"Hein ten Horn" wrote in message
...
Ron Baker, Pluralitas! wrote:
David L. Wilson wrote:
Hein ten Horn wrote:
...
So take another example: 25000 Hz and 25006 Hz.
Again, constructive and destructive interference produce 6 Hz
amplitude variations in the air.
But, as we can't hear ultrasonic frequencies, we will not produce
a 25003 Hz perception in our brain. So there's nothing to hear,
no tone and consequently, no beat.

If one looks at an oscilloscope of the audio converted to voltage, one
still can see the 6Hz variations on the 25003 Hz and still refers to
those
as tone and beat. These exist in mathematically formulation of the
resulting waveforms

Right.

not just as something in the brain.

In this particular example nothing is heard
because 25003 Hz is an ultrasonic frequency.


What is the mathematical formulation?

sin(2 * pi * f_1 * t) + sin(2 * pi * f_2 * t)
or
2 * cos( pi * (f_1 - f_2) * t ) * sin( pi * (f_1 + f_2) * t )

So every cubic micrometre of the air (or another medium)
is vibrating in accordance with
2 * cos( 2 * pi * 3 * t ) * sin(2 * pi * 25003 * t ),
thus having a beat frequency of 2*3 = 6 Hz

How do you arrive at a "beat"?
Hint: Any such assessment is nonlinear.
(And kudos to you that you can do the math.)

Simplifying the math:
x = cos(a) * cos(b) = 0.5 * (cos[a+b] + cos[a-b])
(Where a = 2 * pi * f_1 * t and b = same but f_2.)
All three of the above are equivalent. There is no difference.
You get x if you add two sine waves or if you
multiply two (different) sine waves.
So which is it really? Hint: If all you have is x then
you can't tell how it was generated.

What you do with it afterwards can make a
difference.

and a vibration frequency of 25003 Hz
(let alone phase differences of neighbouring
vibrating elements).

gr, Hein