"Hein ten Horn" <tenhornRemovE@ThiSraketnet.nl> wrote in message
news:469943cd$0$322$e4fe514c@news.xs4all.nl...
> Ron Baker, Pluralitas! wrote:
>> Hein ten Horn wrote:
>>> Ron Baker, Pluralitas! wrote:
>>>>
>>>> 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?
>>
>> Another way to phrase the question would have been:
>> Given a waveform x(t) representing the sound wave
>> in the air how do you decide whether there is a
>> beat in it?
>
> Oh, nice question. Well, usually (in my case) the functions
> are quite simple (like the ones we're here discussing) so that
> I see the beat in a picture of a rough plot in my mind.

Sorry, but your mental images are hardly linear
or authoritative on the laws of physics.

>
>>> 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.
>
> Mathematical terms like linear, logarithmic, etc. are familiar
> to me, but the guys here use linear and nonlinear in another
> sense. Something to do with harmonics or so? Anyway,
> that's why the hint isn't working here.

You have a system decribed by a function f( ) with
input x(t) and output y(t).
y(t) = f( x(t) )
It is linear if
a * f( x(t) ) = f( a*x(t) )
and
f( x1(t) ) + f( x2(t) ) = f( x1(t) + x2(t) )

A linear amplifier is
y(t) = f( x(t) ) = K * x(t)
A ("double balanced") mixer is
y(t) = f( x1(t), x2(t) ) = x1(t) * x2(t)
(which is not linear.)

>
>>>> (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)
>>
>> It would have been more proper of me to say
>> "sinusoid" rather than "sine wave". I called
>> cos() a "sine wave". If you look at cos(2pi f1 t)
>> on an oscilloscope it looks the same as sin(2pi f t).
>> In that case there is essentially no difference.
>> Yes, there are cases where it makes a difference.
>> But at the beginning of an analysis it is rather
>> arbitrary and the math is less cluttered with
>> cos().
>
> Got the (co)sin-stuff. But the unequallities are still there.
> It's easy to understand: the left-hand term is sooner or later
> greater then one, the right-hand term not (in both unequalities).
> As a consequence we've two different x's.

You left out the "0.5".
cos(a) * cos(b) = 0.5 * (cos(a+b) + cos(a-b))


>
>>>> So which is it really? Hint: If all you have is x then
>>>> you can't tell how it was generated.
>
> Yep.
>
>>>> What you do with it afterwards can make a
>>>> difference.
>
> Sure (but nature doesn't mind).
>
>>> Referring to the physical system, what's now your point?
>>> That system contains elements vibrating at 25003 Hz.
>>> There's no math in it.
>>
>> Whoops. You'll need math to understand it.
>
> I would say we need the math to work with it, to get our
> things done. Understanding nature is not self-evident the
> same thing.

Nature works by laws. We express the laws
in math. Nature is complicated and it may be
difficult to simplify the system one is looking
at or to put together of mathematical model
that is representative of the system but it
is often possible. It is regularly possible to
have a mathematical model of a system that
is accurate to better than 5 decimal places.

> I would really appreciate it if you would take
> the time to read my UTC 9:57 reply to JK once again,
> but then with a more open mind. Thanks.

Hmm. You're definitely not going to like part of it.
(You seem so much more reasonable here.)

>
> gr, Hein
>
>