where

(662) |

Now,

(663) | |||

(664) | |||

(665) | |||

(666) |

Thus, the ray equations, (658)-(660), yield

Note that the frequency of a radio pulse does not change as it propagates through the ionosphere, provided that does not vary in time. It is clear, from Eqs. (667)-(669), and the fact that , that a radio pulse which starts off at ground level propagating in the - plane, say, will continue to propagate in this plane.

For pulse propagation in the - plane, we have

The dispersion relation (661) yields

where is the refractive index.

We assume that at , which is equivalent to the
reasonable assumption that the atmosphere is non-ionized at ground level.
It follows from Eq. (672) that

According to Eq. (671), the plus sign corresponds to the upward trajectory of the pulse, whereas the minus sign corresponds to the downward trajectory. Finally, Eqs. (670), (671), (674), and (675) yield the equations of motion of the pulse:

(676) | |||

(677) |

The pulse attains its maximum altitude, , when

The total distance traveled by the pulse (

(679) |

In the limit in which the radio pulse is launched vertically (*i.e.*,
) into the ionosphere, the turning point condition (678) reduces to
that characteristic of a cutoff (*i.e.*, ). The WKB turning point
described in Eq. (678) is a generalization of the conventional turning point,
which occurs when changes sign. Here, changes sign, whilst
and are constrained by symmetry (*i.e.*, is constant,
and is zero).

According to Eqs. (667)-(669) and (673), the equation of motion of the pulse
can also be written

(680) |