** Next:** Radiation and Scattering
** Up:** Wave Propagation in Inhomogeneous
** Previous:** Jeffries Connection Formula

- Consider an electromagnetic wave propagating through a
nonuniform dielectric medium whose dielectric constant
is a
function of
. Demonstrate that the associated
wave equations take the form

- Suppose that a light-ray is incident on the front (air/glass) interface of a uniform pane
of glass of refractive index
at the Brewster angle. Demonstrate that the refracted ray
is also incident on the rear (glass/air) interface of the pane at the Brewster
angle.

- Consider an electromagnetic wave obliquely incident on a plane
boundary between two transparent magnetic media of permeabilities
and
. Find the coefficients of reflection and transmission
as functions of the angle of incidence for the wave polarizations in
which all electric fields are parallel to the boundary and all magnetic
fields are parallel to the boundary. Is there a Brewster angle? If so, what is it?
Is it possible to obtain total reflection? If so, what is the critical angle of
incidence required to obtain total reflection?

- A medium is such that the product of the phase and group
velocities of electromagnetic waves is equal to
at all wave
frequencies. Demonstrate that the dispersion relation for
electromagnetic waves takes the form

where
is a constant.

- Demonstrate that if the equivalent height of reflection in the ionosphere varies with the angular frequency of the wave as

where
,
, and
are positive constants, then
for
, and

for
. Here,
is a Gamma function.

- Suppose that the refractive index,
, of the ionosphere is given by
for
,
and
for
, where
and
are positive constants, and the Earth's magnetic field and curvature are both neglected.
Here,
measures altitude above the Earth's surface.
- A point transmitter sends up a wave packet at an angle
to the vertical. Show that the packet returns to Earth a
distance

from the transmitter. Demonstrate that if
then for some values of
the previous equation is satisfied
by three different values of
. In other words, wave packets can travel from the transmitter to the receiver via one of
three different paths. Show that the critical case
corresponds to
and
.
- A point
radio transmitter emits a pulse of radio waves uniformly in all directions. Show that the pulse first returns to the Earth a
distance
from the transmitter, provided that
.

** Next:** Radiation and Scattering
** Up:** Wave Propagation in Inhomogeneous
** Previous:** Jeffries Connection Formula
Richard Fitzpatrick
2014-06-27