Doppler Effect Explained: Moving Source & Listener Derivations with Examples

[00:00:00]you may have noticed that while driving your  car past a loud source of music you can hear the pitch of the music changing higher on the  way toward the Sound Source and lower As you move away this is one example of the Doppler  effect we can illustrate this example with a quick schematic animation where a 400 Hertz Sound  Source is stationary so we see waves radiating away from this source as concentric circles  and we see the location of a listener and The Listener is hearing the 400 Hertz waves as they  arrive Now The Listener starts approaching the source on the way toward the source The Listener  runs into wave crests at a faster Pace leading to a higher perceived frequency of 420 Hertz The  Listener pauses for a bit and we're back to 400 Hertz finally while moving away from the source  the wave fronts hit the listener at a slower Pace leading to a lower perceived frequency  of 380 Hertz this phenomenon is what we'll call the moving listener case of the Doppler  effect in this video in another case of the Doppler effect the source of the waves is moving  instead of the listener so you may have noticed that when a propeller driven airplane flies in  your direction the pitch of the motor sounds higher and as the airplane moves away the pitch  is lower and the same is true for loud car motors or cars playing loud music we can illustrate  again with a quick schematic animation where the listener and Source always lie on the same  line for Simplicity and the source moves toward the listener emitting a 400 Hertz tone we see that  the wavelength of the emitted sound is compressed to a shorter wavelength in the direction of  motion this means the perceived frequency will be higher than if the source was stationary and  in this case our listener hears a 420 Hertz tone similarly the wavelength opposites the direction  of motion is stretched out and this means The Listener hears a lower frequency as the source  of the Waves moves away in this case what we hear is a 384 Hertz tone and so we heard what  it sounds like as the source crosses over from moving toward us to moving away and we're reminded  of the sound of a loud car motor as it passes us this phenomenon is what we'll call the moving  Source case of the Doppler effect in this video so we have two distinct examples of the Doppler  effect and in this video we're going to work out the math for the frequency shift of the sound  waves in the moving listener case then the moving Source case and finally we're going to put  both of these formulas together and get a general formula that covers the simultaneous motion of  both the listener and the source and along the way we'll work a series of simple examples to  put our formulas into practice so first we look at the moving listener case and in the diagram  VL is the speed of the listener V simply V is the speed of sound FS is the frequency emitted by  the source and FL is the frequency of sound heard by The Listener and if our listener is headed  toward the source we expect that to be a higher frequency than the source frequency so what we  want to find is the rate at which the listener runs into wave crests in hertz that's the detected  frequency for The Listener and we start by getting the wavelength of the sound waves and that's just  V over f s that's just a quick modification of the wave speed formula V equals F Lambda and I'll  post a link to where that was first derived so Lambda is the distance between the wave crests and  we can figure out the time it takes the listener to cross that distance from Crest to Crest by  turning around our familiar equation distance equals rate times time and instead writing  time equals distance over rate and notice in the diagram there that the distance between two  wave crests is Lambda that's the wavelength of the wave and here's where the speed of the listener  comes in because the listener is headed at the sound waves the relative velocity is V plus VL  so we can write the time or the period between hitting wavecrests as T equals Lambda that's the  distance between Wavecrest divided by V plus svl that's the relative velocity between the listener  and the sound waves now the frequency heard by The Listener is the reciprocal of this period so  the listener is hearing a frequency of FL equals V plus VL over Lambda finally we replace the  wavelength again with v over f s by using the wave speed formula and I've used the fact that dividing  by a fraction is the same as multiplying by the reciprocal there and we get an equation for the  frequency heard by a listener moving at a speed of VL toward the source now this equation needs  to be generalized to the case where the listener could move away from the sound waves in this case  the relative velocity is lower than the speed of sound in other words it's V minus VL so we put  a plus or minus in the equation to generalize it and now we just have to remember to choose a  plus when we expect the frequency to be higher that's when we're going toward the source and a  minus when we expect the frequency to be lower that's going away from the source so here's  a simple example that covers both the moving toward and the moving away cases we're told we  have a stationary Source emitting a frequency of 440 Hertz we're driving toward that Source at  30 meters per second and then we pass the source and drive away continuing at 30 meters per second  we're told we can use about 340 meters per second for the speed of sound so in part A we're asked  for the frequency that we hear while driving toward the source so we apply our new formula  and we're going to choose the plus here because we're moving toward the source so we have a V  plus VL in the numerator and then divided by V all we have to do is plug in the numbers here  and we have a 440 Hertz Source frequency and then our speed of sound is 340. the speed  of our listener is 30 meters per second and we divide by the speed of sound again that's  340. and when we run the numbers on this to three significant digits we get 479 Hertz now we want  to find the frequency that we hear while driving away from the source so we've already passed it  and we're driving away this just requires that we put the minus sign into the formula instead of  the plus sign because now we expect the listener to hear a lower frequency so we Sub in all  the numbers I have 440 for my source frequency 340 for the speed of sound 30 meters per second  for the speed of the listener and when we run the numbers on this we get 401 Hertz so we can animate  our car real quick and hear what this sounds like Now we move on to the moving Source case where  we have a stationary listener L and the source is moving toward the listener with the speed of vs  again we're using just a plain V for the speed of the sound waves and vs is the speed of the source  FS is the frequency emitted by the source and we're trying to compute the frequency heard by The  Listener that's FL now recall that the wavelength of this sound would be Lambda equals V over FS if  the source was stationary but the source is moving a little bit in between the emission of each  wave Crest so the crest spacing is a little bit closer than that well how far does the source move  between the emission of wavecrests the time is one period so the distance moved is again rate times  time in other words our small shift is going to be the speed of the source times one period of the  oscillation for the tone that it's emitting and period is the reciprocal of frequency so we can  write the this small shift this small decrease in the Wavecrest spacing as the speed of the source  divided by the frequency of the source vs over f s this means the resulting wavelength is actually  a bit shorter so we get the original wavelength if it was a stationary Source minus that small  shift vs over FS these have a common denominator already so we can put them together and we find  out that the wavelength in the direction that the source is moving is V minus vs that's the speed  of sound minus the speed of the source divided by The Source frequency so what sound does the  listener hear with this compressed wavelength well the frequency our listener hears is the speed  of sound divided by the wavelength of that sound so the frequency that's heard here FL is going  to be FS that's the emitted frequency times V the speed of sound divided by V minus vs again we  would have ended up with a plus in this derivation if the source was moving away from us instead of  toward us so we can generalize the formula with a plus minus now we have to remember to choose  the minus when the source is headed toward us that makes the denominator smaller which gives us  a higher frequency and we choose the plus when the source is moving away from us that may makes the  denominator larger and gives us a lower frequency as we expect from the diagrams and animations so  in our next example we have a stationary listener and a moving Source we have a car playing of 440  Hertz tone driving toward us at a constant speed of 45 meters per second and it's going to pass us  and then drive away from us and of course what we want is what frequency do we hear while it's  coming at us what frequency do we hear whilst driving away from us so we write down our new  formula and we pick the minus sign for part A because we know as the car is driving at us  that compresses the wavelength in the direction of the source motion so we're going to hear a  higher frequency putting a minus sign in that denominator guarantees that our herd frequency  is going to be higher than the emitted frequency so we just plug in our numbers here we had an  emitted tone of 440 Hertz the speed of sound again we're going to use about 340 meters per  second and the speed of our source is 45 meters per second and when we run the numbers on this  we get about 507 Hertz next we want to find out what this tone sounds like when the car is driving  away from us so this time we choose the plus sign in the denominator which guarantees the frequency  we hear is lower than the emitted frequency and we plug in our 440 Hertz emitted tone and 340 meters  per second again for the speed of sound and now we use the Plus in that denominator plus 45 meters  per second the speed of the source and this gives us a frequency of 389 Hertz and again we can play  a quick animation to hear what this sounds like finally we consider the case where the listener  and the source are both moving and we're using all the same notations here vs is the speed  of the source VL is the speed of the listener the speed of sound is just a plain V FS is the  frequency emitted by the source and FL is the frequency heard by The Listener and so far we've  discovered these two formulas in the first Formula we get the frequency heard by a moving listener  and in the second formula we get the frequency heard by a stationary listener when the source  is moving so we're going to start by figuring out what frequency a stationary listener would hear  and I'll call that F Prime so we're allowing our source to move and that stationary listener would  hear a frequency of FS times V over V plus or minus vs that's just our old moving Source formula  now we imagine that the listener starts to move so we can take this frequency F Prime and then we  plug it into our first Formula as the new source frequency agency again this Source frequency  is higher than the original emitted frequency because the source is coming at us so now we're  going to substitute in the value of f Prime in terms of the speed of the source and this gives  us a quite long expression but the v's cancel and we can write this as a single fraction and this is  what we usually call the Doppler shift formula it includes the possible motion of the listener and  the source so we have the frequency heard by The Listener is given by The Source frequency emitted  and then multiplied by this fraction in the numerator we have the speed of sound plus or minus  speed of the listener in the denominator speed of sound plus or minus the speed of the source those  plus and minus signs need to be chosen so that the correct things happen qualitatively so if  a listener is moving at the source I know that would tend to increase the frequency that we hear  so I would choose a plus in the numerator and vice versa if the source is moving at us I would choose  a minus in the denominator because that increases the frequency that we hear and if the source  is moving away I would choose a plus because that reduces the frequency that we hear so let's  wrap things up with two quick examples of combined Motion in our first example we're told we have a  car playing a 440 Hertz tone it's driving straight at us with a speed of 45 meters per second and at  the same time we're driving straight back at that car with a speed of 30 meters per second and the  question of course is what frequency are we going to hear so we write down our general Doppler  effect formula the frequency we hear is equal to the source frequency times this fraction where  we have V plus or minus the speed of the listener over V plus or minus the speed of the source and  again we have to choose the correct signs as we go so we have a 440 Hertz Source frequency our  speed of sound again we're going to take that to be about 340 meters per second and then we  have to choose the right sign for the speed of the listener well the listener is coming right  at the source here that tends to increase the frequency that we hear so we're going to choose a  plus in that numerator plus 30 meters per second now in the denominator we have 340 for the speed  of sound and we have to make a choice of what sign to use here well the source is coming at us and  that tends to increase the frequency that we hear so we're going to choose the minus sign because it  makes the denominator smaller and therefore makes the frequency we hear larger and the speed of the  source we're told was 45 meters per second and we end up hearing a frequency of 552 Hertz finally  in the second example we're told that instead of driving straight at the other car we're driving  directly away from it at 30 meters per second and we want to know what frequency we hear now  so we use the same general Doppler formula and again our source frequency is 440 Hertz the  speed of sound is about 340 meters per second but now we're driving away from the source that  tends to reduce the frequency we hear and we're going to use the minus sign in the numerator in  the denominator the source is still chasing after us which tends to increase the frequency we hear  so we use the minus sign in the denominator and it turns out we're going to hear a tone of about  462 Hertz and we're done if you enjoyed this video or at least found it useful check out another one  by clicking one of the links on the left or click the zaxlab logo on the right to explore 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