Video: Representing the Doppler Shift Produced by an Object’s Motion Using a Graph

The graphs show the change in frequency with time of the sound of the horn of a fast-moving car that is heard by a person walking slowly by the side of the road as the car approaches and passes them. Which of the graphs (A), (B), (C), and (D) most correctly shows the changes in frequency that the pedestrian hears?

06:44

Video Transcript

The graphs show the change in frequency with time of the Sound of the horn of a fast-moving car that is heard by a person walking slowly by the side of the road as the car approaches and passes them. Which of the graphs A, B, C, and D most correctly shows the changes in frequency that the pedestrian hears?

Okay, so in this question, we’ve got four graphs that show the change in frequency with time of the sound of the horn of a fast-moving car which is heard by a person walking slowly by the side of the road as the car approaches and then passes them. We need to work out which graph most correctly shows the changes in frequency that the pedestrian hears. So let’s draw a quick little diagram first.

Let’s say that we’re looking from above, so this is our car. And let’s say that it’s moving towards the right at a speed 𝑉, capital 𝑉. And it’s also honking its horn as it goes. There we go, beep beep! And as it drives along, it goes past a passenger here on the side of the road, walking also towards the right but slowly. Let’s say at a speed lowercase 𝑣. Now truthfully, it doesn’t matter what the values of capital 𝑉 and lowercase 𝑣 are.

The Doppler effect is going to occur regardless because there’s some relative speed between them. And the relative speed between the car and the pedestrian is capital 𝑉 minus lowercase 𝑣. Now what exactly does this mean?

Well, let’s say that capital 𝑉 is 50 miles per hour. Now this is just a random number that we’ve pulled from nowhere, but it’s a good example to try to explain what we’re considering. Let’s also say that lowercase 𝑣 is two miles per hour. So the pedestrian is walking along towards the right at two miles per hour.

Well, the relative speed between these two, the car and the pedestrian, is capital 𝑉 minus lowercase 𝑣, which happens to be 50 miles per hour minus two miles per hour, which happens to be 48 miles per hour. Now it’s this relative speed that affects the amount of Doppler shift that occurs to the sound emitted by the car, which is the horn. It’s honking its horn.

In other words, the Doppler effect which is what we’re considering here, the change in frequency of the sound, would be exactly the same in this situation as if the car was travelling at 48 miles per hour to the right and the pedestrian was stationary, because still the relative speed between them is 48 miles per hour.

Relative to the pedestrian, in other words, the car is moving to the right at 48 miles per hour. And once again, this speed affects the extent to which the Doppler shift occurs on the sound emitted by the car. Anyway, that’s not super relevant.

It’s just an example to show how the Doppler shift would occur when the car moves past the pedestrian. So we know that the car is moving to the right at a much faster speed than the pedestrian is. And we’re trying to consider the frequency change of the horn that the car is honking, as heard by the pedestrian of course.

And as we’ve already mentioned, we’re looking at the Doppler effect. Now the Doppler effect tells us that when the source of the sound, in this case, is moving towards the observer, the frequency of the sound increases compared to when there’s no relative motion between the source and the observer.

In other words, when the source and the observer are moving either at the same speed or both are stationary, so there’s no relative speed between them, the observer hears the horn at a certain frequency.

But when the source is moving towards the observer, so in this case when the car is moving towards the pedestrian, the observer hears the sound as a higher frequency. And vice versa as well, when the source is moving away from the observer, the observer hears the sound as a lower frequency.

And of course the speed with which the source is moving towards the observer or away from the observer affects how much this frequency changes by.

The faster the source is moving, the larger the shift in frequency. So we know that this car is initially moving towards our observer here, our pedestrian. And it continues to move and passes the pedestrian, at which point it starts moving away from the observer.

So in this region here, the frequency of the sound emitted by the car is going to be high and then it decreases as it starts moving away from the observer. In other words, what the observer hears is [imitates a honk], because of course a high-frequency sound corresponds to a high-pitched sound and a low-frequency sound corresponds to low-pitched sound.

So what we can do is to draw our own graph of how the frequency changes as the car passes the observer. On the vertical axis, we can put the frequency of the horn. And of course we’re going to be drawing it arbitrarily.

So it doesn’t matter if it’s in hertz. But it’s best to keep it in hertz because that’s the standard unit of frequency. However, we don’t need to label any numbers because we don’t have any information about the numbers.

We just need to draw a certain shape. Now on the horizontal axis, we can put the time, once again in seconds because that’s the standard unit. And the time is basically the time elapsed since the car was at this point here. In other words, we can say that at zero seconds the car is here, and then it moves to the right as time passes.

So to recap, the sound that the observer would have heard as the car started approaching them, then pass them, and continue to move away from them. What they would have heard is [imitates a honk], so the sound starts as a high pitch or a high frequency and then ends up as a low pitch or low frequency.

In other words, it’s something like this [imitates a honk]. Now this might seem a little bit intuitive to you if you’ve ever stood by the side of a road and heard cars passing by you. In fact, however, this doesn’t necessarily even have to apply to cars. It can apply to anything that’s moving past you.

And the Doppler shift doesn’t even have to apply to sound. It also applies to the light emitted by a source as it moves past the observer. The frequency of the light emitted changes depending on whether the source is moving towards the observer or away from the observer.

And the faster the source is moving, the larger the shift in frequency. So anyway, coming back to our question, we’ve drawn our own graph now. And it looks something like graph B. So hooray! That’s our final answer. But just for fun, let’s imagine what the other graphs would sound like.

Let’s look at graph A. Graph A starts at a low-frequency then increases to a higher frequency and then comes back down again. That sounds something like [makes sound]. And that’s not quite the sound a car makes when it goes past you, is it?

Anyway, looking at graph A. It starts out with a low frequency and then increases. So it goes [makes sound]. And that’s not what a car sounds like either. And then the very last one, graph D, starts off at a high frequency, then dips, and then increases again. In other words, it goes [makes sound].

And once again, that’s not the sound a car makes when it goes past you. And hence our final answer is that graph B most correctly shows the changes in frequency that the pedestrian hears.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.