I know I'm doing something stupid but I just can't find out where. Suppose that I have a sine wave whose angular frequency varies linearly with time. So:
$\omega = ct$
And the sine wave would be:
$S=sin(\omega t) =sin(ct^2)$
Now suppose I have $S$ and I would like to find its angular frequency. For that, I believe I should differentiate the angle of the sinusoid, which gives
$\omega=\frac{d}{dt}ct^2 = 2ct$
and this is twice the value I started with
$\omega = ct$
Where did the "2" come from?!
You are using $\omega$ in two different ways. There is no relation between the $\omega$ in the first and second equations and the $\omega$ in the third. The $\omega$ in the third is the instantaneous rate of increase of the angle. It is, in fact, twice the $\omega$ of the first two equations, as you have shown. Give it a new name and the problem disappears.