Given a point source of radially expanding wave (2D) I need to change its wavefront shape (partially) to planar by reflection. A simple parabolic reflector will not work since the circular wave precedes the reflected planar part. And there is only one pulse.
Was thinking that some ellipsoidal reflector may work, but I lack the geometric optics background to do the rigorous calculations.
Looking for suggestions or references to the textbooks/papers.
The issue is that there is an infinite number of solutions and I never solved PDEs for boundaries.
Below is the visual representation of what is expected. The question is what shape of green walls to use to change the shape of the pulse to straight.
P.S. I need the planar wavefront to hit the target which is close to the stationery point of circular wave origin. No need for the whole circular front to be converted to planar, just a portion is fine.
Update. It looks like this problem is an Eikonal equation in uniform media.
$$v(\vec{r})(\nabla \phi(\vec{r}))=\frac{d\vec{r}}{ds},$$ where $\vec{r}$ is a position in space, $v(\vec{r})$ velocity of the wavefront, s - parameter (distance travelled by the ray of light/sound), $\phi(\vec{r})$ - wavefront of light/sound.
However, I am struggling with how to formulate the problem. Even for the simple scenario, s.a. wavefront at $t_1$ is a circular arc, and wavefront at $t_2$ is a straight segment.