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# Doppler effect formula for observed frequency

## Video transcript

so I've got this source of a wave right here that's moving to the right at some velocity so let's just say that the velocity of the source velocity of source velocity of source let's call it V sub s to the right so we're really going to do what we did in the last video but we're going to do in more abstract term so we can come with a generalized formula for the observed frequency so that's how far how fast he's moving to the right and he's emitting a wave and let's say that that wave that is he so let's say the wave that he's emitting so the velocity of wave let's call that V sub W radially outward got to give a magnitude and a direction so radially outward outward that's the velocity of the wave and that way is going to have a period and a frequency but it's going to have a period in frequency associated from the point of view of the source and we're going to do everything this is all classical mechanics we're not going to be talking about relativistic speed so we don't have to worry about all of the strange things that happen is things approach the speed of light so let's just say that it has so the period it has some period of let me write it this way the source period which is the the period of the wave from the perspective of the source so the source period we'll call it T sub source and the source frequency which would just be we've learned hopefully it's intuitive now it would be the inverse of this so the source frequency the source frequency would be we'll call it F sub s and these two things are the inverse of each other the inverse of the period of a wave is its frequency vice versa so let's think about what's going to happen so let's say it's time equal zero he emits that first crest that first pulse so he's just emitted it you can't even see because it just got emitted and now let's go let's fast forward T seconds let's say that this is in second so every T seconds it emits a new pulse so what first of all where is that first pulse after T sub s seconds well you multiply the velocity of that first pulse times the time velocity times time is going to give you a distance right if you don't believe me let's show you an example if I tell you the velocity is five meters per second and let's say that this period is two seconds that's going to give you ten meters the seconds cancel out so to figure out how far that wave will have gone after T sub s seconds you just multiply e sub s times the velocity of the wave and let's say it's gotten over here it's radially outward so I'll draw it radially outward that's my best attempt at a circle and this distance right here this radius this radius right there that is equal to velocity times time the velocity of that first pulse V sub W V sub dog that sacktual e the speed I'm saying it's V sub W radially outward I'm this isn't a vector quantity this is just a number you could imagine V sub W times the period times T sub s times T sub s I know it's abstract but just think this is just a distance times a time if this was moving at 10 meters per second F and it's and if it and if the period is 2 seconds this is how far we will have gone 10 meters after 2 seconds now this thing we said at the beginning of the video is moving so although this is radially outward from the point at which it was emitted this thing isn't standing still we saw this in the last video this thing has also moved how far well we do the same thing we melt we multiply its velocity times the same number of time remember we're saying what does this look like after T sub s seconds or some period of time T sub s well this thing is moving to the right let's say it's here let's say it's moved right over here and we're going in this video we're assuming that the velocity of our source is strictly less than the velocity of the wave some pretty interesting things happen right when they're equal and obviously when it goes the other way but we're going to assume that is strictly less than the source is traveling slower than the actual wave but what is this distance remember we're talking about let me do it in orange well this orange reality is what's happened after T sub s seconds you can say so this distance right here that distance right there I'll do it in a different I'll do it in a different color is going to be the velocity of the source it's going to be V sub s it's going to be V sub s times the amount of time that's gone by and I said at the beginning that amount of time is the period of the wave that's the time in question so period of the wave T sub s so after one period of the wave if that's five seconds and it will say after five seconds the source has moved this far V sub s times T sub s and the first wave sort though that first crest of our wave has moved that far V sub W times T sub s now the time that we're talking about that's the period of the wave being emitted so exactly after that amount of time this guy is ready to emit the next crest he has gone through exactly one cycle so he's going to emit something right right now so it's just getting emitted right at that point so what is the distance between the crest that he emitted T sub s seconds ago or hours ago or microseconds ago we don't know what's the distance between this crest and the one that he's just emitting well they're going to move at the same velocity but this guy is already out here while this guy is starting off from the sources position so the difference in their distance at least when you look at it this way is the distance between the source here and this crest so what is this distance right here what is that distance right there well this whole radial distance we already said the this whole radial distance is V sub W the velocity of the wave times the period of the wave from the perspective of the source and we're going to subtract out how far the source itself has moved the floors has moved in the direction in this case so if we're looking at it from this point of view of that wave front so it's going to be minus minus V sub s the velocity of the source times times the period of the wave from the perspective of the so so let me ask you a question if you're sitting right here if you're the observer you're this guy right here you're sitting right over there and you've just had that first crest at that exact moment that first crest has passed you by how long are you going to have to wait for the next crest how long until this one that this this guy is emitting right now is going to pass you by well it's going to have to cover this distance it's going to have to cover that distance let me write this down so the question I'm asking is what is the period from the point of view of this observer that's right in the direction of the movement of the source so the period from the point of view of the observer is going to be equal to the distance that that next pulse has to travel which is that business up there so let me copy and paste that copy and paste so it's going to be that let me get rid of that shouldn't look like an equal sign so let me delete that right over there that shouldn't be look like it or negative sign so it's going to be this distance that that next pulse is going to travel that one is sickening emitted right at that moment divided by the speed of that pulse and/or the speed of the wave and we know what or the velocity of the wave and we know what that is that is V sub W that is V sub V sub W now this gives us the period of the observation now if we wanted the frequency and we can manipulate this a little bit let's let's let's do that a little bit so we could also write this we could also write this we could factor out the period of the source so T sub s we could factor out and so it becomes T sub s times the velocity of the wave minus the velocity of the source minus the velocity of the source all of that over the velocity of the wave and so just like that we've gotten our formula for the observed period for this observer who's sitting right in the path of this moving object as a function of the actual period of this wave source the waves velocity and the velocity of the source now if we wanted the frequency we just take the inverse of this so let's do that so the frequency of the observer so this is how many seconds it take for him to see the next cycle if you want cycles per second you take the inverse so the frequency of the observer is just going to be the inverse of this so if we take the inverse of this whole expression we're going to get 1 over T sub s times V sub W over the velocity of the wave minus the velocity of the source and of course 1 over the period from the point of view of the source this is the same thing and this right here is the same thing as the frequency of the source so there you have it we have our two relations at least if you are in the path if the velocity of the source is going in your direction then we have our formulas and I'll rewrite them just because the period the observed period of the observer is going to be the period from the point of view of the source times the velocity of the wave minus the velocity of the source that's the velocity of the source divided by the velocity of the wave itself the frequency from the point of view of this observer it's just the inverse of that which is the frequency the inverse of the period is the frequency from the point of view of the source times the velocity of the wave divided by the velocity of the wave minus the velocity of the source in the next video I'll do the exact same exercise but I'll just think about what happens to the observer that's sitting right there