CHAPTER
12
PLANE
WAVES AT
BOUNDARIES
AND IN
DISPERSIVE
MEDIA
In Chapter 11, we considered basic electromagnetic wave principles. We learned
how to mathematically represent waves as functions of frequency, medium prop
erties, and electric field orientation. We also learned how to calculate the wave
velocity, attenuation, and power. In this chapter, we consider wave reflection and
transmission at planar boundaries between media having different properties.
Our study will allow any orientation between the wave and boundary, and will
also include the important cases of multiple boundaries. We will also study the
practical case of waves that carry power over a finite band of frequencies, as
would occur, for example, in a modulated carrier. We will consider such waves in
dispersive
media, in which some parameter that affects propagation (permittivity
for example) varies with frequency. The effect of a dispersive medium on a signal
is of great importance, since the signal envelope will change its shape as it
propagates. This can occur to an extent that at the receiving end, detection
and
faithful
representation
of
the
original
signal
become
problematic.
Dispersion thus becomes an important limiting factor in allowable propagation
distances, in a similar way that we found to be true for attenuation.
387
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12.1
REFLECTION OF UNIFORM PLANE
WAVES AT NORMAL INCIDENCE
In this section we will consider the phenomenon of reflection which occurs when
a uniform plane wave is incident on the boundary between regions composed of
two different materials. The treatment is specialized to the case of
normal inci
dence
, in which the wave propagation direction is perpendicular to the boundary.
In later sections we remove this restriction. We shall establish expressions for the
wave that is reflected from the interface and for that which is transmitted from
one region into the other. These results will be directly applicable to impedance
matching problems in ordinary transmission lines as well as to waveguides and
other more exotic transmission systems.
We again assume that we have only a single vector component of the
electric field intensity. Referring to Fig. 12.1, we define region l
±
1
; ²
1
as the
halfspace for which
z
<
0; region 2
±
2
; ²
2
is the halfspace for which
z
>
0.
Initially we establish the wave traveling in the
z
direction in region l,
E
x
1
z
;
t
E
x
10
e
±
³
1
z
cos
!
t
±
´
1
z
or
E
xs
1
E
x
10
e
±
jk
1
z
1
where we take
E
x
10
as real. The subscript 1 identifies the region and the super
script + indicates a positively traveling wave. Associated with
E
x
1
z
;
t
is a
magnetic field
H
ys
1
1
µ
1
E
x
10
e
±
jk
1
z
2
where
k
1
and
µ
1
are complex unless
±
00
1
(or
¶
1
) is zero. This uniform plane wave in
region l which is traveling toward the boundary surface at
z
0 is called the
incident
wave. Since the direction of propagation of the incident wave is perpen
dicular to the boundary plane, we describe it as normal incidence.
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 Spring '10
 Wilton
 Electromagnet, Frequency, Wave mechanics, Wave propagation, phase velocity

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