Complex rays for radio waves in an absorbing ionosphere

Ray techniques are used to construct an approximate solution for radio waves propagating in the ionosphere when the loss of energy due to collisions is not negligible. For the time-harmonic problem, the solution is in the form of an asymptotic expansion in a large parameter scaling the operating frequency and the natural frequencies characterizing the ionospheric properties. Complex rays are introduced as the characteristics of the resulting eikonal equation, analytically extended to complex space points. The rapidly varying exponential factor and a series for the wave amplitude are both determined by integrating along the rays. The propagation of narrow-band modulation is also determined by integrating along the same complex rays. It is shown that the nature of the initial disturbance plays a vital role in determining the character of the rays required for the solution of a particular problem.