In article ,
"Ron Baker, Pluralitas!" wrote:


--snippety-snip--

You said you are a physicist/engineer.
What does "linear" mean?

Let's not get too far off the subject here. We were discussing whether
the "tuning beat" that you use to tune a musical instrument involved a
nonlinear process (ie. "modulation").

Then linearity is at the core of the matter.
What does "linear" (or "nonlinear") mean to you?

OK, if you insist -- *in this case* it means "linear enough to not
produce IM products of significant amplitude".

I said that it does not, and that
it could be detected by instrumentation which was proveably linear (i.e.
not "perfectly" linear, because that's not required, but certainly
linear enough to discount the requirement for "modulation").

No nonlinearity is necessary in order to hear
a beat?
Where does the beat come from?

As the phase of the two nearly equal waves move past each other, there
is simple vector summation which varies the amplitude.

Consider two sine waves of precisely the same frequency, where one of
them is adjustable in phase -- use a goniometer, for instance. Use a set
of resistors to sum the two signals, and observe the summing point with
a 'scope or a loudspeaker. By altering the phase of one source, you can
get any amplitude you want from zero up to twice the amplitude of either
one.

Now just twiddle that phase knob around and around as fast as you can.

You've just slightly altered the instantaneous frequency of one of the
generators (but only while you twiddle), and accomplished pretty much
the same effect as listening to the beat between two guitar strings at
nearly zero frequency offset. With no nonlinear processes in sight.

Isaac