Hi there,

I am struggling a great deal with this problem, the main issue being where/how to start.

For failsafe long distance communications, the military still uses short-wave radio signals. Thesesignals are bounced off the atmosphere and can skip even the distance between continents.

For a given inclination α and for a givenatmospheric setting (specified below),your task is to find the the skip distance.

1. Physics Department: The radio signal propagates along a curve C that minimises thetime of flight for fixed start and end points (Fermat’s principle).The speed of the signal is given by c = c0/n(x, y), where c0 = 3 × 108 m/s is the vacuumspeed of light and n(x, y) is the position dependent refractive index.

1. Met Department: The refractive index can be modeled by n(x, y) = 1 + ∆n exp{−y/h}with ∆n = 0.0027 and h = 25km.I had an idea that if

c =c0/1 + ∆n exp{−y/h}was used as Lagrange function I could then use the Beltrami Identity. However, this is where I'm stuck, I'm not sure how to get it in that form. Any advice would be much appreciated.