Ionospheric Radio Wave Propagation
To a first approximation, the Earth's ionosphere consists of an unmagnetized,
horizontally stratified, partially ionized gas (Budden 1985). The dispersion
relation for the electromagnetic plasma wave takes the form [see Equation (5.98)]

(6.109) 
where

(6.110) 
Here, is the density of free electrons in the ionosphere, and
is a coordinate that measures height above the surface of the Earth.
(The curvature of the Earth, the Earth's magnetic field, and collisions, are neglected in the following analysis.)
Now,


(6.111) 


(6.112) 


(6.113) 


(6.114) 
Thus, the ray equations, (6.106)–(6.108), yield
Evidently, the frequency of a radio pulse does not change as it
propagates through the ionosphere, provided that does not vary in time.
Furthermore, it follows from Equations (6.115)–(6.117), and the fact that
, that
a radio pulse that 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 (6.109) yields

(6.121) 
where is the refractive index.
Let us assume that at , which is equivalent to the
reasonable assumption that the atmosphere is nonionized at ground level.
It follows from Equation (6.120) that

(6.122) 
where is the sine of the angle of incidence of the pulse, with respect to
the vertical axis, at ground level. Equations (6.121) and (6.122) yield

(6.123) 
According to Equation (6.119), the plus sign corresponds to the upward trajectory
of the pulse, whereas the minus sign corresponds to the downward trajectory.
Finally, Equations (6.118), (6.119), (6.122), and (6.123)
yield the equations of motion of the pulse:
The pulse attains its maximum altitude, , when

(6.126) 
The total distance traveled by the pulse (i.e., the distance
from its launch
point to the point where it intersects the Earth's surface again) is

(6.127) 
In the limit in which the radio pulse is launched vertically (i.e.,
) into the ionosphere, the turning point condition (6.126) reduces to
that characteristic of a cutoff (i.e., ). The WKB turning point
described in Equation (6.126) is a generalization of the conventional turning point,
which occurs when changes sign. Here, changes sign, while
and are constrained by symmetry (i.e., is constant,
and is zero).
According to Equations (6.115)–(6.117) and (6.121), the equation of motion of the pulse
can also be written

(6.128) 
It follows that the trajectory of the pulse is the same as that of a
particle moving in the gravitational potential
. Thus, if
decreases linearly with increasing height above the
ground [which is the case if increases linearly with ] then the
trajectory of the pulse is a parabola.