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

gr, Hein

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.

gr, Hein

You were correct before. It might be correct to say that matter is
vibrating at an average, or effective frequency of 222 Hz. But the
only sine waves present in the air are vibrating at 220 Hz and 224 Hz.
Obviously. It's a very simple matter to verify this by experiment.
You really ought to perform it (as I just did) before posting
further on the subject.

jk

Itsssssss a strange Worldddddddd afterrrrr alllllllll,,,,, its a
strange, strange, Worrrrrrldddddd,,,,,,,
cuhulin

craigm wrote:
Jim Kelley wrote:
David L. Wilson wrote:
Jim Kelley wrote:

At a particular instant in time the period does in fact equal the average
of the two. But this is true only for an instant every 1/(a-b) seconds.

How do you come up with anything but a period of of the average of the
two for the enveloped waveform?

The error here is in assuming that the sin and cos terms in the
equivalent expression are representative of individual waves. They
are not. The resultant wave can only be accurately described as the
sum of the constituent waves sin(a) and sin(b), or as the function
2sin(.5(a+b))cos(.5(a-b)). That function, plotted against time
appears exactly as I have described. I have simply reported what is
readily observable.

I would submit you plotted it wrong and/or misinterpreted the results.

Jim, if you'd like me to send you an Excel sheet about this,
please let me know.

gr, Hein

I've sent this post already once. For some strange reason it didn't
come up in rec.radio.shortwave (craigm?).
I only read rec.radio.shortwave these days.
(repost to: sci.electronics.basics, rec.radio.shortwave,
rec.radio.amateur.antenna, alt.cellular.cingular,
alt.internet.wireless)

"Rich Grise" wrote in message
news
On Wed, 11 Jul 2007 22:52:17 -0700, Jeff Liebermann wrote:

"NotMe" hath wroth:

(Please learn to trim quotations)

Actually the human ear can detect a beat note down to a few cycles.

If you are talking about the beat between two close
audio frequencies then one can easily hear a beat way
below 1 Hz.


No, you cannot. Figure on 20Hz to 20KHz for human hearing:
http://hypertextbook.com/facts/2003/ChrisDAmbrose.shtml

What happens when you zero beat something is that your brain is filling
in
the missing frequencies. As you tune across the frequency, and the beat
note goes down in frequency, most people overshoot to the other side, and
then compensate by splitting the different.

If you are talking about beat frequency heard when
tuning to a carrier with a radio with a BFO or in SSB mode
then one can't hear any beat below 50 Hz or so.
The audio section of the receiver blocks anything
below about 50 Hz.


No, you've got it all wrong. The beat note happens because, when the
signals are close to 180 degrees out of phase, they cancel out such that
there is, in fact, no sound. This is what your ear detects. Now, if
you're zero-beating, say, 400 Hz against 401 Hz, I don't know if the
801 Hz component is audible or if it's even really there, but
mathematically, it kinda has to, doesn't it?

Are you talking radios or guitars?
With a guitar you might beat 400 Hz against 401 Hz.
With a radio you'd more likely beat 455 kHz against
455.001 kHz.


Thanks,
Rich

"Ron Baker, Pluralitas!" wrote in message
...
|
| "Rich Grise" wrote in message
| news | On Wed, 11 Jul 2007 22:52:17 -0700, Jeff Liebermann wrote:
|
| "NotMe" hath wroth:
|
| (Please learn to trim quotations)
|
| Actually the human ear can detect a beat note down to a few cycles.
|
| If you are talking about the beat between two close
| audio frequencies then one can easily hear a beat way
| below 1 Hz.

Based on studies done at Tulane Department of Neurology (mid 60's) the
detection is not in the ear but in the brain. The process can be taught and
refined though bio-feedback.

"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

"Ron Baker, Pluralitas!" wrote in message
...
|
| "Rich Grise" wrote in message
| news | On Wed, 11 Jul 2007 22:52:17 -0700, Jeff Liebermann wrote:
|
| "NotMe" hath wroth:
|
| (Please learn to trim quotations)
|
| Actually the human ear can detect a beat note down to a few cycles.
|
| If you are talking about the beat between two close
| audio frequencies then one can easily hear a beat way
| below 1 Hz.

But what you hear below ~20 Hz is not the beat note, but changes in sound
pressure (volume) as the mixing product goes in and out of phase. This
actually becomes easier to hear as you near zero beat.

Ron Baker, Pluralitas! wrote:
Hein ten Horn wrote:
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.

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"?

Not by train, neither by UFO.
Sorry. English, German and French are only 'second'
languages to me.
Are you after the occurrence of a beat?
Then: a beat appears at constructive interference, thus
when the cosine function becomes maximal (+1 or -1).
Or are you after the beat frequency (6 Hz)?
Then: the cosine function has two maxima per period
(one being positive, one negative) and with three
periodes a second it makes six beats/second.

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.

??
sin(a) + sin(b) sin(a) * sin(b)
In case you mistyped cosine:
sin(a) + sin(b) cos(a) * cos(b)

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.

Referring to the physical system, what's now your point?
That system contains elements vibrating at 25003 Hz.
There's no math in it. More on that in my posting to
JK at nearly the same sending time.

gr, Hein

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