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PROFESSOR: Today we are going
to talk about the vibration

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00:00:32,310 --> 00:00:34,030
of continuous systems.

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00:00:34,030 --> 00:00:37,590
Not covered on
the quiz, but it's

11
00:00:37,590 --> 00:00:41,680
a really important part
of real-world vibration

12
00:00:41,680 --> 00:00:51,080
and the most-- one of the
easiest ones to demonstrate,

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00:00:51,080 --> 00:00:54,290
I've shown you this one
before, is the taut string.

14
00:00:54,290 --> 00:01:00,040
But I want to show you something
unusual about-- something

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00:01:00,040 --> 00:01:01,830
you may not know about strings.

16
00:01:01,830 --> 00:01:03,959
Wait until it calms
down here a little bit.

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00:01:06,940 --> 00:01:11,930
OK, so this is your guitar
string or a piano string.

18
00:01:11,930 --> 00:01:13,360
It's under tension.

19
00:01:13,360 --> 00:01:17,640
We've already seen
that it exhibits

20
00:01:17,640 --> 00:01:19,640
natural frequencies
in mode shape,

21
00:01:19,640 --> 00:01:21,420
so there's the first mode.

22
00:01:21,420 --> 00:01:23,310
Looks like half a sine wave.

23
00:01:23,310 --> 00:01:27,770
Has a particular frequency
associated with it.

24
00:01:27,770 --> 00:01:31,690
Get it to stop doing
that-- but if I excite it

25
00:01:31,690 --> 00:01:35,040
at twice the frequency-- I
don't know if I can do this.

26
00:01:35,040 --> 00:01:36,484
There we go.

27
00:01:40,260 --> 00:01:42,330
That turns out to
be exactly twice

28
00:01:42,330 --> 00:01:43,990
the frequency of the first.

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00:01:43,990 --> 00:01:45,820
The mode shaped
one full sine wave.

30
00:01:45,820 --> 00:01:50,550
The mode shapes for a taut
string are sine n pi x over L.

31
00:01:50,550 --> 00:01:52,860
But strings can do
something else kind of neat.

32
00:01:56,300 --> 00:02:01,281
And that is if I
hit this thing--

33
00:02:01,281 --> 00:02:03,030
I'm going to wait till
it calms down here.

34
00:02:05,950 --> 00:02:09,520
If I give this
thing just a pulse,

35
00:02:09,520 --> 00:02:12,790
what do you expect to see?

36
00:02:12,790 --> 00:02:15,620
Are you going to see vibration?

37
00:02:15,620 --> 00:02:16,750
Tell me what you see.

38
00:02:25,620 --> 00:02:28,310
What do you see happening?

39
00:02:28,310 --> 00:02:29,930
Something running
back and forth.

40
00:02:29,930 --> 00:02:30,920
Right?

41
00:02:30,920 --> 00:02:33,710
What you're seeing
is wave propagation.

42
00:02:33,710 --> 00:02:34,850
It's not really vibration.

43
00:02:34,850 --> 00:02:38,131
Vibration we see of its modes
and standing waves and things

44
00:02:38,131 --> 00:02:38,630
like that.

45
00:02:38,630 --> 00:02:39,180
Right?

46
00:02:39,180 --> 00:02:45,250
So the taut string satisfies
an equation of motion that's

47
00:02:45,250 --> 00:02:46,660
called the wave equation.

48
00:02:46,660 --> 00:02:49,040
We're going to talk quite a
bit about that this morning.

49
00:02:49,040 --> 00:02:53,020
And the wave equation has
its name give something away.

50
00:02:53,020 --> 00:02:56,530
The wave equation describes
continuous systems

51
00:02:56,530 --> 00:03:00,690
of a particular kind that
support travelling waves.

52
00:03:00,690 --> 00:03:03,870
And so the string
will both support--

53
00:03:03,870 --> 00:03:05,875
I can give it a little pluck.

54
00:03:05,875 --> 00:03:10,300
I'll try to just place it in
a particular shape and let go.

55
00:03:10,300 --> 00:03:10,959
There it is.

56
00:03:10,959 --> 00:03:13,500
And that little pluck just goes
back and forth back and forth

57
00:03:13,500 --> 00:03:15,420
at a particular speed.

58
00:03:15,420 --> 00:03:18,190
So is there a relationship
between the speed

59
00:03:18,190 --> 00:03:20,950
at which things can
travel in a string

60
00:03:20,950 --> 00:03:25,040
and the natural
frequencies of the string?

61
00:03:25,040 --> 00:03:26,910
Well, we'll get into that today.

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00:03:30,370 --> 00:03:34,280
And I'm going to start
by just showing you

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00:03:34,280 --> 00:03:37,940
a little something that
comes from my research

64
00:03:37,940 --> 00:03:41,680
and-- let's see.

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00:03:41,680 --> 00:03:42,990
Let me do this.

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00:03:42,990 --> 00:03:44,060
I think this will work.

67
00:03:53,310 --> 00:03:55,750
Hear that?

68
00:03:55,750 --> 00:04:00,380
As I go slower, does
frequency go up or down?

69
00:04:00,380 --> 00:04:02,360
It's kind of slow, and
I'm going to speed up.

70
00:04:05,680 --> 00:04:06,570
Right?

71
00:04:06,570 --> 00:04:08,980
Goes up as the speed goes up.

72
00:04:08,980 --> 00:04:11,520
So that's the result
of the phenomenon

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00:04:11,520 --> 00:04:14,440
called flow-induced vibration.

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00:04:14,440 --> 00:04:20,240
And I'll give you
a very brief intro

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00:04:20,240 --> 00:04:21,619
to flow-induced vibration.

76
00:04:21,619 --> 00:04:25,980
You have a cylinder sitting
still, flow coming by it--

77
00:04:25,980 --> 00:04:29,050
water or air.

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00:04:29,050 --> 00:04:35,070
The cylinder is diameter D,
velocity U, for the flow.

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00:04:35,070 --> 00:04:40,310
What happens in the wake of that
cylinder, vortices are formed.

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00:04:40,310 --> 00:04:42,916
And just like if you're
paddling a canoe or something

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00:04:42,916 --> 00:04:44,540
and stick a paddle
in the water, you'll

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00:04:44,540 --> 00:04:48,730
see vortices shed off the side.

83
00:04:48,730 --> 00:04:50,240
First you get one
that's positive

84
00:04:50,240 --> 00:04:54,300
and then one that's negative
And so one full cycle of this

85
00:04:54,300 --> 00:04:57,800
is from here to here.

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00:04:57,800 --> 00:05:00,765
There's a frequency
to this shedding.

87
00:05:00,765 --> 00:05:04,750
And the shedding
frequency, FS, in hertz,

88
00:05:04,750 --> 00:05:08,530
can be predicted by a simple
dimensionless parameter called

89
00:05:08,530 --> 00:05:15,600
the Strouhal number, St U over
D. And that's approximately 0.2

90
00:05:15,600 --> 00:05:18,360
U/D for stationary cylinders.

91
00:05:18,360 --> 00:05:22,030
You can predict the frequency at
which these vortices are shed.

92
00:05:22,030 --> 00:05:24,830
Now, associated with
the shedding of vortices

93
00:05:24,830 --> 00:05:26,640
is a lift force.

94
00:05:26,640 --> 00:05:32,080
I'll call it some
FL cosine omega

95
00:05:32,080 --> 00:05:35,340
t, which is 2 pi FS, times t.

96
00:05:35,340 --> 00:05:38,420
So at this frequency
of vortex shedding

97
00:05:38,420 --> 00:05:40,780
there is a transverse force.

98
00:05:40,780 --> 00:05:42,860
There's actually an
inline force also,

99
00:05:42,860 --> 00:05:45,220
which I'll call FD for drag.

100
00:05:45,220 --> 00:05:51,360
And it goes like cosine
2 omega s times t.

101
00:05:51,360 --> 00:05:53,780
It's twice the
frequency of that.

102
00:05:53,780 --> 00:05:57,990
And so you'll get some
inline oscillatory excitation

103
00:05:57,990 --> 00:06:01,580
and what we call cross-flow
oscillatory excitation.

104
00:06:01,580 --> 00:06:08,250
And this is the cause
of lots of things

105
00:06:08,250 --> 00:06:12,670
that the people who work on it
call flow-induced vibration.

106
00:06:12,670 --> 00:06:18,030
Now, an amazing thing happens is
if this cylinder is elastically

107
00:06:18,030 --> 00:06:22,760
mounted or is flexible, and
that force starts to act on it,

108
00:06:22,760 --> 00:06:25,620
it will begin to vibrate.

109
00:06:25,620 --> 00:06:28,570
And the amazing thing,
as it begins to vibrate,

110
00:06:28,570 --> 00:06:33,770
it correlates the
shedding of these vortices

111
00:06:33,770 --> 00:06:35,720
all along the cylinder.

112
00:06:35,720 --> 00:06:38,390
So it's like soldiers
marching in step

113
00:06:38,390 --> 00:06:40,360
going across the bridge.

114
00:06:40,360 --> 00:06:43,349
If everybody's walking
randomly, then the bridge

115
00:06:43,349 --> 00:06:44,390
doesn't respond too much.

116
00:06:44,390 --> 00:06:45,960
But if everybody
marches together,

117
00:06:45,960 --> 00:06:47,980
you can put a pretty
good excitation into it.

118
00:06:47,980 --> 00:06:50,250
Well, the motion of
the cylinder itself

119
00:06:50,250 --> 00:06:54,820
organizes these vortex shedding
all along the cylinder,

120
00:06:54,820 --> 00:06:56,540
so they're all marching in step.

121
00:06:56,540 --> 00:07:00,110
And that means the force is
all correlated on the length.

122
00:07:00,110 --> 00:07:02,640
And you can get some pretty
substantial response.

123
00:07:02,640 --> 00:07:05,720
So that's the subject called
flow-induced vibration.

124
00:07:05,720 --> 00:07:10,800
And with that, I'm going
to show you a few slides.

125
00:07:10,800 --> 00:07:15,000
Let's dim the lights a little
bit, if you could, to see this.

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00:07:15,000 --> 00:07:17,761
There's some pictures I
just want you to see better.

127
00:07:17,761 --> 00:07:18,260
All right.

128
00:07:18,260 --> 00:07:19,690
So I do flow-induced vibration.

129
00:07:19,690 --> 00:07:21,630
I've been doing
this-- working on this

130
00:07:21,630 --> 00:07:23,970
for all my professional career.

131
00:07:23,970 --> 00:07:29,060
And it's applied, primarily,
to big, flexible cylinders

132
00:07:29,060 --> 00:07:29,710
in the ocean.

133
00:07:29,710 --> 00:07:33,490
Particularly associated with the
things that the US Navy does.

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00:07:33,490 --> 00:07:36,990
Long cables and things and
also the offshore oil industry.

135
00:07:36,990 --> 00:07:38,174
Next slide.

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00:07:38,174 --> 00:07:39,090
Can we dim the lights?

137
00:07:41,870 --> 00:07:43,195
Can we dim the lights?

138
00:07:48,752 --> 00:07:49,960
I want you to be able to see.

139
00:07:49,960 --> 00:07:52,100
This is a tension leg platform.

140
00:07:52,100 --> 00:07:54,080
It's one of the
structures that's

141
00:07:54,080 --> 00:07:58,070
used in the offshore
industry to produce oil.

142
00:07:58,070 --> 00:08:03,220
And one of these might be
moored in 3,000 feet of water,

143
00:08:03,220 --> 00:08:04,840
1,000 meters of water.

144
00:08:04,840 --> 00:08:07,330
Might weigh 20,000 tons.

145
00:08:07,330 --> 00:08:10,160
And what's connecting
it-- what holds in place--

146
00:08:10,160 --> 00:08:12,230
are steel cylinders
a half a meter

147
00:08:12,230 --> 00:08:15,710
in diameter, 3,000 feet
long, going vertically

148
00:08:15,710 --> 00:08:19,840
down from each of those three
pontoon legs sticking out.

149
00:08:19,840 --> 00:08:21,630
And they're under
a lot of tension.

150
00:08:21,630 --> 00:08:25,140
And in fact, it pulls the
thing down into the water

151
00:08:25,140 --> 00:08:28,060
so the buoyancy
of the whole thing

152
00:08:28,060 --> 00:08:30,240
puts tension on these cylinders.

153
00:08:30,240 --> 00:08:32,669
But now, what happens
if an ocean current

154
00:08:32,669 --> 00:08:34,179
comes by those cylinders?

155
00:08:34,179 --> 00:08:37,330
Vortex shedding, and
the cylinders vibrate.

156
00:08:37,330 --> 00:08:40,289
And if they vibrate, over time
they will fatigue and fail.

157
00:08:40,289 --> 00:08:40,789
OK.

158
00:08:40,789 --> 00:08:41,617
Next slide.

159
00:08:41,617 --> 00:08:42,950
There's a picture of a real one.

160
00:08:42,950 --> 00:08:44,730
That's a bigger one
called Marco Polo.

161
00:08:44,730 --> 00:08:47,476
It's on a launch ship that'll
take it out to the site

162
00:08:47,476 --> 00:08:47,975
that it is.

163
00:08:47,975 --> 00:08:52,780
And the ship will lower
and it will slide off.

164
00:08:52,780 --> 00:08:54,610
So these are big.

165
00:08:54,610 --> 00:08:56,200
Next slide.

166
00:08:56,200 --> 00:08:59,319
This is a diagram of
the Gulf of Mexico.

167
00:08:59,319 --> 00:09:00,610
South America is at the bottom.

168
00:09:00,610 --> 00:09:02,609
The Yucatan Peninsula is
sticking up there right

169
00:09:02,609 --> 00:09:04,010
in the middle of the bottom.

170
00:09:04,010 --> 00:09:06,930
This is a picture of
satellite imagery of currents

171
00:09:06,930 --> 00:09:08,200
in the Gulf of Mexico.

172
00:09:08,200 --> 00:09:11,110
And there's a current that
flows up off of South America

173
00:09:11,110 --> 00:09:14,530
into the Gulf of Mexico,
goes around in a loop,

174
00:09:14,530 --> 00:09:15,437
and then comes out.

175
00:09:15,437 --> 00:09:17,770
You can see Florida sticking
down in there on the right.

176
00:09:17,770 --> 00:09:19,890
That current comes
out of the Gulf,

177
00:09:19,890 --> 00:09:22,805
goes around the tip of Florida,
and goes up the Atlantic Coast,

178
00:09:22,805 --> 00:09:25,190
and is known as the Gulf Stream.

179
00:09:25,190 --> 00:09:27,060
But it starts as
a big current that

180
00:09:27,060 --> 00:09:28,350
comes into the Gulf of Mexic.

181
00:09:28,350 --> 00:09:31,060
And, every now and then, that
current pinches off an eddy.

182
00:09:31,060 --> 00:09:33,530
And that's what that red
circle is in the middle.

183
00:09:33,530 --> 00:09:37,510
And it's an eddy that's many,
many kilometers in diameter

184
00:09:37,510 --> 00:09:40,880
with surface currents on the
order of a meter per second

185
00:09:40,880 --> 00:09:41,790
or more.

186
00:09:41,790 --> 00:09:48,360
And those are the biggest
threat for causing

187
00:09:48,360 --> 00:09:51,880
flow-induced vibration
failures of long members

188
00:09:51,880 --> 00:09:55,340
from hanging off of
offshore structures.

189
00:09:55,340 --> 00:09:56,660
Next.

190
00:09:56,660 --> 00:09:59,500
So I've been doing research
in this area for a long time.

191
00:09:59,500 --> 00:10:03,120
This is a picture taken
in the summer of 1981.

192
00:10:03,120 --> 00:10:06,920
It is a piece of steel
pipe about 2 inches

193
00:10:06,920 --> 00:10:09,390
in diameter and 75 feet long.

194
00:10:09,390 --> 00:10:12,130
It's under 750
pounds of tension,

195
00:10:12,130 --> 00:10:13,630
and it's pinned at each end.

196
00:10:13,630 --> 00:10:18,650
It behaves almost exactly
like my rubber cord here.

197
00:10:18,650 --> 00:10:22,440
It has natural frequencies,
and it will vibrate

198
00:10:22,440 --> 00:10:23,910
if a current comes by it.

199
00:10:23,910 --> 00:10:26,100
So this is actually a sandbar.

200
00:10:26,100 --> 00:10:29,040
And at low tide, we'd do
all the work putting it up.

201
00:10:29,040 --> 00:10:31,280
Then, as the tide
comes in, the flow

202
00:10:31,280 --> 00:10:33,670
is perpendicular
to the cylinder,

203
00:10:33,670 --> 00:10:37,590
and vortices start shedding.

204
00:10:37,590 --> 00:10:39,100
And as the pipe
begins to move, they

205
00:10:39,100 --> 00:10:41,150
get organized all
along the length.

206
00:10:41,150 --> 00:10:45,300
And a typical
response mode was when

207
00:10:45,300 --> 00:10:48,720
the vortex shedding frequency,
therefore the lift force

208
00:10:48,720 --> 00:10:52,470
frequency, coincided with
the natural frequency.

209
00:10:52,470 --> 00:10:56,210
Then you'd expect it to give
quite a bit of response.

210
00:10:56,210 --> 00:11:00,740
The diagram on the left
is if you cut the cylinder

211
00:11:00,740 --> 00:11:03,750
and looked down its axis,
this is the trajectory

212
00:11:03,750 --> 00:11:05,290
that you'd see
the cylinder make.

213
00:11:05,290 --> 00:11:08,030
It would sit there and just
make big figure eights.

214
00:11:08,030 --> 00:11:11,100
So up and down vertical
is its vertical motion.

215
00:11:11,100 --> 00:11:13,445
Flow's coming from,
say, left to right.

216
00:11:13,445 --> 00:11:15,340
Its vertical motion
is up and down.

217
00:11:15,340 --> 00:11:16,770
In-line motion's like this.

218
00:11:16,770 --> 00:11:20,500
And exactly such a phase it
just makes big beautiful figure

219
00:11:20,500 --> 00:11:21,180
eights.

220
00:11:21,180 --> 00:11:23,140
That's the kind of
motion you'd see.

221
00:11:23,140 --> 00:11:24,170
OK?

222
00:11:24,170 --> 00:11:28,780
So then, very much what I was
talking about a minute ago,

223
00:11:28,780 --> 00:11:32,700
very much behavior
dominated by vibration.

224
00:11:32,700 --> 00:11:36,560
Vibration in the third mode,
cross flow, was a typical one.

225
00:11:36,560 --> 00:11:39,660
And fifth mode,
inline, was typical.

226
00:11:39,660 --> 00:11:42,740
But as cylinders
go in the ocean,

227
00:11:42,740 --> 00:11:43,900
that one's kind of short.

228
00:11:43,900 --> 00:11:46,840
Third mode vibration
is sort of low.

229
00:11:46,840 --> 00:11:51,670
So as years have
gone by and oil is

230
00:11:51,670 --> 00:11:54,000
being produced in deeper
and deeper and deeper water,

231
00:11:54,000 --> 00:11:55,541
the cylinders we're
putting out there

232
00:11:55,541 --> 00:11:57,420
get longer and longer
and longer and longer.

233
00:11:57,420 --> 00:12:00,240
And the modes that are
excited by currents coming by

234
00:12:00,240 --> 00:12:01,887
get quite high.

235
00:12:01,887 --> 00:12:03,220
So this is an experiment we did.

236
00:12:03,220 --> 00:12:05,580
It was roughly a
1/10 scale model.

237
00:12:05,580 --> 00:12:09,280
Model is almost 2 inches
in diameter, 500 feet long.

238
00:12:09,280 --> 00:12:10,950
Scale that up by a
factor of 10, you're

239
00:12:10,950 --> 00:12:13,350
up around 20 inches in
diameter and 5,000 feet

240
00:12:13,350 --> 00:12:17,520
long, which is exactly the
size of the drilling riser

241
00:12:17,520 --> 00:12:20,960
that BP had hung
off the drilling

242
00:12:20,960 --> 00:12:24,800
ship when the blowout occurred.

243
00:12:24,800 --> 00:12:27,780
It's a piece of steel pipe,
21 inches in diameter,

244
00:12:27,780 --> 00:12:30,720
3/4 of an inch wall
thickness, 5,000 feet long,

245
00:12:30,720 --> 00:12:32,320
under a lot of tension.

246
00:12:32,320 --> 00:12:34,340
And when ocean
currents come by, it

247
00:12:34,340 --> 00:12:37,424
behaves just like this string.

248
00:12:37,424 --> 00:12:39,340
And so we're out-- this
is a 1/10 scale model.

249
00:12:39,340 --> 00:12:42,040
So we put a big weight on
the bottom of the cylinder,

250
00:12:42,040 --> 00:12:45,060
put it behind a boat, and
towed it in the Gulf Stream.

251
00:12:45,060 --> 00:12:47,130
Next picture.

252
00:12:47,130 --> 00:12:47,992
So there's the boat.

253
00:12:47,992 --> 00:12:49,200
It's an oceanographic vessel.

254
00:12:49,200 --> 00:12:50,830
It's actually a catamaran.

255
00:12:50,830 --> 00:12:52,140
Next.

256
00:12:52,140 --> 00:12:55,860
This is a spool that had
our test cylinder on it.

257
00:12:55,860 --> 00:12:59,340
There's a reddish object
down on the bottom which

258
00:12:59,340 --> 00:13:08,500
is-- that's a 750-pound
piece of railroad wheel,

259
00:13:08,500 --> 00:13:09,970
and it's the weight
on the bottom.

260
00:13:09,970 --> 00:13:13,510
And so you'd spool this
thing off, lower it down,

261
00:13:13,510 --> 00:13:14,770
and then do your tests.

262
00:13:14,770 --> 00:13:16,280
Next.

263
00:13:16,280 --> 00:13:18,203
Top, we measured
tension inclination.

264
00:13:21,140 --> 00:13:25,630
And then we also had-- it's
a pin joint at the top,

265
00:13:25,630 --> 00:13:27,550
so it would vibrate freely.

266
00:13:27,550 --> 00:13:30,430
Inside, though,
was fiber optics.

267
00:13:30,430 --> 00:13:32,090
Next.

268
00:13:32,090 --> 00:13:34,030
We had eight optical fibers.

269
00:13:34,030 --> 00:13:36,070
And in those optical
fibers were what

270
00:13:36,070 --> 00:13:38,260
we call optical strain gauges.

271
00:13:38,260 --> 00:13:42,850
So we had 280 optical strain
gauges instrumented up and down

272
00:13:42,850 --> 00:13:44,932
that pipe so we could
measure its vibration.

273
00:13:44,932 --> 00:13:47,140
And so you're looking at a
cross section of the pipe.

274
00:13:47,140 --> 00:13:50,060
There were two optical
fibers in each quadrant,

275
00:13:50,060 --> 00:13:55,300
and each one of those
fibers had 35 sensors on it.

276
00:13:55,300 --> 00:13:56,470
Next.

277
00:13:56,470 --> 00:14:00,330
This is typical
experimental case.

278
00:14:00,330 --> 00:14:01,370
This is the surface.

279
00:14:01,370 --> 00:14:03,200
This is 500 feet down.

280
00:14:03,200 --> 00:14:04,730
This is the current profile.

281
00:14:04,730 --> 00:14:07,630
So the flow velocity
is about 2 feet

282
00:14:07,630 --> 00:14:09,860
per second near the
surface, up to 4 feet

283
00:14:09,860 --> 00:14:11,600
per second down on the bottom.

284
00:14:11,600 --> 00:14:14,700
And this is the region
where most of the excitation

285
00:14:14,700 --> 00:14:18,570
was coming from that would drive
the flow-induced vibration.

286
00:14:18,570 --> 00:14:25,270
This is measured RMS strain
caused by the bending vibration

287
00:14:25,270 --> 00:14:25,980
in the cylinders.

288
00:14:25,980 --> 00:14:28,950
And peak-- the maximum
strain-- is about right there.

289
00:14:28,950 --> 00:14:30,200
Next.

290
00:14:30,200 --> 00:14:32,060
Typical response spectrum.

291
00:14:32,060 --> 00:14:35,090
Basically, the frequency content
at three different locations.

292
00:14:35,090 --> 00:14:37,190
Down deep, in the
middle, near the top.

293
00:14:37,190 --> 00:14:38,830
This is frequency.

294
00:14:38,830 --> 00:14:42,660
So this would be the
peak that describes

295
00:14:42,660 --> 00:14:46,300
the principal cross-flow
vibration at the vortex

296
00:14:46,300 --> 00:14:47,640
shedding frequency.

297
00:14:47,640 --> 00:14:48,140
Next.

298
00:14:51,140 --> 00:14:54,990
This is position, bottom to top.

299
00:14:54,990 --> 00:14:59,310
This is time, and these
are strain records from all

300
00:14:59,310 --> 00:15:00,620
of those strain sensors.

301
00:15:00,620 --> 00:15:05,770
There's a strain sensor about
every 2 meters along here.

302
00:15:05,770 --> 00:15:08,140
But what you're seeing
is-- this is evidence.

303
00:15:08,140 --> 00:15:10,720
The red is the
amplitude and red--

304
00:15:10,720 --> 00:15:15,790
let's say red is positive strain
and blue is negative strain.

305
00:15:15,790 --> 00:15:19,190
And so at any location on the
pipe where it's vibrating,

306
00:15:19,190 --> 00:15:22,660
it's going to go from red to
blue, red to blue, red to blue.

307
00:15:22,660 --> 00:15:25,284
But it's showing you that
they're highly correlated

308
00:15:25,284 --> 00:15:26,700
all along the
length, that there's

309
00:15:26,700 --> 00:15:31,240
a red streak all lined up, but
it's not parallel to the pipe.

310
00:15:31,240 --> 00:15:32,320
It's inclined.

311
00:15:32,320 --> 00:15:34,400
This is showing you
wave propagation.

312
00:15:34,400 --> 00:15:36,170
The behavior of the
pipe is completely

313
00:15:36,170 --> 00:15:40,350
dominated by wave propagation,
not by standing wave vibration.

314
00:15:40,350 --> 00:15:45,930
So totally different than
that short pipe in 1981.

315
00:15:45,930 --> 00:15:46,716
The wave equation.

316
00:15:57,500 --> 00:16:02,660
Let's imagine we have a long
pipe or a string like that,

317
00:16:02,660 --> 00:16:08,760
and it can carry waves
traveling along it.

318
00:16:08,760 --> 00:16:13,760
The position at any location on
here-- here's a coordinate x.

319
00:16:13,760 --> 00:16:16,040
We describe the
motion at a point

320
00:16:16,040 --> 00:16:20,250
by a coordinate w of x and t.

321
00:16:20,250 --> 00:16:25,640
So it's a function of
where it is and time.

322
00:16:25,640 --> 00:16:28,790
What describes the
motion of something

323
00:16:28,790 --> 00:16:32,800
which obeys the wave equation
is the following equation.

324
00:16:32,800 --> 00:16:38,400
Partial squared w with
respect to x squared

325
00:16:38,400 --> 00:16:44,640
equals 1/c squared partial
squared w with respect

326
00:16:44,640 --> 00:16:46,220
to t squared.

327
00:16:46,220 --> 00:16:49,310
That's what's known as the
one-dimensional wave equation.

328
00:16:49,310 --> 00:16:51,540
And the one-dimensional
wave equation

329
00:16:51,540 --> 00:16:57,240
governs an incredibly broad
category of physical phenomena.

330
00:16:57,240 --> 00:17:02,700
Light behaves according
to the wave equation.

331
00:17:02,700 --> 00:17:04,958
Sound propagating
across the room to you

332
00:17:04,958 --> 00:17:07,069
is governed by
the wave equation.

333
00:17:07,069 --> 00:17:11,859
Longitudinal vibration of rods,
torsional vibration of rods--

334
00:17:11,859 --> 00:17:13,609
all governed by
the wave equation.

335
00:17:13,609 --> 00:17:15,859
So it's worthwhile to know
a little bit about the wave

336
00:17:15,859 --> 00:17:17,609
equation.

337
00:17:17,609 --> 00:17:19,650
And what I showed
you this morning,

338
00:17:19,650 --> 00:17:21,550
it has this kind
of duality to it.

339
00:17:21,550 --> 00:17:24,940
You can have things that vibrate
with standing waves and mode

340
00:17:24,940 --> 00:17:28,400
shapes, but the same
system can support

341
00:17:28,400 --> 00:17:30,390
waves that travel along it.

342
00:17:30,390 --> 00:17:33,370
So let's figure out why that is.

343
00:17:36,250 --> 00:17:41,060
So I'm going to do the
derivation for you of the wave

344
00:17:41,060 --> 00:17:42,539
equation for a
string, just so you

345
00:17:42,539 --> 00:17:44,080
know where it comes
from because then

346
00:17:44,080 --> 00:17:48,500
that general derivation applies
to all these different things.

347
00:17:48,500 --> 00:17:56,540
So imagine you've got now--
we're interested in eventually

348
00:17:56,540 --> 00:17:57,510
getting to vibration.

349
00:17:57,510 --> 00:18:01,150
So I'm going to make this
a finite length string.

350
00:18:01,150 --> 00:18:11,340
And it has this position we'll
describe as a w of x and t.

351
00:18:11,340 --> 00:18:17,920
It has a tension, T, a
mass per unit length, m.

352
00:18:17,920 --> 00:18:22,630
So this is like
kilograms per meter

353
00:18:22,630 --> 00:18:28,510
is the mass per unit length of
this thing which can vibrate.

354
00:18:28,510 --> 00:18:29,500
So tension.

355
00:18:29,500 --> 00:18:31,970
Mass per unit length.

356
00:18:31,970 --> 00:18:35,480
L, the length of it.

357
00:18:35,480 --> 00:18:37,820
What other parameters
do we need?

358
00:18:37,820 --> 00:18:39,980
That'll do for the moment.

359
00:18:39,980 --> 00:18:43,255
Now-- so let's draw
it again without.

360
00:18:46,730 --> 00:18:52,010
In some displaced position
and what's exciting

361
00:18:52,010 --> 00:18:54,060
it may be my vortex
shedding, and so I'm going

362
00:18:54,060 --> 00:18:57,560
to draw that excitation here.

363
00:18:57,560 --> 00:19:00,980
And that we'll describe
as F of x and t,

364
00:19:00,980 --> 00:19:05,310
some force per unit length.

365
00:19:05,310 --> 00:19:07,665
So this has units of
newtons per meter.

366
00:19:11,750 --> 00:19:15,190
Now, in that little--
there may also

367
00:19:15,190 --> 00:19:17,105
be drag forces,
the fluid damping.

368
00:19:19,880 --> 00:19:24,950
So I'm going to cut out a
little piece of this cylinder

369
00:19:24,950 --> 00:19:33,810
and do a force balance on
that piece of cylinder.

370
00:19:33,810 --> 00:19:35,350
So basically, F equals ma.

371
00:19:35,350 --> 00:19:37,990
We're just applying Newton
to this piece of cylinder.

372
00:19:37,990 --> 00:19:42,320
And I'll draw it right here.

373
00:19:42,320 --> 00:19:45,580
A little section
of it is curved.

374
00:19:45,580 --> 00:19:47,910
Here's horizontal.

375
00:19:47,910 --> 00:19:48,770
There's horizontal.

376
00:19:48,770 --> 00:19:53,130
We need to evaluate
all the forces on it.

377
00:19:53,130 --> 00:19:59,020
So the tension on this
end-- so like that.

378
00:19:59,020 --> 00:20:01,475
And the tension on this end
is some different angle.

379
00:20:04,070 --> 00:20:07,440
This we'll call theta 1.

380
00:20:07,440 --> 00:20:10,290
This we'll call theta 2.

381
00:20:10,290 --> 00:20:19,150
And along here are my
excitation forces, F of x and t.

382
00:20:19,150 --> 00:20:23,360
There may be some resistance--
drag forces, damping.

383
00:20:23,360 --> 00:20:27,380
That'll be a damping
constant, R of x,

384
00:20:27,380 --> 00:20:30,940
which is force per unit
length per unit velocity,

385
00:20:30,940 --> 00:20:36,120
times-- the force
on this would have

386
00:20:36,120 --> 00:20:37,970
to be multiplied
by the velocity, so

387
00:20:37,970 --> 00:20:42,600
the derivative of this
displacement with respect

388
00:20:42,600 --> 00:20:43,280
to time.

389
00:20:43,280 --> 00:20:49,040
That's the force along here,
and it can vary with position.

390
00:20:49,040 --> 00:20:50,420
Have we accounted
for everything?

391
00:20:50,420 --> 00:20:58,840
Ah, well, this is position
x, and this is at x plus dx.

392
00:20:58,840 --> 00:21:01,685
So this little element
is dx in length.

393
00:21:04,290 --> 00:21:07,265
And this is all
for small motions.

394
00:21:33,470 --> 00:21:35,820
And if you assume
small motions, then you

395
00:21:35,820 --> 00:21:43,630
can say theta 1 is approximately
equal to sine theta 1.

396
00:21:43,630 --> 00:21:48,400
That's also approximately
equal to tan theta 1.

397
00:21:48,400 --> 00:21:51,480
And that's equal to the
derivative of w with respect

398
00:21:51,480 --> 00:21:53,870
to x, just the slope.

399
00:21:53,870 --> 00:21:56,070
We're going to take
advantage of that.

400
00:21:56,070 --> 00:21:59,205
Theta 2, same thing.

401
00:21:59,205 --> 00:22:05,660
It's approximately equal to
tan theta 2 here, and sin

402
00:22:05,660 --> 00:22:07,130
and all those things.

403
00:22:07,130 --> 00:22:11,100
But that, then-- the slope
has changed a little bit

404
00:22:11,100 --> 00:22:13,270
when you go through dx.

405
00:22:13,270 --> 00:22:18,950
And this is equal to the
slope on the left-hand side

406
00:22:18,950 --> 00:22:28,860
plus the rate of change
of the slope times dx.

407
00:22:31,930 --> 00:22:34,340
So the slope on the left,
this is now the slope

408
00:22:34,340 --> 00:22:36,490
on the right-hand side.

409
00:22:36,490 --> 00:22:38,290
And so now, all
that's left to do

410
00:22:38,290 --> 00:22:54,510
is to write a force balance
for that little piece

411
00:22:54,510 --> 00:22:55,410
on the element dx.

412
00:22:58,570 --> 00:23:01,510
So if positive, upward.

413
00:23:01,510 --> 00:23:07,240
We have a T sine theta.

414
00:23:07,240 --> 00:23:09,340
But because sine theta
is approximately tan

415
00:23:09,340 --> 00:23:12,310
theta is equal to
dw dx, then there's

416
00:23:12,310 --> 00:23:29,900
an upward force on the
right-hand side, which is T.

417
00:23:29,900 --> 00:23:33,660
And this turns into partial
squared w with respect

418
00:23:33,660 --> 00:23:37,180
to x squared dx.

419
00:23:37,180 --> 00:23:39,530
So on the right-hand
side-- positive

420
00:23:39,530 --> 00:23:41,710
upwards-- you have
T times the partial

421
00:23:41,710 --> 00:23:49,666
of w with respect to x, plus
partial square w with respect

422
00:23:49,666 --> 00:23:53,300
to x squared dx.

423
00:23:53,300 --> 00:23:57,100
That's the upward force
on the right-hand side.

424
00:23:57,100 --> 00:24:01,460
On the left-hand side, we have a
downward force, minus T partial

425
00:24:01,460 --> 00:24:04,240
of w with respect to x.

426
00:24:04,240 --> 00:24:07,840
And you notice that this one's
going to cancel that one.

427
00:24:07,840 --> 00:24:15,600
We have minus R of x partial
w with respect to t--

428
00:24:15,600 --> 00:24:17,940
that's the velocity-- dx long.

429
00:24:17,940 --> 00:24:21,840
Because that's force
per unit length.

430
00:24:21,840 --> 00:24:24,020
And have we missed anything?

431
00:24:24,020 --> 00:24:26,660
So that's the sum of
the external forces

432
00:24:26,660 --> 00:24:28,530
on this little slice.

433
00:24:28,530 --> 00:24:33,400
And that has to be equal
to-- what did Newton say?

434
00:24:33,400 --> 00:24:39,030
The mass, which is the mass
per unit length, times dx,

435
00:24:39,030 --> 00:24:43,300
is the total mass,
times the acceleration,

436
00:24:43,300 --> 00:24:48,235
partial squared w with
respect to t squared.

437
00:24:51,841 --> 00:24:56,130
So this cancels this term.

438
00:24:56,130 --> 00:24:59,690
And then you notice I'm
left with everything

439
00:24:59,690 --> 00:25:04,960
as just something dx,
something dx, something dx.

440
00:25:04,960 --> 00:25:14,360
Get rid of the dx's,
and I can write-- oh, I

441
00:25:14,360 --> 00:25:16,280
left out something.

442
00:25:16,280 --> 00:25:25,900
I left out my distributed
force, F of x and t dx.

443
00:25:25,900 --> 00:25:28,010
It's positive as it's drawn.

444
00:25:28,010 --> 00:25:29,030
It's over here also.

445
00:25:29,030 --> 00:25:32,190
So this, and I
cancel out that dx.

446
00:25:32,190 --> 00:25:35,710
So I put them all together
now and assemble them.

447
00:25:35,710 --> 00:25:44,410
I can write down the equation
that governs this motion.

448
00:25:44,410 --> 00:25:49,840
So T partial square
w with respect

449
00:25:49,840 --> 00:26:02,700
to x squared minus r of x times
velocity plus f of x and t

450
00:26:02,700 --> 00:26:09,472
equals m partial square w
with respect to t squared.

451
00:26:09,472 --> 00:26:11,930
And that just says that the
sum of the forces on the object

452
00:26:11,930 --> 00:26:13,555
equals its mass times
its acceleration.

453
00:26:18,570 --> 00:26:22,500
Now, if we're interested in
natural frequencies and mode

454
00:26:22,500 --> 00:26:28,970
shapes, when we've been doing
one and two degree of freedom

455
00:26:28,970 --> 00:26:31,480
systems, and we want to get
the natural frequencies in mode

456
00:26:31,480 --> 00:26:35,530
shapes, we temporarily let
the damping be 0 and the force

457
00:26:35,530 --> 00:26:36,341
be 0, right?

458
00:26:36,341 --> 00:26:37,840
So we want to do
the same thing now.

459
00:26:37,840 --> 00:26:46,330
We're interested in how do you
find the omega n's and what I

460
00:26:46,330 --> 00:26:47,880
call the psi n's.

461
00:26:47,880 --> 00:26:50,710
Because now the mode
shapes are functions.

462
00:26:50,710 --> 00:26:52,590
And so this is a
natural frequency

463
00:26:52,590 --> 00:26:55,020
and the mode shape for mode n.

464
00:26:55,020 --> 00:26:56,870
We know there's lots of modes.

465
00:26:56,870 --> 00:27:06,540
So we let r of x and
f of x and t be 0.

466
00:27:06,540 --> 00:27:13,970
And when we do that,
this term goes away.

467
00:27:13,970 --> 00:27:14,840
This term goes away.

468
00:27:14,840 --> 00:27:18,150
I'm just left with T partial
squared w with respect

469
00:27:18,150 --> 00:27:19,450
to x squared equals this.

470
00:27:19,450 --> 00:27:21,340
And I'm going to
divide through by t.

471
00:27:21,340 --> 00:27:29,330
So I get partial squared w with
respect to x squared equals 1

472
00:27:29,330 --> 00:27:43,260
over T over m partial squared
w with respect to t squared.

473
00:27:43,260 --> 00:27:49,390
And this T/m quantity
turns out to be

474
00:27:49,390 --> 00:27:53,840
the speed of wave
propagation in the medium.

475
00:28:01,390 --> 00:28:06,335
And that is the wave equation.

476
00:28:11,440 --> 00:28:14,620
So we've just found the
wave equation for the string

477
00:28:14,620 --> 00:28:19,027
just by applying Newton's law
to a little section of string.

478
00:28:19,027 --> 00:28:20,360
You can do that for the vibrate.

479
00:28:20,360 --> 00:28:21,870
You're going to
do the same thing,

480
00:28:21,870 --> 00:28:24,650
cut out a little
section of a beam,

481
00:28:24,650 --> 00:28:28,110
do the force balance on it,
set it equal to the mass times

482
00:28:28,110 --> 00:28:28,890
acceleration.

483
00:28:28,890 --> 00:28:32,620
And for a beam, you'll get
a fourth order differential

484
00:28:32,620 --> 00:28:34,450
equation.

485
00:28:34,450 --> 00:28:36,860
And it's not the wave equation.

486
00:28:36,860 --> 00:28:41,040
It still vibrates, but
it's not governed by what

487
00:28:41,040 --> 00:28:43,550
we call the wave equation.

488
00:28:43,550 --> 00:28:53,030
OK, so this is the one
dimensional wave equation.

489
00:28:53,030 --> 00:28:57,245
This quantity T/m is
the phase velocity.

490
00:28:57,245 --> 00:28:58,370
It's called phase velocity.

491
00:29:09,304 --> 00:29:10,970
You know, that's a
good one to remember.

492
00:29:10,970 --> 00:29:14,094
For a simple string, the speed
of phenomena running down

493
00:29:14,094 --> 00:29:16,260
the string is the square
root of the tension divided

494
00:29:16,260 --> 00:29:17,426
by the mass per unit length.

495
00:29:22,000 --> 00:29:27,460
And if you had a long string,
I put that little pluck in it,

496
00:29:27,460 --> 00:29:29,910
and you can see that pluck
running back and forth on it.

497
00:29:29,910 --> 00:29:31,201
That's the speed it's going at.

498
00:29:36,010 --> 00:29:39,667
Basically, it's called--
well, so if I have my string,

499
00:29:39,667 --> 00:29:41,500
and I put a little bump
on it, and that bump

500
00:29:41,500 --> 00:29:43,630
goes zipping along,
your eye will see

501
00:29:43,630 --> 00:29:45,670
this thing propagating at c.

502
00:29:53,530 --> 00:29:55,480
So to get natural
frequencies in mode shapes,

503
00:29:55,480 --> 00:30:01,700
we basically need to
solve this equation.

504
00:30:01,700 --> 00:30:03,955
And it's quite
straightforward to do.

505
00:30:06,710 --> 00:30:08,760
And a technique
known as separation

506
00:30:08,760 --> 00:30:14,440
of variables works, which
means that all you're doing

507
00:30:14,440 --> 00:30:17,120
is saying, I believe
that I'm going

508
00:30:17,120 --> 00:30:20,180
to be able to write the
solution as some function of x

509
00:30:20,180 --> 00:30:26,310
only times some function of
time only, product of two terms.

510
00:30:26,310 --> 00:30:41,250
And that in fact-- because
we're interested in vibration.

511
00:30:41,250 --> 00:30:43,720
You can tell me what
the function of time is.

512
00:30:46,444 --> 00:30:49,110
You're going to tell me half the
solution just from observation.

513
00:30:49,110 --> 00:30:51,590
What is it?

514
00:30:51,590 --> 00:30:55,160
Just the time dependent part.

515
00:30:55,160 --> 00:30:57,275
It's the same as anything
else that vibrates.

516
00:30:57,275 --> 00:30:59,370
So a single degree
of freedom system,

517
00:30:59,370 --> 00:31:03,450
what is the time dependent
function that we substitute in

518
00:31:03,450 --> 00:31:06,040
to find the natural frequency?

519
00:31:06,040 --> 00:31:06,915
AUDIENCE: [INAUDIBLE]

520
00:31:10,190 --> 00:31:11,332
PROFESSOR: Say again?

521
00:31:11,332 --> 00:31:12,540
AUDIENCE: e to the i omega t.

522
00:31:12,540 --> 00:31:14,850
PROFESSOR: e to the i
omega t would be just fine.

523
00:31:14,850 --> 00:31:16,160
Cosine omega t works.

524
00:31:16,160 --> 00:31:17,230
Sine omega t works.

525
00:31:17,230 --> 00:31:19,840
But e to the i omega t
is pretty easy to use.

526
00:31:19,840 --> 00:31:22,760
Because it's so simple
to take the derivatives.

527
00:31:22,760 --> 00:31:25,140
So we can guess
that this is going

528
00:31:25,140 --> 00:31:34,430
to be some W of x times
Ae to the i omega t.

529
00:31:34,430 --> 00:31:35,240
And plug it in.

530
00:31:39,540 --> 00:31:42,180
Plug it into our wave
equation over here.

531
00:31:42,180 --> 00:31:47,976
So I'll make sure I
write it consistently.

532
00:31:53,310 --> 00:31:57,110
So we plug this
into the first term.

533
00:31:57,110 --> 00:31:59,790
It's two derivatives
with respect to x.

534
00:31:59,790 --> 00:32:09,270
So this is just-- and the
time-dependent part just

535
00:32:09,270 --> 00:32:09,966
stays outside.

536
00:32:12,570 --> 00:32:18,730
And on the right-hand side, when
we plug it in here, 1 over c

537
00:32:18,730 --> 00:32:21,520
squared, two derivatives
with respect to time,

538
00:32:21,520 --> 00:32:24,120
it's going to give me
minus omega squared,

539
00:32:24,120 --> 00:32:28,430
so minus omega squared
over c squared.

540
00:32:28,430 --> 00:32:37,610
And then it gives me back
W of x Ae to the i omega t.

541
00:32:37,610 --> 00:32:42,650
And now I can get rid of
the Ae to the i omega t's.

542
00:32:42,650 --> 00:32:47,250
And I'm left with just an
equation involving x only.

543
00:32:47,250 --> 00:32:50,250
And it's an ordinary
differential equation

544
00:32:50,250 --> 00:32:51,750
in w of x.

545
00:32:58,940 --> 00:33:07,730
So it turns into d2W dx
squared plus omega squared

546
00:33:07,730 --> 00:33:12,170
over c squared W equals 0.

547
00:33:12,170 --> 00:33:16,500
And you've seen this
equation before.

548
00:33:16,500 --> 00:33:20,000
Does this not look like,
have some similarity to,

549
00:33:20,000 --> 00:33:25,840
Mx double dot plus kx equals 0?

550
00:33:25,840 --> 00:33:28,520
They're basically
the same equation.

551
00:33:28,520 --> 00:33:29,950
This one's a function of x.

552
00:33:29,950 --> 00:33:33,250
That one's a function of time.

553
00:33:33,250 --> 00:33:38,450
And we know the solution
to this one is some x of t

554
00:33:38,450 --> 00:33:43,920
is some amplitude
e to the i omega t.

555
00:33:43,920 --> 00:33:47,710
So therefore, we can guess
that the solution to this one

556
00:33:47,710 --> 00:33:55,605
is W of x is going to be--
I'll write it as some B.

557
00:33:55,605 --> 00:33:57,290
Now I need a function of x.

558
00:33:57,290 --> 00:34:03,392
But it can be just like this--
e to the i, and I'll say kx.

559
00:34:03,392 --> 00:34:04,933
I know that's going
to be a solution.

560
00:34:09,420 --> 00:34:11,870
So let's plug it in.

561
00:34:11,870 --> 00:34:23,850
If I plug that in, I
get minus k squared

562
00:34:23,850 --> 00:34:42,469
Be to the ikx plus omega squared
over c squared Be to the ikx

563
00:34:42,469 --> 00:34:44,260
equals 0.

564
00:34:44,260 --> 00:34:47,349
Well, now I get rid of these.

565
00:34:47,349 --> 00:34:51,860
And what I found out
is that k squared is

566
00:34:51,860 --> 00:34:55,600
omega squared over c squared.

567
00:34:55,600 --> 00:34:59,845
And this has a name-- k.

568
00:35:04,200 --> 00:35:07,920
It's called the wave number.

569
00:35:07,920 --> 00:35:12,050
And it also happens to
be 2 pi over lambda.

570
00:35:12,050 --> 00:35:13,100
We'll come back to that.

571
00:35:13,100 --> 00:35:14,141
Lambda is the wavelength.

572
00:35:14,141 --> 00:35:18,080
You have sinusoidal waves
running through the medium.

573
00:35:18,080 --> 00:35:21,220
2 pi over lambda is the
same as omega over c.

574
00:35:21,220 --> 00:35:28,810
And this is called
the wave number--

575
00:35:28,810 --> 00:35:31,120
really important
quantity if you're

576
00:35:31,120 --> 00:35:34,500
trying to understand wave
propagation in systems.

577
00:35:34,500 --> 00:35:36,840
And actually, this
one, this definition

578
00:35:36,840 --> 00:35:39,150
applies to all wave
bearing systems,

579
00:35:39,150 --> 00:35:42,830
whether or not they
obey the wave equation.

580
00:35:42,830 --> 00:35:45,630
It'll apply to waves
traveling down a beam as well.

581
00:35:45,630 --> 00:35:49,260
So the definition of wave number
is frequency divided by speed,

582
00:35:49,260 --> 00:35:52,900
or 2 pi over the wavelength.

583
00:35:52,900 --> 00:35:55,310
Well, let's see.

584
00:35:55,310 --> 00:36:02,850
We can't go much further with
just the wave equation itself.

585
00:36:02,850 --> 00:36:05,000
In order to get the
natural frequencies,

586
00:36:05,000 --> 00:36:07,645
we have to invoke
other information

587
00:36:07,645 --> 00:36:09,140
that we know in the problem.

588
00:36:09,140 --> 00:36:11,950
In particular, we
know that in order

589
00:36:11,950 --> 00:36:20,350
to get natural frequencies,
we had to create conditions

590
00:36:20,350 --> 00:36:21,660
where this could vibrate.

591
00:36:21,660 --> 00:36:24,820
In particular, I fix that
end, and I fix this end,

592
00:36:24,820 --> 00:36:26,590
and I put some tension on it.

593
00:36:26,590 --> 00:36:28,574
And now it'll vibrate.

594
00:36:28,574 --> 00:36:29,990
But it clearly has
something to do

595
00:36:29,990 --> 00:36:32,030
with its ends and its length.

596
00:36:32,030 --> 00:36:35,900
And so this is a
boundary value problem.

597
00:36:35,900 --> 00:36:39,780
And we have to invoke the
boundary conditions to actually

598
00:36:39,780 --> 00:36:42,550
finish finding the natural
frequencies and mode shapes.

599
00:36:48,960 --> 00:37:00,300
Apply the boundary
conditions-- so I assumed here

600
00:37:00,300 --> 00:37:03,690
that my W of x is going to
look something like that.

601
00:37:03,690 --> 00:37:06,530
In order to get a little
more information out of this,

602
00:37:06,530 --> 00:37:12,290
I'm going to write now W of x
in an alternative form that's

603
00:37:12,290 --> 00:37:13,450
equally valid.

604
00:37:13,450 --> 00:37:26,050
And I'll call it B1 cosine
kx plus a B2 sine kx.

605
00:37:26,050 --> 00:37:29,840
And I could relate that to
e to the ikx, B to the ikx,

606
00:37:29,840 --> 00:37:31,700
by real and imaginary
parts, and so forth.

607
00:37:31,700 --> 00:37:33,710
This is a real part.

608
00:37:33,710 --> 00:37:37,370
I'm saying in general it could
have a cosine part and also

609
00:37:37,370 --> 00:37:38,630
a sine part.

610
00:37:38,630 --> 00:37:43,970
But now I know my boundary
conditions are W at x equals 0.

611
00:37:43,970 --> 00:37:45,620
W of 0 is what?

612
00:37:45,620 --> 00:37:47,445
What's the displacement
at x equals 0?

613
00:37:47,445 --> 00:37:48,320
AUDIENCE: [INAUDIBLE]

614
00:37:52,280 --> 00:37:52,955
PROFESSOR: 0.

615
00:37:52,955 --> 00:37:53,580
That's the pin.

616
00:37:53,580 --> 00:37:55,430
That's the end
where it's fixed at.

617
00:37:55,430 --> 00:37:59,110
And we started out here
with a second order

618
00:37:59,110 --> 00:38:00,550
partial differential equation.

619
00:38:00,550 --> 00:38:03,055
And a second order equation
requires two boundary

620
00:38:03,055 --> 00:38:03,555
conditions.

621
00:38:03,555 --> 00:38:05,440
A fourth order
equation for the beam

622
00:38:05,440 --> 00:38:07,560
will require four
boundary conditions.

623
00:38:07,560 --> 00:38:08,990
We only have to find two.

624
00:38:08,990 --> 00:38:11,720
One of them is it has
no motion on the left.

625
00:38:11,720 --> 00:38:15,080
So you plug in 0 for x.

626
00:38:15,080 --> 00:38:16,540
Cosine of 0 is 1.

627
00:38:16,540 --> 00:38:18,770
Sine of 0 is 0.

628
00:38:18,770 --> 00:38:24,670
So we find out that
this is B1 times 1.

629
00:38:24,670 --> 00:38:26,930
But it has to be 0 as
the boundary condition.

630
00:38:26,930 --> 00:38:30,440
So that implies B1 is 0.

631
00:38:30,440 --> 00:38:32,390
There's no cosines
in this answer.

632
00:38:32,390 --> 00:38:36,990
And W at L is 0.

633
00:38:36,990 --> 00:38:47,160
And so that says B2
sine kL equals 0.

634
00:38:47,160 --> 00:38:48,680
And that's true.

635
00:38:48,680 --> 00:38:57,670
That's only true
if kL equals n pi.

636
00:38:57,670 --> 00:38:59,300
So now I've found
out that there's,

637
00:38:59,300 --> 00:39:03,430
just for vibration of
a finite length string,

638
00:39:03,430 --> 00:39:06,950
only particular
values of k that work.

639
00:39:06,950 --> 00:39:11,630
So that says that there are
special values of k which

640
00:39:11,630 --> 00:39:31,560
I'll call k sub n which
are equal to n pi over L.

641
00:39:31,560 --> 00:39:33,989
And from that, we now
have our mode shapes.

642
00:39:33,989 --> 00:39:35,530
Because we can say,
ah, well, there's

643
00:39:35,530 --> 00:39:41,710
special solution
for this W of x that

644
00:39:41,710 --> 00:39:45,160
applies only when we satisfy
the boundary conditions.

645
00:39:45,160 --> 00:39:48,345
And that will be some
undetermined amplitude.

646
00:39:50,870 --> 00:39:53,110
B2 came from the sine term.

647
00:40:00,590 --> 00:40:02,900
And those are our mode shapes.

648
00:40:02,900 --> 00:40:04,900
And now the natural
frequencies-- once

649
00:40:04,900 --> 00:40:07,180
you know mode shapes,
natural frequencies actually

650
00:40:07,180 --> 00:40:09,010
become pretty trivial to find.

651
00:40:09,010 --> 00:40:13,930
In this case, if we know
that's the mode shape,

652
00:40:13,930 --> 00:40:17,690
then how do we get the
natural frequencies?

653
00:40:17,690 --> 00:40:21,260
Well, we know that--
what's the definition of k?

654
00:40:27,710 --> 00:40:31,880
Therefore, the
particular values of k

655
00:40:31,880 --> 00:40:34,600
that were allowed
solutions here are

656
00:40:34,600 --> 00:40:40,390
going to correspond to
particular values of omega n.

657
00:40:40,390 --> 00:40:49,570
And therefore, omega n squared
is just kn squared c squared.

658
00:40:49,570 --> 00:40:59,720
And that's n pi
over L squared T/m.

659
00:41:04,090 --> 00:41:06,810
That's omega n squared.

660
00:41:06,810 --> 00:41:09,990
So the natural
frequencies of a string

661
00:41:09,990 --> 00:41:16,020
are n pi over L root T/m.

662
00:41:21,210 --> 00:41:25,270
And this is in
radians per second.

663
00:41:25,270 --> 00:41:27,670
And I like to work
in hertz sometimes.

664
00:41:27,670 --> 00:41:33,890
So the natural frequencies
in hertz-- omega n over 2 pi.

665
00:41:33,890 --> 00:41:39,110
And that becomes n
over 2L root T/m.

666
00:41:42,580 --> 00:41:50,840
So the first natural frequency,
f1, is 1 over 2L root T/m.

667
00:41:57,910 --> 00:42:00,630
Now, let's draw.

668
00:42:00,630 --> 00:42:02,480
What's the mode shape
for the first mode?

669
00:42:02,480 --> 00:42:10,960
Well, it's half a sine
wave, vibrates like that.

670
00:42:10,960 --> 00:42:11,900
It's full wavelength.

671
00:42:11,900 --> 00:42:14,120
I didn't leave myself
quite enough room.

672
00:42:14,120 --> 00:42:17,350
That's half a wavelength
of a sine wave.

673
00:42:17,350 --> 00:42:20,510
So the full wavelength
would be like that.

674
00:42:20,510 --> 00:42:25,760
This is of length L. And
so is this piece over here.

675
00:42:25,760 --> 00:42:30,890
So the lambda is 2L for
this particular problem.

676
00:42:35,160 --> 00:42:38,410
Let's see, how do I want
to pose this question?

677
00:42:45,820 --> 00:42:52,780
So how long does it take
for a wave or disturbance

678
00:42:52,780 --> 00:43:01,600
to travel the length
of this finite string?

679
00:43:10,991 --> 00:43:14,372
How long does it take it
to go down there and back?

680
00:43:14,372 --> 00:43:15,580
How would you calculate that?

681
00:43:20,640 --> 00:43:22,190
Distance equals rate times time.

682
00:43:22,190 --> 00:43:24,550
What's the distance?

683
00:43:24,550 --> 00:43:26,530
2L.

684
00:43:26,530 --> 00:43:29,500
What's the speed?

685
00:43:29,500 --> 00:43:30,920
c.

686
00:43:30,920 --> 00:43:34,820
So the length of time ought
to be 2L over c, right?

687
00:43:39,690 --> 00:44:01,710
So the time required-- and
2L divided by T over m.

688
00:44:01,710 --> 00:44:05,573
But f1 is T/m divided by 2L.

689
00:44:05,573 --> 00:44:06,072
Hmm.

690
00:44:11,720 --> 00:44:15,730
So the period-- so there's
a direct connection

691
00:44:15,730 --> 00:44:22,340
between propagation speed,
frequencies, wavelengths.

692
00:44:22,340 --> 00:44:23,750
They're very closely related.

693
00:44:23,750 --> 00:44:33,940
So the natural frequency of
the first mode of this string,

694
00:44:33,940 --> 00:44:38,610
that frequency, is exactly
1 over the length of time

695
00:44:38,610 --> 00:44:41,572
it takes for a disturbance
to travel down and back.

696
00:44:49,950 --> 00:44:52,070
So with that depth
of understanding

697
00:44:52,070 --> 00:44:58,710
of how the wave
equation behaves,

698
00:44:58,710 --> 00:45:04,990
you can guess the behavior
of lots of other things

699
00:45:04,990 --> 00:45:07,711
that behave like that,
like my rod here.

700
00:45:07,711 --> 00:45:09,460
I'll do a little demo
with it in a second.

701
00:45:24,960 --> 00:45:28,780
So for example, the
longitudinal vibration,

702
00:45:28,780 --> 00:45:32,630
stress waves running
up and down this thing,

703
00:45:32,630 --> 00:45:35,150
obey the wave equation.

704
00:45:35,150 --> 00:45:40,395
So if I take this thing
and drop it on the floor,

705
00:45:40,395 --> 00:45:42,720
it'll bounce off the floor.

706
00:45:42,720 --> 00:45:44,890
How long does it take
to bounce off the floor?

707
00:45:50,060 --> 00:45:53,320
So what do you think actually--
what physics has to happen?

708
00:45:53,320 --> 00:45:57,800
What's required to make this
thing bounce off the floor?

709
00:45:57,800 --> 00:46:00,650
So we're going to consider
the floor infinitely rigid.

710
00:46:00,650 --> 00:46:01,880
It hits the floor.

711
00:46:01,880 --> 00:46:04,180
It actually stays there for
some finite length of time,

712
00:46:04,180 --> 00:46:06,430
and then it leaves.

713
00:46:06,430 --> 00:46:09,180
So physically, when I was
holding up my string, if I

714
00:46:09,180 --> 00:46:11,700
smacked the end, what happened?

715
00:46:11,700 --> 00:46:18,420
A pulse took off, ran down
the end, reflected, came back.

716
00:46:18,420 --> 00:46:20,710
And that was one round trip.

717
00:46:20,710 --> 00:46:23,390
What do you suppose
happens here?

718
00:46:23,390 --> 00:46:26,370
I put a pulse into the end.

719
00:46:26,370 --> 00:46:29,555
Is it a tension or compression,
the strain that's felt?

720
00:46:29,555 --> 00:46:30,430
AUDIENCE: Compression

721
00:46:30,430 --> 00:46:31,000
PROFESSOR: Compression.

722
00:46:31,000 --> 00:46:33,130
So a little compression
pulse is put into the end.

723
00:46:33,130 --> 00:46:37,270
That compression pulse
then, when it first hits,

724
00:46:37,270 --> 00:46:41,190
the compression and the speed
of propagation is finite.

725
00:46:41,190 --> 00:46:44,490
So that compression wave
starts traveling up here.

726
00:46:44,490 --> 00:46:49,900
Behind the compression wave,
this rod has come to a stop.

727
00:46:49,900 --> 00:46:51,330
In front of the
compression wave,

728
00:46:51,330 --> 00:46:53,960
the rod doesn't know
it hit the ground yet.

729
00:46:53,960 --> 00:46:56,820
It's still moving down.

730
00:46:56,820 --> 00:46:59,480
So that compression
wave travels up,

731
00:46:59,480 --> 00:47:03,390
and it is decelerating
each little slice of mass

732
00:47:03,390 --> 00:47:04,440
as it passes through.

733
00:47:04,440 --> 00:47:05,760
It brings it to a stop.

734
00:47:05,760 --> 00:47:09,090
And so the compression
reaches the top end.

735
00:47:09,090 --> 00:47:12,520
The cylinder has come to a stop.

736
00:47:12,520 --> 00:47:13,347
The end is free.

737
00:47:13,347 --> 00:47:14,780
It can't take any strain.

738
00:47:14,780 --> 00:47:16,790
So an equal and
opposite tension wave

739
00:47:16,790 --> 00:47:20,100
has to start to make the sum
of them go to 0 at the end.

740
00:47:20,100 --> 00:47:22,710
The boundary condition
at the end is no strain.

741
00:47:22,710 --> 00:47:24,260
So it reflects as
a tension wave.

742
00:47:24,260 --> 00:47:25,900
Now you have a tension
wave going down.

743
00:47:25,900 --> 00:47:29,020
And what it does is it
accelerates every atom as it

744
00:47:29,020 --> 00:47:31,200
goes by, as it goes past it.

745
00:47:31,200 --> 00:47:32,480
So everything is stopped now.

746
00:47:32,480 --> 00:47:36,180
Now it starts down,
and this thing

747
00:47:36,180 --> 00:47:39,750
starts rebounding from-- the
top rebounds from the floor

748
00:47:39,750 --> 00:47:40,850
before the bottom does.

749
00:47:40,850 --> 00:47:43,310
The top starts going up.

750
00:47:43,310 --> 00:47:44,800
All of it-- more
and more goes up.

751
00:47:44,800 --> 00:47:45,800
And one hits the bottom.

752
00:47:45,800 --> 00:47:48,660
The tension wave hits the
floor, and it jumps off.

753
00:47:48,660 --> 00:47:49,930
So how long does it take?

754
00:47:59,430 --> 00:48:00,500
Right?

755
00:48:00,500 --> 00:48:02,930
And what do you guess
the natural frequency

756
00:48:02,930 --> 00:48:07,369
of a free-free rod is?

757
00:48:07,369 --> 00:48:08,660
Now, it has a funny mode shape.

758
00:48:08,660 --> 00:48:11,550
The mode shape is not half
a sine wave like this.

759
00:48:11,550 --> 00:48:14,240
The displacement of the
rod, it has free ends.

760
00:48:14,240 --> 00:48:16,740
The ends are moving a lot.

761
00:48:16,740 --> 00:48:19,713
But I'll give you a clue.

762
00:48:19,713 --> 00:48:23,700
[ROD RINGING]

763
00:48:23,700 --> 00:48:26,820
I can hold it in the
center and not damp it.

764
00:48:26,820 --> 00:48:28,755
What do you think the
mode shape looks like?

765
00:48:32,560 --> 00:48:37,560
Half a wavelength long,
ends are free-- cosine,

766
00:48:37,560 --> 00:48:41,080
maximum displacement,
goes to zero,

767
00:48:41,080 --> 00:48:43,100
maximum negative displacement.

768
00:48:43,100 --> 00:48:45,500
So it's half a
wavelength long, but it's

769
00:48:45,500 --> 00:48:48,560
a cosine half a wavelength.

770
00:48:48,560 --> 00:48:50,800
And the full wavelength is 2L.

771
00:48:55,330 --> 00:48:57,190
So this has mode shapes.

772
00:48:57,190 --> 00:49:00,199
The mode shapes-- I've applied
different boundary conditions.

773
00:49:00,199 --> 00:49:01,865
These are free-free
boundary conditions.

774
00:49:01,865 --> 00:49:07,290
The mode shapes are
cosine n pi x over L.

775
00:49:07,290 --> 00:49:10,120
But they have to obey
a certain other law

776
00:49:10,120 --> 00:49:14,240
that we know about,
conservation of momentum.

777
00:49:14,240 --> 00:49:15,830
Because I've got
gravity to deal with,

778
00:49:15,830 --> 00:49:17,163
I have to hang on to this thing.

779
00:49:17,163 --> 00:49:19,730
But I've picked a place to hang
onto it that you can hear it.

780
00:49:19,730 --> 00:49:21,970
I'm not affecting the motion.

781
00:49:21,970 --> 00:49:24,440
There's no motion
where I'm holding it.

782
00:49:24,440 --> 00:49:26,720
So if I were out in
space, I could do this--

783
00:49:26,720 --> 00:49:27,632
[ROD RINGING]

784
00:49:27,632 --> 00:49:30,250
--and just let it hang
there in space, right?

785
00:49:30,250 --> 00:49:32,020
And it would sit there and ring.

786
00:49:32,020 --> 00:49:35,820
What is happening to the
center of mass of this system

787
00:49:35,820 --> 00:49:37,545
as it vibrates?

788
00:49:37,545 --> 00:49:39,920
AUDIENCE: [INAUDIBLE]

789
00:49:39,920 --> 00:49:41,420
PROFESSOR: Stationary.

790
00:49:41,420 --> 00:49:43,310
So half of the mass
of this thing's

791
00:49:43,310 --> 00:49:44,579
got to be moving that way.

792
00:49:44,579 --> 00:49:46,120
And half of the mass
has to be moving

793
00:49:46,120 --> 00:49:48,660
that way so that the total
center of mass doesn't move.

794
00:49:48,660 --> 00:49:51,700
Well, cosine mode
shape, positive here,

795
00:49:51,700 --> 00:49:54,250
negative there, perfectly
symmetric, center of mass

796
00:49:54,250 --> 00:49:55,360
doesn't move.

797
00:49:55,360 --> 00:49:58,030
So there's all sorts
of neat little problems

798
00:49:58,030 --> 00:50:02,450
that you can solve just by
knowing the wave equation

799
00:50:02,450 --> 00:50:06,220
and figuring out
boundary conditions.

800
00:50:06,220 --> 00:50:08,250
How many of you stand
in the shower at home

801
00:50:08,250 --> 00:50:11,150
and sing, and
every now and then,

802
00:50:11,150 --> 00:50:14,619
you hit a note, man, you
just sound great, right?

803
00:50:14,619 --> 00:50:16,160
And it's just all
this reverberation.

804
00:50:16,160 --> 00:50:17,451
How many of you have done that?

805
00:50:17,451 --> 00:50:20,690
OK, right, what's going on?

806
00:50:20,690 --> 00:50:23,866
AUDIENCE: [INAUDIBLE]
Natural frequency?

807
00:50:23,866 --> 00:50:25,490
PROFESSOR: You've
hit a-- somebody said

808
00:50:25,490 --> 00:50:26,260
natural frequency.

809
00:50:26,260 --> 00:50:27,420
Of what?

810
00:50:27,420 --> 00:50:28,890
AUDIENCE: [INAUDIBLE]

811
00:50:28,890 --> 00:50:29,975
PROFESSOR: Huh?

812
00:50:29,975 --> 00:50:30,850
AUDIENCE: [INAUDIBLE]

813
00:50:33,800 --> 00:50:39,960
PROFESSOR: You've hit the
natural frequency of the shower

814
00:50:39,960 --> 00:50:41,020
stall itself.

815
00:50:41,020 --> 00:50:47,310
If the shower stall
is a meter across,

816
00:50:47,310 --> 00:50:49,780
pressure waves-- and
you plot pressure

817
00:50:49,780 --> 00:50:55,130
inside of the shower,
the lowest mode

818
00:50:55,130 --> 00:50:56,530
if you're plotting pressure.

819
00:50:56,530 --> 00:50:58,750
Well, let's plot actually
molecular movement.

820
00:50:58,750 --> 00:51:00,500
What's the boundary
condition at the wall,

821
00:51:00,500 --> 00:51:03,470
the molecules at the wall?

822
00:51:03,470 --> 00:51:04,610
They can't move, right?

823
00:51:04,610 --> 00:51:05,450
0.

824
00:51:05,450 --> 00:51:07,810
So the molecular
motion at resonance

825
00:51:07,810 --> 00:51:11,400
in the shower stall, the
molecules, the pressures making

826
00:51:11,400 --> 00:51:14,400
them move back and forth, looks
like back to the string again.

827
00:51:14,400 --> 00:51:17,970
This is L. The first
natural frequency

828
00:51:17,970 --> 00:51:21,470
of sound waves bouncing
off the walls in the stall

829
00:51:21,470 --> 00:51:32,030
is 1 over 2L root
times c, whatever c is.

830
00:51:32,030 --> 00:51:36,240
And c is the speed
of sound in air,

831
00:51:36,240 --> 00:51:38,490
which is 340 meters per second.

832
00:51:38,490 --> 00:51:40,985
So 340 meters per
second divided by 2L--

833
00:51:40,985 --> 00:51:44,240
so if it's 1 meter
across here, it's

834
00:51:44,240 --> 00:51:48,500
340 divided by 2, 170 hertz.

835
00:51:48,500 --> 00:51:50,750
So that first note you
can hit in the 1 meter

836
00:51:50,750 --> 00:51:54,784
across shower stall is about
170 hertz-- pretty low.

837
00:51:54,784 --> 00:51:55,950
But you can hit second mode.

838
00:51:55,950 --> 00:51:57,720
It'd be twice
that, and so forth.

839
00:51:57,720 --> 00:52:02,770
OK, what about an organ pipe?

840
00:52:02,770 --> 00:52:05,150
This is an organ pipe, wood.

841
00:52:05,150 --> 00:52:08,225
It's got a stoppered end.

842
00:52:08,225 --> 00:52:09,974
Actually, let's do it
without the stopper.

843
00:52:09,974 --> 00:52:12,030
Now it's an open organ pipe.

844
00:52:12,030 --> 00:52:14,022
[ORGAN NOTE]

845
00:52:16,020 --> 00:52:18,390
Basic wave equation--
how would you

846
00:52:18,390 --> 00:52:20,020
model its boundary conditions?

847
00:52:22,640 --> 00:52:26,080
So you can talk about maybe
particle molecular motion.

848
00:52:26,080 --> 00:52:30,960
This is, now again, just
sound waves, so air particles.

849
00:52:30,960 --> 00:52:32,410
And this is now longitudinal.

850
00:52:32,410 --> 00:52:33,530
Things are moving inside.

851
00:52:33,530 --> 00:52:37,850
So what's the boundary condition
at this end, free or fixed?

852
00:52:37,850 --> 00:52:38,352
Free.

853
00:52:38,352 --> 00:52:40,810
And here it's quite open, so
the boundary condition on here

854
00:52:40,810 --> 00:52:42,550
is free.

855
00:52:42,550 --> 00:52:50,416
So for the molecular motion
in a free-free organ pipe,

856
00:52:50,416 --> 00:52:53,720
you have to get back to that
half a wavelength cosine thing.

857
00:52:53,720 --> 00:52:55,530
And if you wanted to
plot pressure instead,

858
00:52:55,530 --> 00:52:57,880
you can write the wave
equation in terms of pressure.

859
00:52:57,880 --> 00:53:00,735
Pressure is-- this is
pressure relief here

860
00:53:00,735 --> 00:53:02,440
and pressure relief there.

861
00:53:02,440 --> 00:53:11,510
So in fact, if is
displacement of the molecules,

862
00:53:11,510 --> 00:53:14,800
pressure would plot like that.

863
00:53:14,800 --> 00:53:17,670
You'd have what's called
a pressure relief boundary

864
00:53:17,670 --> 00:53:18,480
condition.

865
00:53:18,480 --> 00:53:20,120
But again, it's a
half wavelength long.

866
00:53:20,120 --> 00:53:21,540
What do you think the
first natural frequency

867
00:53:21,540 --> 00:53:22,738
of this organ pipe is?

868
00:53:31,740 --> 00:53:34,220
The period would be 2L over c.

869
00:53:34,220 --> 00:53:35,880
The frequency
would be c over 2L.

870
00:53:38,390 --> 00:53:53,205
So the frequency for the organ
pipe open end f1 is c over 2L.

871
00:53:57,080 --> 00:53:59,410
[ORGAN NOTE]

872
00:53:59,410 --> 00:54:00,340
Check your intuition.

873
00:54:00,340 --> 00:54:03,300
I'm going to close the
end-- still an organ pipe.

874
00:54:03,300 --> 00:54:07,850
Is the frequency now going
to be higher or lower?

875
00:54:07,850 --> 00:54:08,580
Take a vote.

876
00:54:08,580 --> 00:54:11,430
How many think the
frequency is going to go up?

877
00:54:11,430 --> 00:54:13,540
Raise your hands, commit.

878
00:54:13,540 --> 00:54:15,479
All right, down.

879
00:54:15,479 --> 00:54:16,770
We've got a lot of uncertainty.

880
00:54:16,770 --> 00:54:18,808
All right, let's
do the experiment.

881
00:54:18,808 --> 00:54:20,720
[ORGAN NOTE]

882
00:54:23,588 --> 00:54:25,500
[LOWER ORGAN NOTE]

883
00:54:26,940 --> 00:54:28,524
How come?

884
00:54:28,524 --> 00:54:30,440
I find that actually
kind of counterintuitive.

885
00:54:30,440 --> 00:54:33,290
Until I learned this, I would
have guessed the opposite way.

886
00:54:33,290 --> 00:54:36,715
What's going on with
pressure in a closed pipe?

887
00:54:41,610 --> 00:54:44,960
Well, here at the orifice
where the sound is actually

888
00:54:44,960 --> 00:54:46,770
generated, it's the pressure.

889
00:54:46,770 --> 00:54:49,720
If we wanted to plot
pressure at the opening,

890
00:54:49,720 --> 00:54:52,390
that's a pressure relief place.

891
00:54:52,390 --> 00:54:54,060
So it's 0.

892
00:54:54,060 --> 00:54:57,590
But at the other end where
the stopper is, it's maximum.

893
00:55:01,070 --> 00:55:03,620
How many wavelengths is that?

894
00:55:03,620 --> 00:55:04,515
A quarter.

895
00:55:07,090 --> 00:55:12,470
And so the length of time
it takes for the thing

896
00:55:12,470 --> 00:55:15,440
to go through one
complete period

897
00:55:15,440 --> 00:55:31,510
is going to be 4L over c, half
the frequency of the open pipe.

898
00:55:31,510 --> 00:55:35,260
OK, so the wave equation
is really quite powerful,

899
00:55:35,260 --> 00:55:36,665
governs lots of things.

900
00:55:41,330 --> 00:55:45,810
I've got 10, 15
minutes left here.

901
00:55:45,810 --> 00:55:51,510
I don't want you to go away
thinking that the whole world

902
00:55:51,510 --> 00:55:53,160
behaves like the wave equation.

903
00:55:53,160 --> 00:55:58,090
Because there are some
important other physical systems

904
00:55:58,090 --> 00:55:59,900
that we care about.

905
00:55:59,900 --> 00:56:02,800
And I'm going to
show you just one.

906
00:56:02,800 --> 00:56:10,500
And that's the
vibration of the beam.

907
00:56:10,500 --> 00:56:14,030
So here's the cantilever beam.

908
00:56:16,630 --> 00:56:20,290
The whole table is moving.

909
00:56:20,290 --> 00:56:23,770
And you can see it
up on the screen.

910
00:56:23,770 --> 00:56:26,950
OK, so its first mode
vibration, tip moves maximum.

911
00:56:26,950 --> 00:56:30,520
It kind of looks like
a quarter wavelength.

912
00:56:30,520 --> 00:56:32,660
It roughly is, but not exactly.

913
00:56:35,410 --> 00:56:36,785
So let's draw a cantilever.

914
00:56:43,730 --> 00:56:48,110
And most of you have had 2.001.

915
00:56:48,110 --> 00:56:53,310
So if you put a load
P out here-- bends,

916
00:56:53,310 --> 00:56:55,900
goes through a
displacement delta.

917
00:56:55,900 --> 00:57:05,170
So you know that delta equals
Pl cubed over 3EI, right?

918
00:57:05,170 --> 00:57:07,749
And what's this I?

919
00:57:07,749 --> 00:57:10,120
AUDIENCE: [INAUDIBLE]

920
00:57:10,120 --> 00:57:12,020
PROFESSOR: Area
moment of inertia.

921
00:57:12,020 --> 00:57:14,470
Now that you've been
doing dynamics all term,

922
00:57:14,470 --> 00:57:16,249
we talk about mass
moments of inertia.

923
00:57:16,249 --> 00:57:17,790
There's also area
moments of inertia.

924
00:57:17,790 --> 00:57:20,790
So this is the area moment
of inertia of a beam.

925
00:57:20,790 --> 00:57:23,320
In this case, our beam is
a little rectangular cross

926
00:57:23,320 --> 00:57:24,770
section.

927
00:57:24,770 --> 00:57:28,550
And the neutral axis is
here, a little variable y

928
00:57:28,550 --> 00:57:29,810
at displacement.

929
00:57:29,810 --> 00:57:35,130
I is the integral
of y squared dA.

930
00:57:35,130 --> 00:57:40,575
And dA is just a little
slice of area here, dA.

931
00:57:40,575 --> 00:57:43,830
And the integral of y squared
dA is your cross sectional area

932
00:57:43,830 --> 00:57:48,380
moment of inertia in the
direction of bending.

933
00:57:48,380 --> 00:57:55,980
So that is I. You can also
write it as a kappa squared A.

934
00:57:55,980 --> 00:57:58,260
And we ran into
this in dynamics.

935
00:57:58,260 --> 00:57:59,930
We called it the
radius of gyration.

936
00:57:59,930 --> 00:58:03,840
You had the same thing with
area moments of inertia,

937
00:58:03,840 --> 00:58:04,860
the radius of gyration.

938
00:58:04,860 --> 00:58:08,910
This is going to be really
helpful in a second.

939
00:58:08,910 --> 00:58:15,970
So if you solve the
force balance for a beam

940
00:58:15,970 --> 00:58:26,960
like I did for the string,
take a little slice,

941
00:58:26,960 --> 00:58:30,270
do force balance for
transverse motions--

942
00:58:30,270 --> 00:58:32,550
I'm not going to grind it out.

943
00:58:32,550 --> 00:58:35,540
And temporarily neglect
external forces and damping.

944
00:58:35,540 --> 00:58:38,090
I want to get to the natural
frequencies and mode shapes.

945
00:58:38,090 --> 00:58:41,300
So the free vibration,
no damping, equation

946
00:58:41,300 --> 00:58:48,680
looks like EI partial
4 w with respect

947
00:58:48,680 --> 00:58:58,970
to x to the fourth plus rho A
partial squared w with respect

948
00:58:58,970 --> 00:59:01,330
to t squared equals 0.

949
00:59:01,330 --> 00:59:03,615
And now this is
density, mass density.

950
00:59:06,440 --> 00:59:10,930
And the A, this A, is the
area of the cross section.

951
00:59:17,290 --> 00:59:22,090
So it's just some bh with
thickness times the width.

952
00:59:22,090 --> 00:59:26,104
So rho times A is a
mass per unit length.

953
00:59:29,060 --> 00:59:31,190
And so mass per
unit length times

954
00:59:31,190 --> 00:59:33,200
dx would be the
little mass associated

955
00:59:33,200 --> 00:59:35,380
with the element times
the acceleration should

956
00:59:35,380 --> 00:59:37,360
be the forces on the element.

957
00:59:37,360 --> 00:59:40,330
So that's the fourth
order partial differential

958
00:59:40,330 --> 00:59:44,690
equation that describes
the vibration of a beam.

959
00:59:44,690 --> 00:59:50,630
And you have to apply
the boundary conditions.

960
00:59:50,630 --> 00:59:55,150
And for the string, it was
just B1 cosine B2 sine.

961
00:59:55,150 --> 01:00:02,470
For the beam, it's B1 cosine
plus B2 sine plus C2 cosh

962
01:00:02,470 --> 01:00:05,730
plus D2 sinh x.

963
01:00:05,730 --> 01:00:08,160
And then you have to apply
four boundary conditions

964
01:00:08,160 --> 01:00:13,177
and solve for B1, B2, and
so forth, all four of those.

965
01:00:13,177 --> 01:00:13,760
I won't do it.

966
01:00:13,760 --> 01:00:14,801
But that's how you do it.

967
01:00:14,801 --> 01:00:18,110
Separation-- and separation
of variables works again.

968
01:00:18,110 --> 01:00:20,540
So we solve this, apply
the boundary conditions.

969
01:00:26,185 --> 01:00:27,560
What are the
boundary conditions?

970
01:00:27,560 --> 01:00:29,643
Just so you understand
what I mean by the boundary

971
01:00:29,643 --> 01:00:31,990
conditions, what are they
for a free-free beam,

972
01:00:31,990 --> 01:00:33,340
zero motion at the wall?

973
01:00:37,360 --> 01:00:43,210
No strain at the end, no
bending moment at the end,

974
01:00:43,210 --> 01:00:44,940
no sheer force at
the end-- so there's

975
01:00:44,940 --> 01:00:47,315
no second derivative,
no third derivative.

976
01:00:49,990 --> 01:00:56,790
And at the wall, the slope
is 0, the first derivative.

977
01:00:56,790 --> 01:00:59,704
No slope comes into the wall,
but the slope is 0 there.

978
01:00:59,704 --> 01:01:01,620
So those are the different
kind of boundaries.

979
01:01:01,620 --> 01:01:03,980
So if you have a
free-free beam, you

980
01:01:03,980 --> 01:01:09,740
have no bending at either end
and no strain at either end.

981
01:01:09,740 --> 01:01:11,830
Fixed-fixed beam--
no displacement,

982
01:01:11,830 --> 01:01:14,342
zero slopes at both at ends,
and all different combinations.

983
01:01:14,342 --> 01:01:16,050
And every different
combination gives you

984
01:01:16,050 --> 01:01:17,990
different natural frequencies.

985
01:01:17,990 --> 01:01:21,010
So you apply the boundary
conditions, and for a beam,

986
01:01:21,010 --> 01:01:24,230
you find out that
for all beams omega

987
01:01:24,230 --> 01:01:31,790
n can be written as some
beta n, a parameter, squared,

988
01:01:31,790 --> 01:01:39,600
I'll call it, times the
square root of EI over rho A.

989
01:01:39,600 --> 01:01:42,725
And this thing varies according
to the boundary conditions.

990
01:01:45,410 --> 01:01:50,170
Now that's what you get shown
in every textbook in the world.

991
01:01:50,170 --> 01:01:54,810
And I have a very hard
time visualizing this,

992
01:01:54,810 --> 01:01:59,060
getting physical
intuition by that.

993
01:01:59,060 --> 01:02:01,140
So something you never
see in a textbook

994
01:02:01,140 --> 01:02:12,050
but I often do is let's
replace I with kappa squared A.

995
01:02:12,050 --> 01:02:19,310
And you get a square root of E
over rho and a square root of I

996
01:02:19,310 --> 01:02:22,740
over A. But I is kappa
squared A. The A's cancel.

997
01:02:22,740 --> 01:02:29,040
It's the square root
of kappa squared.

998
01:02:29,040 --> 01:02:32,346
So this, you know
what E over rho is?

999
01:02:32,346 --> 01:02:38,960
E over rho, square
root of E over rho,

1000
01:02:38,960 --> 01:02:47,910
is the sound speed
in a solid material.

1001
01:02:47,910 --> 01:02:51,790
So the speed of stress
waves traveling up and down

1002
01:02:51,790 --> 01:02:54,409
this thing is the square
root of E over rho.

1003
01:02:54,409 --> 01:02:54,950
[ROD RINGING]

1004
01:02:54,950 --> 01:02:55,658
This is aluminum.

1005
01:02:55,658 --> 01:03:00,650
It's about 4,000
meters a second.

1006
01:03:00,650 --> 01:03:03,970
So if you know just the
properties of the material,

1007
01:03:03,970 --> 01:03:04,730
you have that.

1008
01:03:04,730 --> 01:03:08,590
And that says then
omega n for beams

1009
01:03:08,590 --> 01:03:16,020
is some beta n squared a
parameter times kappa CL.

1010
01:03:16,020 --> 01:03:18,320
And this thing, this
is often written CL.

1011
01:03:18,320 --> 01:03:20,870
It's the longitudinal
sound speed.

1012
01:03:23,980 --> 01:03:28,420
This is sound speed for waves
traveling through the medium.

1013
01:03:28,420 --> 01:03:30,990
So this tells you if you
make the beam twice as thick,

1014
01:03:30,990 --> 01:03:33,790
what do you do to its
natural frequencies?

1015
01:03:33,790 --> 01:03:37,250
Doubles-- instantly
you know that.

1016
01:03:37,250 --> 01:03:40,970
So bending properties depend a
lot on the radius of gyration.

1017
01:03:40,970 --> 01:03:43,180
And I'll give you a
few natural frequencies

1018
01:03:43,180 --> 01:03:44,770
for different
boundary conditions

1019
01:03:44,770 --> 01:03:46,560
just so you see what
they behave like.

1020
01:03:56,020 --> 01:04:06,680
So a pin-pin beam
looks like that.

1021
01:04:06,680 --> 01:04:14,469
So you put a plank across the
stream, rocks on both sides,

1022
01:04:14,469 --> 01:04:16,010
you've got a pin-pin
beam, basically.

1023
01:04:16,010 --> 01:04:17,360
It's set there in rock.

1024
01:04:17,360 --> 01:04:20,405
So some length L
has properties EI.

1025
01:04:20,405 --> 01:04:23,410
So the natural frequencies
for a pin-pin beam,

1026
01:04:23,410 --> 01:04:29,250
the beta n's, are
just n pi over L.

1027
01:04:29,250 --> 01:04:32,690
And so your natural
frequencies-- omega n

1028
01:04:32,690 --> 01:04:38,960
looks like n pi over L
quantity squared kappa CL.

1029
01:04:43,000 --> 01:04:57,590
And for the cantilever,
the natural frequencies

1030
01:04:57,590 --> 01:05:09,795
look like omega n pi squared
over 4L squared, is the beta n.

1031
01:05:12,740 --> 01:05:16,360
And I'll write it this
way again-- EI over

1032
01:05:16,360 --> 01:05:18,870
rho A. You can always
go back and do that.

1033
01:05:18,870 --> 01:05:20,800
Or you can call it kappa CL.

1034
01:05:20,800 --> 01:05:25,310
This is also kappa CL.

1035
01:05:25,310 --> 01:05:29,360
But then there are
some numbers you've got

1036
01:05:29,360 --> 01:05:33,070
to use here-- 1.194 squared.

1037
01:05:33,070 --> 01:05:34,650
That's the first mode.

1038
01:05:34,650 --> 01:05:39,300
Second mode-- 2.988 squared.

1039
01:05:39,300 --> 01:05:45,900
And then after that-- 5
squared, 7 squared, 9 squared.

1040
01:05:45,900 --> 01:05:50,220
So this is the natural
frequency of a cantilever.

1041
01:05:50,220 --> 01:05:53,880
Pi squared over 4L
squared times 1.194

1042
01:05:53,880 --> 01:05:57,240
squared kappa CL, that's
this natural frequency.

1043
01:06:01,840 --> 01:06:05,590
And one final case,
because I can show it

1044
01:06:05,590 --> 01:06:08,640
to you-- the free-free case.

1045
01:06:08,640 --> 01:06:14,822
So that's a beam bending
that vibrates like that.

1046
01:06:17,770 --> 01:06:22,810
And I happen to know on a
beam for the first mode-- this

1047
01:06:22,810 --> 01:06:24,310
is the first mode of a beam.

1048
01:06:24,310 --> 01:06:26,710
Where these nodes are,
where there's no motion,

1049
01:06:26,710 --> 01:06:29,132
I should be able to hold
it there and not damp it.

1050
01:06:29,132 --> 01:06:31,340
And that turns out to be at
about the quarter points.

1051
01:06:34,010 --> 01:06:36,382
So whack it like that.

1052
01:06:36,382 --> 01:06:40,720
[ROD RINGING]

1053
01:06:40,720 --> 01:06:43,470
And do it again.

1054
01:06:43,470 --> 01:06:44,610
[ROD RINGING]

1055
01:06:44,610 --> 01:06:50,921
All right, so I want you to
hold it about right there.

1056
01:06:50,921 --> 01:06:52,920
Nope, you can't hold it
like that, though-- just

1057
01:06:52,920 --> 01:06:53,690
got to balance it.

1058
01:06:53,690 --> 01:06:56,258
Because you've got to be
right where the node is.

1059
01:06:56,258 --> 01:06:59,914
[ROD RINGING]

1060
01:06:59,914 --> 01:07:01,580
You can hear that
little bit lower tone.

1061
01:07:01,580 --> 01:07:03,160
That's that free-free
bending mode.

1062
01:07:03,160 --> 01:07:03,810
And it's just sitting.

1063
01:07:03,810 --> 01:07:06,060
You can feel it vibrating
a little bit but not much.

1064
01:07:06,060 --> 01:07:08,060
When you're right
in the right spot,

1065
01:07:08,060 --> 01:07:11,670
you're right on the mode shape.

1066
01:07:11,670 --> 01:07:14,120
You can almost see it if
you hit it hard enough.

1067
01:07:14,120 --> 01:07:16,210
So that's the free-free beam.

1068
01:07:16,210 --> 01:07:23,472
And the free-free beam has
natural frequencies omega n,

1069
01:07:23,472 --> 01:07:39,160
again, pi squared over 4L
squared kappa CL 3.0112

1070
01:07:39,160 --> 01:07:47,540
squared, 5 squared, 7 squared,
9 squared, so as you go up in n.

1071
01:07:47,540 --> 01:07:49,132
So those are the
natural frequencies

1072
01:07:49,132 --> 01:07:49,965
of a free-free beam.

1073
01:07:53,060 --> 01:08:05,360
Oh, one last fact about
beams-- so this is now

1074
01:08:05,360 --> 01:08:08,860
a steel beam under no tension.

1075
01:08:08,860 --> 01:08:12,310
It can support its own
weight, long though.

1076
01:08:12,310 --> 01:08:16,180
So can a beam support
waves traveling down

1077
01:08:16,180 --> 01:08:18,979
the beam, transverse waves
traveling down the beam?

1078
01:08:18,979 --> 01:08:19,810
What do you think?

1079
01:08:22,350 --> 01:08:24,330
Well, if it can support
this, it can probably

1080
01:08:24,330 --> 01:08:25,920
support waves, right?

1081
01:08:25,920 --> 01:08:27,500
So waves will
propagate in a beam

1082
01:08:27,500 --> 01:08:30,450
even though this is fourth order
partial differential equation.

1083
01:08:30,450 --> 01:08:33,020
But how fast do they go?

1084
01:08:33,020 --> 01:08:35,819
That's the question.

1085
01:08:35,819 --> 01:08:36,880
So this is a beam.

1086
01:08:36,880 --> 01:08:39,649
And I want to know about
waves traveling down it.

1087
01:08:39,649 --> 01:08:41,590
And I'm not going
to go through-- this

1088
01:08:41,590 --> 01:08:46,540
would take another hour or so to
show you where this comes from.

1089
01:08:46,540 --> 01:08:50,859
But here's my beam.

1090
01:08:50,859 --> 01:08:54,130
Here's a disturbance
traveling along it

1091
01:08:54,130 --> 01:08:57,899
with some speed that
I'm going to call CT.

1092
01:08:57,899 --> 01:09:00,880
It's transverse wave speed.

1093
01:09:00,880 --> 01:09:04,189
It's the speed you'd see
a crest of a wave moving

1094
01:09:04,189 --> 01:09:06,970
at running down that beam.

1095
01:09:06,970 --> 01:09:18,899
CT for a beam-- square
root of omega kappa CL.

1096
01:09:18,899 --> 01:09:23,819
And CL, again, is the
square root of E over rho.

1097
01:09:23,819 --> 01:09:25,790
That's the speed of
sound in the material.

1098
01:09:25,790 --> 01:09:28,920
That just turns up in here.

1099
01:09:28,920 --> 01:09:38,250
So what does this tell you
about the frequency dependence

1100
01:09:38,250 --> 01:09:40,253
of the speed?

1101
01:09:40,253 --> 01:09:41,794
Does the speed change
with frequency?

1102
01:09:44,979 --> 01:09:47,970
Omega kappa CL-- it's
proportional to frequency.

1103
01:09:47,970 --> 01:09:51,910
High frequency waves go faster
than low frequency waves

1104
01:09:51,910 --> 01:09:54,250
in a beam.

1105
01:09:54,250 --> 01:09:56,450
I didn't emphasize it when
we were talking about it.

1106
01:09:56,450 --> 01:10:02,170
But the wave equation,
what was c for the string?

1107
01:10:02,170 --> 01:10:06,220
For the wave equation, the
speed of wave propagation

1108
01:10:06,220 --> 01:10:07,280
was square root of T/m.

1109
01:10:07,280 --> 01:10:10,220
Was it frequency dependent?

1110
01:10:10,220 --> 01:10:12,130
Always traveled
at the same speed.

1111
01:10:12,130 --> 01:10:15,455
And so there's an
important consequence.

1112
01:10:19,120 --> 01:10:24,140
So for anything that
obeys the wave equation,

1113
01:10:24,140 --> 01:10:28,330
the speed of propagation is
a constant and independent

1114
01:10:28,330 --> 01:10:30,450
to frequency.

1115
01:10:30,450 --> 01:10:35,600
So I can make any initial
shape that I make in this thing

1116
01:10:35,600 --> 01:10:36,210
and let it go.

1117
01:10:36,210 --> 01:10:39,260
Its initial disturbance,
that little shape

1118
01:10:39,260 --> 01:10:44,559
will stay that shape and run
up and down the thing forever.

1119
01:10:44,559 --> 01:10:46,600
And that shape-- you could
imagine a little pluck

1120
01:10:46,600 --> 01:10:47,950
like this to start with.

1121
01:10:47,950 --> 01:10:51,370
You could imagine
doing a Fourier

1122
01:10:51,370 --> 01:10:52,720
series to approximate that.

1123
01:10:52,720 --> 01:10:55,740
It would be made up of a
bunch of different Fourier

1124
01:10:55,740 --> 01:10:58,270
components.

1125
01:10:58,270 --> 01:11:00,957
And yet for something that
bears the wave equation,

1126
01:11:00,957 --> 01:11:03,290
that little pluck will just
stay the shape of that pluck

1127
01:11:03,290 --> 01:11:04,890
and run around forever.

1128
01:11:04,890 --> 01:11:07,190
But not so in a beam.

1129
01:11:07,190 --> 01:11:09,810
If you did that in a beam,
if you come up and put

1130
01:11:09,810 --> 01:11:14,640
an impulse into a
beam, all that energy

1131
01:11:14,640 --> 01:11:16,720
would start out together.

1132
01:11:16,720 --> 01:11:20,082
But in very brief time, the
high frequency information

1133
01:11:20,082 --> 01:11:22,415
would get out in front of the
low frequency information.

1134
01:11:22,415 --> 01:11:26,570
And if you were way down this
beam, and somebody up a mile

1135
01:11:26,570 --> 01:11:29,660
away whacks one end, and
you're down further along,

1136
01:11:29,660 --> 01:11:32,550
you'll see high
frequency waves past you,

1137
01:11:32,550 --> 01:11:34,780
and then lower frequency,
and finally really slow

1138
01:11:34,780 --> 01:11:37,300
ones coming by, the
really long waves.

1139
01:11:37,300 --> 01:11:41,560
So that's called dispersion.

1140
01:11:41,560 --> 01:11:45,125
So beam waves are dispersive.

1141
01:11:49,860 --> 01:11:53,100
Things that obey the wave
equation are non-dispersive.

1142
01:11:53,100 --> 01:11:56,670
The energy all travels at
the same speed independent

1143
01:11:56,670 --> 01:11:58,270
of frequency.

1144
01:11:58,270 --> 01:12:03,700
All right, so that's
it for the term.

1145
01:12:03,700 --> 01:12:07,520
I'll see you guys
on next Wednesday.