WEBVTT
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I'd like to talk.
Thank you.
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One of the things I'd like to
give a little insight into today
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is the mathematical basis for
hearing.
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For example,
if a musical tone,
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a pure musical tone would
consist of a pure oscillation in
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terms of the vibration of the
air.
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It would be a pure oscillation.
So, [SINGS],
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and if you superimpose upon
that, suppose you sing a triad,
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[SINGS], those are three tones.
Each has its own period of
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oscillation, and then another
one, which is the top one,
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which is even faster.
The higher it is,
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the faster the thing.
Anyway, what you hear,
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then, is the sum of those
things.
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So, C plus E plus G,
let's say, what you hear is the
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wave form.
It's periodics,
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still, but it's a mess.
I don't know,
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I can't draw it.
So, this is periodic,
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but a mess, some sort of mess.
Now, of course,
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if you hear the three tones
together, most people,
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if they are not tone deaf,
anyway, can hear the three
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tones that make up that.
So, in other words,
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if this is the function which
is the sum of those three,
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some sort of messy function,
f of t,
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you're able to do Fourier
analysis on it,
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and break it up.
You're able to take that f of
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t, and somehow mentally express
it as the sum of three pure
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oscillations.
That's Fourier analysis.
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We've been doing it with an
infinite series,
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but it's okay.
It's still Fourier analysis if
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you do it with just three.
So, in other words,
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the f of t is going to
be the sum of,
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let's say, sine,
I don't know,
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it's going to be the sign of
one frequency plus the sine of
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another frequency plus the sine
of a third, maybe with
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coefficients here.
So, somehow,
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since you were born,
you have been able to take the
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f of t,
and express it as the sum of
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the three signs.
And, here, therefore,
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the three tones that make up
the triad.
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Now, the question is,
how did you do that Fourier
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analysis?
In other words,
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does your brain have a little
integrator in it,
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which calculates the
coefficients of that series?
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Of course, the answer is no.
It has to do something else.
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So, one of the things I'd like
to aim at in this lecture is
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just briefly explaining what,
in fact, actually happens to do
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that.
Now, to do that,
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we'll have to make some little
detours, as always.
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So, first I'm going to,
throughout the lecture,
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in fact, I gave you last time a
couple of shortcuts for
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calculating Fourier series based
on evenness and oddness,
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and also some expansion of the
idea of Fourier series where we
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use the different,
but things didn't have to be
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periodic or period two pi,
but it can have an arbitrary
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period, 2L, and we could still
get a Fourier expansion for it.
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Let me, therefore,
begin just as a problem,
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another type of shortcut
exercise, to do a Fourier
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calculation, which we are going
to be later in the period to
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explain the music problem.
So, let's suppose we're
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starting with the function,
f of t,
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which is a real square wave,
and I'll make its period
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different from the one,
not two pi.
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So, suppose we had a function
like this.
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So, this is one,
and this is one.
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So, the height is one,
and this point is one as well.
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And then, it's periodic ever
after that.
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I'll tell you what,
let's do like the electrical
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engineers do and put these
vertical lines there even though
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they don't exist.
Okay, so the height is one and
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it goes over.
The half period is one.
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This really is a square wave.
I mean, it's really a square,
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not what they usually call a
square wave.
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So, my question is,
what's its Fourier series?
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Well, it's neither even nor
odd.
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That's a little dismaying.
It sounds like we're going to
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have to calculate an's and bn's.
So, the shortcuts I gave you
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last time don't seem to be
applicable.
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Now, of course,
nor is the period two pi,
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but that shouldn't be too bad.
In fact, you ought to look for
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an expansion in terms of things
that look like sine of n,
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well, what should it be?
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Since L is equal to one,
the half period is equal to
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one.
Remember, the period is 2L,
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not L.
It's n pi over L,
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but if L is one,
we should be looking for an
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expansion in terms of functions
that look like this.
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Now, since we've already done
the work for the official square
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way, which looks something like
this, what you always try to do
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is reduce these things to
problems that you've already
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solved.
This is a legitimate one,
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since I solved it in lecture
for you.
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So, we can consider it as
something we know.
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So, I observed that since I am
very lazy, that if I lower this
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function by one half,
it will become an odd function.
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Now it's an odd function.
Okay, I just cut the work in
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half.
So, let's call this function,
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let's call this,
I don't know,
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S of t.
The green one is the one we
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wanted to start with.
So, f of t is a green
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function.
But, I can improve things even
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more because the function that
we calculated in the lecture is
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a lot like this salmon function.
That's why I called it S.
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But, the difference is that the
function we calculated with this
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one.
In the first place,
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it went down further.
It went not to negative one
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half, which is where that one
goes.
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But, it went down to negative
one, and then went up here to
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plus one.
And, it went over to pi.
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So, it came down again,
but not, but at the point,
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pi.
And here, negative pi went up
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again.
Okay, let me remind you what
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this one was.
Suppose we call it,
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O doesn't look good,
I don't know,
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how about g of u?
Let's, for a secret reason,
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call the variable u this time,
okay?
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So, the previous knowledge that
I'm relying on was that I
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derived the Fourier series for
you by an orthodox calculation.
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And, it's not too hard to do
because this is an odd function.
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And therefore,
you only have to calculate the
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bn's.
And, half of them turn out to
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be zero, although you don't know
that in advance.
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But anyway, the answer was four
over pi times the sum
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of just the odd ones,
the sine of n u,
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and that you had to
divide by n.
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So, this is the expansion of g,
this function,
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g of u, the Fourier expansion
of this function.
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Since it's an odd function,
it only involves the signs.
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There's no funny stuff here
because the period is now two
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pi.
And, this came from the first
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lecture on Fourier series,
or from the book,
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wherever you want it,
or solutions to the notes.
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There are lots of sources for
that.
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The solution's in the notes.
Okay, now, that looks so much
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like the salmon function,
---
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-- I ought to be able to
convert one into the other.
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Now, I will do that by
shrinking the axis.
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But, since this can get rather
confusing, what I'll do is
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overlay this.
What I prefer to do is I think
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u, okay, I'm changing,
I'm keeping the thing the same.
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But, I'm going to change the
name of the variable,
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the t, in such a way that on
the t-axis, this becomes the
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point, one.
If I do that,
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then this function will turn
exactly into that one,
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except it will go not from
minus a half to a half,
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but it will go from negative
one to one, since I haven't done
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anything to the vertical axis.
So, how I do that?
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What's the relation between u
and t?
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Well, u is equal to pi times t,
or the other way around.
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You know that it's going to be
approximately this.
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Try one, and then check that it
works.
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When t is equal to one,
u is pi, which is what it's
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supposed to be.
So, this is the relation
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between the two.
And therefore,
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without further ado,
I can say that,
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let's write the relation
between them.
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f of t is what I want.
Well, what's f of t if I
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subtract one half of that?
So, that's going to be equal to
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the salmon function plus one
half, right, or the salmon
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function is f of t lowered by
one half.
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One thing is the same as the
other.
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And, what's the relation
between this salmon function and
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the orange function?
Well, the salmon function is,
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so, let's convert,
so, S of t -- it's more
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convenient, as I wrote the
formula g of u.
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Let's start it from that end.
If I start from g of u,
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what do I have to do to convert
it into S of-- into the salmon
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function?
Well, take one half of it.
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So, if I put them all together,
the conclusion is that f of t
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is equal to one half
plus S of t,
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which is one half of g of u,
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but u is pi t.
So, it's four pi,
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four over pi times the sum of
the sine of n.
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And, for u, I will write pi t
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divided by n.
And, sorry, I forgot to say
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that sum is only over the odd
values of n, not all values of
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n.
So, the sum over n odd of that,
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and, of course,
the two will cancel that.
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So, here we have,
in other words,
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just by this business of
shrinking or just stretching or
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shrinking the axis,
lowering it and squishing it
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that way a little bit.
We get from this Fourier
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series, we get that one just by
this geometric procedure.
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I'd like you to be able to do
that because it saves a lot of
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time.
Okay, so let's put this answer
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up in, I'm going to need it in a
minute, but I don't really want
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to recopy it.
So, let me handle it by
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erasing.
So, let's call that plus two
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over pi,
and there is our formula for
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that green function that we
wrote before.
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So, I'll put that in green.
So, we'll have a color-coded
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lecture again.
Now, what we're going to be
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doing ultimately,
to getting at the music problem
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that I posed at the beginning of
the lecture, is we want to
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solve, and this is what a study
of Fourier series has been
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aiming at, to solve second-order
linear equations with constant
00:14:04.000 --> 00:14:10.000
coefficients were the right-hand
side was a more general function
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than the kind we've been
handling.
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So, now, in order to simplify,
and we don't have a lot of time
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in the course,
I'd have to take another day to
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make more complicated
calculations,
00:14:23.000 --> 00:14:29.000
which I don't want to do since
you will learn a lot from them,
00:14:27.000 --> 00:14:33.000
anyway.
I think you will find you've
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had enough calculation by the
time Friday morning rolls
00:14:32.000 --> 00:14:38.000
around.
So, let's look at the undamped
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case, which is simpler,
or undamped spring,
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or undamped anything because it
doesn't have that extra term,
00:14:44.000 --> 00:14:50.000
which requires extra
calculations.
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So, I'll follow the book now
and some of the notes and the
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visuals, and called the
independent variable-- the
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dependent variable I'm going to
call x now.
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And, the independent variable
is, as usual,
00:15:01.000 --> 00:15:07.000
time.
So, this is going to be,
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in general, f of t,
and I'm going to use it by
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calculating example,
this is the actual f of t I'm
00:15:12.000 --> 00:15:18.000
going to be using.
But, the general problem for a
00:15:16.000 --> 00:15:22.000
general f of t is to solve this,
or at least to find a
00:15:20.000 --> 00:15:26.000
particular solution.
That's what most of the work
00:15:23.000 --> 00:15:29.000
is, because we already know how
from that to get the general
00:15:28.000 --> 00:15:34.000
solution by adding the solution
to the reduced equation,
00:15:32.000 --> 00:15:38.000
the associated homogeneous
equation.
00:15:36.000 --> 00:15:42.000
So, all our work has been,
this past couple of weeks,
00:15:40.000 --> 00:15:46.000
in how you find a particular
solution.
00:15:44.000 --> 00:15:50.000
Now, the case in which we know
what to do is,
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so we can find our particular
solution.
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Let's call that x sub p.
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We could find x sub p if the
right hand side is cosine omega,
00:16:01.000 --> 00:16:07.000
well, in general,
an exponential,
00:16:04.000 --> 00:16:10.000
but since we are not going to
use complex exponentials today,
00:16:09.000 --> 00:16:15.000
all these things are real.
And I'd like to keep them real.
00:16:16.000 --> 00:16:22.000
If it's either cosine omega t
or sine omega t,
00:16:20.000 --> 00:16:26.000
or some multiple of that by
00:16:22.000 --> 00:16:28.000
linearity, it's just as good.
We already know how to find the
00:16:26.000 --> 00:16:32.000
thing, and to find a particular
solution.
00:16:30.000 --> 00:16:36.000
So, the procedure is use
complex exponentials,
00:16:33.000 --> 00:16:39.000
and that magic formula I gave
you.
00:16:36.000 --> 00:16:42.000
But, right now,
just to save a little time,
00:16:39.000 --> 00:16:45.000
since I already did that on the
lecture on resonance,
00:16:43.000 --> 00:16:49.000
I solved it explicitly for
that, and you've had adequate
00:16:48.000 --> 00:16:54.000
practice I think in the problem
sets.
00:16:50.000 --> 00:16:56.000
Let's simply write down the
answer that comes out of that.
00:16:55.000 --> 00:17:01.000
The answer for the particular
solution is cosine omega t
00:16:59.000 --> 00:17:05.000
or sine omega t.
00:17:05.000 --> 00:17:11.000
That's the top.
And, it's over a constant.
00:17:08.000 --> 00:17:14.000
And, the constant is omega
naught squared.
00:17:13.000 --> 00:17:19.000
That's the natural frequency
which comes from the system,
00:17:19.000 --> 00:17:25.000
minus the imposed frequency,
the driving frequency that the
00:17:24.000 --> 00:17:30.000
system, the spring or whatever
it is, undamped spring,
00:17:29.000 --> 00:17:35.000
is being driven with.
Okay, understand the notation.
00:17:34.000 --> 00:17:40.000
Cosine this over that,
or sine, depending on whether
00:17:39.000 --> 00:17:45.000
you started driving it with
cosine or sine.
00:17:43.000 --> 00:17:49.000
So, this is from the lecture,
if you like,
00:17:46.000 --> 00:17:52.000
from the lecture on resonance,
but again it's,
00:17:50.000 --> 00:17:56.000
I hope by now,
a familiar fact.
00:17:53.000 --> 00:17:59.000
Let me remind you what this had
to do with resonance.
00:17:58.000 --> 00:18:04.000
Then, the observation was that
if omega, the driving frequency
00:18:03.000 --> 00:18:09.000
is very close to the natural
frequency, then this is close to
00:18:09.000 --> 00:18:15.000
that.
The denominator is almost zero,
00:18:13.000 --> 00:18:19.000
and that makes the amplitude of
the response very,
00:18:17.000 --> 00:18:23.000
very large.
And, that was the phenomenon of
00:18:20.000 --> 00:18:26.000
resonance.
Okay, now what I'd like to do
00:18:23.000 --> 00:18:29.000
is apply those formulas to
finding out what happens for a
00:18:28.000 --> 00:18:34.000
general f(t),
or in particular this one.
00:18:32.000 --> 00:18:38.000
So, in general,
I'll keep using the notation,
00:18:36.000 --> 00:18:42.000
f of t,
even though I've sorted used it
00:18:41.000 --> 00:18:47.000
for that.
But in general,
00:18:44.000 --> 00:18:50.000
what's the situation?
If f of t is a sine series,
00:18:49.000 --> 00:18:55.000
cosine series,
all right, let's do everything.
00:18:53.000 --> 00:18:59.000
Suppose it's,
in other words,
00:18:56.000 --> 00:19:02.000
the procedure is,
take your f of t,
00:19:00.000 --> 00:19:06.000
expand it in a Fourier series.
Well, doesn't that assume it's
00:19:07.000 --> 00:19:13.000
periodic?
Yes, sort of.
00:19:09.000 --> 00:19:15.000
So, suppose it's a Fourier
series.
00:19:12.000 --> 00:19:18.000
I'll make a very general
Fourier series,
00:19:15.000 --> 00:19:21.000
write it this way:
cosine (omega)n t,
00:19:18.000 --> 00:19:24.000
and then the sine terms,
00:19:22.000 --> 00:19:28.000
sine (omega)n t
from one to infinity where
00:19:27.000 --> 00:19:33.000
the omegas are,
omega n is short for that.
00:19:32.000 --> 00:19:38.000
Well, it's going to have the n
in it, of course,
00:19:35.000 --> 00:19:41.000
but I want, now,
to make the general period to
00:19:39.000 --> 00:19:45.000
be 2L.
So, it would be n pi over L.
00:19:42.000 --> 00:19:48.000
Of course, if L is equal to
00:19:45.000 --> 00:19:51.000
one, then it's n pi.
Or, if L equals pi,
00:19:48.000 --> 00:19:54.000
those are the two most popular
cases, by far.
00:19:52.000 --> 00:19:58.000
Then, it's simply n itself,
the driving frequency.
00:19:56.000 --> 00:20:02.000
But, this would be the general
case, n pi over L
00:20:01.000 --> 00:20:07.000
if the period is the period of f
of t is 2L.
00:20:07.000 --> 00:20:13.000
So, that's what the Fourier
series looks like.
00:20:10.000 --> 00:20:16.000
Okay, then the particular
solution will be what?
00:20:14.000 --> 00:20:20.000
Well, I got these formulas.
In other words,
00:20:18.000 --> 00:20:24.000
what I'm using is superposition
principle.
00:20:21.000 --> 00:20:27.000
If it's just this,
then I know what the answer is
00:20:25.000 --> 00:20:31.000
for the particular solution,
the response.
00:20:30.000 --> 00:20:36.000
So, if you make a sum of these
things, a sum of these inputs,
00:20:34.000 --> 00:20:40.000
you are going to get a sum of
the responses by superposition.
00:20:39.000 --> 00:20:45.000
So, let's write out the ones we
are absolutely certain of.
00:20:43.000 --> 00:20:49.000
What's the response to here?
Well, it's (a)n cosine omega n
00:20:47.000 --> 00:20:53.000
t. The only thing is,
00:20:51.000 --> 00:20:57.000
now it's divided by omega
naught squared.
00:20:55.000 --> 00:21:01.000
This constant has changed,
and the same thing here.
00:21:00.000 --> 00:21:06.000
Of course, by linearity,
if this is multiplied by a,
00:21:03.000 --> 00:21:09.000
then the answer is multiplied
by, the response is also
00:21:06.000 --> 00:21:12.000
multiplied by a.
So, the same thing happens
00:21:09.000 --> 00:21:15.000
here.
Here, it's (b)n and over,
00:21:11.000 --> 00:21:17.000
again, omega naught squared
minus omega times the sine of
00:21:14.000 --> 00:21:20.000
omega t.
00:21:17.000 --> 00:21:23.000
So, in other words,
as soon as you have the Fourier
00:21:20.000 --> 00:21:26.000
expansion, the Fourier series
for the input,
00:21:23.000 --> 00:21:29.000
you automatically get this by
just writing it down the Fourier
00:21:27.000 --> 00:21:33.000
series for the response.
That's the fundamental idea of
00:21:32.000 --> 00:21:38.000
Fourier series,
at least applied in this
00:21:35.000 --> 00:21:41.000
context.
They have many other contexts,
00:21:38.000 --> 00:21:44.000
approximations,
so on and so forth.
00:21:41.000 --> 00:21:47.000
But, that's the idea here.
All right, what about that
00:21:45.000 --> 00:21:51.000
constant term?
Well, this formula still works
00:21:49.000 --> 00:21:55.000
if omega equals zero.
If omega equals zero,
00:21:52.000 --> 00:21:58.000
then this is the constant,
one.
00:21:54.000 --> 00:22:00.000
The formula is still correct.
Omega is zero here.
00:22:00.000 --> 00:22:06.000
The only thing you have to
remember is that the original
00:22:03.000 --> 00:22:09.000
thing is written in this form.
So, the response will be,
00:22:07.000 --> 00:22:13.000
what will it be?
Well, it's one divided by omega
00:22:10.000 --> 00:22:16.000
naught squared,
if I'm in the case omega zero
00:22:12.000 --> 00:22:18.000
is equal to zero.
So, it's a zero divided by two
00:22:16.000 --> 00:22:22.000
omega naught squared.
And, as you will see,
00:22:19.000 --> 00:22:25.000
it looks just like the others.
You're just taking omega,
00:22:23.000 --> 00:22:29.000
and making it equal to zero for
that particular case.
00:22:26.000 --> 00:22:32.000
Sorry, this should be omega n's
all the way through here.
00:22:40.000 --> 00:22:46.000
All right, well,
let's apply this to the green
00:22:45.000 --> 00:22:51.000
function.
So, what have we got?
00:22:49.000 --> 00:22:55.000
We have its Fourier series.
So, if the green function is,
00:22:56.000 --> 00:23:02.000
if the input in other words is
this square wave,
00:23:02.000 --> 00:23:08.000
the green square wave,
so in your notes,
00:23:06.000 --> 00:23:12.000
this guy, this particular f of
t is the input.
00:23:15.000 --> 00:23:21.000
And, the equation is x double
prime plus omega naught squared
00:23:20.000 --> 00:23:26.000
x equals f of t.
00:23:23.000 --> 00:23:29.000
Then, the response is,
well, I can't draw you a
00:23:27.000 --> 00:23:33.000
picture of the response because
I don't know what the Fourier
00:23:32.000 --> 00:23:38.000
series actually looks like.
But, let's at least write down
00:23:37.000 --> 00:23:43.000
what the Fourier series is.
The Fourier series will be,
00:23:43.000 --> 00:23:49.000
well, what is it?
It's one half.
00:23:45.000 --> 00:23:51.000
The constant out front is one
half, except it's one over two
00:23:50.000 --> 00:23:56.000
omega naught squared.
00:23:53.000 --> 00:23:59.000
So, this is my function,
f of t.
00:23:56.000 --> 00:24:02.000
That's the general formula for
how the input is related to the
00:24:01.000 --> 00:24:07.000
response.
And, I'm applying it to this
00:24:06.000 --> 00:24:12.000
particular function,
f of t.
00:24:10.000 --> 00:24:16.000
And, the answer is plus.
Well, my Fourier series
00:24:15.000 --> 00:24:21.000
involves only odd sums,
only the summation over odd,
00:24:21.000 --> 00:24:27.000
and only of the sign.
So, it is going to be two over
00:24:26.000 --> 00:24:32.000
pi,
sorry, so it's going to be two
00:24:31.000 --> 00:24:37.000
over pi out front.
That constant will carry along
00:24:37.000 --> 00:24:43.000
by linearity.
And, I'm going to sum over odd,
00:24:39.000 --> 00:24:45.000
n odd values only.
The basic thing in the upstairs
00:24:43.000 --> 00:24:49.000
is going to be the sine of omega
n t.
00:24:47.000 --> 00:24:53.000
But, what is (omega)n?
Well, (omega)n is n pi.
00:24:50.000 --> 00:24:56.000
So, it's n pi t.
And, how about the bottom?
00:24:53.000 --> 00:24:59.000
The bottom is going to be omega
naught squared minus omega n
00:24:57.000 --> 00:25:03.000
squared.
00:25:01.000 --> 00:25:07.000
And, this is my (omega)n,
minus n pi squared.
00:25:05.000 --> 00:25:11.000
What's that?
00:25:07.000 --> 00:25:13.000
Well, I don't know.
All I could do would be to
00:25:12.000 --> 00:25:18.000
calculate it.
You could put it on MATLAB and
00:25:16.000 --> 00:25:22.000
ask MATLAB to calculate and plot
for you the first few terms,
00:25:22.000 --> 00:25:28.000
and get some vague idea of what
it looks like.
00:25:26.000 --> 00:25:32.000
That's nice,
but it's not what's interesting
00:25:31.000 --> 00:25:37.000
to do.
What's interesting to do is to
00:25:35.000 --> 00:25:41.000
look at the size of the
coefficients.
00:25:38.000 --> 00:25:44.000
And, again, rather than do it
in the abstract,
00:25:42.000 --> 00:25:48.000
let's take a specific value.
Let's suppose that the natural
00:25:46.000 --> 00:25:52.000
frequency of the system,
in other words,
00:25:50.000 --> 00:25:56.000
the frequency at which that
little spring wants to go
00:25:54.000 --> 00:26:00.000
vibrate back and forth,
whatever you got vibrating.
00:25:58.000 --> 00:26:04.000
Let's suppose the natural
frequency that's omega naught is
00:26:03.000 --> 00:26:09.000
ten for the sake of
definiteness,
00:26:05.000 --> 00:26:11.000
as they say.
Okay, if that's ten,
00:26:09.000 --> 00:26:15.000
all I want to do is calculate
in the crudest possible way what
00:26:15.000 --> 00:26:21.000
a few of these terms are.
So, the response is,
00:26:19.000 --> 00:26:25.000
so let's see,
we've got to give that a name.
00:26:23.000 --> 00:26:29.000
The response is (x)p of t.
00:26:26.000 --> 00:26:32.000
What's (x)p of t?
I'm just going to calculate it
00:26:31.000 --> 00:26:37.000
very approximately.
This means, you know,
00:26:35.000 --> 00:26:41.000
throwing caution to the winds
because I don't have a
00:26:39.000 --> 00:26:45.000
calculator with me.
And, I want you to look at this
00:26:43.000 --> 00:26:49.000
thing without a calculator.
The first term is one over 200.
00:26:47.000 --> 00:26:53.000
Okay, that's the only term I
00:26:50.000 --> 00:26:56.000
can get exactly right.
[LAUGHTER] Or,
00:26:52.000 --> 00:26:58.000
I could if I could calculate.
I suppose it's 0.005,
00:26:56.000 --> 00:27:02.000
right?
That's the constant term.
00:27:00.000 --> 00:27:06.000
Okay, so the next term,
let's see, two over pi is two
00:27:04.000 --> 00:27:10.000
thirds.
I'll keep that in mind,
00:27:07.000 --> 00:27:13.000
right?
Plus two thirds,
00:27:09.000 --> 00:27:15.000
0.6, let's say,
that's an indication of the
00:27:13.000 --> 00:27:19.000
accuracy with which these things
are going to be performed.
00:27:19.000 --> 00:27:25.000
I think in Texas for a long
while, the legislature declared
00:27:24.000 --> 00:27:30.000
pi to be three,
anyways.
00:27:27.000 --> 00:27:33.000
One of those states did it to
save calculation time.
00:27:31.000 --> 00:27:37.000
I'm not kidding,
by the way.
00:27:36.000 --> 00:27:42.000
All right, so what's the first
term?
00:27:38.000 --> 00:27:44.000
If n equals one,
I have the sine of pi t.
00:27:42.000 --> 00:27:48.000
That's the n equals one term.
00:27:46.000 --> 00:27:52.000
What's the denominator like?
That's about 100 minus 9
00:27:50.000 --> 00:27:56.000
squared. Let's say it's 91,
00:27:53.000 --> 00:27:59.000
sine t over 91.
What's the next term?
00:27:56.000 --> 00:28:02.000
Sine of three pi t, remember,
00:28:00.000 --> 00:28:06.000
I am omitting,
I'm only using the odd values
00:28:04.000 --> 00:28:10.000
of n because those are the only
ones that enter into the Fourier
00:28:09.000 --> 00:28:15.000
expansion for this function,
which is at the bottom of
00:28:14.000 --> 00:28:20.000
everything.
All right, what's the sine
00:28:19.000 --> 00:28:25.000
three pi t?
Well, now, I've got 100 minus
00:28:26.000 --> 00:28:32.000
three pi, --
-- that's 9 squared is 81.
00:28:32.000 --> 00:28:38.000
So, no, what am I doing?
So, we have 100 minus three
00:28:42.000 --> 00:28:48.000
times pi is 9,
squared.
00:28:46.000 --> 00:28:52.000
Well, let's say a little more.
Let's say 85.
00:28:53.000 --> 00:28:59.000
So, that's 15.
How bout the next one?
00:29:02.000 --> 00:29:08.000
Well, it's sine 5 pi t.
00:29:05.000 --> 00:29:11.000
I think I'll stop here as soon
as we do this one because at
00:29:09.000 --> 00:29:15.000
this point it's clear what's
happening.
00:29:12.000 --> 00:29:18.000
This is 100 squared minus,
that's 15 squared is 225,
00:29:16.000 --> 00:29:22.000
so that's about 125 with a
negative sign.
00:29:19.000 --> 00:29:25.000
So, minus this divided by 125.
And, after this they are going
00:29:24.000 --> 00:29:30.000
to get really quite small
because the next one will be
00:29:28.000 --> 00:29:34.000
seven pi squared.
That's 400, and this is
00:29:33.000 --> 00:29:39.000
becoming negligible.
So, what's happening?
00:29:38.000 --> 00:29:44.000
So, it's approximately,
in other words,
00:29:42.000 --> 00:29:48.000
0.005 plus the next coefficient
is, let's see,
00:29:48.000 --> 00:29:54.000
6/10, let's say 100,
sine pi t.
00:29:51.000 --> 00:29:57.000
And, what comes next?
Well, it's now 1/20th.
00:29:56.000 --> 00:30:02.000
It's about a 20th.
Let's call that 0.005 sine
00:30:03.000 --> 00:30:09.000
three pi t,
and now so small,
00:30:08.000 --> 00:30:14.000
minus 0.01, let's say times
this last one,
00:30:13.000 --> 00:30:19.000
sine 5 pi t.
What you find,
00:30:16.000 --> 00:30:22.000
in other words,
is that the frequencies which
00:30:22.000 --> 00:30:28.000
make up the response do not
occur with the same amplitude.
00:30:30.000 --> 00:30:36.000
What happens is that this
amplitude is roughly five times
00:30:35.000 --> 00:30:41.000
larger than any of the
neighboring ones.
00:30:38.000 --> 00:30:44.000
And after that,
it's a lot larger than the ones
00:30:43.000 --> 00:30:49.000
that come later.
In other words,
00:30:46.000 --> 00:30:52.000
the main frequency which occurs
in the response is the frequency
00:30:52.000 --> 00:30:58.000
three pi.
What's happened is,
00:30:54.000 --> 00:31:00.000
in other words,
near resonance has occurred.
00:31:00.000 --> 00:31:06.000
So, if omega is ten,
very near resonance,
00:31:04.000 --> 00:31:10.000
that is, it's not too close,
but it's not too far away
00:31:10.000 --> 00:31:16.000
either, occurs for the frequency
three pi in the input.
00:31:16.000 --> 00:31:22.000
Now, where's the frequency
three pi in the input?
00:31:22.000 --> 00:31:28.000
It isn't there.
It's just that green thing.
00:31:27.000 --> 00:31:33.000
Where in that is the frequency
three pi?
00:31:33.000 --> 00:31:39.000
I can't answer that for you,
but that's the function of
00:31:37.000 --> 00:31:43.000
Fourier series,
to say that you can decompose
00:31:41.000 --> 00:31:47.000
that green function into a sum
of frequencies,
00:31:45.000 --> 00:31:51.000
as it were, and the Fourier
coefficients tell you how much
00:31:50.000 --> 00:31:56.000
frequency goes into each of
those f of t's.
00:31:54.000 --> 00:32:00.000
Now, so, f of t is decomposed
into the sum of frequencies by
00:31:59.000 --> 00:32:05.000
the Fourier analysis.
But, the system isn't going to
00:32:04.000 --> 00:32:10.000
respond equally to all those
frequencies.
00:32:07.000 --> 00:32:13.000
It's going to pick out and
favor the one which is closest
00:32:12.000 --> 00:32:18.000
to its natural frequency.
So, what's happened,
00:32:15.000 --> 00:32:21.000
these frequencies,
the frequencies and their
00:32:19.000 --> 00:32:25.000
relative importance in f of t
are hidden,
00:32:23.000 --> 00:32:29.000
as it were.
They're hidden because we can't
00:32:26.000 --> 00:32:32.000
see them unless you do the
Fourier analysis,
00:32:30.000 --> 00:32:36.000
and look at the size of the
coefficients.
00:32:35.000 --> 00:32:41.000
But, the system can pick out.
The system picks out and
00:32:44.000 --> 00:32:50.000
favors, picks out for resonance,
or resonates with,
00:32:54.000 --> 00:33:00.000
resonates with the frequencies
closest to its natural
00:33:04.000 --> 00:33:10.000
frequency.
Well, suppose the system had
00:33:09.000 --> 00:33:15.000
natural frequency,
not ten.
00:33:11.000 --> 00:33:17.000
This is a put up job.
Suppose it had natural
00:33:14.000 --> 00:33:20.000
frequency five.
Well, in that case,
00:33:17.000 --> 00:33:23.000
none of them are close to the
hidden frequencies in f of t,
00:33:21.000 --> 00:33:27.000
and there would be no
resonance.
00:33:25.000 --> 00:33:31.000
But, because of the particular
value I gave here,
00:33:29.000 --> 00:33:35.000
I gave the value ten,
it's able to pick out n equals
00:33:33.000 --> 00:33:39.000
three as the most important,
the corresponding three pi as
00:33:37.000 --> 00:33:43.000
the most important frequency in
the input, and respond to that.
00:33:44.000 --> 00:33:50.000
Okay, so this is the way we
hear, give or take a few
00:33:48.000 --> 00:33:54.000
thousand pages.
So, what does the ear do?
00:33:51.000 --> 00:33:57.000
How does the ear,
so, it's got that thing,
00:33:55.000 --> 00:34:01.000
messy curve,
which I erased,
00:33:57.000 --> 00:34:03.000
which has a secret,
which just has three hidden
00:34:01.000 --> 00:34:07.000
frequencies.
Okay, from now on I hand wave,
00:34:05.000 --> 00:34:11.000
right, like they do in other
subjects.
00:34:07.000 --> 00:34:13.000
So, we got our frequency.
So, it's got a [SINGS].
00:34:11.000 --> 00:34:17.000
That's one frequency.
[SINGS] And,
00:34:13.000 --> 00:34:19.000
what goes in there is the sum
of those three,
00:34:16.000 --> 00:34:22.000
and the ear has to do something
to say out of all the
00:34:20.000 --> 00:34:26.000
frequencies in the world,
I'm going to respond to that
00:34:24.000 --> 00:34:30.000
one, that one,
and that one,
00:34:26.000 --> 00:34:32.000
and send a signal to the brain,
which the brain,
00:34:29.000 --> 00:34:35.000
then, will interpret as a
beautiful triad.
00:34:34.000 --> 00:34:40.000
Okay, so what happens is that
the ear, I don't talk
00:34:37.000 --> 00:34:43.000
physiology, and I never will
again.
00:34:39.000 --> 00:34:45.000
I know nothing about it,
but anyway, the ear,
00:34:43.000 --> 00:34:49.000
when you get far enough in
there, there are little three
00:34:46.000 --> 00:34:52.000
bones, bang, bang,
bang; this is the eardrum,
00:34:50.000 --> 00:34:56.000
and then there's the part which
has wax.
00:34:52.000 --> 00:34:58.000
Then, there's the eardrum which
vibrates, at least if there is
00:34:57.000 --> 00:35:03.000
not too much wax in your ear.
And then, the vibrations go
00:35:01.000 --> 00:35:07.000
through three little bones which
send the vibrations to the inner
00:35:05.000 --> 00:35:11.000
ear, which nobody ever sees.
And, the inner ear,
00:35:10.000 --> 00:35:16.000
then, is filled with thick
fluid and a membrane,
00:35:13.000 --> 00:35:19.000
and the last bone hits up
against the membrane,
00:35:16.000 --> 00:35:22.000
and the membrane vibrates.
And, that makes the fluid
00:35:20.000 --> 00:35:26.000
vibrate.
Okay, good.
00:35:21.000 --> 00:35:27.000
So, it's vibrating according to
the function f of t.
00:35:25.000 --> 00:35:31.000
Well, what then?
Well, that's the marvelous
00:35:28.000 --> 00:35:34.000
part.
It's almost impossible to
00:35:31.000 --> 00:35:37.000
believe, but there is this,
sort of like a snail thing
00:35:36.000 --> 00:35:42.000
inside.
I've forgotten the name.
00:35:38.000 --> 00:35:44.000
It's cochlea.
And, it has these hairs.
00:35:41.000 --> 00:35:47.000
They are not hairs really.
I don't know what else to call
00:35:45.000 --> 00:35:51.000
them. They're not hairs.
00:35:47.000 --> 00:35:53.000
But, there are things so long,
you know, they stick up.
00:35:52.000 --> 00:35:58.000
And, there are 20,000 of them.
And, they are of different
00:35:56.000 --> 00:36:02.000
lengths.
And, each one is tuned to a
00:35:59.000 --> 00:36:05.000
certain frequency.
Each one has a certain natural
00:36:05.000 --> 00:36:11.000
frequency, and they are all
different, and they are all
00:36:11.000 --> 00:36:17.000
graded, just like a bunch of
organ pipes.
00:36:16.000 --> 00:36:22.000
And, when that complicated wave
hits, the complicated wave hits,
00:36:23.000 --> 00:36:29.000
each one resonates to a hidden
frequency in the wave,
00:36:29.000 --> 00:36:35.000
which is closest to its natural
frequency.
00:36:35.000 --> 00:36:41.000
Now, most of them won't be
resonating at all.
00:36:37.000 --> 00:36:43.000
Only the ones close to the
frequency [SINGS],
00:36:40.000 --> 00:36:46.000
they'll resonate,
and the nearby guys will
00:36:43.000 --> 00:36:49.000
resonate, too,
because they will be nearby,
00:36:45.000 --> 00:36:51.000
almost have the same natural
frequency.
00:36:48.000 --> 00:36:54.000
And, over here,
there will be a few which
00:36:50.000 --> 00:36:56.000
resonate to [SINGS],
and finally over here a few
00:36:53.000 --> 00:36:59.000
which go [SINGS],
and each of those little hairs,
00:36:56.000 --> 00:37:02.000
little groups of hairs will
signal, send that signal to the
00:37:00.000 --> 00:37:06.000
auditory nerve somehow or other,
which will then carry these
00:37:03.000 --> 00:37:09.000
three inputs to the brain,
and the brain,
00:37:06.000 --> 00:37:12.000
then, will interpret that as
you are hearing [SINGS].
00:37:11.000 --> 00:37:17.000
So, the Fourier analysis is
done by resonance.
00:37:15.000 --> 00:37:21.000
You here resonance because each
of these things has a certain
00:37:21.000 --> 00:37:27.000
natural frequency which is able,
then, to pick out a resonant
00:37:27.000 --> 00:37:33.000
frequency in the input.
I'd like to finish our work on
00:37:32.000 --> 00:37:38.000
Fourier series.
So, for homework I'm asking you
00:37:35.000 --> 00:37:41.000
to do something similar.
Taken an input.
00:37:38.000 --> 00:37:44.000
I gave you a frequency here,
a different omega naught,
00:37:42.000 --> 00:37:48.000
a different input,
as you by means of this Fourier
00:37:46.000 --> 00:37:52.000
analysis to find out which it
will resonate,
00:37:50.000 --> 00:37:56.000
which of the hidden frequencies
in the input the system will
00:37:54.000 --> 00:38:00.000
resonate to, just so you can
work it out yourself and do it.
00:38:00.000 --> 00:38:06.000
Now, I'd like to first try to
match up what I just did by this
00:38:05.000 --> 00:38:11.000
formula with what's in your
book, since your book handles
00:38:10.000 --> 00:38:16.000
the identical problem but a
little differently,
00:38:14.000 --> 00:38:20.000
and it's essentially the same.
But I think I'd better say
00:38:19.000 --> 00:38:25.000
something about it.
So, the book's method,
00:38:22.000 --> 00:38:28.000
and to the extent which any of
these problems are worked out in
00:38:28.000 --> 00:38:34.000
the notes, the notes do this,
too.
00:38:32.000 --> 00:38:38.000
Use substitution.
Base uses differentiation of
00:38:36.000 --> 00:38:42.000
Fourier series term by term.
The work is almost exactly the
00:38:42.000 --> 00:38:48.000
same as here.
And, it has a slight advantage,
00:38:46.000 --> 00:38:52.000
that it allows you,
the book's method has a slight
00:38:51.000 --> 00:38:57.000
advantage that it allows you to
forget this formula.
00:38:56.000 --> 00:39:02.000
You don't have to know this
formula.
00:39:01.000 --> 00:39:07.000
It will come out in the wash.
Now, for some of you,
00:39:04.000 --> 00:39:10.000
that may be of colossal
importance, in which case,
00:39:08.000 --> 00:39:14.000
by all means,
use the book's method,
00:39:10.000 --> 00:39:16.000
term by term.
So, it requires no knowledge of
00:39:14.000 --> 00:39:20.000
this formula because after all,
I base this solution,
00:39:17.000 --> 00:39:23.000
I simply wrote down the
solution and I based it on the
00:39:21.000 --> 00:39:27.000
fact that I was able to write
down immediately the solution to
00:39:26.000 --> 00:39:32.000
this and put as being that
response.
00:39:30.000 --> 00:39:36.000
And for that,
I had to remember it,
00:39:32.000 --> 00:39:38.000
or be willing to use complex
exponentials quickly to remind
00:39:36.000 --> 00:39:42.000
myself.
There's very,
00:39:38.000 --> 00:39:44.000
very little difference between
the two.
00:39:41.000 --> 00:39:47.000
Even if you have to re-derive
that formula,
00:39:44.000 --> 00:39:50.000
the two take almost about the
same length of time.
00:39:48.000 --> 00:39:54.000
But anyway, the idea is simply
this.
00:39:50.000 --> 00:39:56.000
With the book,
you assume.
00:39:52.000 --> 00:39:58.000
In other words,
you take your function,
00:39:55.000 --> 00:40:01.000
f of t.
You expand it in a Fourier
00:39:58.000 --> 00:40:04.000
series.
Of course, which signs and
00:40:01.000 --> 00:40:07.000
cosines you use will depend upon
what the period is.
00:40:07.000 --> 00:40:13.000
So, you assume the solution of
the form-- Well,
00:40:10.000 --> 00:40:16.000
if I, for example,
carried out in this particular
00:40:14.000 --> 00:40:20.000
case, I don't know if I will do
all the work,
00:40:18.000 --> 00:40:24.000
but it would be natural to
assume a solution of the form,
00:40:22.000 --> 00:40:28.000
since the input looks like the
green guy.
00:40:26.000 --> 00:40:32.000
Assume a solution which looks
the same.
00:40:30.000 --> 00:40:36.000
In other words,
it will have a constant term
00:40:33.000 --> 00:40:39.000
because the input does.
But all the rest of the terms
00:40:38.000 --> 00:40:44.000
will be sines.
So, it will be something like
00:40:42.000 --> 00:40:48.000
(c)n times the sine of n pi t.
00:40:46.000 --> 00:40:52.000
The only question is,
what are the (c)n's?
00:40:50.000 --> 00:40:56.000
Well, I found one method up
there.
00:40:53.000 --> 00:40:59.000
But, the general method is just
plug-in.
00:40:56.000 --> 00:41:02.000
Substitute into the ODE.
Substitute into the ODE.
00:41:02.000 --> 00:41:08.000
You differentiate this twice to
do it.
00:41:04.000 --> 00:41:10.000
So, I'll do the double
differentiation and I won't stop
00:41:08.000 --> 00:41:14.000
the lecture there,
but I will stop the calculation
00:41:12.000 --> 00:41:18.000
there because it has nothing new
to offer.
00:41:15.000 --> 00:41:21.000
And, this is the way all the
calculations in the books and
00:41:19.000 --> 00:41:25.000
the solutions and the notes are
carried out.
00:41:22.000 --> 00:41:28.000
So, I don't think you'll have
any trouble.
00:41:25.000 --> 00:41:31.000
Well, this term vanishes.
This term becomes what?
00:41:30.000 --> 00:41:36.000
If I differentiate this twice,
I get summation,
00:41:33.000 --> 00:41:39.000
so, this is one to infinity
because I don't know which of
00:41:37.000 --> 00:41:43.000
these are actually going to
appear.
00:41:40.000 --> 00:41:46.000
Summation one to infinity,
(c)n times, well,
00:41:43.000 --> 00:41:49.000
if you differentiate the sine
twice, you get negative sine,
00:41:48.000 --> 00:41:54.000
right?
Do it once: you get cosine.
00:41:50.000 --> 00:41:56.000
Second time:
you get negative sine.
00:41:53.000 --> 00:41:59.000
But, each time you will get
this extra factor n pi from the
00:41:57.000 --> 00:42:03.000
chain rule.
And so, the answer will be
00:42:00.000 --> 00:42:06.000
negative (c)n times n pi squared
times the sine of n pi t.
00:42:05.000 --> 00:42:11.000
And so, the procedure is,
00:42:10.000 --> 00:42:16.000
very simply,
you substitute (x)p double
00:42:13.000 --> 00:42:19.000
prime into the differential
equation.
00:42:16.000 --> 00:42:22.000
In other words,
if you do it,
00:42:17.000 --> 00:42:23.000
we will multiply this by omega
naught squared.
00:42:22.000 --> 00:42:28.000
And, you add them.
And then, on the left-hand
00:42:25.000 --> 00:42:31.000
side, you are going to get a sum
of terms, sine n pi t
00:42:29.000 --> 00:42:35.000
times coefficients
involving the (c)n's.
00:42:34.000 --> 00:42:40.000
And, on the right,
so, you're going to get a sum
00:42:37.000 --> 00:42:43.000
involving the (c)n's,
and the sines n pi t,
00:42:41.000 --> 00:42:47.000
and on the right,
you're going to get the Fourier
00:42:44.000 --> 00:42:50.000
series for f of t,
which is exactly the same kind
00:42:49.000 --> 00:42:55.000
of expression.
The only difference is,
00:42:52.000 --> 00:42:58.000
now the sines have come with
definite coefficients.
00:42:56.000 --> 00:43:02.000
And then, you simply click the
coefficients on the left and the
00:43:01.000 --> 00:43:07.000
coefficients on the right,
and figure out what the (c)n's
00:43:05.000 --> 00:43:11.000
are.
So, by equating coefficients,
00:43:10.000 --> 00:43:16.000
you get the (c)n's.
Would you like me to carry it
00:43:15.000 --> 00:43:21.000
out?
Yeah, okay, I was going to do
00:43:19.000 --> 00:43:25.000
something else,
but I wouldn't have time to do
00:43:24.000 --> 00:43:30.000
it anyway.
So, why don't I take two
00:43:28.000 --> 00:43:34.000
minutes to complete the
calculation just so you can see
00:43:34.000 --> 00:43:40.000
you get the same answer?
All right, what do we get?
00:43:40.000 --> 00:43:46.000
If you add them up,
you get c naught,
00:43:43.000 --> 00:43:49.000
out front, plus (c)n is
multiplied by what?
00:43:47.000 --> 00:43:53.000
Well, from the top it's
multiplied by omega naught
00:43:52.000 --> 00:43:58.000
squared.
On the bottom,
00:43:55.000 --> 00:44:01.000
it's multiplied by n
pi squared.
00:44:00.000 --> 00:44:06.000
Ah-ha, where have I seen that
combination?
00:44:05.000 --> 00:44:11.000
The sum is equal to,
sorry, one half plus what is
00:44:15.000 --> 00:44:21.000
it, sum over n odd of sine n pi
t over n.
00:44:29.000 --> 00:44:35.000
So, the conclusion is that--
I'm sorry, it should be c naught
00:44:34.000 --> 00:44:40.000
times omega naught squared.
00:44:38.000 --> 00:44:44.000
So, what's the conclusion?
If c zero is one over two omega
00:44:44.000 --> 00:44:50.000
naught squared,
00:44:48.000 --> 00:44:54.000
and that (c)n,
only for n odd,
00:44:51.000 --> 00:44:57.000
the others will be even.
The others will be zero.
00:44:55.000 --> 00:45:01.000
The (c)n is going to be equal
to two over pi here.
00:45:01.000 --> 00:45:07.000
So, it's going to be two pi,
00:45:04.000 --> 00:45:10.000
two over pi times one over n
times one over omega naught
00:45:10.000 --> 00:45:16.000
squared minus n over
pi squared.
00:45:15.000 --> 00:45:21.000
This is terrible,
00:45:20.000 --> 00:45:26.000
which is the same answer we got
before, I hope.
00:45:26.000 --> 00:45:32.000
Did I cover it up?
Same answer.
00:45:30.000 --> 00:45:36.000
So, that answer at the
left-hand end of the board is
00:45:35.000 --> 00:45:41.000
the same one.
I've calculated,
00:45:38.000 --> 00:45:44.000
in other words,
what the c zeros are.
00:45:42.000 --> 00:45:48.000
And, I got the same answer as
before.