The Doppler effect
Doppler effect formula when source is moving away
In the last video, we figured out the formulas for the observed period and frequency for an observer sitting in the path of the source. So the source is moving towards the observer. So this is the example where the train is moving towards you, and you perceive the train's horn as having a higher pitch or a higher frequency. And we were able to do that by doing a thought experiment. Saying, OK, my object starts here. After one period-- a period is just a measure of time, but it's the measure of time over which the source emits a cycle, so it emits a cycle every period. But after one period, we said, OK, where is that first wave front, or that first pulse, or that first crest? And where is the source? Because exactly one period has passed by, and the source will be ready to emit another crest or another cycle. So the distance between where the source is and that front of the crest, or that first crest, that is going to be the wavelength. Because this next thing that emits is going to be traveling at the exact same velocity, and it's going to be separated by that distance, which we saw is this expression. We said how long will it take it to travel that distance? Well, it's traveling at a speed of v sub w. That'll tell you what the observed period would be for this dude over here. We calculated it right here, and then the observed frequency is just the inverse of that. Now let's think about the situation where the observer is over here. So these equations, or these formulas that we came up with right here, this is observer-- or let me say source traveling in direction of observer. Now let's think about the opposite case, where the source is traveling away from the observer, and in this case, the observer is that guy over there. Maybe I'll do it in a different color. He'll be blue. So this is the observer. So when we started off, our source was right here. After exactly one period from the source's point of view, that first crest emitted has traveled radially outward that far. This is the distance. The velocity of the wave times the amount of time that passed-- velocity times time is going to give you distance-- and where the source of the wave will have traveled to the right exactly this distance. It's velocity times the amount of time that's gone by. Now, in the last video, we said, OK, that wave is just passing this guy. How long will it take for that pulse that's being emitted right then to also reach him? And then that tells us the period between two pulses or between two crests. Now let's think about that exact same situation here. That first crest is just passing this guy. And a period, or the t sub s, which is a period of the emitted wave has just passed by. So this guy is just about to emit another wave, So? That other wave is going to be right here. So what is the distance between the crest, or the cycle, or however you want to think about it, the pulse that is passing him by right now, what is the distance between that and the pulse, the front edge of that pulse that is being emitted right at that moment? Right at that moment. Well, it's going to be this radius, which is this value. It'll be v sub w times the period. That is that distance plus the amount of distance that our source has traveled away from this guy. So plus v sub s times the period. So that's what this distance is, and this is how far apart this wave pulse is going to be from that one or this crest is going to be from that one. So if he's seeing this first crest right now, right at this moment, how long will it take him to see the crest that's being emitted right now, that's this far away from him? Well, it's that far away from him. So let me write this down. So the amount of time it takes for him to see the next crest or the same point in the next cycle, that's the period. That's the observed period. That's going to be equal to this distance. The velocity of the wave times the period from the perspective of the source plus the velocity of the source, because the source has gotten that much further away from him, velocity of the source times the period of the source, so that's how far the next crest is. And then you divide it by the speed of the wave, by the speed of each of the crests, which is just the velocity of the wave. And we can just factor out the t sub s's here and say this is t sub s times v sub w, the velocity of the wave plus the velocity of the source divided by the velocity the wave. So this will be a larger observed period than if this guy was stationary and especially if the observer was in the path of the guy. That make sense, because every time this guy issues a cycle, he is moving a little bit further away. So every crest the same point in the cycle is going to be further and further apart, so you're going to have longer wavelengths, longer periods. And if you want the observed frequency for that guy over there-- I'll do it in the same color. The observed frequency for a guy where the source is traveling away from him is just the inverse of that. So one over the period, one over the period, same argument we did there, one over the period from the point of view of the source is the frequency of the source. Let me Color code it. So 1 over t sub s is equal to the frequency of the source. This is the inverse of that. So I'm just taking one over everything here. So 1 over t sub s is the frequency of the source, and then we take the inverse of this over here: the velocity of the wave divided by the velocity of the wave plus the velocity of the source. So we're done, at least for the simple cases. Obviously, it becomes a little more interesting when someone isn't exactly in the direction of the source or exactly being moved away from, but these are kind of the two extreme cases. So this is the situation when it is moving away from you. Now, just to check our math and maybe make it a little bit concrete in relation to the video we did where we introduced to the idea of the Doppler effect, let's actually apply those numbers. So on that video, two videos ago, we had a situation where the velocity of our source was 5 meters per second to the right, and the velocity of the wave was 10 meters per second radially outward, and the period of our wave-- I'll call it the period of our-- let me do it in another color-- from the point of view of the source was 1 second per cycle, and the frequency was just the inverse of that. So 1 cycle per second, or 1 Hertz, which is a cycle per second. So using those numbers, let's see if we get to the exact same answer we got in that first video where we first learned about the Doppler effect. So let's look at the frequency from the point of view of this gentleman right here. So the frequency of the source is going to be 1 cycle per second, 1 Hertz. The velocity of the wave is 10 meters per second. Let me write this. 1 cycle per second. The velocity of the wave is 10 meters per second. The velocity of the wave is 10 meters per second minus the velocity of the source is 5 meters per second. So what's this going to be equal to? The observed frequency for this guy right there, is going to be 1 cycle per second times 10 over 10 minus 5, and the meters per second cancel out. Meters per second in the numerator, meters per second in the denominator. So 10 divided by 10 minus 5, or 10 divided by 5, is going to be 2. So it's going to be 2 cycles per second. And if you want the observed period for this guy, its going to be the inverse of that, or it's going to be 1/2 second per cycle. And this is exactly what we got in the previous video, or actually, it was two videos ago. Now what about the guy who this guy's running away from? Well, we'll do the exact same thing. You have 1 cycle per second, or 1 Hertz. That's the emitted frequency from the point of view of the source times the velocity of the wave divided by the velocity of the wave plus the velocity of the source, because it's moving away from him. So it's 10 over 10 plus 5. That's 10 over 15. That is 2/3, so this is equal to 2/3. The units over here all cancel out. This was cycles per second. So 2/3 cycle per second, which confirms the numbers we got in that first video, so that should make us feel good and it also makes a lot of sense. This guy is going to see the wave crest more frequently. He's going to observe a higher frequency. If this is a sound, a higher pitch. This guy, since each crest or the cycles are getting spread out, he's going to see them less frequently, and if this is sound, he's going to observe a lower pitch.