Re: [Phys-l] Harmonics vs Overtones

[John Denker <jsd@av8n.com>]

On 04/01/2009 01:25 PM, Dan Crowe wrote:
> Are the "cross terms", "difference terms", etc. necessary to describe
> the pressure fluctuations in the air, or do they apply only to the
> perception of the sound?

Cross terms arise from nonlinearity.  The perception is 
far more nonlinear than the air under ordinary conditions.

Perception is very nonlinear:  You can hear beats between 
frequency f1 played in one ear and frequency f2 played in 
the other hear.

The nonlinearity of the propagation is significant only for
very very loud sounds e.g. gunshots, sonic booms, et cetera.
You don't want to be anywhere near something in this regime,
unless it is very small e.g. sonoluminescence.

============================

In various notes today, there has been some sloppiness about
"nonlinear" versus "anharmonic".

  *) Something is harmonic (or not) depending on where you
   find the modes as you go up in _frequency_.

  *) Something is linear (or not) depending on how things
   shift as you go up in _amplitude_.

Important example:  A piano string is anharmonic (because of 
stiffness in the piano wire) even though it is linear to an 
excellent approximation at reasonable amplitudes.  

There are plenty of situations where something becomes anharmonic
because it is nonlinear, but this is not always the case.  The
piano string will go nonlinear if you drive it hard enough, but
this is not typical and certainly not the reason why the octaves
are stretched.

Another example:  It is the easiest thing in the world to make an
organ pipe anharmonic, by making it not quite cylindrical.  This
obviously has nothing to do with nonlinearity;  the pipe remains
anharmonic independent of amplitude at all reasonable amplitudes.

=======

On a related note, there is another important distinction:

 *) Propagation is _dispersive_ depending on the omega versus k
  relationship.

 *) Propagation is _nonlinear_ depending on what happens as a
  function of amplitude.

Example:  If you are trying to form a shock, you need nonlinearity.
Dispersion tends to tear the shock apart.

Example:  Propagation in a waveguide (not too far above cutoff) is
highly dispersive ... but linear for all practical purposes.

Example:  Propagation of light through a lead-glass prism is 
dispersive.  That's usually the whole point of the prism.

=======================================

Also remember Floquet's theorem:  

If you drive a system with a periodic input, you will get a periodic 
output _with the same period_.  

  (Note that multiples of the frequency i.e. submultiples of the 
  period are allowed, because they are in fact periodic with the 
  original period.)

Therefore even a system that is anharmonic and/or linear can be Fourier
analyzed.

This is of great significance to the human voice and to wide classes
of bowed instruments, reed instruments, et cetera.  The spectrum of
a given note will contain lots of lines that are strict multiples of 
the fundamental frequency.  It's very hard for a reed to vibrate at 
more than one frequency.  Typically there is mode locking.  The reed 
vibrates at a definite frequency, and everything else is going to be
a multiple of that frequency.

This leads to the usual "excitation plus filter" model.  The excitation
is periodic, and then the signal gets filtered by the modes of the
instrument.

Of course Floquet has nothing to say about a bell, a drum, or a
plucked string.