1
00:00:02,450 --> 00:00:04,820
The following content is
provided under a Creative
2
00:00:04,820 --> 00:00:06,210
Commons license.
3
00:00:06,210 --> 00:00:08,420
Your support will help
MIT OpenCourseWare
4
00:00:08,420 --> 00:00:12,510
continue to offer high quality
educational resources for free.
5
00:00:12,510 --> 00:00:15,050
To make a donation, or to
view additional materials
6
00:00:15,050 --> 00:00:19,010
from hundreds of MIT courses,
visit MIT OpenCourseWare
7
00:00:19,010 --> 00:00:20,316
at ocw.mit.edu.
8
00:00:24,390 --> 00:00:27,660
YEN-JIE LEE: OK, so
welcome back to 803.
9
00:00:27,660 --> 00:00:30,280
Happy to see you again.
10
00:00:30,280 --> 00:00:32,564
So today, we are going to
continue our discussion
11
00:00:32,564 --> 00:00:34,770
of dispersive medium.
12
00:00:34,770 --> 00:00:36,900
And there are two
questions we are
13
00:00:36,900 --> 00:00:39,420
going to ask in
this lecture, and we
14
00:00:39,420 --> 00:00:44,190
will answer also these two
questions in this lecture.
15
00:00:44,190 --> 00:00:47,440
So just to warn you in
advance, this lecture
16
00:00:47,440 --> 00:00:53,640
will have a lot of mathematics,
so fasten the seatbelt
17
00:00:53,640 --> 00:00:57,380
and follow me.
18
00:00:57,380 --> 00:01:01,320
And stop me any time
you don't feel like you
19
00:01:01,320 --> 00:01:04,560
know you understand something.
20
00:01:04,560 --> 00:01:06,840
So let's get started.
21
00:01:06,840 --> 00:01:11,790
OK, so today, we are going
to talk about phenomena
22
00:01:11,790 --> 00:01:13,890
related to dispersion.
23
00:01:13,890 --> 00:01:16,740
And last time, we
started a discussion
24
00:01:16,740 --> 00:01:19,980
about how to send
information from one
25
00:01:19,980 --> 00:01:21,600
place to the other place right.
26
00:01:21,600 --> 00:01:26,550
So what we should be before
was to send square pulse.
27
00:01:26,550 --> 00:01:33,210
So if I do have a machine which
can produce a square pulse,
28
00:01:33,210 --> 00:01:36,770
then I can define
something like this.
29
00:01:36,770 --> 00:01:40,350
So over some ratio,
which I set, I
30
00:01:40,350 --> 00:01:43,470
can actually separate 0 and 1.
31
00:01:43,470 --> 00:01:45,700
So if I have a pulse
which is actually
32
00:01:45,700 --> 00:01:49,110
having an amplitude greater
than some threshold and I say,
33
00:01:49,110 --> 00:01:52,120
OK, I've got 1, and
if it's actually
34
00:01:52,120 --> 00:01:55,230
below some straight line,
say, OK, I've got a 0.
35
00:01:55,230 --> 00:01:58,350
And with that way, we
actually can send information
36
00:01:58,350 --> 00:02:00,570
from one place to
the other place.
37
00:02:00,570 --> 00:02:03,770
So that sounds really nice.
38
00:02:03,770 --> 00:02:09,130
However, if we work on
a dispersive medium,
39
00:02:09,130 --> 00:02:11,580
which is really very common--
40
00:02:11,580 --> 00:02:15,980
for example, light
and gas is actually--
41
00:02:15,980 --> 00:02:21,810
not all the lights with
different wavelengths are
42
00:02:21,810 --> 00:02:26,340
traveling at the
same speed, and also,
43
00:02:26,340 --> 00:02:30,906
as you've seen before in
the p-set, deep water,
44
00:02:30,906 --> 00:02:36,765
and also the strings,
considering a realistic string
45
00:02:36,765 --> 00:02:40,380
with stiffness, et
cetera, et cetera--
46
00:02:40,380 --> 00:02:47,490
to the wavelength of the
input wave is going to affect
47
00:02:47,490 --> 00:02:50,340
the speed of this
travelling wave.
48
00:02:50,340 --> 00:02:54,540
So in short, the speed
of the wave propagation
49
00:02:54,540 --> 00:02:59,230
in a dispersive medium will
depend on the wavelengths
50
00:02:59,230 --> 00:03:01,410
of this wave.
51
00:03:01,410 --> 00:03:03,780
So that brings us
a lot of trouble
52
00:03:03,780 --> 00:03:08,310
because, for example, here we
are trying to send a Gaussian
53
00:03:08,310 --> 00:03:13,260
pulse through the
medium, but after a while
54
00:03:13,260 --> 00:03:17,130
this pulse actually becomes
wider and wider because
55
00:03:17,130 --> 00:03:18,240
of the dispersion.
56
00:03:18,240 --> 00:03:21,810
Because all the components
with different wavelengths
57
00:03:21,810 --> 00:03:26,140
which actually construct
this narrow pulse,
58
00:03:26,140 --> 00:03:29,920
actually are traveling
at different speeds.
59
00:03:29,920 --> 00:03:32,670
Therefore, if you
wait long enough,
60
00:03:32,670 --> 00:03:40,300
all the different frequencies,
or all the different frequency
61
00:03:40,300 --> 00:03:43,290
harmonic waves are travel
at different speeds,
62
00:03:43,290 --> 00:03:46,080
therefore you get
the, dispersion,
63
00:03:46,080 --> 00:03:51,300
which results in a much
wider pulse in the end.
64
00:03:51,300 --> 00:03:52,950
And at some point,
this pulse is going
65
00:03:52,950 --> 00:03:56,670
to be really wide,
such that it's actually
66
00:03:56,670 --> 00:04:02,460
going to be very difficult
to separate 0 from 1.
67
00:04:02,460 --> 00:04:04,110
So that's the problem.
68
00:04:04,110 --> 00:04:07,260
And we also did some
simulations with computer.
69
00:04:07,260 --> 00:04:12,860
We do see this behavior also
in our computer simulation.
70
00:04:12,860 --> 00:04:16,839
If I put in triangular pulse
and allow it to evolve,
71
00:04:16,839 --> 00:04:19,680
and like what we did
before, we assume
72
00:04:19,680 --> 00:04:24,070
that there's a stiffness
in this string system.
73
00:04:24,070 --> 00:04:27,510
And you will see that,
OK, as a function of time,
74
00:04:27,510 --> 00:04:31,290
this part is now longer
a triangular shape,
75
00:04:31,290 --> 00:04:35,170
but you have a very
complicated structure.
76
00:04:35,170 --> 00:04:37,710
So that is actually
a problem we are
77
00:04:37,710 --> 00:04:40,920
going to try to solve today.
78
00:04:40,920 --> 00:04:44,250
And during that discussion
last time, in the lecture,
79
00:04:44,250 --> 00:04:49,360
we also introduced dispersion
relation omega k and also tried
80
00:04:49,360 --> 00:04:55,920
to overlap two travelling
waves with similar wavelengths.
81
00:04:55,920 --> 00:04:59,520
And that would give
you beat phenomenon.
82
00:04:59,520 --> 00:05:02,852
That probably doesn't
surprise you any more.
83
00:05:02,852 --> 00:05:05,820
As you can see
from this example,
84
00:05:05,820 --> 00:05:07,770
you have the beat
phenomenon, and you
85
00:05:07,770 --> 00:05:12,640
can see the amplitude is
actually variating slowly,
86
00:05:12,640 --> 00:05:15,210
the function of position.
87
00:05:15,210 --> 00:05:18,400
And if you follow the red
point, which is actually
88
00:05:18,400 --> 00:05:23,220
associated with one of the
peak, in the structure called
89
00:05:23,220 --> 00:05:26,690
carrier, OK, it's
actually moving
90
00:05:26,690 --> 00:05:32,390
at the phase velocity, the P,
which we introduced last time.
91
00:05:32,390 --> 00:05:37,540
The formula for BP, which is
actually the speed of harmonic,
92
00:05:37,540 --> 00:05:42,935
oscillating travelling wave is
actually defined as vp equal
93
00:05:42,935 --> 00:05:48,350
to omega over k,
and the green point,
94
00:05:48,350 --> 00:05:53,540
which actually always at the
minima of the distribution,
95
00:05:53,540 --> 00:05:58,730
which is actually associated
with the speed of the envelope.
96
00:05:58,730 --> 00:06:01,520
You can see that,
indeed, it actually
97
00:06:01,520 --> 00:06:04,310
can move at different speeds.
98
00:06:04,310 --> 00:06:07,460
It depends on the
dispersion relation omega
99
00:06:07,460 --> 00:06:10,250
as a function of k you
have in this system.
100
00:06:10,250 --> 00:06:16,100
And we call the speed
of these envelope,
101
00:06:16,100 --> 00:06:20,120
which we construct from these
two overlapping travelling
102
00:06:20,120 --> 00:06:21,630
waves to be--
103
00:06:21,630 --> 00:06:25,130
we call it group velocity.
104
00:06:25,130 --> 00:06:27,130
And the definition
of group velocity
105
00:06:27,130 --> 00:06:31,310
is vg equal to d omega dk.
106
00:06:31,310 --> 00:06:35,480
So that's what we have
learned last time.
107
00:06:35,480 --> 00:06:41,720
OK, you may ask, OK, what
do I mean by group velocity?
108
00:06:41,720 --> 00:06:45,500
And can I use it
beyond what we have
109
00:06:45,500 --> 00:06:48,830
done for the beat phenomena.
110
00:06:48,830 --> 00:06:52,240
But what do I mean
by group velocity?
111
00:06:52,240 --> 00:06:54,270
Is that really useful,
and it's actually
112
00:06:54,270 --> 00:06:57,650
which part of the structure
I was talking about.
113
00:06:57,650 --> 00:07:02,780
So in that case of two
overlapping progressing waves
114
00:07:02,780 --> 00:07:06,410
with similar length
or similar frequency,
115
00:07:06,410 --> 00:07:08,360
when we see that the
group velocity actually
116
00:07:08,360 --> 00:07:12,080
present the speed of
the envelope, right?
117
00:07:12,080 --> 00:07:14,070
Can we actually
learn something more
118
00:07:14,070 --> 00:07:17,570
general about group velocity?
119
00:07:17,570 --> 00:07:19,430
The second question
which we are asking
120
00:07:19,430 --> 00:07:24,800
is, OK, now we have this
problem of dispersion.
121
00:07:24,800 --> 00:07:29,690
This square pulse is going to
be something which is really
122
00:07:29,690 --> 00:07:33,070
wide after some period of time.
123
00:07:33,070 --> 00:07:37,820
So that's clearly a problem,
and how do we actually
124
00:07:37,820 --> 00:07:40,650
solve this problem, and how do
we actually send information
125
00:07:40,650 --> 00:07:45,620
like, for example, music
over a large distance
126
00:07:45,620 --> 00:07:47,660
from one place to another place.
127
00:07:47,660 --> 00:07:49,610
So that's essentially
what we're going
128
00:07:49,610 --> 00:07:53,360
to try to understand today.
129
00:07:53,360 --> 00:07:59,150
So let's start with an
infinitely long string.
130
00:08:02,000 --> 00:08:05,780
And this string is actually very
long, and this began from here,
131
00:08:05,780 --> 00:08:10,220
and it goes to some place which
is really, really far away.
132
00:08:10,220 --> 00:08:12,920
And, of course, as
usually, I can actually
133
00:08:12,920 --> 00:08:17,580
hold one end of this
string and shake it a bit,
134
00:08:17,580 --> 00:08:21,420
then I can actually
create some kind of pulse
135
00:08:21,420 --> 00:08:25,420
which is going to
travel along this string
136
00:08:25,420 --> 00:08:28,340
towards a positive x direction.
137
00:08:28,340 --> 00:08:31,150
In this case, I defined
the x coordinate
138
00:08:31,150 --> 00:08:33,360
will be pointing to a
right hand side, and thus
139
00:08:33,360 --> 00:08:35,450
the positive direction.
140
00:08:35,450 --> 00:08:38,690
So of course I can hold this
string, and I just shake it,
141
00:08:38,690 --> 00:08:43,700
and I would prepare a pulse
on this medium, which is
142
00:08:43,700 --> 00:08:46,520
a string with constant tension.
143
00:08:46,520 --> 00:08:50,750
So I can describe the motion--
144
00:08:50,750 --> 00:08:55,060
you can describe the motion of
Yen-Jie's hand by a function.
145
00:08:55,060 --> 00:09:00,210
So you can say, OK, Yen-Jie is
somehow doing a really nice job
146
00:09:00,210 --> 00:09:05,510
and oscillating at
constant frequency.
147
00:09:05,510 --> 00:09:09,620
Like I can say, OK, Yen-Jie
is shaking this thing
148
00:09:09,620 --> 00:09:12,120
to produce a harmonic
wave, for example.
149
00:09:12,120 --> 00:09:15,890
And that, I can actually
describe the motion of the hand
150
00:09:15,890 --> 00:09:17,440
by f of t.
151
00:09:17,440 --> 00:09:19,400
That's very good.
152
00:09:19,400 --> 00:09:24,690
And from what we have
learned in the last lecture,
153
00:09:24,690 --> 00:09:27,725
we've found that,
basically, waves,
154
00:09:27,725 --> 00:09:31,850
harmonic waves with
different frequency,
155
00:09:31,850 --> 00:09:33,670
or with different
wavelengths, are
156
00:09:33,670 --> 00:09:36,950
traveling at different speeds.
157
00:09:36,950 --> 00:09:43,420
Therefore, we would like to
actually decompose the motion
158
00:09:43,420 --> 00:09:47,780
of Yen-Jie's hand into
many, many harmonic waves--
159
00:09:47,780 --> 00:09:51,740
then attack them one by one,
to follow them one by one,
160
00:09:51,740 --> 00:09:54,020
then I can solve this problem.
161
00:09:54,020 --> 00:09:56,540
So that's actually what
we are going to do.
162
00:09:56,540 --> 00:10:01,400
And that will involve some
math, which we would follow
163
00:10:01,400 --> 00:10:03,830
from the math department.
164
00:10:03,830 --> 00:10:07,430
And before that, I would like to
introduce the imitation first.
165
00:10:07,430 --> 00:10:10,970
As I said f of t is
actually the displacement
166
00:10:10,970 --> 00:10:14,540
as a function of time
as x is equal to 0.
167
00:10:14,540 --> 00:10:16,504
So basically, I'm
holding this string,
168
00:10:16,504 --> 00:10:19,820
and I move things up and
down, so that, actually, I
169
00:10:19,820 --> 00:10:23,490
move this string away from the
equilibrium positive, which
170
00:10:23,490 --> 00:10:27,230
is actually y equal to 0.
171
00:10:27,230 --> 00:10:28,690
Then what is going to happen?
172
00:10:28,690 --> 00:10:30,230
What is going to
happen is that I'm
173
00:10:30,230 --> 00:10:35,270
going to produce
some kind of pulse,
174
00:10:35,270 --> 00:10:38,030
and this pulse, I can
actually describe it
175
00:10:38,030 --> 00:10:42,470
by a function, which
is psi x and p,
176
00:10:42,470 --> 00:10:46,460
this psi is actually
describing the displacement
177
00:10:46,460 --> 00:10:50,030
as a function of x,
and as a function of t.
178
00:10:50,030 --> 00:10:52,940
Apparently, if you
put x equals to zero,
179
00:10:52,940 --> 00:10:56,220
then you go back
to f of t, right?
180
00:10:56,220 --> 00:10:58,640
Basically that's the idea.
181
00:10:58,640 --> 00:11:00,410
OK.
182
00:11:00,410 --> 00:11:06,770
So what we have learned
before we introduce
183
00:11:06,770 --> 00:11:20,890
dispersive medium is that, if
I have a non-dispersive medium,
184
00:11:20,890 --> 00:11:24,080
OK, if I have a
non-dispersive medium,
185
00:11:24,080 --> 00:11:29,300
then things are pretty
simple because omega over K
186
00:11:29,300 --> 00:11:33,870
is actually a constant, which
is the phase velocity, vp.
187
00:11:33,870 --> 00:11:39,355
And omega is actually
just equal to vp times k.
188
00:11:42,610 --> 00:11:46,390
That means, no matter
what kind of wavelength
189
00:11:46,390 --> 00:11:49,600
we are talking about, no matter
what kind of angular frequency
190
00:11:49,600 --> 00:11:55,030
we are talking about,
harmonic progressing wave
191
00:11:55,030 --> 00:11:58,900
is going to travel
at the speed of Vp.
192
00:11:58,900 --> 00:12:02,590
No matter what's the frequency,
or what's the wavelength.
193
00:12:02,590 --> 00:12:05,140
So that makes our
life much simpler
194
00:12:05,140 --> 00:12:07,870
when we work on
non-dispersive medium.
195
00:12:07,870 --> 00:12:11,820
In this case, if I have
a non-dispersive medium,
196
00:12:11,820 --> 00:12:15,670
then psi would be equal to--
197
00:12:15,670 --> 00:12:18,730
maybe I write it here--
198
00:12:18,730 --> 00:12:21,670
if I have non-dispersive
medium where,
199
00:12:21,670 --> 00:12:25,090
no matter what
kind of frequency,
200
00:12:25,090 --> 00:12:27,690
the speed of the
harmonic traveling wave
201
00:12:27,690 --> 00:12:31,210
is a constant, which is actually
Vp, I can write down psi
202
00:12:31,210 --> 00:12:43,420
x t to could be equal to
f of t minus x over v.
203
00:12:43,420 --> 00:12:45,640
Just remember f is
actually describing
204
00:12:45,640 --> 00:12:49,240
how I shake one
end of the string,
205
00:12:49,240 --> 00:12:51,850
and, basically you
can see that ha!
206
00:12:51,850 --> 00:12:54,800
What is happening is
that my hand is actually
207
00:12:54,800 --> 00:13:00,380
generating the shape of the
pulse as a function of x,
208
00:13:00,380 --> 00:13:02,650
as a function of
time, and it can
209
00:13:02,650 --> 00:13:08,410
be described by a really
simple formula here.
210
00:13:08,410 --> 00:13:11,680
So this is actually really
nice for non-dispersive.
211
00:13:11,680 --> 00:13:17,050
As I introduced before, when we
talk about dispersive medium,
212
00:13:17,050 --> 00:13:27,970
then, if I go to
dispersive, omega
213
00:13:27,970 --> 00:13:34,840
is actually a function of k, and
can be a non-linear function.
214
00:13:34,840 --> 00:13:36,060
So what does that mean?
215
00:13:36,060 --> 00:13:41,020
That means, if I evaluate
vp, which is actually
216
00:13:41,020 --> 00:13:45,100
the phase velocity, which
is the formula there,
217
00:13:45,100 --> 00:13:51,800
this is going to be
omega of k divided by k.
218
00:13:51,800 --> 00:13:57,010
That means BP is going
to be a function of k,
219
00:13:57,010 --> 00:13:59,220
the wavelength-- wave number.
220
00:13:59,220 --> 00:14:02,390
It's not going to be a
constant in general--
221
00:14:02,390 --> 00:14:07,680
unless omega is actually
equal to vp times k,
222
00:14:07,680 --> 00:14:13,300
in general, vp can
actually be some quantity
223
00:14:13,300 --> 00:14:15,900
which is variating
as a function of k.
224
00:14:15,900 --> 00:14:16,540
OK?
225
00:14:16,540 --> 00:14:20,770
Then we have trouble
because that means,
226
00:14:20,770 --> 00:14:24,880
when I produce progressing wave
from the left hand side end,
227
00:14:24,880 --> 00:14:29,560
it's actually made of
many, many harmonic waves,
228
00:14:29,560 --> 00:14:32,410
right, with different
angular frequency.
229
00:14:32,410 --> 00:14:37,240
So I can shake this like
[MAKES NOISE],, different speed.
230
00:14:37,240 --> 00:14:41,740
And I can always decompose
the motion of Yen-Jie's hand
231
00:14:41,740 --> 00:14:44,230
into many, many harmonic waves.
232
00:14:44,230 --> 00:14:49,150
The problem is, all those
harmonic waves are going to be
233
00:14:49,150 --> 00:14:52,690
travelling at different speed.
234
00:14:52,690 --> 00:14:56,190
How do we actually
describe this?
235
00:14:56,190 --> 00:14:58,110
So that's the trouble.
236
00:14:58,110 --> 00:15:02,647
And I was really frustrated
when I think about this problem,
237
00:15:02,647 --> 00:15:04,230
and my friend from
the math department
238
00:15:04,230 --> 00:15:07,440
said, hey, we have solved
this problem a long time ago.
239
00:15:07,440 --> 00:15:10,320
[LAUGHTER]
240
00:15:10,320 --> 00:15:11,820
So this is not the
problem anymore.
241
00:15:11,820 --> 00:15:16,360
And I say, oh, what is the
idea you're talking about?
242
00:15:16,360 --> 00:15:18,600
And they actually
told me that you
243
00:15:18,600 --> 00:15:22,960
should use Fourier transform
to attack this problem.
244
00:15:22,960 --> 00:15:23,990
OK?
245
00:15:23,990 --> 00:15:25,840
This is the idea.
246
00:15:25,840 --> 00:15:30,180
The idea is that I
can now write down
247
00:15:30,180 --> 00:15:35,610
f of t, which is motion
of Yen-Jie's hand,
248
00:15:35,610 --> 00:15:41,550
and this can even returned
as a superposition
249
00:15:41,550 --> 00:15:46,890
of infinite number of waves.
250
00:15:46,890 --> 00:15:49,350
I can integrate
from minus infinity
251
00:15:49,350 --> 00:15:55,270
to infinity, t omega, which
is the angular frequency.
252
00:15:55,270 --> 00:16:03,450
And each contributing wave
has an amplitutde associated
253
00:16:03,450 --> 00:16:08,340
with it, which is, as you
see, is a function omega.
254
00:16:08,340 --> 00:16:11,900
And the actual wave is
actually written in terms
255
00:16:11,900 --> 00:16:16,900
of exponential minus i omega t.
256
00:16:16,900 --> 00:16:21,750
So now, let's actually you look
at this thing really carefully.
257
00:16:21,750 --> 00:16:23,340
What am I doing?
258
00:16:23,340 --> 00:16:29,670
I am saying that I, now, can
shake one end of the string up
259
00:16:29,670 --> 00:16:33,950
and down according to my will.
260
00:16:33,950 --> 00:16:36,840
And, if I do this
for a long time,
261
00:16:36,840 --> 00:16:41,640
I can actually describe
the motion of my hand
262
00:16:41,640 --> 00:16:47,070
by infinite number of harmonic
waves, which is actually
263
00:16:47,070 --> 00:16:50,160
kind of like
exponential i omega t
264
00:16:50,160 --> 00:16:55,380
describing the frequency of
these waves, and each of them
265
00:16:55,380 --> 00:17:00,020
got associated amplitude.
266
00:17:00,020 --> 00:17:02,230
And you may ask,
OK, wait a second,
267
00:17:02,230 --> 00:17:07,240
you call this Fourier transform,
and I have learned that before,
268
00:17:07,240 --> 00:17:09,060
but I learned a
different version.
269
00:17:09,060 --> 00:17:12,450
I learned a version
of cosine and sine?
270
00:17:12,450 --> 00:17:15,329
And what is going on?
271
00:17:15,329 --> 00:17:17,336
Actually, they are all the same.
272
00:17:17,336 --> 00:17:18,960
No matter what you
do, you can actually
273
00:17:18,960 --> 00:17:21,780
also do that with
cosine and sine,
274
00:17:21,780 --> 00:17:25,710
but what I actually found
is that it's actually easier
275
00:17:25,710 --> 00:17:29,610
to deal with exponential
functional form
276
00:17:29,610 --> 00:17:33,720
You can always write
exponential i omega
277
00:17:33,720 --> 00:17:40,470
t in terms of sine and
cosine and absorb the i
278
00:17:40,470 --> 00:17:42,890
into a c omega.
279
00:17:42,890 --> 00:17:45,520
Basically, these things are
identical between these two
280
00:17:45,520 --> 00:17:47,130
forms of this answer.
281
00:17:47,130 --> 00:17:49,350
So therefore, in
this lecture, I'm
282
00:17:49,350 --> 00:17:52,790
going to stick with
this functional form.
283
00:17:52,790 --> 00:17:54,232
OK, any questions?
284
00:17:54,232 --> 00:17:59,900
STUDENT: We don't include
dx in the [INAUDIBLE]??
285
00:17:59,900 --> 00:18:01,000
YEN-JIE LEE: Not yet.
286
00:18:01,000 --> 00:18:02,590
We are going to include that.
287
00:18:02,590 --> 00:18:07,590
Because, for that, in
order to actually--
288
00:18:07,590 --> 00:18:11,250
OK, so now I actually
decompose the motion of my hand
289
00:18:11,250 --> 00:18:14,020
into many, many waves--
290
00:18:14,020 --> 00:18:17,020
which should be or is say
it many, many oscillation
291
00:18:17,020 --> 00:18:19,860
with different frequencies.
292
00:18:19,860 --> 00:18:23,590
So I actually
describe the motion
293
00:18:23,590 --> 00:18:26,500
of my hand infinite
number of oscillation
294
00:18:26,500 --> 00:18:28,300
with different frequency.
295
00:18:28,300 --> 00:18:34,900
And the trouble we are facing is
that all those oscillations are
296
00:18:34,900 --> 00:18:37,960
going to be charged
travelling at different speeds
297
00:18:37,960 --> 00:18:42,160
because of the
dispersion relation.
298
00:18:42,160 --> 00:18:45,760
Therefore, what I am
going to do afterwards
299
00:18:45,760 --> 00:18:50,170
is to show you that, OK, I can
write down the functional form
300
00:18:50,170 --> 00:18:54,130
for psi in this general case.
301
00:18:54,130 --> 00:18:58,100
So for that, that's actually
what I'm going to do now.
302
00:18:58,100 --> 00:19:03,730
So now, I would like to know
what would be the psi xt, which
303
00:19:03,730 --> 00:19:06,970
actually the position of the
string as a function of x,
304
00:19:06,970 --> 00:19:11,020
and at some specific
time equal to t.
305
00:19:11,020 --> 00:19:16,690
And that can be written as, I
do the an integration from minus
306
00:19:16,690 --> 00:19:23,380
infinity to infinity
over frequency omega,
307
00:19:23,380 --> 00:19:27,310
and I have the usual
amplitude associated
308
00:19:27,310 --> 00:19:29,770
with the angular
frequency omega,
309
00:19:29,770 --> 00:19:35,810
and the exponential i omega t--
310
00:19:35,810 --> 00:19:38,890
minus i omega t because that's
the convention I'm using here--
311
00:19:38,890 --> 00:19:46,020
and I say, OK, plus ik,
which is a function omega--
312
00:19:46,020 --> 00:19:46,960
x.
313
00:19:46,960 --> 00:19:49,510
So now you can to
see that what I'm
314
00:19:49,510 --> 00:19:53,250
doing here is that I
am now progressing,
315
00:19:53,250 --> 00:19:56,680
I am making infinite number
of progressing waves.
316
00:19:56,680 --> 00:19:59,320
Each of these
exponential functions
317
00:19:59,320 --> 00:20:04,630
is a progressing wave with
angular frequency omega.
318
00:20:04,630 --> 00:20:10,110
And why do I write k as a
function of omega x here?
319
00:20:10,110 --> 00:20:12,572
It's because they are going
to be travelling at the speed
320
00:20:12,572 --> 00:20:15,160
of omega over k.
321
00:20:15,160 --> 00:20:19,630
Therefore, I need to
actually put k here,
322
00:20:19,630 --> 00:20:22,660
and this k is actually--
323
00:20:22,660 --> 00:20:27,040
this k is actually not
the independent parameter.
324
00:20:27,040 --> 00:20:29,800
It's actually a
function of omega.
325
00:20:29,800 --> 00:20:33,730
So we can see that, here, we
do an integration over omega
326
00:20:33,730 --> 00:20:36,960
from minus infinity
to infinity--
327
00:20:36,960 --> 00:20:42,040
for each omega you can actually
find the corresponding k,
328
00:20:42,040 --> 00:20:42,760
right?
329
00:20:42,760 --> 00:20:45,370
Because of the
dispersion relation.
330
00:20:45,370 --> 00:20:47,770
Because omega is
a function of k,
331
00:20:47,770 --> 00:20:52,420
therefore you can always solve
the corresponding k, right?
332
00:20:52,420 --> 00:20:54,130
Then you put it there?
333
00:20:54,130 --> 00:20:58,610
Because you are now
trying to propagate
334
00:20:58,610 --> 00:21:01,660
how many waves with
different angular
335
00:21:01,660 --> 00:21:04,130
frequency at different speed--
336
00:21:04,130 --> 00:21:05,307
then we are done.
337
00:21:08,080 --> 00:21:10,690
That looks like a
wonderful solution,
338
00:21:10,690 --> 00:21:17,150
and we can actually see how
it works for our purpose.
339
00:21:17,150 --> 00:21:18,310
Any questions?
340
00:21:20,950 --> 00:21:21,850
All right.
341
00:21:21,850 --> 00:21:23,090
So that's really nice.
342
00:21:23,090 --> 00:21:26,650
And I can now do a
really simple test
343
00:21:26,650 --> 00:21:29,640
to see if this really works.
344
00:21:29,640 --> 00:21:32,790
Let me try a very simple case.
345
00:21:32,790 --> 00:21:34,358
OK, a spatial case.
346
00:21:37,230 --> 00:21:41,970
If I now go back to
use this description
347
00:21:41,970 --> 00:21:46,080
to describe non-dispersive
medium and see what
348
00:21:46,080 --> 00:21:47,340
will happen.
349
00:21:47,340 --> 00:21:52,210
Now my k as a function omega
is actually rather simple.
350
00:21:52,210 --> 00:21:53,740
It's actually omega over vp--
351
00:21:58,200 --> 00:22:02,760
according to the
dispersion relation here.
352
00:22:02,760 --> 00:22:05,820
I can solve k, as
I was mentioning,
353
00:22:05,820 --> 00:22:08,610
with these dispersion
relation formula.
354
00:22:08,610 --> 00:22:11,250
And then I can conclude
k as a function
355
00:22:11,250 --> 00:22:15,920
omega is omega over vp.
356
00:22:15,920 --> 00:22:18,680
Then I can now put that
into this equation,
357
00:22:18,680 --> 00:22:22,380
and I'm going to
get psi x of t--
358
00:22:22,380 --> 00:22:24,930
this would be equal
to minus infinity
359
00:22:24,930 --> 00:22:33,270
to infinity d omega, c omega,
exponential minus i omega
360
00:22:33,270 --> 00:22:37,140
t minus omega over vx.
361
00:22:41,640 --> 00:22:47,160
And we can actually take omega
out of this, minus infinity
362
00:22:47,160 --> 00:22:54,060
to infinity d omega c omega
exponential minus i omega
363
00:22:54,060 --> 00:23:02,850
t minus x divided by v. And
you can see that, huh, indeed,
364
00:23:02,850 --> 00:23:09,990
this is actually ft
minus x over v. OK.
365
00:23:09,990 --> 00:23:11,670
I'm dropping the vp here.
366
00:23:11,670 --> 00:23:16,030
This should be vp
all over the place.
367
00:23:16,030 --> 00:23:21,450
So you can see that, now, if I
have solved the k as a function
368
00:23:21,450 --> 00:23:25,380
omega, and I plug it in
in this special case,
369
00:23:25,380 --> 00:23:31,590
which is non-dispersive medium,
omega over k equal to vp,
370
00:23:31,590 --> 00:23:36,310
then I really calculate
this integral,
371
00:23:36,310 --> 00:23:38,450
then I can quickly
identify that--
372
00:23:38,450 --> 00:23:43,750
huh, I can write the
functional form in this way.
373
00:23:43,750 --> 00:23:48,300
And this is actually
really familiar to me
374
00:23:48,300 --> 00:23:52,580
because that's actually using
this definition, f is actually
375
00:23:52,580 --> 00:23:56,550
equal to integration minus
infinity to infinity, d omega,
376
00:23:56,550 --> 00:23:59,670
c omega, exponential
minus i omega t.
377
00:23:59,670 --> 00:24:06,280
If I replace t, by t minus
x over vp, then I'm done.
378
00:24:06,280 --> 00:24:10,500
So I have evaluated this
integration, which is actually
379
00:24:10,500 --> 00:24:14,670
just f t minus x over vp.
380
00:24:14,670 --> 00:24:19,740
So that's exactly what
guessed from the beginning.
381
00:24:19,740 --> 00:24:22,170
So if I have a
non-dispersive medium,
382
00:24:22,170 --> 00:24:26,490
then psi xt will be
equal to this function.
383
00:24:26,490 --> 00:24:28,440
So that gives us some
kind of confidence that,
384
00:24:28,440 --> 00:24:33,600
OK, at the easy case, it works.
385
00:24:33,600 --> 00:24:39,510
All right, so that's very
nice, all sounds very good.
386
00:24:39,510 --> 00:24:41,190
But wait a second.
387
00:24:41,190 --> 00:24:43,470
How do I actually
extract this c,
388
00:24:43,470 --> 00:24:45,690
which is a function of omega?
389
00:24:45,690 --> 00:24:50,580
I'm troubled because this
is an infinite integral from
390
00:24:50,580 --> 00:24:52,480
minus infinity to infinity.
391
00:24:52,480 --> 00:24:55,560
And that means I have infinite
number of constants, which I
392
00:24:55,560 --> 00:24:58,830
have to determine the c omega.
393
00:24:58,830 --> 00:24:59,970
How do I actually do this?
394
00:25:02,670 --> 00:25:06,180
So that is another
point which I would
395
00:25:06,180 --> 00:25:09,180
like to discuss before
we actually go ahead
396
00:25:09,180 --> 00:25:15,360
and really use this function
for the dispersive medium case.
397
00:25:15,360 --> 00:25:19,680
So how to we actually extract
c as a function of omega?
398
00:25:22,230 --> 00:25:27,630
So for that, we really
need to employ a few uses
399
00:25:27,630 --> 00:25:31,860
for formula, which are
actually documented here.
400
00:25:31,860 --> 00:25:36,030
How many of you actually have
not heard about delta function
401
00:25:36,030 --> 00:25:37,520
before?
402
00:25:37,520 --> 00:25:41,560
OK, a few of you actually have
not heard about delta function.
403
00:25:41,560 --> 00:25:44,730
So what is actually
your delta function?
404
00:25:44,730 --> 00:25:47,910
This is a delta function.
405
00:25:47,910 --> 00:25:52,155
So a delta function
is actually a notation
406
00:25:52,155 --> 00:25:58,280
which actually shows
you a function, which
407
00:25:58,280 --> 00:26:03,180
is should only be non-zero,
at x equal to zero.
408
00:26:03,180 --> 00:26:07,200
And the x equal to zero,
the size of this function
409
00:26:07,200 --> 00:26:11,290
as you're going to infinity,
and on the other hand,
410
00:26:11,290 --> 00:26:16,520
all the other points
at x not equal to zero,
411
00:26:16,520 --> 00:26:20,980
the delta function
is equal to zero.
412
00:26:20,980 --> 00:26:23,820
So that's actually the kind of
function I was talking about.
413
00:26:23,820 --> 00:26:28,010
And the area of this function,
if you're doing the equation
414
00:26:28,010 --> 00:26:32,040
over minus infinity
infinity over x,
415
00:26:32,040 --> 00:26:38,220
the integration of these delta
m the area is actually 1.
416
00:26:38,220 --> 00:26:40,420
So that is actually
the kind of function.
417
00:26:40,420 --> 00:26:45,870
So essentially, it's a really,
really narrow function, OK,
418
00:26:45,870 --> 00:26:50,040
very narrow, very
narrow, very narrow.
419
00:26:50,040 --> 00:26:54,870
But the area is
finite, which is why.
420
00:26:54,870 --> 00:26:57,870
So you can have a square.
421
00:26:57,870 --> 00:27:01,320
You can actually start
with a square pulse,
422
00:27:01,320 --> 00:27:03,390
or square function,
and you can actually
423
00:27:03,390 --> 00:27:07,670
make the width of the square
narrower, smaller and smaller
424
00:27:07,670 --> 00:27:09,610
and smaller, go to 0.
425
00:27:09,610 --> 00:27:11,680
Then what you are going
to get is essentially
426
00:27:11,680 --> 00:27:13,620
the delta function.
427
00:27:13,620 --> 00:27:18,990
That's actually how we
understand this delta function.
428
00:27:18,990 --> 00:27:20,900
All right, really quickly.
429
00:27:20,900 --> 00:27:24,930
And also, we would like
to use a few formula which
430
00:27:24,930 --> 00:27:26,800
are documented here.
431
00:27:26,800 --> 00:27:32,040
So if I do an integration from
minus infinity to infinity,
432
00:27:32,040 --> 00:27:35,490
exponential i omega
minus omega prime,
433
00:27:35,490 --> 00:27:40,830
t over the t which is
integrating over time, t, here.
434
00:27:40,830 --> 00:27:45,770
And then divide the whole
formula by 1 and over 2pi.
435
00:27:45,770 --> 00:27:49,030
What I'm going to get
is a delta function,
436
00:27:49,030 --> 00:27:53,670
which is a delta function which
is omega minus omega prime.
437
00:27:53,670 --> 00:27:58,590
So that means when this delta
function formula tells us
438
00:27:58,590 --> 00:28:02,610
that omega is equal
to omega prime,
439
00:28:02,610 --> 00:28:06,900
then this function is
actually going to infinity.
440
00:28:06,900 --> 00:28:10,620
And only when omega
equal to omega prime,
441
00:28:10,620 --> 00:28:12,700
this function is not zero.
442
00:28:12,700 --> 00:28:16,750
Any other place, this
function is always zero.
443
00:28:19,290 --> 00:28:22,230
And this strange
integration should give you
444
00:28:22,230 --> 00:28:23,530
this delta function.
445
00:28:23,530 --> 00:28:26,140
So that's the first
thing which we will use,
446
00:28:26,140 --> 00:28:28,350
was one useful formula.
447
00:28:28,350 --> 00:28:32,130
The second thing which
what just I talked about,
448
00:28:32,130 --> 00:28:35,560
if I do an integration over
minus infinity to infinity,
449
00:28:35,560 --> 00:28:40,410
delta x dx, then
basically you get 1.
450
00:28:40,410 --> 00:28:43,290
The third one is actually
kind of interesting.
451
00:28:43,290 --> 00:28:44,400
Let's take a look.
452
00:28:44,400 --> 00:28:48,510
So if I do an integration
over from minus infinity
453
00:28:48,510 --> 00:28:54,630
to infinity, delta
function x minus alpha.
454
00:28:54,630 --> 00:28:57,060
Let's look at this
delta function first.
455
00:28:57,060 --> 00:29:04,596
This function is only non-zero
when x is equal to what?
456
00:29:04,596 --> 00:29:05,262
AUDIENCE: Alpha.
457
00:29:05,262 --> 00:29:06,012
YEN-JIE LEE: Yeah.
458
00:29:06,012 --> 00:29:10,310
When x is equal to alpha,
only when that happen,
459
00:29:10,310 --> 00:29:12,410
this is actually non-zero.
460
00:29:12,410 --> 00:29:18,050
If you multiply this delta
function to some function which
461
00:29:18,050 --> 00:29:22,610
is f of alpha, and integrate
over alpha from minus
462
00:29:22,610 --> 00:29:25,040
infinity to infinity.
463
00:29:25,040 --> 00:29:27,620
And that means
that when alpha is
464
00:29:27,620 --> 00:29:32,000
equal to x, or x equal to
alpha, this integration
465
00:29:32,000 --> 00:29:35,750
give you non-zero result.
All the other ways,
466
00:29:35,750 --> 00:29:38,420
you will get zero.
467
00:29:38,420 --> 00:29:41,470
The interesting thing is that
if you do this integration, what
468
00:29:41,470 --> 00:29:47,316
you are going to get is that
OK, when I integrate over alpha,
469
00:29:47,316 --> 00:29:51,260
only when alpha is equal to
x this thing is non-zero.
470
00:29:51,260 --> 00:29:52,730
Therefore what you
are going to get
471
00:29:52,730 --> 00:29:56,780
is, you get only one point of
the width, which is actually
472
00:29:56,780 --> 00:29:59,180
f of x.
473
00:29:59,180 --> 00:30:02,282
So that's the intuition
about this formula.
474
00:30:02,282 --> 00:30:02,990
That's just fine.
475
00:30:02,990 --> 00:30:05,731
Any questions related
to those formulas?
476
00:30:05,731 --> 00:30:08,677
AUDIENCE: [INAUDIBLE]?
477
00:30:08,677 --> 00:30:10,150
YEN-JIE LEE: Hm?
478
00:30:10,150 --> 00:30:11,270
AUDIENCE: [INAUDIBLE]?
479
00:30:13,599 --> 00:30:15,890
YEN-JIE LEE: Yeah, this is
actually pretty complicated,
480
00:30:15,890 --> 00:30:21,300
so it would take a few 10,
20 minutes to explain that.
481
00:30:21,300 --> 00:30:23,790
But let's just take the words
from the math department--
482
00:30:23,790 --> 00:30:24,802
we trust them.
483
00:30:28,100 --> 00:30:31,040
All right, so once I
have those formula,
484
00:30:31,040 --> 00:30:35,610
I can now demonstrate
you how I can actually
485
00:30:35,610 --> 00:30:38,750
track C as a function omega.
486
00:30:38,750 --> 00:30:41,320
So this is actually
the goal, right?
487
00:30:41,320 --> 00:30:43,480
So don't forget why we are
doing what we are doing,
488
00:30:43,480 --> 00:30:47,810
is to try to extract what
is actually the C omega,
489
00:30:47,810 --> 00:30:54,410
so that we can actually
finish this formula.
490
00:30:54,410 --> 00:30:56,450
So how do we do that?
491
00:30:56,450 --> 00:31:00,170
So suppose, if I evaluate this.
492
00:31:05,120 --> 00:31:12,910
This function, 1 over
2pi, minus infinity
493
00:31:12,910 --> 00:31:23,610
to infinity dt, ft,
exponential i omega t.
494
00:31:23,610 --> 00:31:26,480
If I evaluate this function.
495
00:31:26,480 --> 00:31:28,860
This is coming out
of nowhere, right?
496
00:31:28,860 --> 00:31:34,090
So coming out of Yen-Jie's
hand, maybe, I don't know.
497
00:31:34,090 --> 00:31:38,490
Suppose if I evaluate
this function,
498
00:31:38,490 --> 00:31:40,980
and now I have ft here, right?
499
00:31:40,980 --> 00:31:47,790
I can replace ft by these
interesting formula.
500
00:31:47,790 --> 00:31:54,360
If I do that, then basically I
get 1 over 2pi minus infinity
501
00:31:54,360 --> 00:32:04,010
to infinity dt, minus
infinity infinity C omega
502
00:32:04,010 --> 00:32:10,600
prime, exponential
minus i omega prime t.
503
00:32:10,600 --> 00:32:15,330
And the last is actually
integrating over d omega prime.
504
00:32:17,880 --> 00:32:21,480
So this is actually the f of t.
505
00:32:21,480 --> 00:32:24,490
This is actually f of t.
506
00:32:24,490 --> 00:32:28,380
I'm just replacing that
formula into this integral.
507
00:32:28,380 --> 00:32:33,140
And then I have the rest,
which is exponential i omega t.
508
00:32:36,770 --> 00:32:39,660
And of course, I can
continue and collect
509
00:32:39,660 --> 00:32:42,810
all the relevant terms together.
510
00:32:42,810 --> 00:32:51,930
This is actually equal to 1 over
2i, minus infinity to infinity.
511
00:32:51,930 --> 00:32:57,510
I collect all the terms related
to omega prime to the left hand
512
00:32:57,510 --> 00:32:58,690
side.
513
00:32:58,690 --> 00:33:04,940
Basically what I get is C
omega prime d omega prime.
514
00:33:04,940 --> 00:33:10,920
This is actually coming
from here, except--
515
00:33:10,920 --> 00:33:15,420
yeah, OK, it is actually
coming from here.
516
00:33:15,420 --> 00:33:20,830
And I have another integral
which is from minus infinity
517
00:33:20,830 --> 00:33:26,770
to infinity, this time
integrating over delta dt.
518
00:33:26,770 --> 00:33:36,480
And I have dt here, exponential
i omega minus omega prime t.
519
00:33:36,480 --> 00:33:44,650
So basically I'm collecting
these two terms together.
520
00:33:44,650 --> 00:33:48,660
They now become exponential
i omega minus omega prime t.
521
00:33:51,390 --> 00:33:55,980
So basically, no magic happened,
but I'm just re-writing things
522
00:33:55,980 --> 00:34:01,680
and we are arranging things from
this formula to that formula.
523
00:34:01,680 --> 00:34:08,130
Then if we look at this
formula, this formula here,
524
00:34:08,130 --> 00:34:10,889
and the formula sheet we have.
525
00:34:10,889 --> 00:34:14,840
1 over 2pi minus
infinity to infinity
526
00:34:14,840 --> 00:34:17,920
to this integration over
t, exponential i omega
527
00:34:17,920 --> 00:34:19,940
minus omega prime t.
528
00:34:19,940 --> 00:34:22,510
That will give you
delta function,
529
00:34:22,510 --> 00:34:25,130
which is delta omega
minus omega prime.
530
00:34:27,929 --> 00:34:31,750
Therefore, I can continue
this calculation here.
531
00:34:34,600 --> 00:34:45,800
Thus it's going to give you
minus infinity to infinity.
532
00:34:45,800 --> 00:34:53,870
I identify this part,
this part, and this part,
533
00:34:53,870 --> 00:34:55,440
to be the delta function.
534
00:34:58,330 --> 00:35:03,340
Therefore, what I get is
minus infinity to infinity, C
535
00:35:03,340 --> 00:35:14,850
omega prime, delta omega minus
omega prime, d omega prime.
536
00:35:14,850 --> 00:35:17,450
Am I going too fast?
537
00:35:17,450 --> 00:35:20,637
Everybody's following?
538
00:35:20,637 --> 00:35:22,470
So you can see that
what we have been doing,
539
00:35:22,470 --> 00:35:26,490
I use this formula
coming out of nowhere.
540
00:35:26,490 --> 00:35:31,980
I replace f by the formula
I was writing there.
541
00:35:31,980 --> 00:35:34,920
And then I collect the
terms I like together.
542
00:35:34,920 --> 00:35:36,910
That's all I did.
543
00:35:36,910 --> 00:35:38,790
And then I found, aha!
544
00:35:38,790 --> 00:35:44,240
One part of the formula is
actually the delta function.
545
00:35:44,240 --> 00:35:46,420
Then I put the delta
a function here.
546
00:35:46,420 --> 00:35:50,200
And then finally, I use
the third formula here,
547
00:35:50,200 --> 00:35:54,230
which I have related to delta
function, and I found, aha!
548
00:35:54,230 --> 00:35:56,250
If I do this
integration, I know how
549
00:35:56,250 --> 00:35:58,740
to do this integration
even without knowing
550
00:35:58,740 --> 00:36:02,610
the structure of C. This
is actually just changing
551
00:36:02,610 --> 00:36:06,320
the omega prime to omega.
552
00:36:06,320 --> 00:36:10,650
So that's actually what this
integration actually does.
553
00:36:10,650 --> 00:36:16,500
Therefore, I get C omega.
554
00:36:16,500 --> 00:36:20,190
Look at what we have done.
555
00:36:20,190 --> 00:36:23,350
What we have done is
that, we have proof
556
00:36:23,350 --> 00:36:27,240
that this formula
coming out of nowhere,
557
00:36:27,240 --> 00:36:32,080
to be a continuous
version of mode picker.
558
00:36:32,080 --> 00:36:35,130
You remember the fourth
year decomposition before?
559
00:36:35,130 --> 00:36:39,840
You were using the orthogonality
of the sine function,
560
00:36:39,840 --> 00:36:42,840
and I can do some kind
of fancy integration
561
00:36:42,840 --> 00:36:47,700
to actually extract
a m from one of the--
562
00:36:47,700 --> 00:36:50,440
which is associated with one
of the normal mode, right?
563
00:36:50,440 --> 00:36:53,460
What we are doing here is
actually a continuous version.
564
00:36:53,460 --> 00:36:55,740
Now omega is
actually continuous.
565
00:36:55,740 --> 00:37:02,220
And I'm now using
the orthogonality
566
00:37:02,220 --> 00:37:05,220
of the exponential function.
567
00:37:05,220 --> 00:37:08,130
If I do this integration,
that will only
568
00:37:08,130 --> 00:37:14,020
give you non-zero value when
omega is equal to omega prime.
569
00:37:14,020 --> 00:37:16,210
It's exactly the
same thing, right?
570
00:37:16,210 --> 00:37:21,220
Then I can construct an
integration like this.
571
00:37:21,220 --> 00:37:24,360
And now will give
you the redoubting C
572
00:37:24,360 --> 00:37:29,142
as a function omega, which
is like the amplitude of one
573
00:37:29,142 --> 00:37:34,470
of the associated harmonics
exponential i omega t.
574
00:37:34,470 --> 00:37:37,240
So in short, from
this exercise, we
575
00:37:37,240 --> 00:37:40,330
have shown you
that C of omega can
576
00:37:40,330 --> 00:37:46,360
be extracted using this formula
1 over 2pi, minus infinity
577
00:37:46,360 --> 00:37:54,790
to infinity dt, f of t,
exponential i omega t.
578
00:37:54,790 --> 00:37:58,660
That's actually how we
actually can determine
579
00:37:58,660 --> 00:38:06,100
all the amplitude associated
to a specific exponential
580
00:38:06,100 --> 00:38:07,000
function.
581
00:38:07,000 --> 00:38:08,040
Any questions so far?
582
00:38:11,280 --> 00:38:17,760
OK, so if no question, then
we can actually continue.
583
00:38:17,760 --> 00:38:22,040
So let's actually go back
to the original question,
584
00:38:22,040 --> 00:38:24,710
which we were posting.
585
00:38:24,710 --> 00:38:29,540
So we have a problem
related to the transmission
586
00:38:29,540 --> 00:38:31,830
of information.
587
00:38:31,830 --> 00:38:35,570
So this is actually
where we got started.
588
00:38:35,570 --> 00:38:42,320
If I send a square pulse
on a dispersive median,
589
00:38:42,320 --> 00:38:45,170
then I have some
trouble, which is
590
00:38:45,170 --> 00:38:49,190
that this pulse is going
to disperse and become
591
00:38:49,190 --> 00:38:50,700
wider and wider.
592
00:38:50,700 --> 00:38:53,000
It's changing as a
function of time,
593
00:38:53,000 --> 00:38:55,280
as a function of
distance it travel.
594
00:38:55,280 --> 00:38:57,680
That's not cool.
595
00:38:57,680 --> 00:39:03,130
All right, so therefore what
I am going to do is this.
596
00:39:03,130 --> 00:39:06,330
There was a very
smart idea which
597
00:39:06,330 --> 00:39:10,970
were discovered long time
ago, during maybe World War I,
598
00:39:10,970 --> 00:39:17,665
and widely used in World War
II, which is the AM radio.
599
00:39:17,665 --> 00:39:18,640
What is AM?
600
00:39:18,640 --> 00:39:25,120
Is actually amplitude
modulation radio.
601
00:39:25,120 --> 00:39:28,850
This smart idea
is the following.
602
00:39:28,850 --> 00:39:32,840
I will describe it
before we take a break.
603
00:39:32,840 --> 00:39:41,840
So this smart idea, AM
radio is the following.
604
00:39:41,840 --> 00:39:47,270
If I have some kind of
information which is fs t.
605
00:39:47,270 --> 00:39:50,180
s here means signal.
606
00:39:50,180 --> 00:39:51,890
If I have some
kind of information
607
00:39:51,890 --> 00:39:53,870
I would like to
send, I can send it
608
00:39:53,870 --> 00:39:59,300
by oscillating one
end of the string.
609
00:39:59,300 --> 00:40:03,200
And this is what I want to send.
610
00:40:03,200 --> 00:40:08,430
And there are two ways you
can send this fs function.
611
00:40:08,430 --> 00:40:14,050
The first one is actually what
did before, I send it directly.
612
00:40:14,050 --> 00:40:16,940
I just said OK, if I want
to send this function,
613
00:40:16,940 --> 00:40:19,850
then I just oscillate
the string according
614
00:40:19,850 --> 00:40:22,180
to the functional form.
615
00:40:22,180 --> 00:40:25,130
Yen-Jie just have to be
really careful, right?
616
00:40:25,130 --> 00:40:27,650
So that you can
send this function.
617
00:40:27,650 --> 00:40:31,680
And that fails miserably.
618
00:40:31,680 --> 00:40:32,720
Why?
619
00:40:32,720 --> 00:40:38,580
Because all the components
which actually produce the fs,
620
00:40:38,580 --> 00:40:43,690
in this case the square pulse,
all those components are
621
00:40:43,690 --> 00:40:46,200
travelling at different speed.
622
00:40:46,200 --> 00:40:49,020
Therefore, the information
will never get there,
623
00:40:49,020 --> 00:40:53,460
because of the dispersion.
624
00:40:53,460 --> 00:40:56,840
So now what should
we do instead?
625
00:40:56,840 --> 00:41:02,440
Instead of doing sending fs
as a function of t directly,
626
00:41:02,440 --> 00:41:05,040
what you could do
is that I can now
627
00:41:05,040 --> 00:41:15,940
send f of t, which is equal
to f of s t cosine omega0t.
628
00:41:19,630 --> 00:41:26,610
Where omega0 is a very,
very large number.
629
00:41:26,610 --> 00:41:29,310
And basically, look at
what we have been doing.
630
00:41:29,310 --> 00:41:34,350
So that means I, instead
of sending fs directly,
631
00:41:34,350 --> 00:41:39,960
I send fs, but modulated
by a really high frequency
632
00:41:39,960 --> 00:41:43,410
function, cosine omega0t.
633
00:41:43,410 --> 00:41:45,690
And this will work.
634
00:41:45,690 --> 00:41:48,480
And you will only know
that after we come back
635
00:41:48,480 --> 00:41:53,040
from the break, which
is twenty first.
636
00:41:53,040 --> 00:41:54,900
Let's take five minute break.
637
00:41:54,900 --> 00:41:58,380
And if you have any questions,
you can actually ask me here.
638
00:42:06,000 --> 00:42:12,750
So we will continue the
discussion about AM radio.
639
00:42:12,750 --> 00:42:16,890
So before the break, actually
we introduced this one
640
00:42:16,890 --> 00:42:23,220
possible solution to solve
this dispersive median problem,
641
00:42:23,220 --> 00:42:29,450
is that I can now actually send
instead of fs as a function
642
00:42:29,450 --> 00:42:33,200
t, which is actually the
signal I want to send,
643
00:42:33,200 --> 00:42:38,460
I could send fs, but
multiplied by cosine omega0t.
644
00:42:41,960 --> 00:42:48,660
If I assume that fs is
some really slow function,
645
00:42:48,660 --> 00:42:52,290
slowly varying as
a function of time,
646
00:42:52,290 --> 00:42:55,870
compared to cosine omega0t.
647
00:42:55,870 --> 00:43:00,600
Cosine omega0t is a
really fast function,
648
00:43:00,600 --> 00:43:03,930
oscillating up and down
like crazy, really fast.
649
00:43:03,930 --> 00:43:08,190
If I multiply fs
by this function,
650
00:43:08,190 --> 00:43:10,490
what is going to happen?
651
00:43:10,490 --> 00:43:13,030
We are going to show
you that actually that
652
00:43:13,030 --> 00:43:19,410
means I am going to
only have non-zero C
653
00:43:19,410 --> 00:43:23,150
function, or a large
contribution of C,
654
00:43:23,150 --> 00:43:25,610
in a very thin middle
range of omega.
655
00:43:28,440 --> 00:43:29,470
So we'll show that.
656
00:43:29,470 --> 00:43:35,700
So in a typical case, fs is
really slow, which is like,
657
00:43:35,700 --> 00:43:38,620
for example, my
sound, et cetera,
658
00:43:38,620 --> 00:43:41,460
in the label of one kilohertz.
659
00:43:41,460 --> 00:43:43,710
And you can actually
design a system
660
00:43:43,710 --> 00:43:51,570
which will actually multiply
this fs by cosine omega0t.
661
00:43:51,570 --> 00:43:57,960
Omega0 can be as fast
as 1.1 to 30 megahertz.
662
00:43:57,960 --> 00:44:01,470
If you do this
calculation, then you
663
00:44:01,470 --> 00:44:20,210
will find that OK, the range
of omega, with sizable C omega
664
00:44:20,210 --> 00:44:22,500
is small.
665
00:44:26,690 --> 00:44:34,550
It's roughly equal to
omega0 minus omega s,
666
00:44:34,550 --> 00:44:41,470
to omega0 plus omega
s, where omega s is
667
00:44:41,470 --> 00:44:47,780
the typical frequency
in your signal.
668
00:44:47,780 --> 00:44:52,210
And the omega0 is the
typical frequency of--
669
00:44:52,210 --> 00:44:55,820
the frequency of
your cosine omega t
670
00:44:55,820 --> 00:44:58,120
term, which is
actually, later, you
671
00:44:58,120 --> 00:44:59,765
will recognize this as carrier.
672
00:45:03,280 --> 00:45:07,080
So what I want to say is
that if I do this trick,
673
00:45:07,080 --> 00:45:08,870
what is going to happen
is that the range
674
00:45:08,870 --> 00:45:14,450
of omega, which you have sizable
contribution from C omega--
675
00:45:14,450 --> 00:45:17,670
C omega is the
associated amplitude,
676
00:45:17,670 --> 00:45:20,090
associated amplitude.
677
00:45:20,090 --> 00:45:23,450
It's going to be confined
to a really small region
678
00:45:23,450 --> 00:45:28,590
from omega0 minus omega
s, to omega0 plus omega s.
679
00:45:28,590 --> 00:45:33,670
So that's the trick which
actually makes this problem
680
00:45:33,670 --> 00:45:34,170
solvable.
681
00:45:36,710 --> 00:45:40,940
How do we know this?
682
00:45:40,940 --> 00:45:43,760
That is because, if
I now, for example, I
683
00:45:43,760 --> 00:45:51,610
send fs equal to
cosine omega st.
684
00:45:51,610 --> 00:45:54,710
If this is actually the signal
which I would like to send,
685
00:45:54,710 --> 00:45:58,430
just a harmonic wave, then
what is going to happen
686
00:45:58,430 --> 00:46:07,730
is that I'm going to get ft
is equal to cosine omega st
687
00:46:07,730 --> 00:46:09,390
cosine--
688
00:46:09,390 --> 00:46:13,520
so this is actually
multiplied by cosine omega0t.
689
00:46:13,520 --> 00:46:17,960
So I have cosine omega0t here.
690
00:46:17,960 --> 00:46:21,800
I have cosine
multiplied by cosine.
691
00:46:21,800 --> 00:46:24,620
Therefore I have
the question which
692
00:46:24,620 --> 00:46:28,360
I prepare here, the formula,
of cosine alpha times
693
00:46:28,360 --> 00:46:32,540
cosine beta will be equal
to the functional form.
694
00:46:32,540 --> 00:46:35,240
There's a remainder,
therefore I can now
695
00:46:35,240 --> 00:46:42,130
write it as 1 over
2 cosine omega s
696
00:46:42,130 --> 00:46:53,770
minus omega0 t plus cosine
omega s plus omega0.
697
00:46:57,540 --> 00:47:03,640
You can see that when I actually
multiply two cosine functions
698
00:47:03,640 --> 00:47:10,480
together, then what I get is
actually the omega0 minus omega
699
00:47:10,480 --> 00:47:11,580
s.
700
00:47:11,580 --> 00:47:15,420
You can actually put the minus
sign there, it didn't matter.
701
00:47:15,420 --> 00:47:21,930
And cosine omega0
plus omega s times t.
702
00:47:21,930 --> 00:47:24,660
So therefore, you can see that
the frequency, there are only
703
00:47:24,660 --> 00:47:29,250
two frequencies which contribute
to this C of omega, which is
704
00:47:29,250 --> 00:47:31,500
actually these two frequencies.
705
00:47:31,500 --> 00:47:36,280
So that is actually why,
if you do this trick,
706
00:47:36,280 --> 00:47:41,100
you actually try to modulate
your slow signal function
707
00:47:41,100 --> 00:47:45,000
by a fast carrier frequency.
708
00:47:45,000 --> 00:47:49,950
Then you are going to confine
the effective range of omega
709
00:47:49,950 --> 00:47:53,460
into a very small range.
710
00:47:53,460 --> 00:47:56,660
Why is that useful?
711
00:47:56,660 --> 00:48:00,240
That's actually what I
want to answer to you.
712
00:48:00,240 --> 00:48:06,060
Suppose I have this crazy
dispersion relation, which
713
00:48:06,060 --> 00:48:10,940
is omega as a function
of K. You can graph it,
714
00:48:10,940 --> 00:48:14,350
and suppose it looks
really crazy like this.
715
00:48:18,660 --> 00:48:26,010
And if I set my carrier
oscillation frequency
716
00:48:26,010 --> 00:48:35,130
to be omega0, and that will
give you a corresponding wave
717
00:48:35,130 --> 00:48:39,240
number which is K0.
718
00:48:39,240 --> 00:48:40,290
I hope you can see it.
719
00:48:43,060 --> 00:48:47,130
That's the corresponding K0.
720
00:48:47,130 --> 00:48:54,990
Before we actually multiply this
function, it's a slow function.
721
00:48:54,990 --> 00:48:57,220
It's not exactly
one cosine function.
722
00:48:57,220 --> 00:49:00,270
So if you just have a cosine
function harmonic wave,
723
00:49:00,270 --> 00:49:02,086
then you don't really
need this trick,
724
00:49:02,086 --> 00:49:03,460
because it's
actually going to be
725
00:49:03,460 --> 00:49:07,620
traveling at a speed
of some constant speed.
726
00:49:07,620 --> 00:49:09,690
It's a harmonic traveling wave.
727
00:49:09,690 --> 00:49:14,100
But if this is actually
a slow function, but not
728
00:49:14,100 --> 00:49:18,780
really a single harmonic wave,
then what is going to happen
729
00:49:18,780 --> 00:49:23,700
is that you are going to need a
wide range of K value or omega
730
00:49:23,700 --> 00:49:28,470
value to describe fs.
731
00:49:28,470 --> 00:49:31,360
Then you are in
trouble because now,
732
00:49:31,360 --> 00:49:35,260
all the waves with different
wavelengths are going to be
733
00:49:35,260 --> 00:49:37,290
travelling at different speed.
734
00:49:37,290 --> 00:49:40,900
Then you have this
dispersion problem.
735
00:49:40,900 --> 00:49:44,460
On the other hand, if I
multiply this function,
736
00:49:44,460 --> 00:49:48,940
this slow function, by a
fast oscillating function,
737
00:49:48,940 --> 00:49:55,420
I am confining the
effective range of omega
738
00:49:55,420 --> 00:49:58,690
into this small box,
which is actually
739
00:49:58,690 --> 00:50:06,100
between omega0 minus
or plus omega s.
740
00:50:06,100 --> 00:50:09,870
This is actually omega0
plus minus omega s.
741
00:50:09,870 --> 00:50:13,830
This is actually the range
of the possible omega,
742
00:50:13,830 --> 00:50:17,650
which contribute to
this resulting f of t.
743
00:50:17,650 --> 00:50:21,870
Therefore, the behavior
of this function
744
00:50:21,870 --> 00:50:26,470
is actually much
easier to understand.
745
00:50:26,470 --> 00:50:33,370
So with that given
there, suppose now I have
746
00:50:33,370 --> 00:50:37,330
a large difference between--
747
00:50:37,330 --> 00:50:40,945
suppose I have a large
difference between omega s
748
00:50:40,945 --> 00:50:42,490
and omega0.
749
00:50:42,490 --> 00:50:46,780
Then I can actually focus
on a very small range
750
00:50:46,780 --> 00:50:53,530
in this dispersion
relation diagram.
751
00:50:53,530 --> 00:50:56,830
Then I can write omega
as a function of K,
752
00:50:56,830 --> 00:51:03,790
the dispersion relation
equal to omega0 plus K
753
00:51:03,790 --> 00:51:10,400
minus K0, partial omega,
partial K. Evaluate
754
00:51:10,400 --> 00:51:15,700
it at a equal to K0,
plus higher order term.
755
00:51:15,700 --> 00:51:20,470
Basically I can do
this Taylor expansion.
756
00:51:20,470 --> 00:51:26,640
And maybe it surprised you, you
can immediately identify, ha!
757
00:51:26,640 --> 00:51:35,060
This is delta d omega dk,
is the group velocity.
758
00:51:35,060 --> 00:51:39,210
Suddenly it show up in
the Taylor expansion
759
00:51:39,210 --> 00:51:42,550
of the dispersion relation.
760
00:51:42,550 --> 00:51:48,760
so if I focus on the region
which is around omega0,
761
00:51:48,760 --> 00:51:53,750
then I can actually re-write
omega in this functional form.
762
00:51:53,750 --> 00:51:59,500
Omega is actually
equal to omega0 plus K
763
00:51:59,500 --> 00:52:04,810
minus K0 times Vg, which
is the group velocity.
764
00:52:09,770 --> 00:52:14,570
Suppose this is happening,
then now I can actually
765
00:52:14,570 --> 00:52:23,540
go ahead and really calculate
the functional form for f of t.
766
00:52:23,540 --> 00:52:29,200
So suppose I have
this definition of t.
767
00:52:29,200 --> 00:52:41,330
The definition of t is equal
to fs t times cosine omega0t.
768
00:52:41,330 --> 00:52:45,230
Or say I can actually
write it in a complex form,
769
00:52:45,230 --> 00:52:51,230
instead of writing it in a
cosine omega0t functional form,
770
00:52:51,230 --> 00:52:54,540
I can write it in
exponential functional form.
771
00:52:54,540 --> 00:52:59,750
Exponential minus i omega0t,
which is actually more
772
00:52:59,750 --> 00:53:04,130
convenient for the discussion.
773
00:53:04,130 --> 00:53:09,190
So what is going to happen if
I actually do this calculation?
774
00:53:09,190 --> 00:53:12,080
Then basically that's
one example signal
775
00:53:12,080 --> 00:53:14,990
which I would like to
send on the slides.
776
00:53:14,990 --> 00:53:21,680
So if I am trying to send a
progressing harmonic wave,
777
00:53:21,680 --> 00:53:27,740
then after multiplying by
this exponential i omega0t
778
00:53:27,740 --> 00:53:30,380
function, or a cosine
function, basically
779
00:53:30,380 --> 00:53:32,990
you get something which is
actually oscillating really
780
00:53:32,990 --> 00:53:37,100
fast, which is actually
the AM signal we are trying
781
00:53:37,100 --> 00:53:41,030
to send through this media.
782
00:53:41,030 --> 00:53:43,640
So we can actually
identify, this
783
00:53:43,640 --> 00:53:47,300
is actually the structure of
this, actually the carrier.
784
00:53:50,360 --> 00:53:55,700
And this signal
become the analogue,
785
00:53:55,700 --> 00:54:00,630
in analogy to what we actually
have discussed for that beat
786
00:54:00,630 --> 00:54:01,710
phenomenon case.
787
00:54:04,650 --> 00:54:14,810
So now, if the omega
range is really small,
788
00:54:14,810 --> 00:54:18,700
then I can actually
write this down.
789
00:54:18,700 --> 00:54:23,060
Write a functional form of
omega in this functional form.
790
00:54:23,060 --> 00:54:29,090
Or I can actually
take out the kVg term,
791
00:54:29,090 --> 00:54:31,910
and the rest is actually
going to be something
792
00:54:31,910 --> 00:54:39,650
like some constant a, where
a is actually equal to omega0
793
00:54:39,650 --> 00:54:43,220
minus Vg times K0.
794
00:54:43,220 --> 00:54:47,960
So basically I'm just
taking out this K term here,
795
00:54:47,960 --> 00:54:51,620
and this become this term.
796
00:54:51,620 --> 00:54:54,290
With this formula,
I can solve what
797
00:54:54,290 --> 00:54:58,350
would be the
functional form for K,
798
00:54:58,350 --> 00:55:01,860
as a preparation for what
I'm going to do later.
799
00:55:01,860 --> 00:55:08,330
So K can be also expressed
as omega over Vg.
800
00:55:08,330 --> 00:55:12,910
So basically I just
solve the K plus b.
801
00:55:12,910 --> 00:55:16,030
b is actually just
some constant.
802
00:55:16,030 --> 00:55:17,810
Just do it for
convenience, I can
803
00:55:17,810 --> 00:55:25,520
write b equal to K0 minus
omega0 divided by Vg.
804
00:55:25,520 --> 00:55:31,010
What we learned here is that
if the range of effective omega
805
00:55:31,010 --> 00:55:35,710
is really small
around omega zero,
806
00:55:35,710 --> 00:55:39,590
then the relation
between omega and the K
807
00:55:39,590 --> 00:55:42,170
becomes a linear function.
808
00:55:42,170 --> 00:55:47,910
Of course, it's still not like
the case for the non dispersive
809
00:55:47,910 --> 00:55:53,700
median, where omega
over K is a constant.
810
00:55:53,700 --> 00:55:56,922
But at least it becomes
a linear function,
811
00:55:56,922 --> 00:55:58,130
which is actually much nicer.
812
00:56:01,070 --> 00:56:05,780
So finally, with all those
preparation we have done,
813
00:56:05,780 --> 00:56:10,100
we would like to show one
important consequence.
814
00:56:10,100 --> 00:56:17,300
So what we are trying to
do is to show that psi xt.
815
00:56:17,300 --> 00:56:22,130
Now I send, I
oscillate the median,
816
00:56:22,130 --> 00:56:27,450
the string, by this f of
t, which I designed there.
817
00:56:27,450 --> 00:56:33,930
ft is actually fs times
exponential i omega0t.
818
00:56:33,930 --> 00:56:37,460
That's actually designed there.
819
00:56:37,460 --> 00:56:41,810
I would like to show that
the resulting amplitude will
820
00:56:41,810 --> 00:56:56,130
be equal to fs t minus x divided
by Vg, exponential minus i
821
00:56:56,130 --> 00:57:00,460
omega0t minus K0x.
822
00:57:00,460 --> 00:57:03,636
Of course I need to take
the real part of this
823
00:57:03,636 --> 00:57:07,730
in, to go back to the real axis.
824
00:57:07,730 --> 00:57:14,990
Basically I dropped the i
sine omega t contribution.
825
00:57:14,990 --> 00:57:18,500
So this is actually
what I want to show.
826
00:57:18,500 --> 00:57:21,540
Before I go through
all those math,
827
00:57:21,540 --> 00:57:25,070
let's do get the conclusion
which we would like to draw,
828
00:57:25,070 --> 00:57:28,120
before we actually really
go through the math.
829
00:57:28,120 --> 00:57:31,730
The conclusion which I would
like to draw is that, OK,
830
00:57:31,730 --> 00:57:35,540
this is actually my analogue.
831
00:57:35,540 --> 00:57:41,480
My analogue is going to be
travelling at the speed of Vg,
832
00:57:41,480 --> 00:57:43,504
which is the group velocity.
833
00:57:43,504 --> 00:57:45,170
That's the conclusion
which I would like
834
00:57:45,170 --> 00:57:47,660
to draw from this exercise.
835
00:57:47,660 --> 00:57:54,350
And this thing is actually
cosine omega0t minus K0x.
836
00:57:54,350 --> 00:57:57,830
Therefore, this is
actually a harmonic wave.
837
00:57:57,830 --> 00:58:02,540
The carrier is a harmonic
wave travelling at Vp equal
838
00:58:02,540 --> 00:58:06,590
to omega0 divided by K0.
839
00:58:06,590 --> 00:58:08,660
That's the kind of
conclusion which I would like
840
00:58:08,660 --> 00:58:13,050
to draw from this exercise.
841
00:58:13,050 --> 00:58:18,110
Any questions about what
we have discussed so far?
842
00:58:18,110 --> 00:58:23,750
OK, then really you have to
hold tight and follow me really,
843
00:58:23,750 --> 00:58:26,350
100% focus, because
this is actually
844
00:58:26,350 --> 00:58:29,010
a complicated calculation.
845
00:58:29,010 --> 00:58:35,150
So now what I can do is, now I
need to express my fs in terms
846
00:58:35,150 --> 00:58:41,360
of C. So I do integration
from minus infinity
847
00:58:41,360 --> 00:58:50,150
to infinity, d omega, C omega,
exponential minus i omega t.
848
00:58:50,150 --> 00:58:53,870
So basically I can
write my f of s
849
00:58:53,870 --> 00:58:59,690
in a functional form,
which we introduced before.
850
00:58:59,690 --> 00:59:07,640
Then my f function is actually
equal to fs times exponential i
851
00:59:07,640 --> 00:59:10,910
omega minus i omega0t.
852
00:59:10,910 --> 00:59:14,480
So that's actually
what we defined there.
853
00:59:14,480 --> 00:59:19,270
And this would be equal to
minus infinity to infinity.
854
00:59:19,270 --> 00:59:27,090
I do this integration number,
omega C omega exponential minus
855
00:59:27,090 --> 00:59:33,800
i omega plus omega0 times t.
856
00:59:33,800 --> 00:59:38,200
So there's nothing special, I
just take my expression for fs,
857
00:59:38,200 --> 00:59:42,877
multiply that by
exponential minus i omega0t.
858
00:59:42,877 --> 00:59:44,210
Then that's actually what I get.
859
00:59:54,740 --> 00:59:58,320
So since this is actually
integration over omega
860
00:59:58,320 --> 01:00:02,700
from minus infinity to
infinity, therefore I
861
01:00:02,700 --> 01:00:07,410
can always have the freedom
to shift the origin.
862
01:00:07,410 --> 01:00:15,600
So that means f of t can be
returned as minus infinity
863
01:00:15,600 --> 01:00:24,210
to infinity d omega C omega
minus omega0 exponential minus
864
01:00:24,210 --> 01:00:27,440
i omega t.
865
01:00:27,440 --> 01:00:31,950
Then we can see that is
fix a relation between C
866
01:00:31,950 --> 01:00:38,760
of the f function, and
the C of the fs function.
867
01:00:38,760 --> 01:00:41,420
So so far, everything is exact.
868
01:00:41,420 --> 01:00:46,110
I haven't made any
approximation so far.
869
01:00:46,110 --> 01:00:54,660
So now, I can take this function
and propagate that to all x.
870
01:00:54,660 --> 01:00:58,800
In other words, I
can now take this ft,
871
01:00:58,800 --> 01:01:03,570
and write down the psi
as a function of x and t.
872
01:01:03,570 --> 01:01:06,690
So that means all the
different components are
873
01:01:06,690 --> 01:01:10,860
traveling at different speeds.
874
01:01:10,860 --> 01:01:12,570
So basically, I
can write it down
875
01:01:12,570 --> 01:01:20,160
like d omega C omega minus
omega0, exponential minus i
876
01:01:20,160 --> 01:01:24,170
omega t, exponential ikx.
877
01:01:24,170 --> 01:01:28,650
Kx k is actually a
function of omega.
878
01:01:31,180 --> 01:01:32,410
Any questions?
879
01:01:32,410 --> 01:01:35,550
So that's actually just
identical to what we actually
880
01:01:35,550 --> 01:01:36,900
have done before.
881
01:01:36,900 --> 01:01:45,250
So now I can go from f of t to
sine, if you are following me.
882
01:01:45,250 --> 01:01:48,270
So until here, everything exact.
883
01:01:48,270 --> 01:01:50,890
You have all the
problems you have,
884
01:01:50,890 --> 01:01:53,240
like you know this
dispersion essentially,
885
01:01:53,240 --> 01:01:54,730
because all the
little components,
886
01:01:54,730 --> 01:01:59,820
as you can see here, can be
travelling at different speeds.
887
01:01:59,820 --> 01:02:04,430
So now, what I could
do is that if I
888
01:02:04,430 --> 01:02:11,820
assume that C omega is only
sizable at the small range
889
01:02:11,820 --> 01:02:18,130
around, it's only
sizable around omega0.
890
01:02:18,130 --> 01:02:23,040
If now I take this
assumption and propagate
891
01:02:23,040 --> 01:02:27,360
into this formula, then
I can write this psi
892
01:02:27,360 --> 01:02:32,820
function roughly
like minus infinity
893
01:02:32,820 --> 01:02:40,780
to infinity d omega C omega
minus omega0 exponential minus
894
01:02:40,780 --> 01:02:45,880
i omega t exponential i.
895
01:02:45,880 --> 01:02:50,800
Now I can take the
formula which I actually
896
01:02:50,800 --> 01:02:56,040
did an approximation,
around omega0.
897
01:02:56,040 --> 01:03:03,450
Around omega0, K can be returned
us omega over Vg plus b.
898
01:03:03,450 --> 01:03:05,910
This is actually where I
take the approximation.
899
01:03:05,910 --> 01:03:10,320
Only consider the first order
in the Taylor expansion.
900
01:03:10,320 --> 01:03:13,650
So you can see now here,
it's not exact anymore.
901
01:03:13,650 --> 01:03:19,110
But now I write approximate
function of form for K omega.
902
01:03:19,110 --> 01:03:25,180
So what I'm going to
get is omega over Vg
903
01:03:25,180 --> 01:03:28,170
plus b, multiplied by x.
904
01:03:32,980 --> 01:03:33,931
Any questions?
905
01:03:36,510 --> 01:03:38,350
Now I have the approximation.
906
01:03:38,350 --> 01:03:42,760
And of course now I can gather
all the terms related to omega
907
01:03:42,760 --> 01:03:44,530
together.
908
01:03:44,530 --> 01:03:47,650
I'm getting minus
infinity to infinity
909
01:03:47,650 --> 01:03:56,830
d omega C omega minus omega0
exponential minus i omega t
910
01:03:56,830 --> 01:04:05,560
minus x over Vg,
exponential ibx.
911
01:04:05,560 --> 01:04:09,550
So basically, I am merging
this term and that term.
912
01:04:09,550 --> 01:04:14,040
This term and that term
will give you this term.
913
01:04:14,040 --> 01:04:18,160
And what is essentially the
rest is the exponential ibx.
914
01:04:21,520 --> 01:04:22,750
We are almost there.
915
01:04:27,530 --> 01:04:32,570
So now I would like to use this
board, so I need to erase that.
916
01:04:41,050 --> 01:04:48,160
So now I continue from
here, and I can now again,
917
01:04:48,160 --> 01:04:52,750
I can again change the origin
of this infinite integral
918
01:04:52,750 --> 01:04:58,720
so that this can be
written as minus infinity
919
01:04:58,720 --> 01:05:04,400
to infinity, d omega
C function of omega,
920
01:05:04,400 --> 01:05:16,960
exponential minus i omega
plus omega0, t minus x divided
921
01:05:16,960 --> 01:05:22,320
by Vg, and exponential ibx.
922
01:05:25,100 --> 01:05:29,320
So what I come from this
board to that formula,
923
01:05:29,320 --> 01:05:32,260
if you are following
me we are almost there,
924
01:05:32,260 --> 01:05:34,900
because I am changing
the origin again,
925
01:05:34,900 --> 01:05:38,200
so that omega minus
omega0 becomes
926
01:05:38,200 --> 01:05:40,510
omega, become a new omega.
927
01:05:40,510 --> 01:05:43,840
Is everybody
accepting this fact?
928
01:05:43,840 --> 01:05:46,100
And that means
the original omega
929
01:05:46,100 --> 01:05:52,360
will become omega plus omega0.
930
01:05:52,360 --> 01:05:58,120
I'm trying to go really slow,
so that everybody can follow.
931
01:05:58,120 --> 01:05:59,980
I hope you are following.
932
01:05:59,980 --> 01:06:02,470
All right, then
now I can actually
933
01:06:02,470 --> 01:06:05,200
redistribute, arrange
all those terms
934
01:06:05,200 --> 01:06:07,930
and the magic will happen.
935
01:06:07,930 --> 01:06:10,680
So that means rearrange
all those terms,
936
01:06:10,680 --> 01:06:16,480
minus infinity to
infinity d omega C omega
937
01:06:16,480 --> 01:06:22,450
exponential minus i
omega t minus x divided
938
01:06:22,450 --> 01:06:33,760
by Vg, exponential minus i
omega0t, exponential i omega0
939
01:06:33,760 --> 01:06:39,310
over Vg plus b x.
940
01:06:39,310 --> 01:06:42,070
So basically, there's
really no magic.
941
01:06:42,070 --> 01:06:45,940
What I'm doing is really to
rearrange all those terms,
942
01:06:45,940 --> 01:06:49,010
so that this term is
actually rearranged
943
01:06:49,010 --> 01:06:53,150
so that it's now omega
times t minus x over Vg.
944
01:06:53,150 --> 01:06:55,090
It's an independent
exponential term.
945
01:06:58,080 --> 01:07:06,310
And I actually extract this term
times t to be returned here.
946
01:07:06,310 --> 01:07:08,850
I'm just rearranging things, OK?
947
01:07:08,850 --> 01:07:12,220
I'm not changing anything.
948
01:07:12,220 --> 01:07:17,800
And finally, I can merge
this term and that term,
949
01:07:17,800 --> 01:07:20,900
and become this function field.
950
01:07:23,740 --> 01:07:30,420
I can immediately recognize
that after this rearrangement,
951
01:07:30,420 --> 01:07:32,800
this is just
re-writing the formula,
952
01:07:32,800 --> 01:07:36,460
putting all those terms
in different place.
953
01:07:36,460 --> 01:07:38,400
Of course, you can
actually review
954
01:07:38,400 --> 01:07:42,160
this part of the lecture
in the lecture notes later.
955
01:07:42,160 --> 01:07:46,000
But basically, we're not doing
anything fancy but rearranging
956
01:07:46,000 --> 01:07:50,050
things over in different place.
957
01:07:50,050 --> 01:07:52,840
Then I can actually
quickly identify
958
01:07:52,840 --> 01:07:55,750
what I am trying to integrate.
959
01:07:55,750 --> 01:07:59,590
So this integration
is over omega.
960
01:07:59,590 --> 01:08:03,730
Therefore all those terms
are now related to omega.
961
01:08:03,730 --> 01:08:07,240
Therefore, they are just some
terms which are sitting there,
962
01:08:07,240 --> 01:08:08,840
they don't participate.
963
01:08:08,840 --> 01:08:16,479
And if you focus on
this part, what is this?
964
01:08:16,479 --> 01:08:21,880
If you compare that to the
original equation of which
965
01:08:21,880 --> 01:08:23,500
I have here.
966
01:08:23,500 --> 01:08:27,399
If you compare that to the
original fs equation here,
967
01:08:27,399 --> 01:08:29,710
you can't immediately
identify that actually that's
968
01:08:29,710 --> 01:08:31,580
a function of fs.
969
01:08:31,580 --> 01:08:37,490
Originally this function
fs is a function of t.
970
01:08:37,490 --> 01:08:40,149
And I'm going to that board now.
971
01:08:40,149 --> 01:08:48,290
This is actually fs
with t minus x over Vg.
972
01:08:48,290 --> 01:08:49,960
Surprisingly simple.
973
01:08:53,380 --> 01:08:58,510
Now let's look at the right
hand side, this mass here.
974
01:08:58,510 --> 01:09:05,899
This is actually K0, which
actually you cannot see
975
01:09:05,899 --> 01:09:06,399
anymore.
976
01:09:06,399 --> 01:09:10,700
It's in the back of this board.
977
01:09:10,700 --> 01:09:13,189
And then if you combine
these two terms,
978
01:09:13,189 --> 01:09:15,810
basically what you
get is exponential
979
01:09:15,810 --> 01:09:19,720
minus i omega0t minus K0x.
980
01:09:22,880 --> 01:09:24,229
So look at what we have done.
981
01:09:26,760 --> 01:09:33,660
I got started with this
Fourier transform functional
982
01:09:33,660 --> 01:09:36,810
form of fs.
983
01:09:36,810 --> 01:09:40,800
I multiplied fs
by cosine omega0t
984
01:09:40,800 --> 01:09:43,210
and go to the complex notation.
985
01:09:43,210 --> 01:09:47,660
It becomes exponential
minus i omega0t.
986
01:09:47,660 --> 01:09:53,250
If I multiplied that, I get my
f function, which is like this.
987
01:09:53,250 --> 01:09:55,910
You get additional term there.
988
01:09:55,910 --> 01:10:00,030
I rearrange things
and change the origin,
989
01:10:00,030 --> 01:10:03,750
and I can rewrite ft in
this functional form.
990
01:10:03,750 --> 01:10:08,700
And I can have a relation
between the C related to fs
991
01:10:08,700 --> 01:10:12,930
to the C related to f of t.
992
01:10:12,930 --> 01:10:18,120
I propagate ft over
the full space,
993
01:10:18,120 --> 01:10:21,690
and attain my sine, which is
the amplitude as a function
994
01:10:21,690 --> 01:10:25,910
of place and the time.
995
01:10:25,910 --> 01:10:29,060
Until here, everything is exact.
996
01:10:29,060 --> 01:10:31,750
Then I have
introduced assumption,
997
01:10:31,750 --> 01:10:36,850
which is C of omega is only
sizable, only contributing,
998
01:10:36,850 --> 01:10:41,570
around omega zero, therefore
I can do approximation form
999
01:10:41,570 --> 01:10:46,520
for the K function, which
is this functional form.
1000
01:10:46,520 --> 01:10:48,770
Then I just do the integration.
1001
01:10:48,770 --> 01:10:50,690
Then I found that, interesting!
1002
01:10:55,450 --> 01:10:58,020
This side is-- you
should be taking
1003
01:10:58,020 --> 01:11:01,130
the real part of this thing.
1004
01:11:01,130 --> 01:11:03,860
This side have two components.
1005
01:11:03,860 --> 01:11:06,770
The first component
is fs, which is
1006
01:11:06,770 --> 01:11:12,630
the original signal you put in,
the signal you want to send.
1007
01:11:12,630 --> 01:11:18,080
It's actually progressing at
the speed of group velocity.
1008
01:11:18,080 --> 01:11:22,440
So now you understand what
this group velocity means.
1009
01:11:22,440 --> 01:11:25,190
That's the speed
of the signal you
1010
01:11:25,190 --> 01:11:29,510
want to send in the AM radio.
1011
01:11:29,510 --> 01:11:32,460
And this thing is
actually modulated
1012
01:11:32,460 --> 01:11:36,270
by exponential function,
which is actually
1013
01:11:36,270 --> 01:11:44,720
the propagating at the speed
of Vp, equal to omega0 over K0.
1014
01:11:44,720 --> 01:11:48,620
So the carrier still,
after you actually
1015
01:11:48,620 --> 01:11:53,750
include many, many
terms contracting
1016
01:11:53,750 --> 01:11:59,900
the f function, the sine
which is the amplitude,
1017
01:11:59,900 --> 01:12:05,162
the trick is that only the
omega value around omega0
1018
01:12:05,162 --> 01:12:06,460
contributes.
1019
01:12:06,460 --> 01:12:10,550
If that happen, then
you can see that there
1020
01:12:10,550 --> 01:12:13,610
are two structures
actually propagating
1021
01:12:13,610 --> 01:12:16,520
at different speeds, and that
you can actually understand
1022
01:12:16,520 --> 01:12:20,620
the structure independently.
1023
01:12:20,620 --> 01:12:25,610
That means your signal
will not be distorted
1024
01:12:25,610 --> 01:12:27,620
if you're sending it this way.
1025
01:12:27,620 --> 01:12:30,020
But the difference
is that the speed
1026
01:12:30,020 --> 01:12:34,350
of the signal you are
sending is actually
1027
01:12:34,350 --> 01:12:37,860
at the speed of group velocity.
1028
01:12:37,860 --> 01:12:40,250
That is actually
the amazing fact
1029
01:12:40,250 --> 01:12:44,230
which actually enables
us to send signal
1030
01:12:44,230 --> 01:12:49,640
over thousands and thousands
of miles away from the source.
1031
01:12:49,640 --> 01:12:53,470
So what is actually
done is actually that,
1032
01:12:53,470 --> 01:12:56,810
suppose you have some
kind of radio station.
1033
01:12:56,810 --> 01:13:01,860
You can send the radio, and the
radio will go over the place,
1034
01:13:01,860 --> 01:13:05,450
and got refracted
by atmosphere--
1035
01:13:05,450 --> 01:13:08,490
the atmosphere on Earth.
1036
01:13:08,490 --> 01:13:13,350
Got refracted, and the receiver
from some place which is really
1037
01:13:13,350 --> 01:13:15,710
distant from the
source can still
1038
01:13:15,710 --> 01:13:21,170
see it without any
dispersion, as we show here.
1039
01:13:21,170 --> 01:13:23,330
And it's actually
going to be propagating
1040
01:13:23,330 --> 01:13:27,870
at the speed of group velocity.
1041
01:13:27,870 --> 01:13:33,620
So you may not
actually believe that.
1042
01:13:33,620 --> 01:13:37,360
How about we do a
simulation like what
1043
01:13:37,360 --> 01:13:39,950
we did before with MIT wave?
1044
01:13:39,950 --> 01:13:43,730
So this is actually the
example which we did last time.
1045
01:13:43,730 --> 01:13:45,450
We have an nit wave.
1046
01:13:45,450 --> 01:13:48,950
We can compose that
into many, many pieces.
1047
01:13:48,950 --> 01:13:52,730
And then see how it evolved
as a function of time.
1048
01:13:52,730 --> 01:13:54,590
This is actually
without dispersion,
1049
01:13:54,590 --> 01:13:57,770
therefore everything is perfect.
1050
01:13:57,770 --> 01:14:00,760
So now I would like to
introduce some excitement there.
1051
01:14:03,820 --> 01:14:10,440
If I have dispersion, like
0.1, alpha is equal to 0.1,
1052
01:14:10,440 --> 01:14:11,860
and see will happen.
1053
01:14:11,860 --> 01:14:16,690
Then just a reminder that
things will not go super well.
1054
01:14:16,690 --> 01:14:19,140
Wait a second, what am I doing?
1055
01:14:19,140 --> 01:14:23,080
This is actually still
without dispersion.
1056
01:14:23,080 --> 01:14:24,230
Sorry for that.
1057
01:14:28,840 --> 01:14:31,730
It should be--
1058
01:14:31,730 --> 01:14:35,180
OK, so let's take a look
at the triangular case.
1059
01:14:35,180 --> 01:14:39,550
This is now with dispersion.
1060
01:14:39,550 --> 01:14:43,390
And you can see that as a
reminder as a function of time,
1061
01:14:43,390 --> 01:14:46,190
the shape of the signal
which you would like to send
1062
01:14:46,190 --> 01:14:49,160
is actually changing
as a function of time.
1063
01:14:49,160 --> 01:14:53,020
And after a few
thousands of miles,
1064
01:14:53,020 --> 01:14:57,310
you will not even recognize the
original structure we put in.
1065
01:14:57,310 --> 01:15:03,600
So that's the trouble
we are actually facing.
1066
01:15:03,600 --> 01:15:07,790
You can see that it's getting
wider and wider et cetera.
1067
01:15:07,790 --> 01:15:18,250
So now what will happen if
I send this kind of signal.
1068
01:15:18,250 --> 01:15:20,820
This is a signal which you
have some kind of shape.
1069
01:15:20,820 --> 01:15:23,880
You can imagine that there's
sounds kind of analogue.
1070
01:15:23,880 --> 01:15:27,540
And I am now doing the
calculation to actually map
1071
01:15:27,540 --> 01:15:29,870
all the individual components.
1072
01:15:29,870 --> 01:15:32,820
And now I'm going to
propagate through the median.
1073
01:15:32,820 --> 01:15:39,060
And the blue is the original
non-dispersive median
1074
01:15:39,060 --> 01:15:40,350
situation.
1075
01:15:40,350 --> 01:15:44,360
And the red is actually
the propagation
1076
01:15:44,360 --> 01:15:45,910
in a dispersive median.
1077
01:15:45,910 --> 01:15:48,770
You can see the propagation
in a dispersive median
1078
01:15:48,770 --> 01:15:53,160
is faster, because alpha is
actually larger than one--
1079
01:15:53,160 --> 01:15:54,270
larger than zero.
1080
01:15:54,270 --> 01:15:56,310
So it's actually, in
this case it's 0.1.
1081
01:15:56,310 --> 01:16:03,975
And you can see that the red
cosine omega0t modulated signal
1082
01:16:03,975 --> 01:16:12,480
is progressing, and the shape
of the analogue is not changing.
1083
01:16:12,480 --> 01:16:13,770
You can see that, right?
1084
01:16:13,770 --> 01:16:17,380
So it's very different from
what we actually see before
1085
01:16:17,380 --> 01:16:20,130
with a single triangular pulse.
1086
01:16:20,130 --> 01:16:21,420
Now you can see that, ha!
1087
01:16:21,420 --> 01:16:24,360
Only when n gets
very large, I start
1088
01:16:24,360 --> 01:16:27,150
to be able to feed all
those little structures.
1089
01:16:27,150 --> 01:16:31,740
That means the end value, or say
the omega value, which I need
1090
01:16:31,740 --> 01:16:34,429
is you really narrow,
a very narrow range,
1091
01:16:34,429 --> 01:16:36,720
which will actually match
with what we have been doing.
1092
01:16:36,720 --> 01:16:39,460
And now I start to
propagate all those things.
1093
01:16:39,460 --> 01:16:43,530
And you can see that the red
is actually traveling faster
1094
01:16:43,530 --> 01:16:46,470
than the blue, which
is what we expect.
1095
01:16:46,470 --> 01:16:48,730
And you can see
now, in the instance
1096
01:16:48,730 --> 01:16:51,120
they actually
overlap each other,
1097
01:16:51,120 --> 01:16:54,420
you can see that envelope,
the shape of the envelope,
1098
01:16:54,420 --> 01:16:55,230
is still the same.
1099
01:16:55,230 --> 01:16:57,970
It's exactly what
we actually printed.
1100
01:16:57,970 --> 01:17:01,800
And that actually brings me
to the end of my lecture.
1101
01:17:01,800 --> 01:17:06,960
We have on understood how
the AM radio actually works.
1102
01:17:06,960 --> 01:17:08,820
And next time, we
are going to talk
1103
01:17:08,820 --> 01:17:11,170
about uncertainty principles.
1104
01:17:11,170 --> 01:17:12,150
What the hell?
1105
01:17:12,150 --> 01:17:14,520
What happened?
1106
01:17:14,520 --> 01:17:17,460
And believe me,
they are actually
1107
01:17:17,460 --> 01:17:19,272
connected to each other.
1108
01:17:19,272 --> 01:17:21,970
Uncertainty principle is
actually highly related
1109
01:17:21,970 --> 01:17:24,640
to wave and the vibrations.
1110
01:17:24,640 --> 01:17:29,510
Thank you very much, and let me
know if you have any questions.