### Video Transcript

In this video, we’re going to learn about the Doppler effect. We’ll see what this effect is, what causes it, and how it changes the sounds that we hear and the light that we see.

To get started, imagine that you are riding with a friend in a car that they’re driving quickly down the road. While your friend drives the car, you hold up a radio receiver in your hand to try to get a signal from your favorite radio station. Even though you know the frequency at which this radio station transmits, when you tune your receiver to exactly that frequency, somehow it doesn’t pick up the signal. To understand why this might be happening, it will be helpful to know about the Doppler effect.

The Doppler effect has to do with the frequency of waves that we perceive when we’re moving relative to the source of those waves. Imagine that you’re standing still, fixed in one place. And while you do this, you blow on a whistle, producing a steady constant tone. If we drew in the sound wavefronts that are produced as you blow on this whistle, they would look like concentric circles emanating from the source of the sound. Once these wavefronts are produced, they move outward from the source at the speed of sound.

Imagine after doing this for a bit that you start to move while continuing to blow this steady note. If we tracked the source of the sound, your throat and mouth, over time as you moved at a steady rate to the right, we would see that waves are produced periodically along this line. But at the moment in time when this last wave to the far right is created, the waves behind it to the left wouldn’t look like what we’ve drawn. That’s because as time passes, these waves expand radially outward at the speed of sound.

This means that once our waves are expanded, the wavefronts would look something like this. Notice that these circles are no longer concentric. The center of the largest circle is somewhere behind the center of the second largest, which is somewhere behind the center of the third largest, and so on. That’s because sound waves, when they’re produced, move radially outward from their point of origin. And that point doesn’t change over time. It’s a bit like if someone was running along beside a body of water and dropped a stone they were carrying into the water as they ran.

Because the stone is initially moving along with the person, it has a speed in the direction of the person’s motion and so enters the water with a nonzero horizontal velocity. But once the stone enters the water, it creates concentric rings of water waves from that point of entry. Though the waves move as they spread out across the body of water, their center, or their origin, doesn’t.

Looking back at the sound waves produced by our walking whistler, we see that, in the direction of the walker’s motion, the wave density is greater than opposite that direction of motion. Here’s something interesting about that. We perceive sound according to the frequency of the sound waves as they reach our ear. The higher the frequency, the higher the tone we perceive, and vice versa.

This means that if we stood in front of the walking whistler and listened to the sound produced, we would hear a frequency we could call 𝑓 one. But then, if we went and stood on the other side behind the walker, we would hear a different frequency, 𝑓 two. And 𝑓 two would be less than 𝑓 one. We can see that 𝑓 two would be less than 𝑓 one because the number of wavefronts that reach our ear every second when we stand behind the walker is less than the number of wavefronts that reach our ear every second if we stand in front.

Not only is this fact interesting, but also the actual frequency of sound being produced by our whistler that we can call 𝑓 is equal neither to 𝑓 one or 𝑓 two. Even though the frequency that’s being produced, what we could call the source frequency, is not equal to what we could call the perceived frequency, 𝑓 two and 𝑓 one, depending on where we’re standing, there is a mathematical relationship that connects these frequencies. This effect is known as the Doppler effect. So, the equation is known as the Doppler equation.

Here’s what this equation says. Let’s say we have a source of sound which is being produced at a steady constant frequency we can call 𝑓. We also have an observer who hears the sound being produced by the source. We’ll say that both the source and the observer are able to move. But they’re only able to move in one dimension; we’ll say to the left or to the right. The Doppler effect tells us that if the source and the observer are in motion relative to one another, then the frequency of sound heard by the observer will not be equal to 𝑓. It will be equal to some other frequency we can call 𝑓 prime.

The question is how do 𝑓 prime and 𝑓 relate to one another mathematically? We can say that if the source of the sound as well as the person hearing the sound are in motion, then the rate at which the wavefronts of sound wash over the ear of the observer will be the same as the rate at which they’re produced. In other words, 𝑓 prime will be equal to 𝑓. But what if the source, say, starts to walk toward the observer with a speed we can call 𝑣 sub 𝑠? In that case, we would take this speed of the walker, 𝑣 sub 𝑠, along with the speed of the expanding sound waves, which move at the speed of sound that we can call 𝑣, and input that into our equation like this.

If the source of sound was walking toward the observer, then the observed frequency would be equal to the speed of sound divided by the speed of sound minus the speed of the walker times the original frequency produced, 𝑓. And then, we might wonder, okay, but what if the observer was also in motion? Say the observer was walking toward the source at a speed we call 𝑣 sub 𝑜. In that case, in our numerator, we would have 𝑣 plus the speed of the observer, 𝑣 sub 𝑜.

Looking at this equation, the question might come up why we have a plus sign in the numerator and a minus sign in the denominator. If we consider the plus sign in the numerator first, we see that if our observer is moving toward the source, then the sound wavefronts will reach the observer’s ear at a higher rate than if the observer were standing still. This means the perceived frequency, 𝑓 prime, will be higher than 𝑓. And this plus sign tends to make that happen. That makes the numerator of our fraction bigger, and therefore 𝑓 prime increases. That makes sense physically if our observer is walking towards the source.

In the same way, the minus sign in the denominator comes from the fact that if our source is moving toward our observer, then that will overall have the effect of increasing the observed frequency, 𝑓 prime. Having a minus sign in our denominator, 𝑣 minus 𝑣 sub 𝑠, overall makes this fraction larger. And so, it makes the observed frequency higher, which is consistent with what physically happens. We might then wonder, what if our source instead of moving towards the observer moves away? And what if our observer moves away from the source?

In those cases, we switch the sign that we use in our numerator and denominator of this fraction, multiplying the source frequency 𝑓. The key to remembering which sign to use, positive or negative, is to think the situation through physically. If the source is moving away from the observer, will that raise or lower the observed frequency? The source moving away from the observer means that fewer wavefronts per second will reach the observer’s ear. Which means that, overall, our frequency shift will be less, which means we want to use the positive sign in our denominator to make the overall fraction smaller. Let’s try a quick Doppler effect example to see how this equation works with real numbers.

Given the setup that we have shown here, let’s imagine that our source frequency being produced is a sound at two hertz. This means that if we picked a point outside these advancing wavefronts, every half second, one wavefront will pass that point. If we heard this sound as a tone, it might sound something like this. Beep-beep-beep-beep-beep-beep-beep-beep, something at that rate or thereabouts. We’ll say that this source of sound isn’t stationary, but it’s in motion. And it has a speed we’ll call 𝑣 sub 𝑠. And we’ll let 𝑣 sub 𝑠 be the somewhat unrealistically high speed of 100 meters per second.

Not only is our source in motion though, we’ll say also that our observer of that source is in motion too, with a speed we’ll call 𝑣 sub 𝑜. And we’ll let 𝑣 sub 𝑜 be the even less realistic speed of 140 meters per second. Maybe the source and the observer are in rocket-powered cars or some such thing. Knowing the source frequency, the source speed, and the observer’s speed, there’s only one other factor we’ll want to know before we can solve for 𝑓 prime. That’s the speed of sound that we’ve called 𝑣. It’s the speed at which these wavefronts advance out in space.

One common value for the speed of sound at standard atmospheric pressure is 340 meters per second. We’ll use that for 𝑣. When we plug these numbers in to our Doppler effect equation, there’s only one last question we’ll want to answer before calculating our result. And that is, which sign do we use in our numerator and denominator?

Looking first at the numerator, which has to do with the speed of our observer relative to the source, we see that that motion will tend to make more wavefronts per second pass over the observer’s ear. This means the frequency shift from 𝑓 to 𝑓 prime will be greater. And that means we choose the positive sign in our numerator. Moving on to our denominator, the movement of our source will again have that same effect. It will make more wavefronts pass over the observer’s ear per second. So, to help that happen mathematically, we choose the negative sign in between our two values in the denominator.

So, our fraction simplifies to 480 meters per second over 240 meters per second, or simply a factor of two. This means that, at these speeds for the observer and the source and at this source frequency of two hertz, our observer would find a frequency of four hertz. So, whereas the source frequency might sound like beep-beep-beep-beep-beep-beep-beep-beep, the observed frequency would sound like beep-beep-beep-beep-beep-beep-beep-beep-beep-beep-beep-beep-beep-beep-beep-beep, something like that, a very great difference between the observed and source frequencies.

And this difference is entirely due to the Doppler effect, that is, the relative motion between the source and the observer. So far, as we’ve talked about the Doppler effect, we’ve considered it entirely as it relates to sound waves, perceived frequencies of sound. But the Doppler effect applies to any periodically produced wave from a source that’s in motion relative to an observer. If we consider astronomical bodies such as stars, which give off light themselves, these are also in motion relative to one another in many cases.

If a given star is in motion relative to the Earth, where we would measure the radiation received from that star, we see a similar effect. We see a shift in the observed frequencies produced by that star. When it comes to the light that our eyes are capable of seeing, we observe wavelengths of light along a spectrum called the visible spectrum. At one end of this spectrum is blue light, which has the highest frequency of light our eyes can see. And at the other end is red light, which has the lowest frequency.

This means that if a star is emitting radiation and moving away from us observing the star on Earth, then, consistent with the Doppler effect, the frequency of emitted radiation we observe is less, since the star is moving away from us. We say then that the frequency observed from that star has been red-shifted. Its frequency has decreased from source to observer.

On the other hand, if the star is moving towards the Earth, then the observed frequency we notice is higher than the source frequency. We say this the frequency has been blue-shifted. So, though we often encounter the Doppler effect with sound waves, keep in mind that it applies to any periodically produced wave.

Let’s summarize what we’ve learned so far about the Doppler effect. We’ve seen that the Doppler effect describes the frequency shift that happens when a periodic wave source and an observer are in motion relative to one another. Written as an equation, we can say that the observed frequency, 𝑓 prime, is equal to the speed of the waves being produced plus or minus the speed of the observer experiencing the waves divided by the wave speed minus or plus the speed of the source of the waves, all multiplied by the original frequency, or source frequency, 𝑓.

We also saw that the choice of positive or negative signs in the numerator and denominator of this fraction have to do with whether our source and observer motion will create a greater or lesser frequency shift. And finally, we saw that the Doppler effect is not just for sound waves. It also occurs for light waves or any periodically produced wave source.