For a pursuit starting with a non-zero crosswind component there will be at most only one instant when the headwind component is zero. The crosswind component will only be zero if the beacon is reached.
Nahin, referenced in the Wiki page on radiodromes, for a terrestrial coordinate system with the beacon at the origin and the wind acting along the positive y-axis, provides this function [eqn 1.2.8] for the radiodrome which he refers to as the "wind blown plane's path":
The wind speed is
w and the true airspeed is
v. At
x =
a, where the path crosses the x-axis, the heading is orthogonal to the wind. If
n < 1, i.e. if the beacon can be reached, then the y-axis is an asymptote for y'(
x). Therefore the beacon is reached directly from the downwind position unless it is initially approached directly from the upwind position.
KayPam,
That is a famous variational problem of similar vintage and one that took Newton half a day to solve.