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I've got this source of a wave
right here that's moving to

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the right at some velocity.

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So let's just say that the
velocity of the source-- let's

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call it v sub s to the right--
so we're really going to do

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what we do in the last video,
but we're going to do it in

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more abstract terms so we can
come up with a generalized

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formula for the observed
frequency.

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So that's how fast he's moving
to the right, and he's

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emitting a wave. Let's say the
wave that he's emitting-- so

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the velocity of wave--
let's call that v

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sub w radially outward.

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We've got to give a magnitude
and a direction.

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So radially outward.

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That's the velocity of the wave,
and that wave is going

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to have a period and a
frequency, but it's going to

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have a period and a frequency
associated from the point of

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view of the source.

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And we're going to
do everything.

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This is all classical
mechanics.

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We're not going to be talking
about relativistic speed, so

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we don't have to worry about all
of the strange things that

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happen as things approach
the speed of light.

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So let's just say it has
some period of-- let me

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write it this way.

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The source period, which is the
period of the wave from

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the perspective of the source,
so the source period, we'll

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call it t sub source, And the
source frequency, which would

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just be-- we've learned,
hopefully it's intuitive now--

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would be the inverse of this.

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So the source frequency would
be-- we'll call it f sub s.

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And these two things are the
inverse of each other.

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The inverse of the period
of a wave is its

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frequency, vice versa.

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So let's think about what's
going to happen.

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Let's say at time equal zero,
he emits that first crest,

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that first pulse, so he's
just emitted it.

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You can't even see it because
it just got emitted.

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And now let's fast forward
t seconds.

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Let's say that this is in
seconds, so every t seconds,

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it emits a new pulse.

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First of all, where is
that first pulse

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after t sub s seconds?

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Well, you multiply the velocity
of that first pulse

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times the time.

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Velocity times time is going
to give you a distance.

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If you don't believe me, I'll
show you an example.

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If I tell you the velocity is 5
meters per second, and let's

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say that this period is 2
seconds, that's going to give

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you 10 meters.

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The seconds cancel out.

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So to figure out how far that
wave will have gone after t

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sub s seconds, you just multiply
t sub s times the

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velocity of the wave.

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And let's say it's
gotten over here.

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It's radially outward.

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So, I'll draw it radially
outward.

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That's my best attempt
at a circle.

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And this distance right here,
this radius right there, that

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is equal to velocity
times time.

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The velocity of that first
pulse, v sub w, that's

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actually the speed.

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I'm saying it's v sub
w radially outward.

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This isn't a vector quantity.

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This is just a number
you can imagine.

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v sub w times the period,
times t of s.

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I know it's abstract, but just
think, this is just the

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distance times the time.

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If this was moving at 10 meters
per second and if the

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period is 2 seconds,
this is how far.

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It will have gone 10 meters
after 2 seconds.

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Now, this thing we said
at the beginning of

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the video is moving.

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So although this is radially
outward from the point at

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which it was emitted, this thing
isn't standing still.

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We saw this in the last video.

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This thing has also moved.

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How far?

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Well, we do the same thing.

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We multiply its velocity times
the same number of time.

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Remember, we're saying what does
this look like after t

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sub s seconds, or some period
of time t sub s.

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Well, this thing is moving
to the right.

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Let's say it's here.

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Let's say it's moved
right over here.

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In this video, we're assuming
that the velocity of our

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source is strictly less than the
velocity of the wave. Some

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pretty interesting things happen
right when they're

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equal, and, obviously, when
it goes the other way.

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But we're going to assume that
it's strictly less than.

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The source is traveling slower
than the actual wave.

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But what is this distance?

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Remember, we're talking
about-- let me do it

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in orange as well.

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This orange reality is what's
happened after t sub s

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seconds, you can say.

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So this distance right here.

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That distance right there--
I'll do it in a different

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color-- is going to be the
velocity of the source.

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It's going to be v sub
s times the amount of

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time that's gone by.

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And I said at the beginning,
that amount of time is the

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period of the wave. That's
the time in question.

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So period of the wave t sub s.

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So after one period of the wave,
if that's 5 seconds,

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then we'll say, after 5 seconds,
the source has moved

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this far, v sub s times t sub s,
and that first crest of our

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wave has moved that far,
V sub w times t sub s.

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Now, the time that we're talking
about, that's the

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period of the wave
being emitted.

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So exactly after that amount of
time, this guy is ready to

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emit the next crest.
He has gone

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through exactly one cycle.

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So he is going to emit
something right now.

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So it's just getting emitted
right at that point.

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So what is the distance between
the crest that he

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emitted t sub s seconds ago or
hours ago or microseconds ago,

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we don't know.

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What's the distance between this
crest and the one that

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he's just emitting?

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Well, they're going to move at
the same velocity, but this

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guy is already out here, while
this guy is starting off from

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the source's position.

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So the difference in their
distance, at least when you

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look at it this way, is the
distance between the source

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here and this crest.

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So what is this distance
right here?

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What is that distance
right there?

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Well, this whole radial
distance, we already said,

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this whole radial distance is
v sub w, the velocity of the

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wave, times the period of the
wave from the perspective of

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the source, and we're going to
subtract out how far the

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source itself has moved.

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The source has moved in the
direction, in this case, if

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we're looking at it from
this point of view,

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of that wave front.

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So it's going to be minus v
sub s, the velocity of the

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source, times the period of the
wave from the perspective

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of the source.

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So let me ask you a question.

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If you're sitting right here, if
you're the observer, you're

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this guy right here, you're
sitting right over there, and

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you've just had that first
crest, at that exact moment

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that first crest has passed you
by, how long are you going

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to have to wait for
the next crest?

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How long until this one that
this guy's emitting right now

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is going to pass you by?

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Well, it's going to have
to cover this distance.

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It's going to have to
cover that distance.

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Let me write this down.

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So the question I'm asking is
what is the period from the

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point of view of this observer
that's right in the direction

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of the movement of the source?

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So the period from the point
of view of the observer is

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going to be equal to the
distance that the next pulse

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has to travel, which is that
business up there.

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So let me copy and paste that.

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So it's going to be that.

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Let me get rid of that.

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It shouldn't look like an equal
sign, so I can delete

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that right over there.

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Or a negative sign.

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So it's going to be this
distance that the next pulse

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is going to travel, that one
that's going to be emitted

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right at that moment, divided by
the speed of that pulse, or

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the speed of the wave, or the
velocity the wave, and we know

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what that is.

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That is v sub w.

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Now this gives us the period
of the observation.

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00:08:01,600 --> 00:08:01,770
Now.

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If we wanted the frequency-- and
we can manipulate this a

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little bit.

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Let's do that a little bit.

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So we can also write this.

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We could factor out the
period of the source.

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00:08:12,390 --> 00:08:14,630
So t sub s we could
factor out.

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00:08:14,630 --> 00:08:20,440
So it becomes t sub s times the
velocity of the wave minus

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00:08:20,440 --> 00:08:26,690
the velocity of the source, all
of that over the velocity

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00:08:26,690 --> 00:08:30,230
of the wave. And so just like
that, we've gotten our formula

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for the observed period for this
observer who's sitting

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00:08:33,308 --> 00:08:38,270
right in the path of this moving
object as a function of

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00:08:38,270 --> 00:08:42,270
the actual period of this wave
source, the wave's velocity

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00:08:42,270 --> 00:08:44,930
and the velocity
of the source.

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00:08:44,930 --> 00:08:46,670
Now, if we wanted the frequency,
we just take the

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00:08:46,670 --> 00:08:48,140
inverse of this.

185
00:08:48,140 --> 00:08:49,340
So let's do that.

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00:08:49,340 --> 00:08:52,640
So the frequency of the
observer-- so this is how many

187
00:08:52,640 --> 00:08:54,630
seconds it takes for him
to see the next cycle.

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00:08:54,630 --> 00:08:57,110
If you want cycles per second,
you take the inverse.

189
00:08:57,110 --> 00:08:58,880
So the frequency of the observer
is just going to be

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00:08:58,880 --> 00:08:59,600
the inverse of this.

191
00:08:59,600 --> 00:09:02,060
So if we take the inverse of
this whole expression, we're

192
00:09:02,060 --> 00:09:07,800
going to get 1 over t sub
s times v sub w over the

193
00:09:07,800 --> 00:09:10,840
velocity of the wave minus
the velocity the source.

194
00:09:10,840 --> 00:09:13,740
And of course, 1 over the period
from the point of view

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00:09:13,740 --> 00:09:16,580
of the source, this
is the same thing.

196
00:09:16,580 --> 00:09:18,840
This right here is the
same thing as the

197
00:09:18,840 --> 00:09:20,260
frequency of the source.

198
00:09:20,260 --> 00:09:21,070
So there you have it.

199
00:09:21,070 --> 00:09:22,160
We have our two relations.

200
00:09:22,160 --> 00:09:26,010
At least if you are in the path,
if the velocity of the

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00:09:26,010 --> 00:09:28,690
source is going in
your direction,

202
00:09:28,690 --> 00:09:30,310
then we have our formulas.

203
00:09:30,310 --> 00:09:34,190
And I'll rewrite them, just
because the observed period of

204
00:09:34,190 --> 00:09:37,600
the observer is going to be the
period from the point of

205
00:09:37,600 --> 00:09:42,060
view of the source times the
velocity of the wave minus the

206
00:09:42,060 --> 00:09:44,080
velocity of the source-- that's
the velocity of the

207
00:09:44,080 --> 00:09:47,650
source-- divided by the velocity
of the wave itself.

208
00:09:47,650 --> 00:09:51,640
The frequency, from the point
of view of this observer, is

209
00:09:51,640 --> 00:09:53,560
just the inverse of that,
which is the frequency.

210
00:09:53,560 --> 00:09:56,590
The inverse of the period is the
frequency from the point

211
00:09:56,590 --> 00:10:00,840
of view of the source times
the velocity of the wave

212
00:10:00,840 --> 00:10:03,730
divided by the velocity
of the wave minus the

213
00:10:03,730 --> 00:10:05,450
velocity of the source.

214
00:10:05,450 --> 00:10:07,640
In the next video, I'll do the
exact same exercise, but I'll

215
00:10:07,640 --> 00:10:10,400
just think about what happens to
the observer that's sitting

216
00:10:10,400 --> 00:10:11,960
right there.