CHAPTER
11
THE
UNIFORM
PLANE
WAVE
In this chapter we shall apply Maxwell's equations to introduce the fundamental
theory of wave motion. The uniform plane represents one of the simplest appli
cations of Maxwell's equations, and yet it is of profound importance, since it is a
basic entity by which energy is propagated. We shall explore the physical pro
cesses that determine the speed of propagation and the extent to which attenua
tion may occur. We shall derive and make use of the Poynting theorem to find
the power carried by a wave. Finally, we shall learn how to describe wave
polarization. This chapter is the foundation for our explorations in later chapters
which will include wave reflection, basic transmission line and waveguiding con
cepts, and wave generation through antennas.
11.1
WAVE PROPAGATION IN FREE SPACE
As we indicated in our discussion of boundary conditions in the previous chap
ter, the solution of Maxwell's equations without the application of any boundary
conditions at all represents a very special type of problem. Although we restrict
our attention to a solution in rectangular coordinates, it may seem even then that
we are solving several different problems as we consider various special cases in
this chapter. Solutions are obtained first for freespace conditions, then for
perfect dielectrics, next for lossy dielectrics, and finally for the good conductor.
We do this to take advantage of the approximations that are applicable to each
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special case and to emphasize the special characteristics of wave propagation in
these media, but it is not necessary to use a separate treatment; it is possible (and
not very difficult) to solve the general problem once and for all.
To consider wave motion in free space first, Maxwell's equations may be
written in terms of
E
and
H
only as
r ±
H
±
0
@
E
@
t
1
r ±
E
²
²
0
@
H
@
t
2
r ³
E
0
3
r ³
H
0
4
Now let us see whether wave motion can be inferred from these four equa
tions without actually solving them. The first equation states that if
E
is changing
with time at some point, then
H
has curl at that point and thus can be considered
as forming a small closed loop linking the changing
E
field. Also, if
E
is changing
with time, then
H
will in general also change with time, although not necessarily
in the same way. Next, we see from the second equation that this changing
H
produces an electric field which forms small closed loops about the
H
lines. We
now have once more a changing electric field, our original hypothesis, but this
field is present a small distance away from the point of the original disturbance.
We might guess (correctly) that the velocity with which the effect moves away
from the original point is the velocity of light, but this must be checked by a more
quantitative examination of Maxwell's equations.
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 Spring '10
 Wilton
 Electromagnet, Magnetic Field, Engineering Electromagnetics, uniform plane wave, Exs

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