1
00:00:10 --> 00:00:16
I'd like to talk.
Thank you.
2
00:00:15 --> 00:00:21
One of the things I'd like to
give a little insight into today
3
00:00:28 --> 00:00:34
is the mathematical basis for
hearing.
4
00:00:38 --> 00:00:44
For example,
if a musical tone,
5
00:00:41 --> 00:00:47
a pure musical tone would
consist of a pure oscillation in
6
00:00:49 --> 00:00:55
terms of the vibration of the
air.
7
00:00:53 --> 00:00:59
It would be a pure oscillation.
So, [SINGS],
8
00:00:59 --> 00:01:05
and if you superimpose upon
that, suppose you sing a triad,
9
00:01:06 --> 00:01:12
[SINGS], those are three tones.
Each has its own period of
10
00:01:14 --> 00:01:20
oscillation, and then another
one, which is the top one,
11
00:01:19 --> 00:01:25
which is even faster.
The higher it is,
12
00:01:22 --> 00:01:28
the faster the thing.
Anyway, what you hear,
13
00:01:26 --> 00:01:32
then, is the sum of those
things.
14
00:01:29 --> 00:01:35
So, C plus E plus G,
let's say, what you hear is the
15
00:01:34 --> 00:01:40
wave form.
It's periodics,
16
00:01:37 --> 00:01:43
still, but it's a mess.
I don't know,
17
00:01:40 --> 00:01:46
I can't draw it.
So, this is periodic,
18
00:01:43 --> 00:01:49
but a mess, some sort of mess.
Now, of course,
19
00:01:47 --> 00:01:53
if you hear the three tones
together, most people,
20
00:01:52 --> 00:01:58
if they are not tone deaf,
anyway, can hear the three
21
00:01:56 --> 00:02:02
tones that make up that.
So, in other words,
22
00:02:01 --> 00:02:07
if this is the function which
is the sum of those three,
23
00:02:05 --> 00:02:11
some sort of messy function,
f of t,
24
00:02:09 --> 00:02:15
you're able to do Fourier
analysis on it,
25
00:02:12 --> 00:02:18
and break it up.
You're able to take that f of
26
00:02:15 --> 00:02:21
t, and somehow mentally express
it as the sum of three pure
27
00:02:20 --> 00:02:26
oscillations.
That's Fourier analysis.
28
00:02:22 --> 00:02:28
We've been doing it with an
infinite series,
29
00:02:26 --> 00:02:32
but it's okay.
It's still Fourier analysis if
30
00:02:29 --> 00:02:35
you do it with just three.
So, in other words,
31
00:02:34 --> 00:02:40
the f of t is going to
be the sum of,
32
00:02:38 --> 00:02:44
let's say, sine,
I don't know,
33
00:02:41 --> 00:02:47
it's going to be the sign of
one frequency plus the sine of
34
00:02:46 --> 00:02:52
another frequency plus the sine
of a third, maybe with
35
00:02:51 --> 00:02:57
coefficients here.
So, somehow,
36
00:02:53 --> 00:02:59
since you were born,
you have been able to take the
37
00:02:58 --> 00:03:04
f of t,
and express it as the sum of
38
00:03:02 --> 00:03:08
the three signs.
And, here, therefore,
39
00:03:07 --> 00:03:13
the three tones that make up
the triad.
40
00:03:11 --> 00:03:17
Now, the question is,
how did you do that Fourier
41
00:03:15 --> 00:03:21
analysis?
In other words,
42
00:03:18 --> 00:03:24
does your brain have a little
integrator in it,
43
00:03:23 --> 00:03:29
which calculates the
coefficients of that series?
44
00:03:28 --> 00:03:34
Of course, the answer is no.
It has to do something else.
45
00:03:34 --> 00:03:40
So, one of the things I'd like
to aim at in this lecture is
46
00:03:39 --> 00:03:45
just briefly explaining what,
in fact, actually happens to do
47
00:03:44 --> 00:03:50
that.
Now, to do that,
48
00:03:46 --> 00:03:52
we'll have to make some little
detours, as always.
49
00:03:50 --> 00:03:56
So, first I'm going to,
throughout the lecture,
50
00:03:54 --> 00:04:00
in fact, I gave you last time a
couple of shortcuts for
51
00:03:59 --> 00:04:05
calculating Fourier series based
on evenness and oddness,
52
00:04:04 --> 00:04:10
and also some expansion of the
idea of Fourier series where we
53
00:04:09 --> 00:04:15
use the different,
but things didn't have to be
54
00:04:13 --> 00:04:19
periodic or period two pi,
but it can have an arbitrary
55
00:04:17 --> 00:04:23
period, 2L, and we could still
get a Fourier expansion for it.
56
00:04:25 --> 00:04:31
Let me, therefore,
begin just as a problem,
57
00:04:28 --> 00:04:34
another type of shortcut
exercise, to do a Fourier
58
00:04:33 --> 00:04:39
calculation, which we are going
to be later in the period to
59
00:04:38 --> 00:04:44
explain the music problem.
So, let's suppose we're
60
00:04:43 --> 00:04:49
starting with the function,
f of t,
61
00:04:48 --> 00:04:54
which is a real square wave,
and I'll make its period
62
00:04:55 --> 00:05:01
different from the one,
not two pi.
63
00:05:00 --> 00:05:06
So, suppose we had a function
like this.
64
00:05:02 --> 00:05:08
So, this is one,
and this is one.
65
00:05:04 --> 00:05:10
So, the height is one,
and this point is one as well.
66
00:05:08 --> 00:05:14
And then, it's periodic ever
after that.
67
00:05:10 --> 00:05:16
I'll tell you what,
let's do like the electrical
68
00:05:13 --> 00:05:19
engineers do and put these
vertical lines there even though
69
00:05:17 --> 00:05:23
they don't exist.
Okay, so the height is one and
70
00:05:20 --> 00:05:26
it goes over.
The half period is one.
71
00:05:23 --> 00:05:29
This really is a square wave.
I mean, it's really a square,
72
00:05:27 --> 00:05:33
not what they usually call a
square wave.
73
00:05:31 --> 00:05:37
So, my question is,
what's its Fourier series?
74
00:05:36 --> 00:05:42
Well, it's neither even nor
odd.
75
00:05:40 --> 00:05:46
That's a little dismaying.
It sounds like we're going to
76
00:05:47 --> 00:05:53
have to calculate an's and bn's.
So, the shortcuts I gave you
77
00:05:54 --> 00:06:00
last time don't seem to be
applicable.
78
00:06:00 --> 00:06:06
Now, of course,
nor is the period two pi,
79
00:06:03 --> 00:06:09
but that shouldn't be too bad.
In fact, you ought to look for
80
00:06:07 --> 00:06:13
an expansion in terms of things
that look like sine of n,
81
00:06:11 --> 00:06:17
well, what should it be?
82
00:06:14 --> 00:06:20
Since L is equal to one,
the half period is equal to
83
00:06:18 --> 00:06:24
one.
Remember, the period is 2L,
84
00:06:20 --> 00:06:26
not L.
It's n pi over L,
85
00:06:22 --> 00:06:28
but if L is one,
we should be looking for an
86
00:06:26 --> 00:06:32
expansion in terms of functions
that look like this.
87
00:06:30 --> 00:06:36
Now, since we've already done
the work for the official square
88
00:06:34 --> 00:06:40
way, which looks something like
this, what you always try to do
89
00:06:39 --> 00:06:45
is reduce these things to
problems that you've already
90
00:06:43 --> 00:06:49
solved.
This is a legitimate one,
91
00:06:48 --> 00:06:54
since I solved it in lecture
for you.
92
00:06:52 --> 00:06:58
So, we can consider it as
something we know.
93
00:06:56 --> 00:07:02
So, I observed that since I am
very lazy, that if I lower this
94
00:07:02 --> 00:07:08
function by one half,
it will become an odd function.
95
00:07:09 --> 00:07:15
Now it's an odd function.
Okay, I just cut the work in
96
00:07:17 --> 00:07:23
half.
So, let's call this function,
97
00:07:22 --> 00:07:28
let's call this,
I don't know,
98
00:07:26 --> 00:07:32
S of t.
The green one is the one we
99
00:07:32 --> 00:07:38
wanted to start with.
So, f of t is a green
100
00:07:37 --> 00:07:43
function.
But, I can improve things even
101
00:07:40 --> 00:07:46
more because the function that
we calculated in the lecture is
102
00:07:46 --> 00:07:52
a lot like this salmon function.
That's why I called it S.
103
00:07:51 --> 00:07:57
But, the difference is that the
function we calculated with this
104
00:07:56 --> 00:08:02
one.
In the first place,
105
00:07:58 --> 00:08:04
it went down further.
It went not to negative one
106
00:08:04 --> 00:08:10
half, which is where that one
goes.
107
00:08:08 --> 00:08:14
But, it went down to negative
one, and then went up here to
108
00:08:15 --> 00:08:21
plus one.
And, it went over to pi.
109
00:08:18 --> 00:08:24
So, it came down again,
but not, but at the point,
110
00:08:23 --> 00:08:29
pi.
And here, negative pi went up
111
00:08:27 --> 00:08:33
again.
Okay, let me remind you what
112
00:08:31 --> 00:08:37
this one was.
Suppose we call it,
113
00:08:34 --> 00:08:40
O doesn't look good,
I don't know,
114
00:08:38 --> 00:08:44
how about g of u?
Let's, for a secret reason,
115
00:08:44 --> 00:08:50
call the variable u this time,
okay?
116
00:08:47 --> 00:08:53
So, the previous knowledge that
I'm relying on was that I
117
00:08:50 --> 00:08:56
derived the Fourier series for
you by an orthodox calculation.
118
00:08:54 --> 00:09:00
And, it's not too hard to do
because this is an odd function.
119
00:08:58 --> 00:09:04
And therefore,
you only have to calculate the
120
00:09:01 --> 00:09:07
bn's.
And, half of them turn out to
121
00:09:05 --> 00:09:11
be zero, although you don't know
that in advance.
122
00:09:09 --> 00:09:15
But anyway, the answer was four
over pi times the sum
123
00:09:14 --> 00:09:20
of just the odd ones,
the sine of n u,
124
00:09:17 --> 00:09:23
and that you had to
divide by n.
125
00:09:21 --> 00:09:27
So, this is the expansion of g,
this function,
126
00:09:25 --> 00:09:31
g of u, the Fourier expansion
of this function.
127
00:09:30 --> 00:09:36
Since it's an odd function,
it only involves the signs.
128
00:09:34 --> 00:09:40
There's no funny stuff here
because the period is now two
129
00:09:39 --> 00:09:45
pi.
And, this came from the first
130
00:09:41 --> 00:09:47
lecture on Fourier series,
or from the book,
131
00:09:45 --> 00:09:51
wherever you want it,
or solutions to the notes.
132
00:09:49 --> 00:09:55
There are lots of sources for
that.
133
00:09:51 --> 00:09:57
The solution's in the notes.
Okay, now, that looks so much
134
00:09:56 --> 00:10:02
like the salmon function,
---
135
00:10:00 --> 00:10:06
-- I ought to be able to
convert one into the other.
136
00:10:04 --> 00:10:10
Now, I will do that by
shrinking the axis.
137
00:10:07 --> 00:10:13
But, since this can get rather
confusing, what I'll do is
138
00:10:11 --> 00:10:17
overlay this.
What I prefer to do is I think
139
00:10:15 --> 00:10:21
u, okay, I'm changing,
I'm keeping the thing the same.
140
00:10:19 --> 00:10:25
But, I'm going to change the
name of the variable,
141
00:10:23 --> 00:10:29
the t, in such a way that on
the t-axis, this becomes the
142
00:10:27 --> 00:10:33
point, one.
If I do that,
143
00:10:30 --> 00:10:36
then this function will turn
exactly into that one,
144
00:10:34 --> 00:10:40
except it will go not from
minus a half to a half,
145
00:10:37 --> 00:10:43
but it will go from negative
one to one, since I haven't done
146
00:10:41 --> 00:10:47
anything to the vertical axis.
So, how I do that?
147
00:10:45 --> 00:10:51
What's the relation between u
and t?
148
00:10:47 --> 00:10:53
Well, u is equal to pi times t,
or the other way around.
149
00:10:51 --> 00:10:57
You know that it's going to be
approximately this.
150
00:10:54 --> 00:11:00
Try one, and then check that it
works.
151
00:10:57 --> 00:11:03
When t is equal to one,
u is pi, which is what it's
152
00:11:00 --> 00:11:06
supposed to be.
So, this is the relation
153
00:11:04 --> 00:11:10
between the two.
And therefore,
154
00:11:06 --> 00:11:12
without further ado,
I can say that,
155
00:11:09 --> 00:11:15
let's write the relation
between them.
156
00:11:11 --> 00:11:17
f of t is what I want.
Well, what's f of t if I
157
00:11:15 --> 00:11:21
subtract one half of that?
So, that's going to be equal to
158
00:11:19 --> 00:11:25
the salmon function plus one
half, right, or the salmon
159
00:11:23 --> 00:11:29
function is f of t lowered by
one half.
160
00:11:26 --> 00:11:32
One thing is the same as the
other.
161
00:11:30 --> 00:11:36
And, what's the relation
between this salmon function and
162
00:11:35 --> 00:11:41
the orange function?
Well, the salmon function is,
163
00:11:40 --> 00:11:46
so, let's convert,
so, S of t -- it's more
164
00:11:45 --> 00:11:51
convenient, as I wrote the
formula g of u.
165
00:11:49 --> 00:11:55
Let's start it from that end.
If I start from g of u,
166
00:11:54 --> 00:12:00
what do I have to do to convert
it into S of-- into the salmon
167
00:12:00 --> 00:12:06
function?
Well, take one half of it.
168
00:12:05 --> 00:12:11
So, if I put them all together,
the conclusion is that f of t
169
00:12:12 --> 00:12:18
is equal to one half
plus S of t,
170
00:12:18 --> 00:12:24
which is one half of g of u,
171
00:12:22 --> 00:12:28
but u is pi t.
So, it's four pi,
172
00:12:25 --> 00:12:31
four over pi times the sum of
the sine of n.
173
00:12:31 --> 00:12:37
And, for u, I will write pi t
174
00:12:36 --> 00:12:42
divided by n.
And, sorry, I forgot to say
175
00:12:40 --> 00:12:46
that sum is only over the odd
values of n, not all values of
176
00:12:44 --> 00:12:50
n.
So, the sum over n odd of that,
177
00:12:47 --> 00:12:53
and, of course,
the two will cancel that.
178
00:12:50 --> 00:12:56
So, here we have,
in other words,
179
00:12:53 --> 00:12:59
just by this business of
shrinking or just stretching or
180
00:12:57 --> 00:13:03
shrinking the axis,
lowering it and squishing it
181
00:13:01 --> 00:13:07
that way a little bit.
We get from this Fourier
182
00:13:06 --> 00:13:12
series, we get that one just by
this geometric procedure.
183
00:13:11 --> 00:13:17
I'd like you to be able to do
that because it saves a lot of
184
00:13:16 --> 00:13:22
time.
Okay, so let's put this answer
185
00:13:19 --> 00:13:25
up in, I'm going to need it in a
minute, but I don't really want
186
00:13:25 --> 00:13:31
to recopy it.
So, let me handle it by
187
00:13:28 --> 00:13:34
erasing.
So, let's call that plus two
188
00:13:32 --> 00:13:38
over pi,
and there is our formula for
189
00:13:36 --> 00:13:42
that green function that we
wrote before.
190
00:13:39 --> 00:13:45
So, I'll put that in green.
So, we'll have a color-coded
191
00:13:43 --> 00:13:49
lecture again.
Now, what we're going to be
192
00:13:46 --> 00:13:52
doing ultimately,
to getting at the music problem
193
00:13:50 --> 00:13:56
that I posed at the beginning of
the lecture, is we want to
194
00:13:55 --> 00:14:01
solve, and this is what a study
of Fourier series has been
195
00:13:59 --> 00:14:05
aiming at, to solve second-order
linear equations with constant
196
00:14:04 --> 00:14:10
coefficients were the right-hand
side was a more general function
197
00:14:09 --> 00:14:15
than the kind we've been
handling.
198
00:14:14 --> 00:14:20
So, now, in order to simplify,
and we don't have a lot of time
199
00:14:17 --> 00:14:23
in the course,
I'd have to take another day to
200
00:14:20 --> 00:14:26
make more complicated
calculations,
201
00:14:23 --> 00:14:29
which I don't want to do since
you will learn a lot from them,
202
00:14:27 --> 00:14:33
anyway.
I think you will find you've
203
00:14:29 --> 00:14:35
had enough calculation by the
time Friday morning rolls
204
00:14:32 --> 00:14:38
around.
So, let's look at the undamped
205
00:14:36 --> 00:14:42
case, which is simpler,
or undamped spring,
206
00:14:39 --> 00:14:45
or undamped anything because it
doesn't have that extra term,
207
00:14:44 --> 00:14:50
which requires extra
calculations.
208
00:14:46 --> 00:14:52
So, I'll follow the book now
and some of the notes and the
209
00:14:51 --> 00:14:57
visuals, and called the
independent variable-- the
210
00:14:54 --> 00:15:00
dependent variable I'm going to
call x now.
211
00:14:58 --> 00:15:04
And, the independent variable
is, as usual,
212
00:15:01 --> 00:15:07
time.
So, this is going to be,
213
00:15:04 --> 00:15:10
in general, f of t,
and I'm going to use it by
214
00:15:08 --> 00:15:14
calculating example,
this is the actual f of t I'm
215
00:15:12 --> 00:15:18
going to be using.
But, the general problem for a
216
00:15:16 --> 00:15:22
general f of t is to solve this,
or at least to find a
217
00:15:20 --> 00:15:26
particular solution.
That's what most of the work
218
00:15:23 --> 00:15:29
is, because we already know how
from that to get the general
219
00:15:28 --> 00:15:34
solution by adding the solution
to the reduced equation,
220
00:15:32 --> 00:15:38
the associated homogeneous
equation.
221
00:15:36 --> 00:15:42
So, all our work has been,
this past couple of weeks,
222
00:15:40 --> 00:15:46
in how you find a particular
solution.
223
00:15:44 --> 00:15:50
Now, the case in which we know
what to do is,
224
00:15:48 --> 00:15:54
so we can find our particular
solution.
225
00:15:52 --> 00:15:58
Let's call that x sub p.
226
00:15:55 --> 00:16:01
We could find x sub p if the
right hand side is cosine omega,
227
00:16:01 --> 00:16:07
well, in general,
an exponential,
228
00:16:04 --> 00:16:10
but since we are not going to
use complex exponentials today,
229
00:16:09 --> 00:16:15
all these things are real.
And I'd like to keep them real.
230
00:16:16 --> 00:16:22
If it's either cosine omega t
or sine omega t,
231
00:16:20 --> 00:16:26
or some multiple of that by
232
00:16:22 --> 00:16:28
linearity, it's just as good.
We already know how to find the
233
00:16:26 --> 00:16:32
thing, and to find a particular
solution.
234
00:16:30 --> 00:16:36
So, the procedure is use
complex exponentials,
235
00:16:33 --> 00:16:39
and that magic formula I gave
you.
236
00:16:36 --> 00:16:42
But, right now,
just to save a little time,
237
00:16:39 --> 00:16:45
since I already did that on the
lecture on resonance,
238
00:16:43 --> 00:16:49
I solved it explicitly for
that, and you've had adequate
239
00:16:48 --> 00:16:54
practice I think in the problem
sets.
240
00:16:50 --> 00:16:56
Let's simply write down the
answer that comes out of that.
241
00:16:55 --> 00:17:01
The answer for the particular
solution is cosine omega t
242
00:16:59 --> 00:17:05
or sine omega t.
243
00:17:05 --> 00:17:11
That's the top.
And, it's over a constant.
244
00:17:08 --> 00:17:14
And, the constant is omega
naught squared.
245
00:17:13 --> 00:17:19
That's the natural frequency
which comes from the system,
246
00:17:19 --> 00:17:25
minus the imposed frequency,
the driving frequency that the
247
00:17:24 --> 00:17:30
system, the spring or whatever
it is, undamped spring,
248
00:17:29 --> 00:17:35
is being driven with.
Okay, understand the notation.
249
00:17:34 --> 00:17:40
Cosine this over that,
or sine, depending on whether
250
00:17:39 --> 00:17:45
you started driving it with
cosine or sine.
251
00:17:43 --> 00:17:49
So, this is from the lecture,
if you like,
252
00:17:46 --> 00:17:52
from the lecture on resonance,
but again it's,
253
00:17:50 --> 00:17:56
I hope by now,
a familiar fact.
254
00:17:53 --> 00:17:59
Let me remind you what this had
to do with resonance.
255
00:17:58 --> 00:18:04
Then, the observation was that
if omega, the driving frequency
256
00:18:03 --> 00:18:09
is very close to the natural
frequency, then this is close to
257
00:18:09 --> 00:18:15
that.
The denominator is almost zero,
258
00:18:13 --> 00:18:19
and that makes the amplitude of
the response very,
259
00:18:17 --> 00:18:23
very large.
And, that was the phenomenon of
260
00:18:20 --> 00:18:26
resonance.
Okay, now what I'd like to do
261
00:18:23 --> 00:18:29
is apply those formulas to
finding out what happens for a
262
00:18:28 --> 00:18:34
general f(t),
or in particular this one.
263
00:18:32 --> 00:18:38
So, in general,
I'll keep using the notation,
264
00:18:36 --> 00:18:42
f of t,
even though I've sorted used it
265
00:18:41 --> 00:18:47
for that.
But in general,
266
00:18:44 --> 00:18:50
what's the situation?
If f of t is a sine series,
267
00:18:49 --> 00:18:55
cosine series,
all right, let's do everything.
268
00:18:53 --> 00:18:59
Suppose it's,
in other words,
269
00:18:56 --> 00:19:02
the procedure is,
take your f of t,
270
00:19:00 --> 00:19:06
expand it in a Fourier series.
Well, doesn't that assume it's
271
00:19:07 --> 00:19:13
periodic?
Yes, sort of.
272
00:19:09 --> 00:19:15
So, suppose it's a Fourier
series.
273
00:19:12 --> 00:19:18
I'll make a very general
Fourier series,
274
00:19:15 --> 00:19:21
write it this way:
cosine (omega)n t,
275
00:19:18 --> 00:19:24
and then the sine terms,
276
00:19:22 --> 00:19:28
sine (omega)n t
from one to infinity where
277
00:19:27 --> 00:19:33
the omegas are,
omega n is short for that.
278
00:19:32 --> 00:19:38
Well, it's going to have the n
in it, of course,
279
00:19:35 --> 00:19:41
but I want, now,
to make the general period to
280
00:19:39 --> 00:19:45
be 2L.
So, it would be n pi over L.
281
00:19:42 --> 00:19:48
Of course, if L is equal to
282
00:19:45 --> 00:19:51
one, then it's n pi.
Or, if L equals pi,
283
00:19:48 --> 00:19:54
those are the two most popular
cases, by far.
284
00:19:52 --> 00:19:58
Then, it's simply n itself,
the driving frequency.
285
00:19:56 --> 00:20:02
But, this would be the general
case, n pi over L
286
00:20:01 --> 00:20:07
if the period is the period of f
of t is 2L.
287
00:20:07 --> 00:20:13
So, that's what the Fourier
series looks like.
288
00:20:10 --> 00:20:16
Okay, then the particular
solution will be what?
289
00:20:14 --> 00:20:20
Well, I got these formulas.
In other words,
290
00:20:18 --> 00:20:24
what I'm using is superposition
principle.
291
00:20:21 --> 00:20:27
If it's just this,
then I know what the answer is
292
00:20:25 --> 00:20:31
for the particular solution,
the response.
293
00:20:30 --> 00:20:36
So, if you make a sum of these
things, a sum of these inputs,
294
00:20:34 --> 00:20:40
you are going to get a sum of
the responses by superposition.
295
00:20:39 --> 00:20:45
So, let's write out the ones we
are absolutely certain of.
296
00:20:43 --> 00:20:49
What's the response to here?
Well, it's (a)n cosine omega n
297
00:20:47 --> 00:20:53
t. The only thing is,
298
00:20:51 --> 00:20:57
now it's divided by omega
naught squared.
299
00:20:55 --> 00:21:01
This constant has changed,
and the same thing here.
300
00:21:00 --> 00:21:06
Of course, by linearity,
if this is multiplied by a,
301
00:21:03 --> 00:21:09
then the answer is multiplied
by, the response is also
302
00:21:06 --> 00:21:12
multiplied by a.
So, the same thing happens
303
00:21:09 --> 00:21:15
here.
Here, it's (b)n and over,
304
00:21:11 --> 00:21:17
again, omega naught squared
minus omega times the sine of
305
00:21:14 --> 00:21:20
omega t.
306
00:21:17 --> 00:21:23
So, in other words,
as soon as you have the Fourier
307
00:21:20 --> 00:21:26
expansion, the Fourier series
for the input,
308
00:21:23 --> 00:21:29
you automatically get this by
just writing it down the Fourier
309
00:21:27 --> 00:21:33
series for the response.
That's the fundamental idea of
310
00:21:32 --> 00:21:38
Fourier series,
at least applied in this
311
00:21:35 --> 00:21:41
context.
They have many other contexts,
312
00:21:38 --> 00:21:44
approximations,
so on and so forth.
313
00:21:41 --> 00:21:47
But, that's the idea here.
All right, what about that
314
00:21:45 --> 00:21:51
constant term?
Well, this formula still works
315
00:21:49 --> 00:21:55
if omega equals zero.
If omega equals zero,
316
00:21:52 --> 00:21:58
then this is the constant,
one.
317
00:21:54 --> 00:22:00
The formula is still correct.
Omega is zero here.
318
00:22:00 --> 00:22:06
The only thing you have to
remember is that the original
319
00:22:03 --> 00:22:09
thing is written in this form.
So, the response will be,
320
00:22:07 --> 00:22:13
what will it be?
Well, it's one divided by omega
321
00:22:10 --> 00:22:16
naught squared,
if I'm in the case omega zero
322
00:22:12 --> 00:22:18
is equal to zero.
So, it's a zero divided by two
323
00:22:16 --> 00:22:22
omega naught squared.
And, as you will see,
324
00:22:19 --> 00:22:25
it looks just like the others.
You're just taking omega,
325
00:22:23 --> 00:22:29
and making it equal to zero for
that particular case.
326
00:22:26 --> 00:22:32
Sorry, this should be omega n's
all the way through here.
327
00:22:31 --> 00:22:37
328
00:22:40 --> 00:22:46
All right, well,
let's apply this to the green
329
00:22:45 --> 00:22:51
function.
So, what have we got?
330
00:22:49 --> 00:22:55
We have its Fourier series.
So, if the green function is,
331
00:22:56 --> 00:23:02
if the input in other words is
this square wave,
332
00:23:02 --> 00:23:08
the green square wave,
so in your notes,
333
00:23:06 --> 00:23:12
this guy, this particular f of
t is the input.
334
00:23:15 --> 00:23:21
And, the equation is x double
prime plus omega naught squared
335
00:23:20 --> 00:23:26
x equals f of t.
336
00:23:23 --> 00:23:29
Then, the response is,
well, I can't draw you a
337
00:23:27 --> 00:23:33
picture of the response because
I don't know what the Fourier
338
00:23:32 --> 00:23:38
series actually looks like.
But, let's at least write down
339
00:23:37 --> 00:23:43
what the Fourier series is.
The Fourier series will be,
340
00:23:43 --> 00:23:49
well, what is it?
It's one half.
341
00:23:45 --> 00:23:51
The constant out front is one
half, except it's one over two
342
00:23:50 --> 00:23:56
omega naught squared.
343
00:23:53 --> 00:23:59
So, this is my function,
f of t.
344
00:23:56 --> 00:24:02
That's the general formula for
how the input is related to the
345
00:24:01 --> 00:24:07
response.
And, I'm applying it to this
346
00:24:06 --> 00:24:12
particular function,
f of t.
347
00:24:10 --> 00:24:16
And, the answer is plus.
Well, my Fourier series
348
00:24:15 --> 00:24:21
involves only odd sums,
only the summation over odd,
349
00:24:21 --> 00:24:27
and only of the sign.
So, it is going to be two over
350
00:24:26 --> 00:24:32
pi,
sorry, so it's going to be two
351
00:24:31 --> 00:24:37
over pi out front.
That constant will carry along
352
00:24:37 --> 00:24:43
by linearity.
And, I'm going to sum over odd,
353
00:24:39 --> 00:24:45
n odd values only.
The basic thing in the upstairs
354
00:24:43 --> 00:24:49
is going to be the sine of omega
n t.
355
00:24:47 --> 00:24:53
But, what is (omega)n?
Well, (omega)n is n pi.
356
00:24:50 --> 00:24:56
So, it's n pi t.
And, how about the bottom?
357
00:24:53 --> 00:24:59
The bottom is going to be omega
naught squared minus omega n
358
00:24:57 --> 00:25:03
squared.
359
00:25:01 --> 00:25:07
And, this is my (omega)n,
minus n pi squared.
360
00:25:05 --> 00:25:11
What's that?
361
00:25:07 --> 00:25:13
Well, I don't know.
All I could do would be to
362
00:25:12 --> 00:25:18
calculate it.
You could put it on MATLAB and
363
00:25:16 --> 00:25:22
ask MATLAB to calculate and plot
for you the first few terms,
364
00:25:22 --> 00:25:28
and get some vague idea of what
it looks like.
365
00:25:26 --> 00:25:32
That's nice,
but it's not what's interesting
366
00:25:31 --> 00:25:37
to do.
What's interesting to do is to
367
00:25:35 --> 00:25:41
look at the size of the
coefficients.
368
00:25:38 --> 00:25:44
And, again, rather than do it
in the abstract,
369
00:25:42 --> 00:25:48
let's take a specific value.
Let's suppose that the natural
370
00:25:46 --> 00:25:52
frequency of the system,
in other words,
371
00:25:50 --> 00:25:56
the frequency at which that
little spring wants to go
372
00:25:54 --> 00:26:00
vibrate back and forth,
whatever you got vibrating.
373
00:25:58 --> 00:26:04
Let's suppose the natural
frequency that's omega naught is
374
00:26:03 --> 00:26:09
ten for the sake of
definiteness,
375
00:26:05 --> 00:26:11
as they say.
Okay, if that's ten,
376
00:26:09 --> 00:26:15
all I want to do is calculate
in the crudest possible way what
377
00:26:15 --> 00:26:21
a few of these terms are.
So, the response is,
378
00:26:19 --> 00:26:25
so let's see,
we've got to give that a name.
379
00:26:23 --> 00:26:29
The response is (x)p of t.
380
00:26:26 --> 00:26:32
What's (x)p of t?
I'm just going to calculate it
381
00:26:31 --> 00:26:37
very approximately.
This means, you know,
382
00:26:35 --> 00:26:41
throwing caution to the winds
because I don't have a
383
00:26:39 --> 00:26:45
calculator with me.
And, I want you to look at this
384
00:26:43 --> 00:26:49
thing without a calculator.
The first term is one over 200.
385
00:26:47 --> 00:26:53
Okay, that's the only term I
386
00:26:50 --> 00:26:56
can get exactly right.
[LAUGHTER] Or,
387
00:26:52 --> 00:26:58
I could if I could calculate.
I suppose it's 0.005,
388
00:26:56 --> 00:27:02
right?
That's the constant term.
389
00:27:00 --> 00:27:06
Okay, so the next term,
let's see, two over pi is two
390
00:27:04 --> 00:27:10
thirds.
I'll keep that in mind,
391
00:27:07 --> 00:27:13
right?
Plus two thirds,
392
00:27:09 --> 00:27:15
0.6, let's say,
that's an indication of the
393
00:27:13 --> 00:27:19
accuracy with which these things
are going to be performed.
394
00:27:19 --> 00:27:25
I think in Texas for a long
while, the legislature declared
395
00:27:24 --> 00:27:30
pi to be three,
anyways.
396
00:27:27 --> 00:27:33
One of those states did it to
save calculation time.
397
00:27:31 --> 00:27:37
I'm not kidding,
by the way.
398
00:27:36 --> 00:27:42
All right, so what's the first
term?
399
00:27:38 --> 00:27:44
If n equals one,
I have the sine of pi t.
400
00:27:42 --> 00:27:48
That's the n equals one term.
401
00:27:46 --> 00:27:52
What's the denominator like?
That's about 100 minus 9
402
00:27:50 --> 00:27:56
squared. Let's say it's 91,
403
00:27:53 --> 00:27:59
sine t over 91.
What's the next term?
404
00:27:56 --> 00:28:02
Sine of three pi t, remember,
405
00:28:00 --> 00:28:06
I am omitting,
I'm only using the odd values
406
00:28:04 --> 00:28:10
of n because those are the only
ones that enter into the Fourier
407
00:28:09 --> 00:28:15
expansion for this function,
which is at the bottom of
408
00:28:14 --> 00:28:20
everything.
All right, what's the sine
409
00:28:19 --> 00:28:25
three pi t?
Well, now, I've got 100 minus
410
00:28:26 --> 00:28:32
three pi, --
-- that's 9 squared is 81.
411
00:28:32 --> 00:28:38
So, no, what am I doing?
So, we have 100 minus three
412
00:28:42 --> 00:28:48
times pi is 9,
squared.
413
00:28:46 --> 00:28:52
Well, let's say a little more.
Let's say 85.
414
00:28:53 --> 00:28:59
So, that's 15.
How bout the next one?
415
00:29:02 --> 00:29:08
Well, it's sine 5 pi t.
416
00:29:05 --> 00:29:11
I think I'll stop here as soon
as we do this one because at
417
00:29:09 --> 00:29:15
this point it's clear what's
happening.
418
00:29:12 --> 00:29:18
This is 100 squared minus,
that's 15 squared is 225,
419
00:29:16 --> 00:29:22
so that's about 125 with a
negative sign.
420
00:29:19 --> 00:29:25
So, minus this divided by 125.
And, after this they are going
421
00:29:24 --> 00:29:30
to get really quite small
because the next one will be
422
00:29:28 --> 00:29:34
seven pi squared.
That's 400, and this is
423
00:29:33 --> 00:29:39
becoming negligible.
So, what's happening?
424
00:29:38 --> 00:29:44
So, it's approximately,
in other words,
425
00:29:42 --> 00:29:48
0.005 plus the next coefficient
is, let's see,
426
00:29:48 --> 00:29:54
6/10, let's say 100,
sine pi t.
427
00:29:51 --> 00:29:57
And, what comes next?
Well, it's now 1/20th.
428
00:29:56 --> 00:30:02
It's about a 20th.
Let's call that 0.005 sine
429
00:30:03 --> 00:30:09
three pi t,
and now so small,
430
00:30:08 --> 00:30:14
minus 0.01, let's say times
this last one,
431
00:30:13 --> 00:30:19
sine 5 pi t.
What you find,
432
00:30:16 --> 00:30:22
in other words,
is that the frequencies which
433
00:30:22 --> 00:30:28
make up the response do not
occur with the same amplitude.
434
00:30:30 --> 00:30:36
What happens is that this
amplitude is roughly five times
435
00:30:35 --> 00:30:41
larger than any of the
neighboring ones.
436
00:30:38 --> 00:30:44
And after that,
it's a lot larger than the ones
437
00:30:43 --> 00:30:49
that come later.
In other words,
438
00:30:46 --> 00:30:52
the main frequency which occurs
in the response is the frequency
439
00:30:52 --> 00:30:58
three pi.
What's happened is,
440
00:30:54 --> 00:31:00
in other words,
near resonance has occurred.
441
00:31:00 --> 00:31:06
So, if omega is ten,
very near resonance,
442
00:31:04 --> 00:31:10
that is, it's not too close,
but it's not too far away
443
00:31:10 --> 00:31:16
either, occurs for the frequency
three pi in the input.
444
00:31:16 --> 00:31:22
Now, where's the frequency
three pi in the input?
445
00:31:22 --> 00:31:28
It isn't there.
It's just that green thing.
446
00:31:27 --> 00:31:33
Where in that is the frequency
three pi?
447
00:31:33 --> 00:31:39
I can't answer that for you,
but that's the function of
448
00:31:37 --> 00:31:43
Fourier series,
to say that you can decompose
449
00:31:41 --> 00:31:47
that green function into a sum
of frequencies,
450
00:31:45 --> 00:31:51
as it were, and the Fourier
coefficients tell you how much
451
00:31:50 --> 00:31:56
frequency goes into each of
those f of t's.
452
00:31:54 --> 00:32:00
Now, so, f of t is decomposed
into the sum of frequencies by
453
00:31:59 --> 00:32:05
the Fourier analysis.
But, the system isn't going to
454
00:32:04 --> 00:32:10
respond equally to all those
frequencies.
455
00:32:07 --> 00:32:13
It's going to pick out and
favor the one which is closest
456
00:32:12 --> 00:32:18
to its natural frequency.
So, what's happened,
457
00:32:15 --> 00:32:21
these frequencies,
the frequencies and their
458
00:32:19 --> 00:32:25
relative importance in f of t
are hidden,
459
00:32:23 --> 00:32:29
as it were.
They're hidden because we can't
460
00:32:26 --> 00:32:32
see them unless you do the
Fourier analysis,
461
00:32:30 --> 00:32:36
and look at the size of the
coefficients.
462
00:32:35 --> 00:32:41
But, the system can pick out.
The system picks out and
463
00:32:44 --> 00:32:50
favors, picks out for resonance,
or resonates with,
464
00:32:54 --> 00:33:00
resonates with the frequencies
closest to its natural
465
00:33:04 --> 00:33:10
frequency.
Well, suppose the system had
466
00:33:09 --> 00:33:15
natural frequency,
not ten.
467
00:33:11 --> 00:33:17
This is a put up job.
Suppose it had natural
468
00:33:14 --> 00:33:20
frequency five.
Well, in that case,
469
00:33:17 --> 00:33:23
none of them are close to the
hidden frequencies in f of t,
470
00:33:21 --> 00:33:27
and there would be no
resonance.
471
00:33:25 --> 00:33:31
But, because of the particular
value I gave here,
472
00:33:29 --> 00:33:35
I gave the value ten,
it's able to pick out n equals
473
00:33:33 --> 00:33:39
three as the most important,
the corresponding three pi as
474
00:33:37 --> 00:33:43
the most important frequency in
the input, and respond to that.
475
00:33:44 --> 00:33:50
Okay, so this is the way we
hear, give or take a few
476
00:33:48 --> 00:33:54
thousand pages.
So, what does the ear do?
477
00:33:51 --> 00:33:57
How does the ear,
so, it's got that thing,
478
00:33:55 --> 00:34:01
messy curve,
which I erased,
479
00:33:57 --> 00:34:03
which has a secret,
which just has three hidden
480
00:34:01 --> 00:34:07
frequencies.
Okay, from now on I hand wave,
481
00:34:05 --> 00:34:11
right, like they do in other
subjects.
482
00:34:07 --> 00:34:13
So, we got our frequency.
So, it's got a [SINGS].
483
00:34:11 --> 00:34:17
That's one frequency.
[SINGS] And,
484
00:34:13 --> 00:34:19
what goes in there is the sum
of those three,
485
00:34:16 --> 00:34:22
and the ear has to do something
to say out of all the
486
00:34:20 --> 00:34:26
frequencies in the world,
I'm going to respond to that
487
00:34:24 --> 00:34:30
one, that one,
and that one,
488
00:34:26 --> 00:34:32
and send a signal to the brain,
which the brain,
489
00:34:29 --> 00:34:35
then, will interpret as a
beautiful triad.
490
00:34:34 --> 00:34:40
Okay, so what happens is that
the ear, I don't talk
491
00:34:37 --> 00:34:43
physiology, and I never will
again.
492
00:34:39 --> 00:34:45
I know nothing about it,
but anyway, the ear,
493
00:34:43 --> 00:34:49
when you get far enough in
there, there are little three
494
00:34:46 --> 00:34:52
bones, bang, bang,
bang; this is the eardrum,
495
00:34:50 --> 00:34:56
and then there's the part which
has wax.
496
00:34:52 --> 00:34:58
Then, there's the eardrum which
vibrates, at least if there is
497
00:34:57 --> 00:35:03
not too much wax in your ear.
And then, the vibrations go
498
00:35:01 --> 00:35:07
through three little bones which
send the vibrations to the inner
499
00:35:05 --> 00:35:11
ear, which nobody ever sees.
And, the inner ear,
500
00:35:10 --> 00:35:16
then, is filled with thick
fluid and a membrane,
501
00:35:13 --> 00:35:19
and the last bone hits up
against the membrane,
502
00:35:16 --> 00:35:22
and the membrane vibrates.
And, that makes the fluid
503
00:35:20 --> 00:35:26
vibrate.
Okay, good.
504
00:35:21 --> 00:35:27
So, it's vibrating according to
the function f of t.
505
00:35:25 --> 00:35:31
Well, what then?
Well, that's the marvelous
506
00:35:28 --> 00:35:34
part.
It's almost impossible to
507
00:35:31 --> 00:35:37
believe, but there is this,
sort of like a snail thing
508
00:35:36 --> 00:35:42
inside.
I've forgotten the name.
509
00:35:38 --> 00:35:44
It's cochlea.
And, it has these hairs.
510
00:35:41 --> 00:35:47
They are not hairs really.
I don't know what else to call
511
00:35:45 --> 00:35:51
them. They're not hairs.
512
00:35:47 --> 00:35:53
But, there are things so long,
you know, they stick up.
513
00:35:52 --> 00:35:58
And, there are 20,000 of them.
And, they are of different
514
00:35:56 --> 00:36:02
lengths.
And, each one is tuned to a
515
00:35:59 --> 00:36:05
certain frequency.
Each one has a certain natural
516
00:36:05 --> 00:36:11
frequency, and they are all
different, and they are all
517
00:36:11 --> 00:36:17
graded, just like a bunch of
organ pipes.
518
00:36:16 --> 00:36:22
And, when that complicated wave
hits, the complicated wave hits,
519
00:36:23 --> 00:36:29
each one resonates to a hidden
frequency in the wave,
520
00:36:29 --> 00:36:35
which is closest to its natural
frequency.
521
00:36:35 --> 00:36:41
Now, most of them won't be
resonating at all.
522
00:36:37 --> 00:36:43
Only the ones close to the
frequency [SINGS],
523
00:36:40 --> 00:36:46
they'll resonate,
and the nearby guys will
524
00:36:43 --> 00:36:49
resonate, too,
because they will be nearby,
525
00:36:45 --> 00:36:51
almost have the same natural
frequency.
526
00:36:48 --> 00:36:54
And, over here,
there will be a few which
527
00:36:50 --> 00:36:56
resonate to [SINGS],
and finally over here a few
528
00:36:53 --> 00:36:59
which go [SINGS],
and each of those little hairs,
529
00:36:56 --> 00:37:02
little groups of hairs will
signal, send that signal to the
530
00:37:00 --> 00:37:06
auditory nerve somehow or other,
which will then carry these
531
00:37:03 --> 00:37:09
three inputs to the brain,
and the brain,
532
00:37:06 --> 00:37:12
then, will interpret that as
you are hearing [SINGS].
533
00:37:11 --> 00:37:17
So, the Fourier analysis is
done by resonance.
534
00:37:15 --> 00:37:21
You here resonance because each
of these things has a certain
535
00:37:21 --> 00:37:27
natural frequency which is able,
then, to pick out a resonant
536
00:37:27 --> 00:37:33
frequency in the input.
I'd like to finish our work on
537
00:37:32 --> 00:37:38
Fourier series.
So, for homework I'm asking you
538
00:37:35 --> 00:37:41
to do something similar.
Taken an input.
539
00:37:38 --> 00:37:44
I gave you a frequency here,
a different omega naught,
540
00:37:42 --> 00:37:48
a different input,
as you by means of this Fourier
541
00:37:46 --> 00:37:52
analysis to find out which it
will resonate,
542
00:37:50 --> 00:37:56
which of the hidden frequencies
in the input the system will
543
00:37:54 --> 00:38:00
resonate to, just so you can
work it out yourself and do it.
544
00:38:00 --> 00:38:06
Now, I'd like to first try to
match up what I just did by this
545
00:38:05 --> 00:38:11
formula with what's in your
book, since your book handles
546
00:38:10 --> 00:38:16
the identical problem but a
little differently,
547
00:38:14 --> 00:38:20
and it's essentially the same.
But I think I'd better say
548
00:38:19 --> 00:38:25
something about it.
So, the book's method,
549
00:38:22 --> 00:38:28
and to the extent which any of
these problems are worked out in
550
00:38:28 --> 00:38:34
the notes, the notes do this,
too.
551
00:38:32 --> 00:38:38
Use substitution.
Base uses differentiation of
552
00:38:36 --> 00:38:42
Fourier series term by term.
The work is almost exactly the
553
00:38:42 --> 00:38:48
same as here.
And, it has a slight advantage,
554
00:38:46 --> 00:38:52
that it allows you,
the book's method has a slight
555
00:38:51 --> 00:38:57
advantage that it allows you to
forget this formula.
556
00:38:56 --> 00:39:02
You don't have to know this
formula.
557
00:39:01 --> 00:39:07
It will come out in the wash.
Now, for some of you,
558
00:39:04 --> 00:39:10
that may be of colossal
importance, in which case,
559
00:39:08 --> 00:39:14
by all means,
use the book's method,
560
00:39:10 --> 00:39:16
term by term.
So, it requires no knowledge of
561
00:39:14 --> 00:39:20
this formula because after all,
I base this solution,
562
00:39:17 --> 00:39:23
I simply wrote down the
solution and I based it on the
563
00:39:21 --> 00:39:27
fact that I was able to write
down immediately the solution to
564
00:39:26 --> 00:39:32
this and put as being that
response.
565
00:39:30 --> 00:39:36
And for that,
I had to remember it,
566
00:39:32 --> 00:39:38
or be willing to use complex
exponentials quickly to remind
567
00:39:36 --> 00:39:42
myself.
There's very,
568
00:39:38 --> 00:39:44
very little difference between
the two.
569
00:39:41 --> 00:39:47
Even if you have to re-derive
that formula,
570
00:39:44 --> 00:39:50
the two take almost about the
same length of time.
571
00:39:48 --> 00:39:54
But anyway, the idea is simply
this.
572
00:39:50 --> 00:39:56
With the book,
you assume.
573
00:39:52 --> 00:39:58
In other words,
you take your function,
574
00:39:55 --> 00:40:01
f of t.
You expand it in a Fourier
575
00:39:58 --> 00:40:04
series.
Of course, which signs and
576
00:40:01 --> 00:40:07
cosines you use will depend upon
what the period is.
577
00:40:07 --> 00:40:13
So, you assume the solution of
the form-- Well,
578
00:40:10 --> 00:40:16
if I, for example,
carried out in this particular
579
00:40:14 --> 00:40:20
case, I don't know if I will do
all the work,
580
00:40:18 --> 00:40:24
but it would be natural to
assume a solution of the form,
581
00:40:22 --> 00:40:28
since the input looks like the
green guy.
582
00:40:26 --> 00:40:32
Assume a solution which looks
the same.
583
00:40:30 --> 00:40:36
In other words,
it will have a constant term
584
00:40:33 --> 00:40:39
because the input does.
But all the rest of the terms
585
00:40:38 --> 00:40:44
will be sines.
So, it will be something like
586
00:40:42 --> 00:40:48
(c)n times the sine of n pi t.
587
00:40:46 --> 00:40:52
The only question is,
what are the (c)n's?
588
00:40:50 --> 00:40:56
Well, I found one method up
there.
589
00:40:53 --> 00:40:59
But, the general method is just
plug-in.
590
00:40:56 --> 00:41:02
Substitute into the ODE.
Substitute into the ODE.
591
00:41:02 --> 00:41:08
You differentiate this twice to
do it.
592
00:41:04 --> 00:41:10
So, I'll do the double
differentiation and I won't stop
593
00:41:08 --> 00:41:14
the lecture there,
but I will stop the calculation
594
00:41:12 --> 00:41:18
there because it has nothing new
to offer.
595
00:41:15 --> 00:41:21
And, this is the way all the
calculations in the books and
596
00:41:19 --> 00:41:25
the solutions and the notes are
carried out.
597
00:41:22 --> 00:41:28
So, I don't think you'll have
any trouble.
598
00:41:25 --> 00:41:31
Well, this term vanishes.
This term becomes what?
599
00:41:30 --> 00:41:36
If I differentiate this twice,
I get summation,
600
00:41:33 --> 00:41:39
so, this is one to infinity
because I don't know which of
601
00:41:37 --> 00:41:43
these are actually going to
appear.
602
00:41:40 --> 00:41:46
Summation one to infinity,
(c)n times, well,
603
00:41:43 --> 00:41:49
if you differentiate the sine
twice, you get negative sine,
604
00:41:48 --> 00:41:54
right?
Do it once: you get cosine.
605
00:41:50 --> 00:41:56
Second time:
you get negative sine.
606
00:41:53 --> 00:41:59
But, each time you will get
this extra factor n pi from the
607
00:41:57 --> 00:42:03
chain rule.
And so, the answer will be
608
00:42:00 --> 00:42:06
negative (c)n times n pi squared
times the sine of n pi t.
609
00:42:05 --> 00:42:11
And so, the procedure is,
610
00:42:10 --> 00:42:16
very simply,
you substitute (x)p double
611
00:42:13 --> 00:42:19
prime into the differential
equation.
612
00:42:16 --> 00:42:22
In other words,
if you do it,
613
00:42:17 --> 00:42:23
we will multiply this by omega
naught squared.
614
00:42:22 --> 00:42:28
And, you add them.
And then, on the left-hand
615
00:42:25 --> 00:42:31
side, you are going to get a sum
of terms, sine n pi t
616
00:42:29 --> 00:42:35
times coefficients
involving the (c)n's.
617
00:42:34 --> 00:42:40
And, on the right,
so, you're going to get a sum
618
00:42:37 --> 00:42:43
involving the (c)n's,
and the sines n pi t,
619
00:42:41 --> 00:42:47
and on the right,
you're going to get the Fourier
620
00:42:44 --> 00:42:50
series for f of t,
which is exactly the same kind
621
00:42:49 --> 00:42:55
of expression.
The only difference is,
622
00:42:52 --> 00:42:58
now the sines have come with
definite coefficients.
623
00:42:56 --> 00:43:02
And then, you simply click the
coefficients on the left and the
624
00:43:01 --> 00:43:07
coefficients on the right,
and figure out what the (c)n's
625
00:43:05 --> 00:43:11
are.
So, by equating coefficients,
626
00:43:10 --> 00:43:16
you get the (c)n's.
Would you like me to carry it
627
00:43:15 --> 00:43:21
out?
Yeah, okay, I was going to do
628
00:43:19 --> 00:43:25
something else,
but I wouldn't have time to do
629
00:43:24 --> 00:43:30
it anyway.
So, why don't I take two
630
00:43:28 --> 00:43:34
minutes to complete the
calculation just so you can see
631
00:43:34 --> 00:43:40
you get the same answer?
All right, what do we get?
632
00:43:40 --> 00:43:46
If you add them up,
you get c naught,
633
00:43:43 --> 00:43:49
out front, plus (c)n is
multiplied by what?
634
00:43:47 --> 00:43:53
Well, from the top it's
multiplied by omega naught
635
00:43:52 --> 00:43:58
squared.
On the bottom,
636
00:43:55 --> 00:44:01
it's multiplied by n
pi squared.
637
00:44:00 --> 00:44:06
Ah-ha, where have I seen that
combination?
638
00:44:05 --> 00:44:11
The sum is equal to,
sorry, one half plus what is
639
00:44:15 --> 00:44:21
it, sum over n odd of sine n pi
t over n.
640
00:44:29 --> 00:44:35
So, the conclusion is that--
I'm sorry, it should be c naught
641
00:44:34 --> 00:44:40
times omega naught squared.
642
00:44:38 --> 00:44:44
So, what's the conclusion?
If c zero is one over two omega
643
00:44:44 --> 00:44:50
naught squared,
644
00:44:48 --> 00:44:54
and that (c)n,
only for n odd,
645
00:44:51 --> 00:44:57
the others will be even.
The others will be zero.
646
00:44:55 --> 00:45:01
The (c)n is going to be equal
to two over pi here.
647
00:45:01 --> 00:45:07
So, it's going to be two pi,
648
00:45:04 --> 00:45:10
two over pi times one over n
times one over omega naught
649
00:45:10 --> 00:45:16
squared minus n over
pi squared.
650
00:45:15 --> 00:45:21
This is terrible,
651
00:45:20 --> 00:45:26
which is the same answer we got
before, I hope.
652
00:45:26 --> 00:45:32
Did I cover it up?
Same answer.
653
00:45:30 --> 00:45:36
So, that answer at the
left-hand end of the board is
654
00:45:35 --> 00:45:41
the same one.
I've calculated,
655
00:45:38 --> 00:45:44
in other words,
what the c zeros are.
656
00:45:42 --> 00:45:48
And, I got the same answer as
before.