2.57 NanotoMacro Transport Processes
Fall 2004
Lecture 16
Review of last lecture
2
Q
1>2
1
1
2
1
2
Q
2>1
z
∑
vEf
τ
1
→
2
JJG
1
v
x
>
0
In Lecture 14, we have discussed the energy exchange between two points. We have
spent time on calculating the transmissivity
for cases in the right two figures. The
12
→
velocity v
1
is the group velocity, not the phase velocity determined by
v
=
ω
/
k
(derived
x
tk
x
.)
from constant phase
Φ=
−
x
Consider a plane traveling along xdirection
A
e
−
i
(
t
−
kx
)
.
The velocity
v
=
/
k
is NOT the speed of signal or energy propagation. We see that the
x
plane wave represented by above equation extends from minus infinite to plus infinite in
both time and space. There is no start or finish and it does not represent any meaningful
signal. In practice, a signal has a starting point and an ending point in time. Let’s suppose
that a harmonic signal at frequency
ω
o
is generated during a time period [0,t
o
], as shown
in the following figure (a).
∞
→
t
−
∞
→
t
ω
o
t
o
o
t
2
π
Amplitude
Frequency
(a)
(b)
(c)
Such a finitetime harmonic signal can be decomposed through Fourier series into the
summation of truly plane waves with time extending from minus infinite to plus infinite,
as shown in figure (b). The frequencies of these plane waves are centered around
ω
o
and
their amplitudes decay as frequency moves away from
ω
o
, as illustrated in figure (c).
2.57 Fall 2004 – Lecture 16
1
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View Full DocumentOne can better understand these pictures by actually carrying out the Fourier expansion
(
∑
a
sin
2
π
n
t
). Because each of the plane waves in such a series expansion is at a
n
T
frequency slight different from the central frequency
ω
o
, it also has a corresponding
wavevector that is different from k
o
, as determined by the dispersion relation between
ω(
k). The subsequent propagation of the signal can be obtained from tracing the spatial
evolution of all these Fourier components as a function of time.
For simplicity, let’s consider that the signal is an electromagnetic wave.
We pick up only
two of the Fourier components and consider their superposition, one at frequency
ω
ο
−∆ω/2
and another at frequency
ω
ο
+
∆ω/2
propagating along positive xdirection
[figure (a)]. The superposition of these two waves gives the electric fields as
y
(
,
Ex
t
)
=
a
e
x
p
{
−
i
⎡
(
ω
−
∆
)
t
−(
k
o
−
∆
k
x
⎤
}
+
a
e
x
{
−
i
⎡
(
+
∆
)
t
−
(
k
−
∆
⎤
}
⎣
o
)
⎦
⎣
o
o
)
⎦
=
2c
o
s
(
∆ω
•
t
−
∆
kx
)
e
x
⎡−
i
(
k
)
⎤
a
•
⎣
o
−
o
⎦
ω
o

∆ω
ω
o
+
∆ω
(b)
(a)
(c)
The electric field is schematically shown in above figures. There appears to be two waves,
one is the carrier wave at central frequency
ω
ο
(
term
exp
i
(
k
)
⎦
⎤
), another is the
⎣
o
−
o
modulation of the carrier wave by a wave at frequency
∆ω
(the amplitude term
a
t
•
y
(
,
o
(
∆ω
•−
∆
)
in
)
)
. The latter one changes much slower compared with
the former one.
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 Spring '10
 WRIGHT,J
 Fundamental physics concepts, wave packet, group velocity, ωo

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