The solution outline:
Step 1. Verify, that all the initial (at the moment $t=0$) wave circles don't contain your starting point $(0,0)$. If yes, then continue - otherwise exit, no escape.
Step 2. For each wave circle $W_i$ you need to find an escape sector - the range of directions, where the escape is guaranteed.
Imagine a circle of your possible ...
I only noticed after posting that my first answer does not directly address your
specific question. So now my answer is in two parts.
(1) As suggested by @HEKTO, to find the closest segment,
you could first compute the Voronoi diagram:
CGAL 2D Segment Voronoi Diagrams.
Then next you need to perform point location within the (non-rectilinear) cells of the ...