WEBVTT
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GILBERT STRANG: OK.
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This video is about the
third of the great trio
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of partial
differential equations.
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Laplace's equation
was number one.
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That's called an
elliptic equation.
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The heat equation
was number two.
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That's called a
parabolic equation.
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Now we reach the wave equation.
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That's number three, and it's
called a hyperbolic equation.
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So somehow the three
equations remind us
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of ellipses, parabolas,
and hyperbolas.
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They have different
types of solutions.
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Laplace's equation, you
solve it inside a circle
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or inside some closed region.
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The heat equation and
the wave equation, time
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enters, and you're
going forward in time.
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The heat equation is first
order in time, du dt.
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And the wave equation, the
full-scale wave equation,
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is second order in time.
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That stands for the second
derivative, d second u
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dt squared.
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And it matches the
second derivative
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in space with a velocity
coefficient c squared.
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I'm in one-dimensional space.
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If I were in three
dimensions, where we really
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have sound waves and light
waves and all the most important
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things in life, then there
would be a uxx and a uyy
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and a uzz, second derivatives
in all the space directions.
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But that's good enough to do 1D.
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So what are the
differences, first of all,
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between the heat equation
and wave equation?
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So I'll say heat
versus wave equations.
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What are the sort of
biggest differences?
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The heat, the signal
travels infinitely fast.
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Under the wave
equation, the signal
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travels with finite
velocity, and that velocity
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is that number c with speed c.
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So fortunately, for sound
waves, the wave comes to us,
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or when I'm speaking
to you, my voice
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is traveling out
to the microphone
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at the speed of
sound, the speed c.
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And actually, another good
thing is after it gets there,
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it goes on, and it
goes by the point
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and doesn't just stay there
messing up the future sounds.
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It travels.
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It dies out.
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Whereas-- well, the heat.
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So let me try to
give an example.
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Suppose the initial condition
is a delta function.
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So if u equals a delta
function, a point source,
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which is quite normal.
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A point source of heat,
something really hot,
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or a point source
of sound, my voice.
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So it's that at t equals 0.
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Then for the heat
equation, there's
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a famous solution to
the heat equation.
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Remember, the heat equation
is du dt equal uxx,
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first derivative in time.
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And the solution that starts
from the delta function?
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Oh, do I know what it is?
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I think there's a 1 over--
there's a square root of 4
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pi t.
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There's an e to the
minus x squared over 4t.
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I think perhaps that's it.
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Perhaps that's it.
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So what do I see from that?
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I see big damping out.
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I see immediate travel.
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As soon as time is just
a little beyond 0, then
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for every x we get an answer,
but it's an extremely,
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extremely small answer.
e to the minus x squared
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is tailing off to
0 incredibly fast.
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So that's a very small answer.
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Very little of the heat
immediately gets very,
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very quickly across the ocean.
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But in theory, a
little bit does.
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Whereas for the
wave equation, it
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takes time to cross the ocean.
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We have a tsunami.
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We have a wave.
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It gets there.
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It reaches the other side.
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And actually, at the--
it's very important.
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What is the speed of
that wave to tell people
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about a tsunami that's
coming, and you can actually
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do it, which you couldn't
do for the heat equation.
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So for the wave
equation, what comes out
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of a delta function in 1D?
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Well, a wave goes to the right,
and a wave goes to the left.
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That's what happens.
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And those waves are 1/2 of
a delta function each way.
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So the solution is 1/2
of a delta function
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that's traveling.
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I see that-- let me write
down the other half that's
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traveling the other
way-- delta at x plus ct.
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That's a cool solution.
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So that means the
sound in 1D, the sound,
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half of it takes
off in one direction
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and half in the other direction.
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And what happens in each
direction is a spike of sound.
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You don't hear anything,
then at a particular time,
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depending on your
position x, there's
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a particular time when
you get 0 in there,
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and you hear the signal.
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And then as soon as time goes
past that, it's past you.
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So you get a big shock, and
it comes to you with speed c.
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If you look at that
expression x minus ct,
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it's telling you that
the speed of the wave
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in dx dt for the wave is c.
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OK.
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So that's a contrast for a very
particular initial condition--
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a big wall of water, a
big noise, a big bang.
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OK.
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I want to solve
the wave equation,
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study it further for
other initial conditions.
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And of course, initial
condition's plural
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because the wave equation
is second order in time.
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So I'm given u at t equals
0 and all x and du dt.
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I'm given an
initial distribution
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of the wall of
water, shall we say,
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and its velocity,
the normal thing.
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When I have a
second-order equation,
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I'm given an initial condition
and an initial velocity.
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And of course, because it's a
partial differential equation,
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I'm given those for every x.
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So we have functions
instead of just two numbers,
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and that's where a Fourier
series can come in.
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So we can solve that
by a Fourier series
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if we're on a finite--
like a violin string.
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You've plucked a
violin string that
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starts waves going back
and forth in the string.
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They solve the wave equation.
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You hear music, great
music if it's a good sound.
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Or we could solve it on an
infinite line with no boundary,
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as in an essentially infinite
ocean or waves in space.
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Light waves in space are
solving the wave equation
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with no boundary,
as far as we know.
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OK.
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So which shall I do?
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I'll write the solution
down in free space,
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and then I'll write one
down for a violin string.
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So in space-- well, it's a
one-dimensional space here.
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So this would be minus infinity
less than x less than infinity.
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Then what does the solution to
the wave equation look like?
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It looks like some function
of plus some function
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of x plus ct.
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Well, that's exactly the
form that we had here
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when they were delta functions.
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Here, in general, they don't
have to be delta functions.
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I chose that function f and
that function g so that at t
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equals 0, I'm good.
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So at t equals 0--
so I set t equals 0.
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At t equals 0, u of 0 and x
would be f of x plus g of x.
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Good.
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But that's only one condition,
and I have f and g to find.
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So I would also use
du dt at the start.
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What's the time derivative?
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It's minus c f prime at x.
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And the time derivative of this
will be a plus c g prime at x.
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OK.
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No big deal.
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I'm given two functions.
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I've got two functions
to find in the answer,
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and I've got two equations.
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I could solve those for f
and g, and it gives a formula
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called d'Alembert's formula,
named after the person who
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put this together.
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I'd rather go on to the
violin, to a finite string.
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OK.
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So now I have a
finite string, and I'm
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holding u equals 0 at the ends.
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OK.
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And my solution will
still be functions that
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depend on x minus ct waves.
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My solution will still be waves.
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The solution that I'm
going to write down
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comes from a very,
very important method
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called separation of variables.
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I want to separate x from t.
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I need to give you a full video
about separation of variables.
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That's the best tool we
have to get solutions
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to lots of equations.
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So let me just jump to the
form of the solution here.
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I'm imagining the violin
string is at rest.
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So it starts with--
I'm imagining,
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let's say in our
particular problem,
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this is 0, the initial velocity.
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This is plucked.
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The starting position
is-- with your finger
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you've moved it off 0.
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OK.
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What do the solutions
look like, u of t and x?
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OK.
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They will be a sum of neat,
special, convenient, separated
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solutions, with separated
meaning t separated from x.
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And I think we would
have cosine of nct.
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I'm going to have a sum.
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Oh, we'll have a coefficient,
of course, a sub-- well, or b,
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or c, or even d.
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How about d for a
new letter-- d sub n.
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And I think-- well,
let me finish here.
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I'm separating that from
I think it would probably
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be the sine of nx.
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OK.
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If we just look at
that for a little bit,
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that's the point of this video.
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So this is x equals 0.
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This is, say, x equal pi.
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That's for convenience.
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Suppose our violin
string has length pi,
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then, I think, that's what
the solution looks like.
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It starts from at t equals 0.
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At t equals 0, the cosine is 1.
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So at t equals 0, this
is an initial condition
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that we have to match.
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This'll tell us the d's will
be the sum of dn sine nx.
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That tells us the d's.
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And then we have our answer.
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So the initial
condition-- remember,
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it's starting at rest.
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So the initial velocity
is 0, and that's
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why I don't have any sine
t's because I'm starting
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with initial velocity 0.
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I only have cosines
in the t direction.
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But I only have sines
in the x direction
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because the violin
string is being held down
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at the two ends, and
it's the sine function
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that matches that perfectly.
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So this is separation of
variables, t separated from x.
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And I really have to
do a proper explanation
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of that highly important method
of getting solutions like this.
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Thank you.