What happens when you pluck a guitar string?

[physickkksss]

Hi guys, I know the basics of waves and standing waves, but I am trying to understand what exactly happens when you pluck a guitar string...

So, due to standing waves, a string that is clamped down on both ends needs to vibrate in one of its resonant frequencies:

f = (harmonic number)* v/2L


But I've read that when you pluck a guitar string, the sound produced is a combination of ALL of the fundamental frequencies.

That does not make sense to me. When you pluck ONE string, can't it only vibrate at ONE frequency, and thus produce a pitch of that frequency?

 

[olivermsun]

Think this one may have been discussed in the past, but basically think of the disturbance you make when you pluck the string. Instead of a sinusoid, it's more of a kink, right? So, it contains lots of sinusoidal components; the ones which survive are the fundamental wavelength of the string and integer multiples which also "fit" on the string, thereby producing the harmonics.

 

[physickkksss]

Thanks for the reply

So, it contains lots of sinusoidal components

Ok I understand this part...one actually imparts many different frequencies on the string, but it can only vibrate at its fundamental harmonic (or a multiple)

the ones which survive are the fundamental wavelength of the string and integer multiples which also "fit" on the string, thereby producing the harmonics.

This is the part I don't get. Cant ONE string only vibrate at ONE frequency?

So shouldn't the string settle at one of its harmonic frequencies and then produce a pitch at that frequency? How can it produce all of them at once?

 

[bp_psy]

This is the part I don't get. Cant ONE string only vibrate at ONE frequency?

So shouldn't the string settle at one of its harmonic frequencies and then produce a pitch at that frequency? How can it produce all of them at once?

A string on a guitar can be considered as a series of damped oscillators. Each of these oscillators has a natural frequency. If you want to make a oscillator vibrate continuously at a frequency other than its natural frequency you must supply it with with energy ,this is called driven oscillator.If you stop the driving force the oscillator it will eventually damp the driving frequency and vibrate at its natural frequency and its harmonics until it completely stops .For an oscillator the natural frequency is given by its constant physical properties like spring constant and mass, inductance and capacitance.When you pluck a string you make all parts of the guitar vibrate at multiple frequencies but since the pluck originates from a string that is composed from a series of oscillators that have the same natural frequency, most of the frequencies will be damped quickly.What will have the most energy when the vibration will final propagate trough the whole guitar is the note you plucked and its harmonics. This is pretty much what makes a musical instrument possible.
You could make a string vibrate at just some frequency but just by driving it with some single frequency driving force. With a pluck of a string this can't be achieved because what you do is make the string vibrate at multiple frequencies but it chooses at what frequencies to vibrate .
For the question "How can it produce all of them at once?" i recommend you play with this :
http://www.falstad.com/loadedstring/

 

[I like Serena]

I believe you can get a string to vibrate at (almost) exactly one frequency if you enforce it by for instance connecting the string to an electronic sinusoidal wave generator.

As for how multiple harmonics can be in a string at once, look for instance at this picture:
http://www.coe.drexel.edu/ret/personalsites/2007/Dirnbach/curriculum_files//harmonics.png

The various waves are superimposed on each other.

 

[physickkksss]

cool, those replies helped a lot

Thanks :)

 

[rcgldr]

It might help to visualize a string with a combination of fundamental and double frequency. The double frequency on it's own would have a staionary node at the mid point of the string, but combined with the fundamental frequency, that node oscillates at the fundamental frequency. A guitar player can place his finger lightly on the mid point of the string to prevent node movement and pluck it at about 1/4 the way down the string to get a mostly double frequency sound.

Although the overall output of a guitar string is a composite waveform, the entire string movement does not follow that composite waveform. Instead the actual string movement at any point on the string is affected by the nearness of the nodes and peaks related to the frequencies produced by the string. From a 2d side view at any moment in time, you would see peaks and valleys along the length of the string due to the combined frequencies.

The soundboard of the guitar that actually produces the sound, also has pockets of peaks and valleys related to the frequency being produce.

Link to a video someone made of guitar string movement, the first one is affected by the rotating shutter, so the second one is a better example.

http://createdigitalmusic.com/2011/...l-guitar-string-movement-and-iphone-shutters/

 

[I like Serena]

Link to a video someone made of guitar string movement, the first one is affected by the rotating shutter, so the second one is a better example.

http://createdigitalmusic.com/2011/...l-guitar-string-movement-and-iphone-shutters/

Nice video!
It's one thing to more or less know how things work, but it's quite another to actually see it!

 

[physickkksss]

Also I wanted to follow up about the soundbox...

I understand that the string itself does not move a lot of air, so you cannot hear it
It transfers its vibrations to the soundboard, which moves much more air
This leaves the sound hole, and we get an amplified sound

My question is that wouldn't the wood need to have the same harmonic frequencies as the string? (ie. a tuning fork can only transfer its sound by resonance to a tuning fork with the same natural frequency).

If the wood is indeed selected to have the same natural frequency of the string, then how does it resonate with all of the different strings on the guitar?

 

[I like Serena]

A string is designed to have a specific resonance frequency determined by its length.

A soundboard is designed to not have a resonance frequency.
It will vibrate with any frequency.

Btw, the majority of the sound does not leave the soundboard by the sound hole.

 

[olivermsun]

I don't know how a guitar works specifically, but I have seen analysis of the body of stringed instruments such as the violin, and actually there are many vibrating modes. Here the varying thicknesses of the wood, the shape of the body, and the cavity contained in the body, allow much more complicated resonances than the resonances in the string. So, I would guess that the guitar soundboard and body also have many distinct resonances rather than none.

 

[physickkksss]

A string is designed to have a specific resonance frequency determined by its length.

A soundboard is designed to not have a resonance frequency.
It will vibrate with any frequency.

Btw, the majority of the sound does not leave the soundboard by the sound hole.

How does that fit with the statement:
all objects have a natural frequency or set of frequencies at which they vibrate when struck, plucked, strummed or somehow disturbed

Maybe its because normally objects quickly damp out frequencies other than its natural frequency, so the soundbox just does not damp out all those other frequencies too much?

 

[I like Serena]

How does that fit with the statement:
all objects have a natural frequency or set of frequencies at which they vibrate when struck, plucked, strummed or somehow disturbed

Maybe its because normally objects quickly damp out frequencies other than its natural frequency, so the soundbox just does not damp out all those other frequencies too much?

Yeah, well, if you knock on the soundboard it will indeed vibrate with a characteristic set of frequencies.
But it can also vibrate at other frequencies without really dampening out.

However, if you make a string vibrate with a frequency that does not match its length, it will dampen out immediately.

 

[physickkksss]

Yeah so I guess that must be it...

The soundboard, like any other object, does have its own natural frequency that it is prone to vibrate at. However, it must have the quality of not quickly dampening out other frequences, as would most other objects. That would make it good at transferring the vibrational frequencies of all the guitar strings.

 

[chingel]

I think the soundboard is just carrying the sound waves and vibrating along with it. Just like air can vibrate at all sorts of frequencies and carry all sorts of sound waves. The string is against the bridge, which is against the soundboard, the string gives it's vibrations to the soundboard, which carries them and vibrates along.

Why does a string dampen out immediately if I make it vibrate at a frequency that does not match it's length?

 

[sophiecentaur]

Why does a string dampen out immediately if I make it vibrate at a frequency that does not match it's length?

Say you excite the string with a nearby loudspeaker. The energy from the loudspeaker, whatever frequency, will hit the string and make it vibrate and it will dissipate. However, if the frequency of sound from the LS happens to be at a suitable frequency, the vibrations, being in step, will add up (build up) and cause the string to vibrate with a high amplitude because some of the energy is being absorbed into the resonant system. Once enough energy has been absorbed onto the system, it will, as before, dissipate at the rate it is being supplied and the string will be vibrating at the maximum amplitude. The resonant system can only store energy at certain frequencies, which is why it only 'responds' to those frequencies. A string on a solid bodied guitar will have fewer losses than on an acoustic guitar so the resonance will be more 'frequency selective' and the note will sustain for longer when the excitation is removed and the energy gradually dissipates. (Hence the Brian May effect)

 

[Pythagorean]

See "harmonic series (music)" as opposed to the mathematical concept.

 

[sophiecentaur]

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.

 

[rcgldr]

Why does a string dampen out immediately if I make it vibrate at a frequency that does not match it's length?

I'm not sure it does. The main difference in the case of a frequency that is not an integer multiple of the fundamental frequency is that the nodes travel back and forth along the string, as opposed to being fixed in place. The harmonic frequencies will sustain longer due to resonance, but non-harmonic frequencies will decay depending on how much the string retains energy. I'm not aware of any additional dampening factor for non-harmonic frequencies.

The soundboard is being driven by the strings, so it's harmonics aren't as much of an issue, as long as they don't over amplify a particular frequency A soundboard or a speaker can handle a range of frequencies, and similar to the string, non-harmonic frequencies result in the hills and valleys moving around a 2d plane, while harmonic frequencies result are mostly fixed in place.

 

[sophiecentaur]

Why does a string dampen out immediately if I make it vibrate at a frequency that does not match it's length?

How would you make it do that in the first place? If you had a mechanical vibrator to drive the string, then it's hard to say what it would do if you suddenly removed the drive. I imagine that the string would start to vibrate at its fundamental frequency (plus harmonics) because it would be as if you had just plucked it. It would have no memory of how it got to the shape that it was displaced by the drive and impulses would move along the string, redistributing the energy (a bit of BS there, I'm afraid) as the waves settled down such that the remaining oscillations were just the natural ones - which would decay relatively slowly because of the energy stored in the resonance. The other components of the energy in the system would release in a 'snap'. which is also what you get when you first pluck a string.
So, unless you are quoting an actual experiment (?), I suggest that your thought experiment wouldn't go as you say. If you excited the string off-frequency with a weak coupling, then a resonance just wouldn't have built up in the first place.

 

[sophiecentaur]

The soundboard is being driven by the strings, so it's harmonics aren't as much of an issue, as long as they don't over amplify a particular frequency A soundboard or a speaker can handle a range of frequencies, and similar to the string, non-harmonic frequencies result in the hills and valleys moving around a 2d plane, while harmonic frequencies result are mostly fixed in place.

Yes, the soundboard is, essentially, an impedance matching unit. A string, vibrating in the air, held between two rigid points, will not produce much sound but keep vibrating for ages because it can't get rid of the energy. The soundboard, having a large area, can move a lot of air and couple the energy away as sound. Ideally, a soundboard would not exhibit any resonance (a wideband match) but, as with many systems, you can get a better coupling if you have some degree of tuning (as with some loudspeaker cabinets). I think that accounts for the general 'waisted' shape of many stringed instrument bodies and the sizes chosen, which are appropriate for the range of pitch of the instrument.

 

[rcgldr]

Why does a string dampen out immediately if I make it vibrate at a frequency that does not match it's length?

How would you make it do that in the first place?

By plucking it. The initial shape is somewhat triangular, which would represent the sum of a continuous series of frequencies. Looking at that strobe light video of a string, in some of the sequences, a moving wave can be seen, and a moving wave would correspond to a frequency that is not a multiple of the fundamental frequency. I don't know how quickly such a wave would be dampened out, or if it would last long enough to produce a significant audio effect on a guitar.

 

[sophiecentaur]

But a shape is not a set of frequencies. It is just a displacement. The string is going to move towards its unstretched position.
And there is initially just energy. When you stretch a spring, it will oscillate at just one frequency. that's the only mode that the initial energy can go into. A similar thing must happen with a string. It can only vibrate at certain frequencies which are the natural frequency and overtones so that's where the energy goes.
I imagine that's not the answer you wanted but I'll have to think deeper for another one in your terms.

 

[mikeph]

Looking at that strobe light video of a string, in some of the sequences, a moving wave can be seen, and a moving wave would correspond to a frequency that is not a multiple of the fundamental frequency.

The moving wave is an illusion caused by the shutter speed of the camera. It's sampling the displacement of a standing wave at regular intervals in time which makes it look like the wave is moving, but it's not.

 

[olivermsun]

The moving wave is an illusion caused by the shutter speed of the camera. It's sampling the displacement of a standing wave at regular intervals in time which makes it look like the wave is moving, but it's not.

How about this one: http://www.youtube.com/watch?v=_X72on6CSL0", which should be more like the second video at the page that was linked before.

 

[chingel]

How would you make it do that in the first place? If you had a mechanical vibrator to drive the string, then it's hard to say what it would do if you suddenly removed the drive. I imagine that the string would start to vibrate at its fundamental frequency (plus harmonics) because it would be as if you had just plucked it. It would have no memory of how it got to the shape that it was displaced by the drive and impulses would move along the string, redistributing the energy (a bit of BS there, I'm afraid) as the waves settled down such that the remaining oscillations were just the natural ones - which would decay relatively slowly because of the energy stored in the resonance. The other components of the energy in the system would release in a 'snap'. which is also what you get when you first pluck a string.
So, unless you are quoting an actual experiment (?), I suggest that your thought experiment wouldn't go as you say. If you excited the string off-frequency with a weak coupling, then a resonance just wouldn't have built up in the first place.

I got the impression that if you pluck a string, it will start vibrating at various frequencies and the fundamental and the harmonics survive. Is this a wrong impression?

I understood your argument about exciting the string with a loudspeaker and that only at the resonant frequency the energy in the string would build up. But I don't understand how would an frequency that is not a multiple of the fundamental get damped, because it is not in resonance and the nodes move back and forth. How does it work?

 

[olivermsun]

The string fundamental frequency is set by the properties of the string and the boundary conditions: at the ends of the length L, the string is held by supports and can't move. Since the traveling waves on the string go at some fixed speed c, and the frequency is just the number of wiggles that go by per time, the frequencies correspond to one (half) sine wave, two (half) sine waves, etc...

When you pluck a string, you are giving the string an initial condition which has some nonzero shape (and extra tension). No matter how complicated this shape might be, it still has to fulfill the boundary conditions at the ends of the string. Therefore, every sinusoidal component (and hence frequency) corresponding to the shape turns out also to be a harmonic.

The nodes moving back and forth thing is not a problem, since you can see from some of the videos above that a plucked string typically does not exhibit classical standing wave patterns and nodes.

 

[chingel]

I can understand that when you change the string's shape and then release it, the frequency is determined basically by how fast it goes down the other side and comes back again, and that is determined by the mass of the string and the tension pulling it down or up.

But the harmonics I don't really understand. Does a vibrating string divide itself into nodes or not? If not, what is creating the harmonics? If the moving wave means that it's frequency is not a multiple of the fundamental, does it mean the string will vibrate at all sorts of frequencies that are not multiples of the fundamental when it is plucked? Why do the non-harmonic frequencies decay faster?

 

[mikeph]

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).

 

[olivermsun]

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

Because you don't typically pluck the string in the middle. Plucking close to one end results as usual in a pair of traveling wave solutions moving in opposite directions, but both of them start "tilted" to the same side.

Did you watch the movies done with proper high-speed photography?

 

[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.