### Video Transcript

In this video, we’re talking about the Doppler shift. We’ll see that this shift has to do with waves, in our case sound waves, that are produced by a moving source. Whether we know it or not, each one of us has firsthand experience with this shift.

In this lesson, we’ll come to understand why Doppler shifts occur and what effects they have. Before we start talking about a sound source that’s its motion, let’s consider one that’s stationary. Let’s say you’re standing in place and listening to some music through a boombox. And just for the sake of this demonstration, let’s say that this music is some very uninteresting music. It’s a single constant tone, one at the same frequency. And let’s say further that as the boombox produces this sound, the single tone, that the sound waves move out equally in all directions.

One way for us to show these sound waves in our sketch is to indicate where the peaks, the areas of maximum air compression, are on those waves. Say that we start playing this tone. And the first wave represented by this hashed line starts to move out from the speaker. This wave, by the way, will be moving at the speed of sound, very quickly, about 340 meters per second.

So as time passes, this wavefront, this compression point on our sound wave, will move outward from the source. After a bit of time has passed, the wave might look like this, more spread out than it was before. It’s important to see though that while this particular wavefront was expanding out from the source, the source was continuing to emit sound waves. So that means where this original wavefront used to be there’s now a new wavefront that’s been emitted by that source. And this new wavefront is also moving outward at the same speed that the original wavefront moved.

So over a bit of time, here’s what will have happened. Our original wavefront will have moved out even further from the source. And what we could call the second wavefront has moved out to where the first one was just a moment ago. And then on top of all this, our source continues to emit sound waves. And a new wavefront comes to replace the second one.

Once more, all these waves will continue to move outward at the speed of sound. And as our boombox continues to produce sound at this one steady frequency, we’ll see all these rings, these concentric circles, where the circles represent wavefronts moving outward from that source.

Taking a look at these rings, notice something interesting about them. If we were to measure the closest distance in space between one ring and another one of the rings, then that distance will be equal to the wavelength of this sound wave. Remember that each one of these rings represents a compression, a maximum, on that sound wave. If we travel from one maximum point on the wave to an adjacent maximum, another high point, then that distance is equal to the wavelength of the wave.

If we were to measure out that distance between each pair of rings that are next to one another, we’d find the distance is constant. It’s always the same. That is, the wavelength of the wave is constant as it moves out. Not only that, but the speed of these wavefronts is also constant as they move out from the source. And it’s at this point we can recall an equation for the speed of a wave, called 𝑣, in terms of its wavelength, 𝜆, and its frequency, 𝑓.

Wave speed is equal to the product of wavelength and frequency. For this particular wave, we’ve just seen that the wavelength is a constant value as well as the speed of the wave. And this tells us that the frequency of the wave is fixed too. That means no matter where we put our ear, whether over here or up here or over here on the other side, no matter where we listen to this wave, we always hear the same frequency, the same tone.

The reason for that, the reason the frequency of this sound is always the same no matter where we listen to it, is because the source of that sound isn’t in motion. It’s stationary. Since the source isn’t moving, that means the distance between adjacent wavefronts, what we’ve called the wavelength of the wave, is constant everywhere. But interestingly, if the source of the sound does start to move, that’s no longer true.

To see how, let’s imagine we stop playing that tone for a moment. If we do that, the sound waves that were already produced fly off in every direction. Then imagine we start walking at a steady rate over in this direction. And after we’ve started to move, we begin to play this tone once more.

So here we go. We’re moving left to right. And the same sound frequency is being produced by our boombox. The first wavefront is emitted, sketched in here. This original wavefront, the first one produced, moves outward from the point at which it was emitted from the boombox. And then from the same point on the boombox that emitted the first wavefront, a second wavefront is emitted.

At this point, there’s something very important we should bring up. Imagine you were running along the ground. And in one hand, you’re holding a stone. Now if you run along with a constant speed and then drop the stone, what will happen to it? The stone of course will fall to the ground. But that won’t be the only motion that it has. It also moves left to right. That’s because while it was being held, the stone picked up the same speed as its environment. So if we were running along, say at three meters per second, then the stone at the instant we release it would have that same speed, three meters per second, to the right. That’s why, instead of falling straight down to the ground as soon as we release it, the stone moves to the ground in an arc like this. Because we’d given it a horizontal speed that matches our speed as we run.

Well, we bring all this up to say that sound waves are not like this stone. Sound waves do not pick up the speed of any object they’re emitted from. As soon as a sound wavefront is emitted, the center of that wavefront, wherever it is, remains the center of the wavefront as it expands out. We can see this in our walker holding a boombox. The first wavefront of this sound was emitted from a point here. The wavefront then expands outward from that point in all directions. But then as the source of sound moves, as the walker moves ahead, the point source for these wavefronts moves as well.

Notice that our second wavefront originates not where the first one did, but at this point here. The gap or the distance between these two points is thanks to the fact that our sound source is in motion. But the sound wavefronts aren′t affected by this motion. Each one is emitted at a particular point in space and then expands outward at the speed of sound, always keeping that point as its center.

Thanks to this fact, as the sound source continues to move, our source will keep emitting new wavefronts, which look a bit lopsided compared to the other wavefronts that have already been given off. And the reason for that is our sound source is in motion. And yet the wavefronts produced in that sound wave keep their same center, even after they’ve been emitted.

Looking at the sketch, we can see that the most recent wavefront was emitted from a point here, whereas this slightly expanded one was emitted from this point. That’s its center. While the front that’s one further out than that was emitted at this point. And that continues to be its center. And then our largest wavefront drawn in has a center point right here. These points — one, two, three, four — show the position of the boombox’s sound source over time.

Now notice something interesting. Let’s say we once again take a look at the distance between subsequent wavefronts in this wave. Starting out on the right-hand side, we see that this is that distance. But then if we started from the left-hand side, that distance is much different. It’s significantly greater. Here’s what this means.

If we were to put an observer over here and have them listen to this sound wave, this person will be reached by a relatively high number of wavefronts every second. That is, the wavefronts are closer together at this front part of the wave than at the back part.

Now we’ve said that each one of these wavefronts represents a peak in the sound wave. And the more of these wavefronts that pass a given point every second, the higher the frequency of that wave. So for an observer standing here, ahead of this sound source, since more wavefronts pass over their ear per second, they will perceive a higher frequency to this sound as compared to an observer back here behind the motion of the source.

So if these two observers were to count the number of wavefronts that pass by their ear every second and then use that count to determine the frequency of the wave, the observer out front of the source might report a frequency 𝑓 one. While the observer behind the source would report a frequency 𝑓 two. And these two frequencies wouldn’t be the same. In fact, because we can already see that more wavefronts per unit time passed by the observer in front of the source than the one behind it, then the frequency reported by the observer in front of the source would be greater than the frequency reported by the person behind it. And this is because more wavefronts per second are washing over the ears of the observer out front.

And then, on top of all this, here’s another interesting fact. Remember that we started out with a stationary source that was broadcasting sound at a certain particular frequency. If we call that frequency 𝑓 sub zero, the original frequency, then it turns out that 𝑓 sub one is greater than that original broadcast frequency, which itself is greater than the frequency observed behind the motion of the source. The fact that this original frequency, 𝑓 sub zero, is shifted either up or down depending on where we stand in relation to the motion of the source is the effect known as the Doppler shift.

We can write it this way. We can say that a Doppler shift is when a constant wave source is in motion and therefore observers detect higher or lower frequencies depending on their position relative to that source. We saw that if a sound source is moving toward an observer, then that observer measures a relatively higher frequency than is actually being emitted by the source. And then, on the other hand, if the source is moving away from us, we measure a relatively lower frequency than is actually being given off.

This entire effect of the Doppler shift comes back to the whole notion of wavefronts passing a given point over some unit of time. The more of them there are, the higher the frequency. And the fewer of them there are, the lower the frequency. Knowing all this, here’s a question. Say we consider our observer out front of the moving source. Is there anything this observer could do to increase the measured frequency of these wavefronts even more?

The answer is yes. One way for even more of these wavefronts to pass by this observer over some unit of time is to move into them, say to walk toward the source. If this observer did that, from their perspective, the wavefronts will get even more compressed together than they already are. And similar to this, if we wanted our observer behind the sound source to measure an even lower frequency than they originally did while they’re stationary, we could accomplish this by having them move away from the source, to the left in this case. If they did that, even a fewer wavefronts per second, per unit of time, would pass by them. They would measure an even lower frequency.

What we’re seeing then is not only can the source of the sound be in motion and that caused a Doppler shift, a frequency shift, but also the observers of that sound can be in motion relative to the sound. And that causes a further shift. And just as a side note, even if our sound source was stationary, we can still have a Doppler shift going on so long as our observers were in motion relative to it. Either source or observer could be in motion, and either one causes such a shift. Now knowing all this about Doppler shifts, let’s try out an example on this topic.

A person moves toward a source of sound waves as shown in the diagram. The sound source starts to emit sound at the same moment that the person starts to move. The sound from the source moves a distance equal to the wavelength of the sound wave in a time interval 𝑡. How many wavefronts from the sound source will have reached or passed the person by the time that the person has moved from their initial position to the point 𝑃?

Taking a look at this diagram, we see on the left-hand side this gold dot that represents the person in motion to the right. And then on the right-hand side, we have our sound source emitting these sound waves to the left. And then midway in between these two points is the point 𝑃. Along with this, we see this blue arrow, which represents the motion of the person over some time interval 𝑡. And then the magenta arrow, which represents the distance a wavefront will move over that same time interval.

So basically, however long 𝑡 is, and we don’t know how long. But we know that, in that time interval, the person will move this distance shown here, whereas a sound wavefront will move this distance shown here. So and we have these two things in motion, the person and the sound waves. And we want to know by the time the person reaches this midpoint, point 𝑃, how many sound wavefronts will have reached or passed the person?

To figure out the answer of this question, we can sketch in the wavefronts produced by our sound source as well as the position of this person as they move towards point 𝑃. Starting out at what we could call time is equal to zero, we see the person’s position here. And we know the initial wavefront from our sound source is being emitted. We could say its position is there.

Then after a time interval 𝑡, we see our updated person’s position. They moved that distance towards point 𝑃. And our original sound wavefront will have reached to this point. And then in addition to that, a second wavefront will be at that moment emitted by our sound source. So by the time the person has reached this point here, along this dashed line, we now have two wavefronts in motion toward them. So let’s continue on.

Let’s let another time interval 𝑡 elapse. Over this second time interval of 𝑡, the person will now have progressed to this location. And as well as this, our leading wavefront will now be positioned here, in line with point 𝑃. And then if we let a time interval 𝑡 elapse once more, the person will have progressed to this location. And our leading wavefront will now be here. It will have passed this person. So let’s count that because we want to know how many wavefronts from the sound source reach or pass this person by the time they get to point 𝑃. So far, one wavefront has passed them. So we’ll record that at the bottom of our screen.

Now if we start to let another time interval 𝑡 begin to pass, we see that, before that interval has passed, something interesting happens. Before the person has reached point 𝑃, when they’re perhaps there say, this wavefront, which at this time interval began along that point 𝑃, will have moved far enough to the left so that it reaches and then passes this person. So that’s the second wavefront that passes then. We’ll record that in our tally at the bottom of the screen.

If we continue to let the clock run, after a completed time interval 𝑡, the person will end up at point 𝑃. And by that point, the sound wave, which started out here when the person was here, will have crossed over this distance and will now be in line with point 𝑃. Therefore, we can say it will have reached this person. So that means a third wavefront has reached or passed this person over this time. And since the person is now at point 𝑃, that’s all the wavefronts we’ll count. In the time it took for them to move from their initial position to point 𝑃, three wavefronts reached or passed them.

Let’s take a moment now to summarize what we’ve learned about the Doppler shift. We’ve seen here that when a source of sound is in motion relative to an observer, the perceived sound frequency changes. That’s called a Doppler shift. Given a source of sound with a baseline or original frequency we could call 𝑓 sub zero, if an observer is moving toward that source, then they would measure or observe a higher frequency. We could call it 𝑓 sub one.

But on the other hand, for an observer moving away from the source, they would measure a different frequency — we could call it 𝑓 sub two — which is less than the original frequency. And we saw that all of this goes back to the fact that observed frequency is equal to the number of wavefronts that pass by an observer over a time interval of one second.