20 Monochromatic wave solutions and phasor
notation
•
Recall that we reached the travelingwave d’Alembert solutions
E
,
H
∝
f
(
t
∓
z
v
)
via the superposition of timeshifted and amplitudescaled versions of
f
(
t
) = cos(
ωt
)
,
namely the
monochromatic waves
A
cos[
ω
(
t
∓
z
v
)] =
A
cos(
ωt
∓
βz
)
,
with amplitudes
A
where
β
≡
ω
v
=
ω
√
μ±
can be called
wavenumber
in analogy with
wavefrequency
ω
.
T
=
2
π
ω
cos(
ωt
)
t
1
1
Period
λ
=
2
π
β
cos(
βz
)
z
1
1
Wavelength
–
As depicted in the margin, monochromatic solutions
A
cos(
ωt
∓
βz
)
are periodic in position and time, with the
wavenumber
β
being
essentially a
spatialfrequency
, the spatial counterpart of
ω
.
This is an important point that you should try to understand
well — it has implications for signal processing courses related
to images and vision.
1
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In general
, monochromatic solutions of
1D waveequations
ob
tained in various branches of science and engineering can all be rep
resented in the same format as above in terms of wavefrequency
/ wavewavenumber pairs
ω
and
β
having a ratio
v
≡
ω
β
recognized as the
wavespeed
and speciFc
dispersion relations
such as:
T
=
2
π
ω
cos(
ωt
)
t
1
1
Period
λ
=
2
π
β
cos(
βz
)
z
1
1
Wavelength
Dispersion relations
between
wavefrequency
ω
and
wavenumber
β
determine the
propagation
veloc
ity
v
=
ω
β
=
λf
for all types of
wave motions.
1.
TEM waves
in perfect dielectrics:
β
=
ω
√
μ±,
2.
Acoustic waves
in monoatomic gases with temperature
T
(K)
and atomic mass
m
(kg):
β
=
ω
±
m
5
3
KT
,
3. TEM waves in collisionless
plasmas
(ionized gases) with plasma
frequency
ω
p
=
²
Ne
2
m±
o
:
β
=
1
c
²
ω
2

ω
2
p
.
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 Summer '10
 KUDEKI
 Cos, group velocity, Wave propagation, dispersion relation, TEM waves, Monochromatic wave solutions

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