Work Step by Step
The waves are travelling at the same speed. If $d_1$ is the distance between a focus and the closed portion of the hyperbola, and $d_2$ is the distance between a focus and the same section of the hyperbola, then $d_2$ - $d_1$ is constant. As you go around the hyperbola, $d_2$ and $d_1$ increase/decrease at the same pace. Because the waves' speeds are $d_2$ and $d_1$, the paths of their intersections will follow a hyperbola's path.