Video: Calculating the Wave Fronts of a Sound from a Moving Source Passes by a Moving Observer

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 ๐‘ก. 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 ๐‘ƒ?

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Video Transcript

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 ๐‘ก. 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 ๐‘ƒ?

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 ๐‘ƒ. Along with this, we see this blue arrow, which represents the motion of the person over some time interval ๐‘ก. And then the magenta arrow, which represents the distance a wavefront will move over that same time interval.

So basically, however long ๐‘ก is, and we donโ€™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 ๐‘ƒ, 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 ๐‘ƒ. Starting out at what we could call time is equal to zero, we see the personโ€™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 ๐‘ก, we see our updated personโ€™s position. They moved that distance towards point ๐‘ƒ. 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 letโ€™s continue on.

Letโ€™s let another time interval ๐‘ก elapse. Over this second time interval of ๐‘ก, 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 ๐‘ƒ. And then if we let a time interval ๐‘ก 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 letโ€™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 ๐‘ƒ. So far, one wavefront has passed them. So weโ€™ll record that at the bottom of our screen.

Now if we start to let another time interval ๐‘ก begin to pass, we see that, before that interval has passed, something interesting happens. Before the person has reached point ๐‘ƒ, when theyโ€™re perhaps there say, this wavefront, which at this time interval began along that point ๐‘ƒ, will have moved far enough to the left so that it reaches and then passes this person. So thatโ€™s the second wavefront that passes then. Weโ€™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 ๐‘ก, the person will end up at point ๐‘ƒ. 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 ๐‘ƒ. 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 ๐‘ƒ, thatโ€™s all the wavefronts weโ€™ll count. In the time it took for them to move from their initial position to point ๐‘ƒ, three wavefronts reached or passed them.

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