I've got this source of a wave right here that's moving to
the right at some velocity.
So let's just say that the velocity of the source-- let's
call it v sub s to the right-- so we're really going to do
what we do in the last video, but we're going to do it in
more abstract terms so we can come up with a generalized
formula for the observed frequency.
So that's how fast he's moving to the right, and he's
emitting a wave. Let's say the wave that he's emitting-- so
the velocity of wave-- let's call that v
sub w radially outward.
We've got to give a magnitude and a direction.
So radially outward.
That's the velocity of the wave, and that wave is going
to have a period and a frequency, but it's going to
have a period and a frequency associated from the point of
view of the source.
And we're going to do everything.
This is all classical mechanics.
We're not going to be talking about relativistic speed, so
we don't have to worry about all of the strange things that
happen as things approach the speed of light.
So let's just say it has some period of-- let me
write it this way.
The source period, which is the period of the wave from
the perspective of the source, so the source period, we'll
call it t sub source, And the source frequency, which would
just be-- we've learned, hopefully it's intuitive now--
would be the inverse of this.
So the source frequency would be-- we'll call it f sub s.
And these two things are the inverse of each other.
The inverse of the period of a wave is its
frequency, vice versa.
So let's think about what's going to happen.
Let's say at time equal zero, he emits that first crest,
that first pulse, so he's just emitted it.
You can't even see it because it just got emitted.
And now let's fast forward t seconds.
Let's say that this is in seconds, so every t seconds,
it emits a new pulse.
First of all, where is that first pulse
after t sub s seconds?
Well, you multiply the velocity of that first pulse
times the time.
Velocity times time is going to give you a distance.
If you don't believe me, I'll show you an example.
If I tell you the velocity is 5 meters per second, and let's
say that this period is 2 seconds, that's going to give
you 10 meters.
The seconds cancel out.
So to figure out how far that wave will have gone after t
sub s seconds, you just multiply t sub s times the
velocity of the wave.
And let's say it's gotten over here.
It's radially outward.
So, I'll draw it radially outward.
That's my best attempt at a circle.
And this distance right here, this radius right there, that
is equal to velocity times time.
The velocity of that first pulse, v sub w, that's
actually the speed.
I'm saying it's v sub w radially outward.
This isn't a vector quantity.
This is just a number you can imagine.
v sub w times the period, times t of s.
I know it's abstract, but just think, this is just the
distance times the time.
If this was moving at 10 meters per second and if the
period is 2 seconds, this is how far.
It will have gone 10 meters after 2 seconds.
Now, this thing we said at the beginning of
the video is moving.
So although this is radially outward from the point at
which it was emitted, this thing isn't standing still.
We saw this in the last video.
This thing has also moved.
How far?
Well, we do the same thing.
We multiply its velocity times the same number of time.
Remember, we're saying what does this look like after t
sub s seconds, or some period of time t sub s.
Well, this thing is moving to the right.
Let's say it's here.
Let's say it's moved right over here.
In this video, we're assuming that the velocity of our
source is strictly less than the velocity of the wave. Some
pretty interesting things happen right when they're
equal, and, obviously, when it goes the other way.
But we're going to assume that it's strictly less than.
The source is traveling slower than the actual wave.
But what is this distance?
Remember, we're talking about-- let me do it
in orange as well.
This orange reality is what's happened after t sub s
seconds, you can say.
So this distance right here.
That distance right there-- I'll do it in a different
color-- is going to be the velocity of the source.
It's going to be v sub s times the amount of
time that's gone by.
And I said at the beginning, that amount of time is the
period of the wave. That's the time in question.
So period of the wave t sub s.
So after one period of the wave, if that's 5 seconds,
then we'll say, after 5 seconds, the source has moved
this far, v sub s times t sub s, and that first crest of our
wave has moved that far, V sub w times t sub s.
Now, the time that we're talking about, that's the
period of the wave being emitted.
So exactly after that amount of time, this guy is ready to
emit the next crest. He has gone
through exactly one cycle.
So he is going to emit something right now.
So it's just getting emitted right at that point.
So what is the distance between the crest that he
emitted t sub s seconds ago or hours ago or microseconds ago,
we don't know.
What's the distance between this crest and the one that
he's just emitting?
Well, they're going to move at the same velocity, but this
guy is already out here, while this guy is starting off from
the source's position.
So the difference in their distance, at least when you
look at it this way, is the distance between the source
here and this crest.
So what is this distance right here?
What is that distance right there?
Well, this whole radial distance, we already said,
this whole radial distance is v sub w, the velocity of the
wave, times the period of the wave from the perspective of
the source, and we're going to subtract out how far the
source itself has moved.
The source has moved in the direction, in this case, if
we're looking at it from this point of view,
of that wave front.
So it's going to be minus v sub s, the velocity of the
source, times the period of the wave from the perspective
of the source.
So let me ask you a question.
If you're sitting right here, if you're the observer, you're
this guy right here, you're sitting right over there, and
you've just had that first crest, at that exact moment
that first crest has passed you by, how long are you going
to have to wait for the next crest?
How long until this one that this guy's emitting right now
is going to pass you by?
Well, it's going to have to cover this distance.
It's going to have to cover that distance.
Let me write this down.
So the question I'm asking is what is the period from the
point of view of this observer that's right in the direction
of the movement of the source?
So the period from the point of view of the observer is
going to be equal to the distance that the next pulse
has to travel, which is that business up there.
So let me copy and paste that.
So it's going to be that.
Let me get rid of that.
It shouldn't look like an equal sign, so I can delete
that right over there.
Or a negative sign.
So it's going to be this distance that the next pulse
is going to travel, that one that's going to be emitted
right at that moment, divided by the speed of that pulse, or
the speed of the wave, or the velocity the wave, and we know
what that is.
That is v sub w.
Now this gives us the period of the observation.
Now.
If we wanted the frequency-- and we can manipulate this a
little bit.
Let's do that a little bit.
So we can also write this.
We could factor out the period of the source.
So t sub s we could factor out.
So it becomes t sub s times the velocity of the wave minus
the velocity of the source, all of that over the velocity
of the wave. And so just like that, we've gotten our formula
for the observed period for this observer who's sitting
right in the path of this moving object as a function of
the actual period of this wave source, the wave's velocity
and the velocity of the source.
Now, if we wanted the frequency, we just take the
inverse of this.
So let's do that.
So the frequency of the observer-- so this is how many
seconds it takes for him to see the next cycle.
If you want cycles per second, you take the inverse.
So the frequency of the observer is just going to be
the inverse of this.
So if we take the inverse of this whole expression, we're
going to get 1 over t sub s times v sub w over the
velocity of the wave minus the velocity the source.
And of course, 1 over the period from the point of view
of the source, this is the same thing.
This right here is the same thing as the
frequency of the source.
So there you have it.
We have our two relations.
At least if you are in the path, if the velocity of the
source is going in your direction,
then we have our formulas.
And I'll rewrite them, just because the observed period of
the observer is going to be the period from the point of
view of the source times the velocity of the wave minus the
velocity of the source-- that's the velocity of the
source-- divided by the velocity of the wave itself.
The frequency, from the point of view of this observer, is
just the inverse of that, which is the frequency.
The inverse of the period is the frequency from the point
of view of the source times the velocity of the wave
divided by the velocity of the wave minus the
velocity of the source.
In the next video, I'll do the exact same exercise, but I'll
just think about what happens to the observer that's sitting
right there.