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In the last several videos, we
assumed that the velocity of

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the source, of the object that's
emitting the wave,

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we've assumed that that's less
than the velocity of the wave.

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And we saw what happens
with the Doppler

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effect and all of that.

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But what I want to do in this
video is not make this

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assumption.

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In particular, let's see what
happens, at least, at first

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for our formulas, and then get
a conceptual understanding.

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Let's see what happens when the
velocity of the source is

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equal to the velocity
of the wave.

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The first thing we might try to
do is just apply this new

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assumption into the formulas
that we had in the last video.

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And those formulas were here.

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These are the observed period
and frequency for an observer

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that's in the direction
of the object.

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And if we make this assumption,
that the velocity

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of the sound-- and the velocity
of the source-- we're

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not necessarily dealing with
soundwaves, although that

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might be an easy visualization
for you.

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That tends to be how
I visualize it.

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But what happens to these
formulas when the velocity of

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the source is equal to the
velocity of the wave?

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If these two quantities are
equal up here, you have

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something, subtract the
same thing from it.

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This numerator right
here becomes 0.

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So it'll turn this whole
thing into being 0.

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So the period, or the observed
period, will be 0, which means

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you don't have to wait
any time at all

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between successive crests.

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The entire waveform just gets
infinitely bunched together.

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So it's kind of like
one impulse.

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And if we look at the frequency,
we can either look

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directly at the formula, and
you'll see, you have something

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divided by 0 right here.

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So you can say this is 1 over 0,
or you could just say that

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the frequency is 1 over the
period, and you get this thing

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that's undefined.

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But if you want to think about,
what does the frequency

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approach as the velocity of
the source approaches the

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velocity of the wave, if this
thing is only a little bit

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less that thing, this is going
to be a very, very, very

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small, very, very small
positive number.

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So when you divide these
quantities by that very, very,

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very small positive
number, you're

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going to approach infinity.

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So the frequency is undefined at
the speed, at the velocity

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of the wave, but it's going
to approach infinity.

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It will approach infinity as
the source approaches the

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velocity of the wave. Not
necessarily a soundwave.

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I keep using soundwaves, because
that's how I tend to

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visualize things.

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And we'll talk in future videos
specifically about

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soundwaves.

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And we'll touch on it a little
bit in this video.

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So what is this telling us?

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Does this make any sense?

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And if you think about it, at
least to me, it starts to make

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sense just what you saw in the
last couple of videos.

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The last couple of videos,
when something was moving

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slower than the speed
of sound, you had,

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OK, I'm here now.

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And I'm about to release the
next crest. If I go one period

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ago, maybe I was right there.

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And the crest that I had
released at that time period

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maybe has traveled this
far, just like that.

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If you go a period before
that, I would

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have been over there.

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And crest that I released then
would have traveled that far.

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We saw this in the
last two videos.

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And if you go the period
before that, I

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would have been there.

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And the crest that I would
have released would

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have been that far.

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This was the whole reason why
the Doppler effect happens.

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Because the observer sitting
right here-- let me do this in

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a separate color-- the observer
sitting right here is

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going to experience these crests
more frequently than an

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observer sitting out here.

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Because the wavelength gets
compressed, because every time

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this guy releases a new
crest, or a new cycle,

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he has moved forward.

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He's moved forward in the
direction of this

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motion right here.

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So let's think about what
happens when he is exactly

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moving at the speed of the wave.
So let's say that the

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source is here now.

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This is right where he is.

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And he's right about to release
a new crest. So where

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was he at one period ago?

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So let's say he was here
one period ago.

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So one period ago, if he's right
going to release crest

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or cycle right now,
one period ago, he

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released another cycle.

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And where has that cycle gone?

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Well, we're assuming that the
wave is traveling at the same

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velocity as this guy.

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But it's going radially
outward.

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So whatever he released then,
it will have traveled at the

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same velocity as himself.

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So it will have gotten
this far.

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He released it one period ago,
and that's where he was one

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period ago.

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Over the course of the next
period, he traveled there and

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so did the wave. The wave
also traveled there.

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Now, where was this character
two periods ago?

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When I talk about the period,
I'm talking about the actual

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period of the wave. Every
period, or how long does it

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take between similar points
in the cycle?

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And I like to think of them as
the crests in the cycle.

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So two periods ago,
he was here.

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And he released a-- you can
imagine, a pulse, or crest.

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And where will that be now?

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Well that will have traveled
as far as he did.

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He traveled that far,
and so will the

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pulse that he released.

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It will have traveled-- actually
I have to make it a

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little bit more symmetric.

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It will have traveled
that far.

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And if you go three periods ago,
I think you get the idea.

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If you go three periods
ago, he was here.

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And he released a pulse then,
or crest, or a cycle of the

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wave. And where will
that be now?

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Well, it will have traveled
as fast as he's gotten.

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So it will have gotten
this far.

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Of course, it's traveling that
velocity in every direction,

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radially outward.

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Now think about the situation
for the observer.

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Think, in particular, about
the observer who's sitting

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right here.

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Let's say he's just out of the
way so that this thing doesn't

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run into him and kill him,
and what whatever else.

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But he's just out of the way,
just enough to experience the

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sound, but not directly collide
with this object

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that's emitting the-- well, I
shouldn't say sound-- emitting

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the wave. I want to be
general right here.

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We're not assuming that
this is a soundwave.

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What's he going to experience?

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Well, he's not going to-- well,
if we assume this is a

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soundwave, he's not going to
hear anything until the thing

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passes right there.

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And right when the thing passes,
it has all of the

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sound that it generated coming
with it at exactly

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that moment in time.

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Instead of hearing things
periodically, all of the wave

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fronts smack this
guy all at once.

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And perceptually, instead of
hearing a pitch, because

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you're hearing something
periodic, you're just going to

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hear a big thump, because all
of that sound energy is

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getting to you at
the same time.

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You're just going
to hear a thump.

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Because it's no longer
really a frequency.

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All of the energy is coming
to you at the same time.

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And when you are talking about
sound, specifically, and

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especially, when you are
trans-sonic, which means

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you're around the speed of
sound, or parts of you are

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above or below the speed of
sound, and you move into

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supersonic speeds, that's
what people relate

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to the sonic boom.

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And we're going to talk a little
bit more about that in

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the next video and mach numbers,
all that, because I

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just find all of that
fascinating.

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But I think this is intuitive.

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Because when you look at this,
everything is just reaching

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you at exactly the same time.

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And this was the case of
soundwaves, but it would be

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true of any type of waves.

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