Jim Kelley wrote:
Hein ten Horn wrote:
Hein ten Horn wrote:

quote
We hear the average of two frequencies if both frequencies
are indistinguishably close, say with a difference of some few
hertz. For example, the combination of a 220 Hz signal and
a 224 Hz signal with the same amplitude will be perceived as
a 4 Hz beat of a 222 Hz tone.
unquote

From the example: there's no 222 Hz tone in the air.

That one I'd like to take back.
Obviously the superposition didn't cross my mind.
The matter is actually vibrating at the frequency
of 222 Hz. Not at 220 Hz or 224 Hz.

You were correct before.

That's a misunderstanding.
A vibrating element here (such as a cubic micrometre
of matter) experiences different changing forces. Yet
the element cannot follow all of them at the same time.
As a matter of fact the resulting force (the resultant) is
fully determining the change of the velocity (vector) of
the element.
The resulting force on our element is changing at the
frequency of 222 Hz, so the matter is vibrating at the
one and only 222 Hz.

It might be correct to say that matter is vibrating at an
average, or effective frequency of 222 Hz.

No, it is correct. A particle cannot follow two different
harmonic oscillations (220 Hz and 224 Hz) at the same
time.

But the only sine waves present in the air are vibrating
at 220 Hz and 224 Hz.

If so, we have a very interesting question...
What is waving here? A vacuum?
But don't take the trouble to answer.
You'd better distinguish the behaviour of nature and the
way we try to understand and describe all things.
As long as both sound sources are vibrating there are
no sine waves (220 Hz, 224 Hz) present, yet we do
use them to find the frequency of 222 Hz (and the
displacement of a vibrating element at a particular
location in space on a particular point in time).

Obviously. It's a very simple matter to verify this by experiment.

Indeed, it is. But watch out for misinterpretations of
the measuring results! For example, if a spectrum
analyzer, being fed with the 222 Hz signal, shows
that the signal can be composed from a 220 Hz and
a 224 Hz signal, then that won't mean the matter is
actually vibrating at those frequencies.

You really ought to perform it (as I just did) before
posting further on the subject.

I did happen to see interference of waterwaves
including some beautiful (changing) hyperbolic structures,
but no sign of any sine wave at all. So, with your kind
permission, here's my posting. ;-)

gr, Hein