21 Monochromatic waves 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(
ω
t
)
,
namely the
monochromatic waves
A
cos[
ω
(
t
∓
z
v
)] =
A
cos(
ω
t
∓
β
z
)
,
with amplitudes
A
where
β
≡
ω
v
=
ω
√
μ
can be called
wavenumber
in analogy with
wavefrequency
ω
.
T
=
2
π
ω
cos(
ω
t
)
t
1
1
Period
λ
=
2
π
β
cos(
β
z
)
z
1
1
Wavelength
–
As depicted in the margin, monochromatic solutions
A
cos(
ω
t
∓
β
z
)
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 specific
dispersion relations
such as:
T
=
2
π
ω
cos(
ω
t
)
t
1
1
Period
λ
=
2
π
β
cos(
β
z
)
z
1
1
Wavelength
Dispersion relations
between
wavefrequency
ω
and
wavenumber
β
determine the
propagation
veloc
ity
v
=
ω
β
=
λ
f
for all types of
wave motions.
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 Spring '08
 Kim
 Cos, Ω, group velocity, Wave propagation, HY, ej Ωt

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