I am experiencing what I think is really simple confusion.
Take $y(t) = \sin(2 \cdot \pi \cdot t \cdot\omega(t))$ and $\omega(t) = a \cdot t+b$ for $t \in [0,p)$ and let $\omega(t)$ have a periodic extension with period $p$. The values $a,b,p$ are parameters
$\omega(t)$ looks like the blue function here
with $\omega$ as such, $y(t)$ should be two chirps -- a sine wave whose frequency sweeps. My intuition is that the two chirps should be identical, but they're not. The second chirp is higher in frequency. The argument to $\sin$ , the quantity $2 \cdot \pi \cdot t \cdot \omega(t)$ looks like the blue function here:
Two piecewise quadratics. The second quadratic should just be a shifted version of the first one, which is what is depicted in red. I can't argue with the math, but there is something very simple wrong with my intuition. The function $\omega$ should modulate the "instantaneous frequency" of the sinewave.