I have a sine wave $w_0$ with a linearly increasing frequency $f_0$
$f_0(t) = r_0 \cdot (t - c_0)$
$w_0(t) = a \cdot sin(2 \pi \cdot f(t) \cdot t + \phi) $
I want to glue that sine wave to another sine wave at the moment $c_1$:
$f_1(t) = r_1 \cdot (t - c_1)$
$w_1(t) = a \cdot sin(2 \pi \cdot f(t) \cdot t + x) $ (NB $x$ instead of $\phi$)
at the point $c_1$ where $f_0(c_1) = f_1(c_1)$
But at that point the $w_0$ is in a certain phase. I need to plug that phase into $x$ (or at least in a way that the function stays continuous). How do I calculate the phase of $w_0(c_1)$?