Visualizing Standing Wave Created by Plucking a String

[Tranceform]

When plucking a string once on a string instrument (a guitar for example), I can understand that this creates a disturbance in the form of a (actually two) pulses that travels along the string. It makes sense to visualize this.

Let's for simplicitys sake say that the string is 1 meter long parallell to the x-axis and the y-axis is perpendicular to it. The left end is x=0 and one plucks it at x = 10 cm. Then I would see that there is one disturbance traveling to the right of this point and one to the left. The disturbance traveling left will hit the left end (x=0) of the string first, and then it will move to the right but it will look like it's turned upside down in the y-direction. Later the disturbance traveling to the right will hit the right end of the string (x = 1 m) and when it does it will also be turned upside down in the y-direction. Then soon after that the two disturbances will meet and create an interference. Then they will continue along the string in the x-directions they were going before, switching y-direction at the end points and causing interference whenever they meet, but essentially keep going back and forth the same way until damping causes the string to be still again.

That's how I visualize it. If I were to describe the motion, I would say that the disturbances are TRAVELLING along the string.

However, it seems some descriptions of this phenomenon say that rather than a traveling disturbance, this would cause a STANDING wave? This I don't understand. I know what standing waves are and I can see how standing waves occur in a rope which is fixed in one end and you move the other end CONTINUOUSLY upside down, but not for a string being plucked once. I can't see (visualize) why it would happen.

So my question is - when plucking a string, does this actually cause a STANDING wave, and if so, how can that happen? Can someone explain - intuitively/visually - (not mathematically) why this happens? Secondly, if this actually happens, does that mean the description of the phenomenon I gave above is incorrect, since I describe it as a traveling disturbance, rather than standing?

 

[UltrafastPED]

Maybe this helps: http://www.phys.unsw.edu.au/jw/strings.html

 

[Tranceform]

I read what it said in that link but unfortunately my question wasn't explained there. About standing waves it said the following: "An interesting effect occurs if you try to send a simple wave along the string by repeatedly waving one end up and down." This is exactly what I already understood by standing waves, that if you have a rope with one fixed end and CONTINUOSLY pull the other up and down, you get a standing wave. That makes sense.

But consider this: if you only pull the rope up (and down) ONCE, there is NO standing wave formed, only a traveling wave. The standing wave comes from the CONTINUOUS movement of the rope that interfers with the traveling wave. Now moving up (and down) once can be compared to plucking a string ONCE. So my question is, that if you only pluck a string once, how can a standing wave be formed from that?

 

[jtbell]

You do not get a "pure" standing wave with a single wavelength and frequency, like the pictures you see in most books. You get a superposition (linear combination or "sum") of standing waves, from the fundamental frequency on upwards through the harmonics, with different amplitudes. The amplitudes of the individual component standing waves depend on where you pluck the string. If you have a formula that describes the initial configuration of the string (just before you release it), you can calculate these amplitudes using Fourier analysis. When you have the amplitudes, you can predict the details of how the string will move.

The resulting wave is neither a "pure" standing wave nor traveling wave. It has a complicated motion which can "slosh" back and forth along the string.

 

[Tranceform]

I understand what you wrote, but even if there is not just one pure standing wave, how does this explain how these (albeit manifold and impure) standing waves appear in the first place?

I mean when moving a rope (fixed in one end) up and down with my hand it's easy to see the standing waves appear because of the motion I make with my hand creates a pulse that combinates (interfers) with the returning pulse (caused by an earlier hand motion) i.e. the standing wave is caused by superposition of two pulses that meet. But if I only move the rope up and down once and then hold my hand still, this doesn't happen - no standing waves are formed, but only one pulse propagates throughout the rope.

So how is moving a rope up and down once different from plucking a string once? I mean what is it that makes standing waves appear in the string, when they don't appear in the rope?

 

[BOAS]

I understand what you wrote, but even if there is not just one pure standing wave, how does this explain how these (albeit manifold and impure) standing waves appear in the first place?

I mean when moving a rope (fixed in one end) up and down with my hand it's easy to see the standing waves appear because of the motion I make with my hand creates a pulse that combinates (interfers) with the returning pulse (caused by an earlier hand motion) i.e. the standing wave is caused by superposition of two pulses that meet. But if I only move the rope up and down once and then hold my hand still, this doesn't happen - no standing waves are formed, but only one pulse propagates throughout the rope.

So how is moving a rope up and down once different from plucking a string once? I mean what is it that makes standing waves appear in the string, when they don't appear in the rope?

A standing is an oscillation pattern with a stationary outline that results from the superposition of two identical waves traveling in opposite directions.

When you pluck a guitar string in the middle, two waves propagate, one to the left and one to the right. These waves can become superposed and a standing wave is set up.

If you send one pulse down a length of rope, it can reflect, but there isn't another wave for it to interact with.

Edited to add - I'm looking at a very good illustration of this in "Optical Physics" - Ariel, Stephen and Henry Lipson. I'll try to post it here.

 

[Tranceform]

I see. So the standing waves are caused by superposition of the two pulses going to the left and right after the string is plucked. That makes sense, although I imagine that would theoretically only create one single standing wave. I understand in reality you get a combination of harmonics, like jtbell wrote. What is the reason for this exactly? I guess it has to do with the material and composition of the string, as well as how it is plucked? Am I right to assume that for a "perfect" string that was plucked "perfectly" only one single standing wave would occur?

Is the book you are referring to this one http://f3.tiera.ru/2/P_Physics/PE_Electromagnetism/PEo_Optics/Lipson%20A.,%20Lipson%20S.G.,%20Lipson%20H.%20Optical%20Physics%20%284ed.,%20CUP,%202010%29%28ISBN%209780521493451%29%28O%29%28592s%29_PEo_.pdf ? If so on which page number may I find the good illustration you are referring to?

 

[BOAS]

I see. So the standing waves are caused by superposition of the two pulses going to the left and right after the string is plucked. That makes sense, although I imagine that would theoretically only create one single standing wave. I understand in reality you get a combination of harmonics, like jtbell wrote. What is the reason for this exactly? I guess it has to do with the material and composition of the string, as well as how it is plucked? Am I right to assume that for a "perfect" string that was plucked "perfectly" only one single standing wave would occur?

Frankly, you're better off asking someone more knowledgeable than me about that - I've only just started teaching myself from this book for a paper, and the topic is further along than my actual education.

Is the book you are referring to this one http://f3.tiera.ru/2/P_Physics/PE_Electromagnetism/PEo_Optics/Lipson%20A.,%20Lipson%20S.G.,%20Lipson%20H.%20Optical%20Physics%20%284ed.,%20CUP,%202010%29%28ISBN%209780521493451%29%28O%29%28592s%29_PEo_.pdf ? If so on which page number may I find the good illustration you are referring to?

Yes that's the book (sorry, got distracted and never pulled out the image for you). Page 23 has a discussion of non dispersive waves on a guitar string and page 24 has a good diagram and some high speed photography that might shock you...

 

[Tranceform]

That was quite a good description and picture in that book, thanks.

Frankly, you're better off asking someone more knowledgeable than me about that - I've only just started teaching myself from this book for a paper, and the topic is further along than my actual education.

I didn't ask you specifically, but like an open question (that anyone in the forum can answer). I will write it again, hopefully someone else could explain.

So the question is: what is the reason that plucking a string creates such a myriad of harmonics? I guess it has to do with the material and composition of the string, as well as how it is plucked? Am I right to assume that for a "perfect" string that was plucked "perfectly" only one single standing wave would occur?

 

[AlephZero]

So the question is: what is the reason that plucking a string creates such a myriad of harmonics? I guess it has to do with the material and composition of the string, as well as how it is plucked? Am I right to assume that for a "perfect" string that was plucked "perfectly" only one single standing wave would occur?

The large number of harmonics are caused by the shape of the string when it starts vibrating. Usually you can assume the plucking process is very slow compared with the vibration frequency of the string. So it is equivalent to displacing the string at one point, so its shape looks line two sloping straight lines, and then releasing it. The vibrating shapes for the separate harmonics are sine waves. If you calculate the Fourier series that represents the initial shape as an infinite set of sine waves, you get the amplitudes of the different harmonics.

Note that for some plucking points, not every harmonic is excited. For example if you pluck the string at its mid point, you only excite the odd numbered harmonics, not the even numbered ones which have zero amplitude at the mid point. This has practical relevance to musical instruments where the strings are plucked or struck (guitar, piano, etc). Planos are designed so the hammer hits the string at about 1/7 of its length, so the 7th harmonic (which is the first one that sounds discordant) is not excited.

The description of the traveling waves in the OP ignores the effect of damping on the motion. If no energy was lost from the string, the guitar would make no sound, so the damping is rather important! The high frequency harmonics are damped out faster than the low frequencies, so after a short time the shape of the vibrating string is mostly just the fundamental frequency, and the vibration looks like a stationary sine wave, not a traveling triangular shape.

 

[Tranceform]

AlephZero, you explain very well, it makes me want to learn more about the subject.

Note that for some plucking points, not every harmonic is excited. For example if you pluck the string at its mid point, you only excite the odd numbered harmonics, not the even numbered ones which have zero amplitude at the mid point.

That's a very good observation. I understand that when plucking the mid point the harmonics that have zero amplitude there won't be excited there, since that's a point where the two pulses (travelling from the center to the right and left at the same speed) will always meet and superposition each other at the mid point.

When plucking the string at this point, the initial positioning of the string will look like two equally sloping lines, like "^". Am I right to assume that the harmonics are then caused by the fact that these initial "string coordinates" can be described as a Fourier series with many terms, each term representing a harmonic?

In other words, if one was able to initially displace the string in a way that shaped it as a pure sine wave of the fundamental frequency, there wouldn't be any harmonics formed above the first, when letting go of the string?

This has practical relevance to musical instruments where the strings are plucked or struck (guitar, piano, etc). Planos are designed so the hammer hits the string at about 1/7 of its length, so the 7th harmonic (which is the first one that sounds discordant) is not excited.

That's interesting. Is there any explanation why the 7th harmonic is the first to sound discordant? I guess it may have to do with human auditory perception rather than physics, but it would still be interesting to know. I assume other string instruments are formed with the same idea, for example that guitars, cellos, violins have their resonance holes at that point also.

The description of the traveling waves in the OP ignores the effect of damping on the motion. If no energy was lost from the string, the guitar would make no sound, so the damping is rather important! The high frequency harmonics are damped out faster than the low frequencies, so after a short time the shape of the vibrating string is mostly just the fundamental frequency, and the vibration looks like a stationary sine wave, not a traveling triangular shape.

That makes me wonder, how come no damping would make no sound? Wouldn't no damping be equivalent to a tone generator constantly generating the same tone for an infinite time?

 

[AlephZero]

When plucking the string at this point, the initial positioning of the string will look like two equally sloping lines, like "^". Am I right to assume that the harmonics are then caused by the fact that these initial "string coordinates" can be described as a Fourier series with many terms, each term representing a harmonic?

Yes, and the series only contains the odd harmonics. It is something like $\sin x + \frac{1}{9}\sin 3x + \frac{1}{25}\sin 5x + \frac{1}{49}\sin 7x + \dots$ (warning, the signs may be wrong here, but that gives the general idea).

In other words, if one was able to initially displace the string in a way that shaped it as a pure sine wave of the fundamental frequency, there wouldn't be any harmonics formed above the first, when letting go of the string?

Correct, but of course it is hard to do that in real life.

Is there any explanation why the 7th harmonic is the first to sound discordant?

Basically, because it doesn't fit into the standard western music scale. Starting from C, the pitch of the first 6 harmonics match a major chord. C,C,G,C,E,G. The 7th is somewhere between A and B flat. Numbers 8 to 10 are C, D, E, then 11 is somewhere between F and F sharp.

In a single note, you can't usually hear separate pitches of each harmonic, and the 7th doesn't sound "discordant", but if you play chords it does affect the sound.

That makes me wonder, how come no damping would make no sound? Wouldn't no damping be equivalent to a tone generator constantly generating the same tone for an infinite time?

The sound energy has to come from somewhere. A tone generator is converting another energy source (e.g. electricity) into sound energy to produce a continuous sound. When you pluck a string, you store energy in the string when you pluck it, and the energy is slowly converted into sound as the vibration amplitude decays.

 

[Integral]

Generating sound requires energy transfer to the air. Energy loss implies damping. So no damping no sound.