John Fields wrote:
> On Sat, 14 Jul 2007 23:43:55 +0200, "Hein ten Horn" wrote:
>>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.
>
> And what does it look like, then?

Roughly like the ones in your Excel(lent) plots. :)

>>>> 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.
>
> Where is "here"?

In this thread.

> I'm writing from sci.electronics.basics

Subscribing to that group would be a good
thing to do, I suspect.

> and, classically, a device
> with a linear response will provide an output signal change over its
> linear dynamic range which varies as a function of an input signal
> amplitude change and some system constants and is described by:
>
>
> Y = mx+b
>
>
> Where Y is the output of the system, and is the distance traversed
> by the output signal along the ordinate of a Cartesian plot,
>
> m is a constant describing the slope (gain) of the system,
>
> x is the input to the system, is the distance traversed by
> the input signal along the abscissa of a Cartesian plot, and
>
> b is the DC offset of the output, plotted on the ordinate.
>
> In the context of this thread, then, if a couple of AC signals are
> injected into a linear system, which adds them, what will emerge
> from the output will be an AC signal which will be the instantaneous
> arithmetic sum of the amplitudes of both signals, as time goes by.

In general: that sum times a constant factor.
Perhaps the factor being one is usually tacitly assumed.

> As nature would have it, if the system was perfectly linear, the
> spectrum of the output would contain only the lines occupied by the
> two inputs.
>
> Kinda like if we listened to some perfectly recorded and played back
> music...
>
> If the system is non-linear, however, what will appear on the output
> will be the AC signals input to the system as well as some new
> companions.
>
> Those companions will be new, real frequencies which will be located
> spectrally at the sum of the frequencies of the two AC signals and
> also at their difference.

From physics (and my good old radio hobby)
I'm familiar with the phenomenon. The meanwhile
cleared using of the word non-linear in a narrower
sense made me sometimes too careful, I guess.

>> Something to do with harmonics or so? Anyway,
>> that's why the hint isn't working here.
>
> Harmonics _and_ heterodynes.
>
> If the hint isn't working then you must confess ignorance, yes?

The continuous thread was clear to me.

Thanks.

gr, Hein