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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.