# I Are harmonics "real" in a vibrating string?

#### dRic2

Gold Member
Summary
When solving the wave equation for an oscillating string we look for a solution that is the linear combination/superposition of an infinite serie of "harmonics", i.e. sine and cosine functions (classical Fourier series). But is that a mere mathematical trick or do these states have physical meaning ?
This question reminds me of the interpretation of the "wave packet" in QM for a free particle moving freely in the whole space; but in QM it is obvious that plane waves can't be of physical meaning. Now consider a guitar string (fixed at both ends) and suppose the solution to the wave equation is some function $u(x, t)$. Assuming $u(x, t)$ to be a "nice" function, a Fourier serie representation could look like this $u(x,t) = \sum_{n} c_n \sin(\frac { \pi n}{L} x)$. From Fourier's work we know that ${c_n}$ is a monotonically (fast) decreasing succession, so we hear mainly the first (principal) harmonic which define the note we are hearing. The other contributions give a sort of "background" which characterizes the particular sound of an instrument (in this case a guitar). Up to this point it doesn't really matter if each harmonic really exists or if it is just a mathematical representation, but now consider the following technique: string harmonics. For those of you who are not familiar with this beautiful technique it goes like this (there are tones of different ways of doing it, if you're curious just refer to the wikipedia page, I'll use a specific version which suits my problem better):
- play an open string, for example the low E
- gently put you finger on the metal strip corresponding to the 5th, 7th, or 12th fret. Be careful, you don't press the fret, you simply lay your finger on the metal strip!
- you'll hear a very high pitched tone, much higher then the tone you would expect for the corresponding fret. (I'll try to record something or post a video if I have the time... it's 2:28 am here now )

Explanation
Basically the most common explanation (also reported by wikipedia) is that, by doing that "movement" with your finger you are muting all the various harmonics of the vibrating string except those having a node at that particular point. Since all the higher harmonics have higher frequencies that the fundamental one the high pitched tone is no more surprising. You can also do the math (it's super easy) and check that this explanation correctly predicts the right note (frequency). For instance the harmonic on the fifth fret corresponds to play the open string 2 octave higher (check with the math or with your ear... it's fun!)

Then my question is: if you "mute" all the harmonics except for those having a node at that particular point, it means that they really do exist and that they're not just a mathematical trick. On the other hand this seems very strange to me because the string oscillates according to the function $u(x, t)$ which is something $per sé$, something real and physically observable: it is the "true" motion of the string.

Sorry if it took me so long to get to my question, I hope I didn't bother you too much. If you'll find my question too silly (it might be...) at least I hope you'll find this as an interesting fact.

Thanks
Ric

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#### scottdave

Homework Helper
This is visible with a string bass (especially upright bass) where the frequencies are lower (slower vibrations) and larger amplitude (string deviates farther from the center). If the string is bowed rather than plucked the effect can be sustained.

#### dRic2

Gold Member
Yes, sorry if the title is misleading. I don't doubt the existence of the phenomenon: when I ask if "harmonics" are real I am referring to each term in the series expansion of the solution

#### sophiecentaur

Gold Member
Then my question is: if you "mute" all the harmonics except for those having a node at that particular point, it means that they really do exist
The motion of the string 'is what it is', however it is described. If you start with the equation of motion of elemental parts of the string and impose the boundary conditions (length etc.), then you can solve the equation and predict the motion of the string. This can work pretty well in practice and it's common to express this solution in terms of the frequency spectrum of the oscillations (frequency domain). But that is no more fundamental than describing the motion in terms of the shape of the string in time (time / spatial domain).
I mentioned "boundary conditions" above and they are the things that determine the result. Firm, massive supports at each end ensure no displacement at any time. Placing a finger at a place in the string adds another boundary. If it's in a suitable place then some of the modes of the string can be sustained and others not. To return to your question about whether these modes "really do exist", the possible modes are determined by the boundary conditions. Those modes don't exist when the boundary conditions don't support them.

Guitar tuning: People often use "harmonics' to get the tuning right between the different strings. This works ok for a start but good players usually tiffle the tuning afterwards to make chords to sound just right. That's partly because of the frequency difference between harmonics and overtones and it ends up as a compromise.

Incidentally, people tend to talk in terms of "harmonics" of vibrating physical systems but it's only in ideal systems, like an ideal string that the higher frequencies are actually harmonically related. The modes I referred to are overtones. Sometimes they approximate to harmonics but they may be way off from harmonic frequencies in many musical instruments. It's worth bearing this in mind when you try to imagine what's going on in a Brass wind instrument in which the waveform is certainly not full of 'harmonics'. (This is a personal gripe of mine and I keep trying to spread the gospel about it.)

#### ZapperZ

Staff Emeritus
2018 Award
Make a measurement and see if they are "real".

Zz.

#### sophiecentaur

Gold Member
Make a measurement and see if they are "real".

Zz.
Good article. The result curves for the 'higher harmonics' seem to support my objection to calling the modes harmonics.
I guess that would be reasonably easy to achieve for oneself with spectrum analysis on a PC these days. But the behaviour of a plucked string is very transitory and it could be hard to identify what's going on. Swept frequency analysis could give an answer, though, but you would need to excite the string with a tone input.

#### dRic2

Gold Member
I mentioned "boundary conditions" above and they are the things that determine the result. Firm, massive supports at each end ensure no displacement at any time. Placing a finger at a place in the string adds another boundary. If it's in a suitable place then some of the modes of the string can be sustained and others not. To return to your question about whether these modes "really do exist", the possible modes are determined by the boundary conditions. Those modes don't exist when the boundary conditions don't support them.
Nice. I didn't think about the change in the boundary condition because I assumed that letting the string vibrate "before" adding the finger implied that the string would be subjected to the same boundary condition. Wrong, of course. When I add my finger the boundary condition changes to u(x=fret,t′)=0u(x=fret,t′)=0 and the problem is solved. I'm the first one who "hates" to think bout this quasi-"philosophical" questions, but I really couldn't get my head around it.

The modes I referred to are overtones. Sometimes they approximate to harmonics but they may be way off from harmonic frequencies in many musical instruments.
Yeah, that's the point of equal temperament. The only "true" harmonic we have in a string instrument I think is the octave (which doubles the frequencies by halving the length) but all the others are approximation. I suppose in a fretless instrument (like a violin) you could try to get better harmonics, but, in the end, it all depends on how you define the notes of the scale in the first place.

Thanks to everyone! :)

#### sophiecentaur

Gold Member
The only "true" harmonic we have in a string instrument I think is the octave
Even that assumes the string has no width. There will new an end effect which is different for all wavelengths because the terminantion is not zero length but a different fraction of a wavelength.

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#### dRic2

Gold Member
There will new an end effect which is different for all wavelengths because the terminantion is not zero length but a different fraction of eye wavelength.
Sorry, but my English is failing me here... I don't quite understand the meaning of the sentence :(

Anyway, what I meant is the the octave is the most accurate of all the harmonics you might hope to get. I'm sure there will be more than one source of inaccuracy.

#### sophiecentaur

Gold Member
Sorry, but my English is failing me here... I don't quite understand the meaning of the sentence :(

Anyway, what I meant is the the octave is the most accurate of all the harmonics you might hope to get. I'm sure there will be more than one source of inaccuracy.
Sorry. Autospell’s fault and I didn’t check.
There will be a so-called end effect because, at the end of a thick string, there is not a true node. There is a short length of the string, inside which is carrying the wave. The mode in the section is different and the speed will vary with frequency. Strings are nearest to perfection. Also, of course, a musical instrument has to have movement of the bridge or it will be silent.

#### rcgldr

Homework Helper
Adding to the complexity here is that there can be traveling waves along a string that are not true harmonics. These would normally quickly dampen out, except as noted above, the bridge vibrates a little, and could be feeding in a frequency into a string that is not a natural harmonic of the string.

As a loose analogy, consider a 2D case such as a vibrating plate or speaker. If the driven frequency is a natural harmonic, the result is a static set of vibrating sub-areas of the plate or speaker that don't move (with a plate, you can get a cool looking pattern by putting some type of powder on the plate). If the driven frequency is not a harmonic, the the vibrating sub-areas of the plate or speaker move around.

#### sophiecentaur

Gold Member
Adding to the complexity here is that there can be traveling waves along a string that are not true harmonics.
Except that they are not harmonics; they are modes (overtones)
If the driven frequency is a natural harmonic,
Again, you are really referring to modes. This particularly important to get right when you deal with a two dimensional plate.
The situation with a driven string is very different from a plucked string because a driving waveform can consist of any set of frequencies (even harmonics). A plucked string can only produce a waveform that consists of its natural modes.

A real (acoustic) guitar string will have a length of 'uncluttered' string and a bridge which is loaded with a mechanical Impedance Matching system - the table which supports the bridge and which has a number of resonances (2D oscillations - or more). They present a load on the string with possibly a wildly varying impedance. Where to put a finger is hard to predict. Matters can be simpler with an electric guitar which is very lightly loaded and which has a much longer sustain (higher Q).

#### DennisN

I guess that would be reasonably easy to achieve for oneself with spectrum analysis on a PC these days.
Yes.
And there are even free apps (Android applications) for mobile phones/tablets available :

Spectroid (Spectrum analyzer and spectrogram/sonogram):

and this is also a nifty audio app (though not spectrum analysis):

Function Generator:

EDIT:
Also, the quite cool Physics Toolbox Sensor Suite includes the following functions (among others):

"(16) Sound Meter - sound intensity
(17) Tone Detector - frequency and musical tone
(18) Oscilloscope (audio) - wave shape and relative amplitude
(19) Spectrum Analyzer (audio) - graphical FFT
(20) Spectrogram (audio) - waterfall FFT"

Physics Toolbox Sensor Suite on Google play:

#### Stephen Tashi

Summary: When solving the wave equation for an oscillating string we look for a solution that is the linear combination/superposition of an infinite serie of "harmonics", i.e. sine and cosine functions (classical Fourier series). But is that a mere mathematical trick or do these states have physical meaning ?
A waveform can be expressed as a sum of other waveforms in many different ways. Why would we say that some of these methods use summands that have physical meaning and other do not? I suppose if you could examine an object generating the waveform and recognize it is composed of individual objects each generating different waveforms then the those individual wave forms would have "physical meaning" insofar as they could be assigned to distinct objects. However in the familiar example of a single vibrating string, we don't see distinct physical objects producing distinct pure frequencies.

From Fourier's work we know that ${c_n}$ is a monotonically (fast) decreasing succession, so we hear mainly the first (principal) harmonic which define the note we are hearing.
Fourier's work (by itself) doesn't explain what we hear. The fact we recognize frequencies is due to the architecture of our hearing system. For example, parts of it are constructed to pick out different frequencies - e.g. the hair cells https://en.wikipedia.org/wiki/Hair_cell and critical bands http://digitalsoundandmusic.com/4-1-6-sound-perception/

It's possible to imagine a hearing system based on decomposing a waveform in to the sum orthogonal polynomials or wavelets - or a hearing system based on neural nets where separation of a waveform in to components isn't used at all. A being with such a hearing system wouldn't get a primary sensation that harmonics are physically real.

Fourier analysis (done by the conscious mind) is an effective method of mathematical analysis. It's interesting that mammalian hearing systems have evolved to do analog imitations of it. Imagining a waveform as a sum of sine waves is physically real in the sense that both Nature and applied mathematicians have found it useful. However, I don't see any other sense in which a decomposition of a wave in to sine waves is more real than any other type of decomposition.

- play an open string, for example the low E
- gently put you finger on the metal strip corresponding to the 5th, 7th, or 12th fret. Be careful, you don't press the fret, you simply lay your finger on the metal strip!
This type of reasoning demonstrates the usefulness of frequency analysis in modeling certain physical phenomena. As mentioned above, one might consider that utility in applications defines "physical meaning". I don't find anything in this type of argument that supports a more extensive definition of "physical meaning".

#### Mister T

Gold Member
Then my question is: if you "mute" all the harmonics except for those having a node at that particular point, it means that they really do exist and that they're not just a mathematical trick.
It means that the mathematical model is valid. Whether or not that makes them real is a matter of semantics.

Fourier analysis doesn't have to be done with sine (or cosine) functions. Any periodic function can be approximated to any desired accuracy by a series of any other periodic function. So, for example, you could add up a sequence of square waves of the appropriate frequency, amplitude, and phase. It's a model. We judge its validity based on its ability to match what we observe. Whether or not that makes the model real depends on your definition of real. Models are a creation of the human mind, so in that sense they're real.

#### dRic2

Gold Member
@Stephen Tashi @Mister T Thanks for the replies! I don't want to sound rude after you spent the time to answer, but if you read post #7 you can see what I misinterpreted and how I solved my "philosophical" problem. Btw I may have put little care in choosing some words because it was 2.30 a.m. when I wrote the post... I apologize for that.

#### sophiecentaur

Gold Member
Fourier analysis doesn't have to be done with sine (or cosine) functions.
The Fourier Transform of a sinewave is zero everywhere but at the sine wave frequency. In 'reality', any sine wave has to be turned on and off (or the measurement has to be made over a finite time) That means the transform changes from a single value to a narrow, humped curve (frequency is spread), and that is nearer 'real' because it's a practical result. But people tend to read more than they should into the obvious outputs from the most common Discrete Fourier Transform routines. Much of what your computer monitor will show you is artefacts when you sample a waveform over a short period of time. Does that make it real or unreal? I'm not sure.

#### Stephen Tashi

@Stephen Tashi @Mister T but if you read post #7 you can see what I misinterpreted and how I solved my "philosophical" problem.
I don't see that you have defined a philosophical problem or solved one. For example, if I take the sequence of numbers $5, 10, 15, 20, 25$, I can subtract the arithmetic progression $2,4,6,8,10$ from it term-by-term and obtain $3,6,9,12,15$, which is another arithmetic progression. Is this a demonstration the resulting arithmetic sequences are the "real" components of $5,10,15,20,25$?

Making an analogy to your remarks in post #1, I think you would say that if $5,10,15,20,25$ is a waveform associated with a physical process and if a physical processes exists that can implement the term-by-term subtraction of $2,4,6,8,10$ from $5,10,15,20,25$ then $3,6,9,12,15$ is a "physically real" component of $5,10,15,20,15$ Is that a correct analogy?

#### rcgldr

Homework Helper
A plucked string can only produce a waveform that consists of its natural modes.
Example of a moving wave generated from a plucked string:

#### sophiecentaur

Gold Member
However, I don't see any other sense in which a decomposition of a wave in to sine waves is more real than any other type of decomposition.
The frequency domain is just one of a huge set of possible domains which can be used to describe time varying signals. The variation of a voltage with time (or another variable with space) is not any more fundamental than any other description of a signal. The Fourier Transform works both ways.
Example of a moving wave generated from a plucked string:

Good video.
i think it would have to involve some longitudinal displacement by the plucker. Added complication, maybe. The pluck in the video is very close to the bridge so end effect would be significant perhaps.

#### dRic2

Gold Member
that a correct analogy?
No. I'll try to be quick and easy. There is this phenomenon I was talking about and it's called "string harmonic". The explanation goes like this:
"Placing your finger on the metal strip kills some of the harmonics while leaving all the others"

This explanation is a philosophical heresy. How can a physical object (my finger) interact with a mathematical/abstract object (the harmonics)?

There are two solutions:
1) the harmonics are real objects too and so interact with the finger in the physical world
2) the finger is "not real" and thus it interacts with the harmonics in the mathematical/abstract world.

Although one might be tempted to go for the first explanation, it turns out the second one is much simpler. In fact the interaction of the finger with the harmonics can be formulated in the mathematical realm by saying that it changes the boundary conditions of the problem. In this way we have a relation in the abstract mathematical world and the problem is solved without useless speculations about the physical interpretation of a mathematical object (the harmonics).

Hope that clears it out.

Gold Member

#### Stephen Tashi

What's the problem ?
Assuming I understand your question, the problem is to detect your reasoning. Your presentation seems to amount to:

Premise: Using my finger, I can dampen certain harmonics while leaving others
Therefore:
Harmonics are real

This has the logical form:
Premise A
Conclusion: B
without explaining why A should imply B.

#### Ibix

Hope that clears it out.
Um... You seem to be confusing the physical world and the mathematical model we use to describe it. In the real world, your finger is simply stopping the string from moving at a point where it was previously free to move. The real world cares nothing about harmonics - the string just moves the way it does as a result of the way the various atoms interact.

But the beauty of physics is that it turns out to be possible to summarise this complex microscopic behaviour in terms of fairly simple differential equations. And it turns out to be possible to write the solution as a sum of harmonics. In this mathematical model, you model the finger by adding another boundary condition. That changes the valid solutions, but you can still write them as a sum of harmonics.

So whether or not harmonics are "real" isn't the point. It's probably not answerable because you'd first need to define "real" and "not real" in falsifiable terms. The question is, is a mathematical decomposition of the string shape in terms of harmonics useful? Yes it is - because I can describe the motion of the string with and without the finger as a different sum of harmonics. I don't have to do it that way, though. It's just a convenient formalism.

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