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PROFESSOR: I want to use this
applet, Amplitude and Phase:

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First Order, to show you some
of the functionalities that

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are common to the MIT mathlets.

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First of all, if your browser
window looks like this,

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so that the label at
the top is too large,

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or like this, so that it's
missing some of the window,

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it's because you have the zoom
on your browser set wrong.

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And you can adjust it
with command or control

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plus or minus keys, so
that it looks correct.

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So let's see what we
have in this applet.

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At the top, you can see
the differential equation

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that this applet deals with.

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This differential
equation models

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many different
physical phenomena,

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and the story I want to tell
you involves the ocean, measured

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by this blue level
here, connected to a bay

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by this channel.

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And here's a measure of
the water level in the bay.

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As time goes on,
the level of water

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in the ocean changes
because of tides,

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and the water level
in the bay follows

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suit mediated by a slanting
level in the channel.

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I'm dragging this time slider
under the graphing window,

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and you can set it
wherever you want.

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In fact, if you want to
set it at exactly 12,

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because there's a hash mark
here, I can click on this 12,

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and the time slider
moves to that point.

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Or you can return time to
time zero by this arrow key,

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and then animate by pushing
this double right arrow key

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and watch the tide
move up and down.

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There's very carefully
designed color

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coding in all of these applets.

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In this case, blue
represents the ocean,

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and you can see that
it also represents

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the curve traced out by
the height of the ocean

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as time increases.

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And yellow represents the
water level in the bay.

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When I drag the cursor
over this graphing window,

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you can see a crosshair
forming, and below the window,

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you can see readout of time
and the x variable, vertical

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direction.

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You can use this to
make measurements.

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For example, I can see
that the tide in the bay

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seems to have maximal
height of 0.38.

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In many of these applets,
you can make measurements

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by using a rollover over
the graphing window.

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Let's see what else we
have in this applet.

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Down here, there's
a slider marked k.

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This is the coupling constant.

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It reflects the width of
the channel in the story

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that I'm telling you.

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When k is small, the
channel is very narrow

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and the ocean level
has very little effect

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on the water level in the bay.

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In fact, when k is equal to
zero, it has no effect at all.

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On the other hand,
when k becomes larger,

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the water level in the bay
tracks the ocean water level

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very closely.

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Let's set k to one, here.

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And think about another
thing you can vary.

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In reality, you can't change
the period of the tides.

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Even King Canute
couldn't do that.

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But here in this tool, we can.

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And I can vary the circular
frequency of this tidal input

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signal by changing the omega
slider over here on the right.

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I can make it small, and so the
period is very long, or larger

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and make the tides
happen faster.

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So we can animate this again.

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You can see things
happen faster.

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The curve is tighter, and
the effect is different.

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In fact, if we watch what
happens when I move omega

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from one down to
a smaller number,

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you can see the maximum
height of the tide changes.

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The amplitude of
the tide changes.

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It depends upon omega.

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In fact, one of the nice
things about these systems

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is that if you have a sinusoidal
input-- as we do here,

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this blue curve--
then the output signal

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is also going to be sinusoidal
and of the same frequency.

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So there's only
two things we need

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to know about the output
here, this yellow signal.

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Namely, the amplitude
and the phase

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lag behind the input
signal, the blue curve.

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And those two quantities can
be measured, can be graphed,

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against the input frequency,
and we can see those graphs

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by clicking on this check box
called Bode Plots down here.

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So this opens two new windows.

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The upper one
records the amplitude

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as a function of the
angular input frequency,

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and the lower one
records the number

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of degrees behind the input
frequency that the bay falls.

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So as omega changes, you can
see all these various things

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changing simultaneously.

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This is a characteristic
feature of these applets.

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The same information is recorded
in several different places

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and always connected
visually by placement.

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So you can see the
amplitude is 0.71 here.

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This yellow horizontal
line connects it

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with the maxima of
the output curve.

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And also by color.

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What else do we see here?

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There's a red line segment
here on this curve, which

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I can make a little bit bigger
if I decrease the k a bit

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and decrease the
frequency a little bit.

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What is this red curve
here, this red line here?

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Well, it connects to
this vertical strut

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which goes up to the
maximum of the output curve.

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And it begins at the
maximum of the input curve.

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In other words, it's
the amount of time

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that the output falls
behind the input.

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It's the time lag,
and that time lag

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is recorded
numerically down here.

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It's 1.96 in this example.

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There's one more
check box to explore

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called Nyquist Plot here.

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Let's click it and
see what happens.

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This opens a window
at the bottom here

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which records a complex number.

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That complex number contains
a magnitude and an angle.

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And the magnitude is the
amplitude of the system

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response, and the angle
represents the phase lag

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of the system response.

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You can see these things
changing together when

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I move the omega slider here.

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Watch both the windows above
the slider and the window

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below the slider, and you can
see that they change together.

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This bottom representation
is a complex number.

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It's very useful in
understanding the relationship

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between phase lag and
amplitude, and it also

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represents the way we
solve these differential

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equations in the Differential
Equations course at MIT.

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This is just a
beginning and indication

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of some of the functionalities
of these MIT mathlets.

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If you want to see
a list of them that

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are associated any
one of the applets,

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there's always a Help key in the
upper right hand corner which

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will open a page
that simply describes

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the functionalities present
in that particular applet.

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So have fun with these.

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You can play around with them.

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You can't break anything
by clicking buttons

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and experimenting with rollovers
and moving the sliders.