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Firstly, I'm not a mathematician, I'm an engineer, so you can freely make fun of the question.

I have the following counter-intuitive behaviour in a sweep function. I have a sweep sine function (something like $$x(t) = \sin(\omega(t) \times t)$$, where $\omega(t)$ is a linear function of time, so it could be said that the function is $$x(t) = \sin(\omega t^2)$$ When I plot the function and take some instant in time and measure the frequency of the response (counting the time elapsed between two consecutive peaks of the wave) this do not give the supposed instant frequency of the wave, namely $f = \omega t$.

In fact, in my example, the instant frequency of the wave is approximately 2 times $\omega t$.

Of course, I feel bad about asking this trivial question here, but I could not find light anywhere.

Thanks !

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  • $\begingroup$ $t$ is not constant across the distance between two consecutive peaks. $\endgroup$ – Pockets Apr 21 '16 at 16:15
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Your function can be written in the form $$ x(t) = \sin(\phi(t)) $$ in this function, $\phi(t)$ tells you which "phase angle" (in radians) you hit at any particular time $t$.

Now, a question you should ask yourself is: "what exactly does frequency measure, anyway"? I would posit that the frequency for a non-constant $\phi(t)$ measures how quickly the phase angle changes. That is, the frequency at time $t$ is given by $\frac{d\phi}{dt}$.

So, if $\phi(t) = \omega t^2$, then the frequency at a time $t$ will be $\phi'(t) = 2\omega t$.

Of course, if $\phi(t) = \omega t$, then we get the expected constant frequency of $\phi'(t) = \omega$.

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  • $\begingroup$ This is so simple and true. Not much more to add my friend... Thanks very much for your explanation, I can not emphasize enough how silly I feel write now. $\endgroup$ – nodarkside Apr 21 '16 at 16:35
  • $\begingroup$ No problem. This is an important point if you study how FM radio works $\endgroup$ – Ben Grossmann Apr 21 '16 at 17:16

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