otherwise, they would be terribly hard to play.

Fair enough.