Waves and optics
Doppler effect formula for observed frequency Doppler effect formula for observed frequency
Doppler effect formula for observed frequency
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- 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.
- 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.
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