Are harmonics "real" in a vibrating string?

[fizzle]

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's a mathematical trick. The guitar string is not a sum or superposition of "substrings" vibrating at different frequencies with different amplitudes. You can *approximate* the string's behavior with Fourier or other mathematical analysis techniques ... but the string does what it does independent of your analysis!

Many moons ago, when I was a young student filled with mathematical techniques, I built a circuit that produced a square wave. When I slapped a scope on it, I somewhat expected to see harmonics in the output ... but the output simply went up and down, with some ringing on the transitions. The ringing was intrinsic to the circuit itself and not actually in the input. That really got me thinking about a how a narrow bandpass filter really works. You think of it "extracting" a signal from the input but it's actually a pseudo-oscillator responding to the input, i.e. it doesn't "extract" anything.

The next step is to wonder whether superposition is ever "real"...

 

[sophiecentaur]

When I slapped a scope on it, I somewhat expected to see harmonics in the output

Without a fair amount of dispersion and a non flat frequency response, it is not surprising that the scope produced a 'good' picture of the variation with time of the waveform. It was doing its job!

To display the frequency spectrum of a waveform requires (in effect) a swept filter with a display of what the filter is passing. Alternatively (but totally equivalent) you could do a Discrete Fourier Transform of a long sequence of samples of the time variation of the
waveform. The FT is effectively the result of finding the correlation between the signal and each of the harmonics of 1/(the time period of the sample).

 

[Ralph Dratman]

I agree with Ibix that the answer depends on what you take as "real" or "not real". Nevertheless I think I understand the motivation of the question, as I have often wondered whether, for example, a sharp click sound is "really" a combination of many different high frequency "tones" -- as the click might be portrayed using Fourier analysis. There is no single correct answer to that and similar questions, but it would be reasonable to study whether a few of the tiny hairs in our ears that sense pitch are in fact activated by a particular click sound.

 

[sophiecentaur]

It is not true to say that what we are aware of , as a ‘sound’, can be fully represented by a simple sum of single tones. Our ears work on how those hairs react to the vibrations. The response of a hair is certainly not ‘ideal’ but it’s perhaps the nearest thing to ‘real’ that you can get. Which makes a bit of a nonsense of the idea of ‘reality’ being applied to Science.

 

[Swamp Thing]

it would be reasonable to study whether a few of the tiny hairs in our ears that sense pitch are in fact activated by a particular click sound.

IIRC, the frequency analysis mechanism in the ear is the cochlea, which is a nonlinear(?) tapered transmission line. Different frequencies create different standing wave patterns along the length of the line, and each hair is just sensing the intensity of the standing wave at its own location in the cochlea. The brain then takes the entire standing wave's amplitude profile and works out its frequency spectrum. The brain probably also looks at the transient changes in the wave profile over tens to hundreds of milliseconds.

In the case of a sharp click, I would imagine a transient bouncing back and forth in the cochlea a few times, exciting all the hairs with pretty much the same amplitude. So if the brain got an input saying that all the hairs quivered equally during a span of a few tens of milliseconds, then it would conclude, "ah, yes, that must have been a click".

 

[sophiecentaur]

So if the brain got an input saying that all the hairs quivered equally during a span of a few tens of milliseconds, then it would conclude, "ah, yes, that must have been a click".

This would depend on a good timing mechanism in the brain itself and the ability to hear binaurally and image locate, supports this idea. It's clear that we use both frequency and time domain analysis. Breathtaking.

 

[Nik_2213]

Late to the party, but answer is 'YES'.

I could see the standing waves on my wife's viola's strings.
This took me back to fun college physics 'practicals', with paper 'riders' etc etc.

I'd been totally hopeless at music but, when my wife needed a 'Second Viola' and 'Pianist' for concert practice, I bought a PC, taught myself enough theory to run 'Cakewalk Home Studio', drove her Casio piano and a good 'Strings' sound-font via MIDI...

 

[tech99]

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.

An analog of the string is perhaps a metal box used as a cavity resonator for EM waves.
We can drive the cavity with a sine wave and many modes will spring up. If we insert a probe and connect it to a spectrum analyser we do not see any harmonics as a result of the various modes.
On the other hand, if we drive the resonator with a harmonic rich waveform, such as a square wave, our probe shows the harmonics, but altered in relative amplitude by the various resonances of the cavity.

 

[dRic2]

* To all the new posters please see post #4 by @sophiecentaur *
It's a very very good explanation of what I was trying to understand