[Music] section 16.9 the doppler effect this is something you've all heard at some point the doppler effect is the change in frequency or pitch of the sound detected by an observer because the sound source and the observer have different velocities with respect to the medium of the sound propagation so for instance if you've uh had a fire engine that's making noise right like it's putting its siren out there you can hear that in all directions the sound goes out and the condensations are shown as these circles right where those are the sound going out we can imagine being a single pitch the wavelength is the distance between those condensations so if the truck is stationary you just everywhere around you hear the same thing you hear that siren if that fire truck is moving you start to hear a different thing depending on where you're standing if you're standing behind the fire truck then because it's moving away from you the wavelength appears to be longer longer wavelength based upon our basic equation that the speed of the wave is equal to the frequency times the wavelength the speed of the wave is set just by the error so if the wavelength goes up that means the frequency needs to go down because it's a constant speed so as the true the siren goes by you or even just a car drives by you hear this frequency drops versus if you are in front of the fire truck as it's coming towards you it's moving and it's catching up with those condensations so they're getting closer together or compressed in that case the wavelength is getting smaller so what happens to the frequency it's going to go up that's what you hear as it starts to approach where it goes right that increase in frequency is the doppler effect and we can actually quantify it in physics so we'll first look at a moving source like this fire truck so at rest it produces this normal wavelength lambda if it's moving then the wavelength is modified and the modification is going to depend on how fast the truck is moving so that's what this is trying to show that it's the original wavelength minus the speed of the source times the time uh the period of it which we could instead make into frequency and so the frequency that's observed by the stationary observer is equal to the speed of sound divided by this modified wavelength so the modified wavelength lambda minus vs period instead of working with wavelength and period we're going to switch to frequency because we're talking about sound and we usually describe sound and frequency so the lambda becomes the speed of sound divided by the frequency of the source and then vs is the speed of the source right so that's an important note here our basic v is the speed of sound versus the speed of the source and the vs is the speed of source so if we then do some algebra we can cancel out one of those v's one of the speeds of sound and we can solve for the frequency observed that is equal to the frequency of the source what the fire truck what environment or fire woman would detect times 1 minus the speed of the source divided by the speed of sound this is when the truck is moving towards the observer right it turns out if instead the source is moving away from the stationary observer we switch from having a negative sign to a positive sign in the denominator so that's a key detail here if the source is moving toward we have this negative sign down here if it's moving away from then we have a positive sign here so these are important notes to include with these equations so you can keep them straight because that's one of the most important parts of solving these problems and we can see that in this next example so example 10 the sound of a passing train a high-speed train is traveling at a speed of 44.7 meters per second that is fast when the engineer sounds the 415 hertz warning horn right boop the speed of sound is 343 meters per second where the frequency and wavelength of the sound is perceived by a person standing at the crossing when the train is a approaching b leaving the crossing so remember we need to decide for approaching and crossing all the to get or leaving the crossing all that's going to change is the positive or negative sign so remember approaching is the negative sign and leaving moving away from is the positive sign once we have that we are in good shape right we already have our speed of sound which is the v variable we have the speed of the train which is our source of sound so that would be our vs and then the 415 hertz which frequency is that is that fo or fs this is what is generating the noise this is the source so i'd call that fs and then perceived by the person who is not at the source that is the observer so that is what we are trying to solve for so now we can plug and chug so we find that approaching is the minus case we can plug in those numbers and approaching it's 477 hertz if instead the train is leaving we still plug in 415 hertz times 1 divided by 1 plus 44.7 divided by 343 meters per second and this the frequency is now 367 hertz notice this matches our common experience right relative to the frequency of the source 415 when it's approaching the frequency is higher when it's leaving the frequency is lower [Music] right so those are our two extremes there if we then wanted to get the wavelength of the sound we could go to our classic v is equal to frequency times the wavelength and so then we could plug in and solve for the wavelength and it would just be the speed of sound divided by the frequency that is observed and that would work right so this describes if the source is moving but with the doppler effect just occurs any time the source or the observer is moving relative to the medium so the other case we have is if we have a moving observer so here's a case where the observer is moving toward a stationary source the source is now not moving but the person is you'll still have a doppler effect here so as the observer moves towards they encounter the crests more frequently and the frequency goes up i won't go through the derivation here but the frequency observed is now the source frequency plus their speed divided by the speed of sound so it's a slightly higher frequency so this is our grand take away observer is moving towards the stationary source we have this positive sign it is moving away from the stationary sources the observer is now a minus sign because this is reducing the frequency as you're moving away from it so this is something you could hear if you yourself drove by say a fire truck but the fire truck is stationary you're now moving you could hear the frequency change as you're moving towards you're going to hear a higher frequency as you're moving away you'll hear a lower frequency you could sum all four of these equations up into one massive equation if you want so the most general case is that the observed frequency is equal to the source frequency times the quantity of one plus or minus the speed of the observer divided by the speed of sound divided by one minus plus the speed of the source divided by the speed of sound so in the numerator the plus sign applies when the observer moves toward the source that's the default in the denominator the minus sign is the default that applies when the source moves toward the observer and if either one is not moving either the observer or the source then that second term would disappear like if the observer's not moving this term disappears and you're left with the equation that we already had so this isn't anything new but if you like it it's a more compact way to think about it so i'll leave that there for you to check out another thing to check out this is an optional video but this is a favorite and it highlights the doppler effect it's the apparent change in frequency of a wave caused by relative motion between the source of the wave and the observer so there you have it that is the doppler effect you can hopefully watch that video i'll add that to the playlist and then there's some cool sound applications to the medical field in the very last video so that is the end of 16.9 the doppler effect