Jim Kelley wrote
> Hein ten Horn wrote:
>
>> 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.
>
> Under the stated conditions there is no sine wave oscillating at 222 Hz.
> The wave has a complex shape and contains spectral components at two
> distinct frequencies (neither of which is 222Hz).

Not a pure sine oscillation (rather than wave), but a near sine
oscillation at an exact period of 1/222 s. The closer the source
frequenties, the better the sine fits a pure sine. Thus if you
wish to get a sufficient near harmonic oscillation, conditions like
"slow changing envelope" are essential.

>>>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.
>
> The particle also does not average the two frequencies.

Hmm, let's examine this.
From the two composing oscillations you get the overall
displacement:
y(t) = sin(2 pi 220 t) + sin(2 pi 224 t)
From the points of intersection of y(t) at the time-axes you
can find the period of the function, so examine when y(t) = 0.
sin(2 pi 220 t) + sin(2 pi 224 t) = 0
(..)
(Assuming you can do the math.)
(..)
The solutions are:
t = k/(220+224) with k = 0, 1, 2, 3, etc.
so the time between two successive intersections is
Dt = 1/(220+224) s.
With two intersections per period, the period is
twice as large, thus
T = 2/(220+224) s,
hence the frequency is
f = (220+224)/2 = 222 Hz,
which is the arithmetic average of both composing
frequencies.

> The waveform which results from the sum of two pure sine waves is not a pure
> sine wave, and therefore cannot be accurately described at any single
> frequency.

As seen above, the particle oscillates (or vibrates) at 222 Hz.
Since the oscillation is non-harmonic (not a pure sine),
it needs several harmonic oscillations (frequencies,
here 220 Hz and 224 Hz) to compose the oscillation at 222 Hz.

>>>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.
>
> :-) Matter would move in the same way the sound pressure wave does,

To be precise, this is nonsense, but I suspect you're trying
to state somewhat else, and since I'm not able to read your
mind today, I skip that part.

> the amplitude of which is easily plotted versus time using Mathematica,
> Mathcad, Sigma Plot, and even Excel. I think you should still give that a
> try.

No peculiarities found.

gr, Hein