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19 d’Alembert wave solutions, radiation from
current sheets, Poynting theorem
•
d’Alembert wave solutions of Maxwell’s equations for homogeneous and
sourcefree regions obtained in the last lecture having the forms
E
,
H
∝
f
(
t
∓
z
v
)
are classiFed as
uniform planeTEM waves
.
–
TEM stands for
T
ransverse
E
lectro
M
agnetic, and the reason for
this designation is:
x
y
z
H
E
=ˆ
xf
(
t

z
v
)
E
×
H
x
y
z
H
E
xf
(
t
+
z
v
)
E
×
H
viable solutions satisfying
∇ ·
E
=
H
=0
conditions have their
E
and
H
vectors
transverse
to the
direction of propagation
which always
coincides with the direction of vector
S
≡
E
×
H
known as
Poynting
vector
— more on this later on.
Poynting vector
E
×
H
–
d’Alembert wave solutions such as
E
xf
(
t

z
v
)
and
H
y
f
(
t

z
v
)
η
are also designated as
uniform plane
waves because:
these waveFelds are constant (have the same vector value) at
planes
of constant phase
, e.g., on planes deFned by
t

z
v
=
const.,
1
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View Full Documentwhich are planes transverse to the propagation direction (direction of
vector
E
×
H
).
Not all waves solutions of Maxwell’s equations are uniform plane — for in
stance nonuniform TEM waves with spherical surfaces of constant phase are
ubiquitous, but they will be examined later on (in ECE 450, mainly).
After the next set of examples we will examine how uniform plane waves
can be radiated by inFnite planes of surface currents. By contrast, spherical
waves are produced by compact antennas having Fnite dimensions.
Example 1:
Let
E
=ˆ
x
±
(
t

y/c
τ
)
be a wave solution in free space where
±
(
t
τ
)
is a triangular waveform of duration
τ
peaking at
t
=0
(deFned in ECE 210). We will next provide two di±erent
solutions demonstrating how the wave Feld
B
accompanying
E
can be found.
Solution 1:
We recognize the given wave Feld
E
as a TEM uniform plane wave travel
ing in
y
direction given the
t

y/c
dependence of phase. Consequently, we obtain
H
by dividing
E
with
η
=
η
o
and rotating it by
90
◦
from
ˆ
x
direction to coalign
it with
E
×
H
vector. As a result,
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 Summer '10
 KUDEKI

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