What is the upper limit of harmonics in standing waves?

[Theseus]

As an example of a standing wave we have a musical tone, which is a combination of a fundamental pitch and a series of harmonics. Is there anything to suggest that additional harmonics don't continue up far beyond the range of hearing, perhaps even approaching infinity? Or is there some point where the vibrations begin to fizzle out?

And my question applies to all standing waves, not just musical tones.

 

[Redbelly98]

The harmonics can go well past the range of hearing, but not to infinity. When half the wavelength equals the spacing between adjacent atoms, that is the highest frequency that can be produced.

You could picture such a standing wave like this:

_o ^o _o ^o _o ^o _o ^o _o ^o _o ^o _o​

Hope that helps.

 

[Vanadium 50]

Also the idea of a "harmonic" is an approximation (or maybe idealization is a better word). Think about a guitar string. The harmonic series for a string assumes that the string's width can be ignored - an idealization that breaks down if the wavelength of the harmonic is comparable to the string's width.

 

[Theseus]

When half the wavelength equals the spacing between adjacent atoms, that is the highest frequency that can be produced.

Are we talking atoms of "air"? Any idea what the frequency would be at that level?
Also if we look at, say, a sine wave for alternating current, would we use the space between electrons as opposed to atoms to set the maximum frequency? Or is this a whole other subject?

Also the idea of a "harmonic" is an approximation (or maybe idealization is a better word). Think about a guitar string. The harmonic series for a string assumes that the string's width can be ignored - an idealization that breaks down if the wavelength of the harmonic is comparable to the string's width.

Coming from a practical orientation on this subject (music), what you say is interesting although I'm afraid I'm not tracking with you. When you say "idealization" do you mean the general knowledge about harmonics is accurate unless we have certain conditions, such as the wavelength is comparable to string width? Can you please amplify?

I guess at my stage more answers lead to more questions, thanks for bearing with me.

 

[Redbelly98]

The harmonics can go well past the range of hearing, but not to infinity. When half the wavelength equals the spacing between adjacent atoms, that is the highest frequency that can be produced.

Are we talking atoms of "air"?

Yes (though I should have said molecules) -- or whatever material is transmitting the sound wave. Could also be the atoms in a guitar string. It's a little more complicated, since there is randomness in the spacing between atoms, but that is the general idea.

Any idea what the frequency would be at that level?

For a ballpark figure (won't be exact), we can use the relation taught in introductory physics,

speed of sound = frequency × wavelength​

For air:

The speed of sound is 340 m/s.
[EDIT: see note below] [STRIKE]Atoms are roughly 3 nm apart on average, so the wavelength is twice that spacing or 6×10^-9 m​

Plug the numbers for speed of sound and wavelength into the equation, and see what frequency you get.[/STRIKE]

Disclaimer: I imagine the actual frequency limit quite less than this, because the spacing between molecules is random, the air will dissipate/absorb sound energy, or reasons I'm not even aware of -- I am no accoustics expert, and am just seeing how far introductory physics goes toward getting an answer here.

EDIT:
After some more google searching, I'm find that for a gas we use the mean free path of the molecules, i.e. the average distance a molecule travels before colliding with another molecule, which is (apparently) 70 nm. So a wavelength of 70×10^-9 m, and 340 m/s for the speed of sound, will give a rough idea of the maximum frequency for "sound" in air.

 

[Theseus]

OK, if I have the decimal point in the right place, that would put the upper limit at about 4.8 GHz which is some 25 octaves above middle C. I guess there's a lot going on up there that we don't hear.

I think the equivalent in the EMF spectrum would be the HF radio band. I wonder, do sound and EM waves ever influence each other such as by resonance or some other property?

Sound systems often produce the dreaded 60Hz hum which is caused by the standard AC frequency, but I think you only hear it because it is electronically converted to sound through the transducer. But do the vibrations/oscillations of sound and EM waves ever directly influence each other?

 

[Redbelly98]

OK, if I have the decimal point in the right place, that would put the upper limit at about 4.8 GHz which is some 25 octaves above middle C. I guess there's a lot going on up there that we don't hear.

Yup, but again it's just a ballpark figure, so I'd round that to 5 GHz.

I think the equivalent in the EMF spectrum would be the HF radio band. I wonder, do sound and EM waves ever influence each other such as by resonance or some other property?

The only way I know is that sound, being a pressure wave, causes a change in the density (and therefore the refractive index) of a material. This can cause an EM wave to change direction. Or in the case of a so-called acousto-optic modulator, a sound wave behaves similar to a diffraction grating for a laser beam (click http://en.wikipedia.org/wiki/Acousto-optic_modulator" for wikipedia article).

Sound systems often produce the dreaded 60Hz hum which is caused by the standard AC frequency, but I think you only hear it because it is electronically converted to sound through the transducer.

Yes.

But do the vibrations/oscillations of sound and EM waves ever directly influence each other?

See my comment above.