Doppler effect for a moving observer

the frequency that you'll observe when standing next to a speaker is determined by the rate at which wave crests strike your location if the speaker moves toward you you'll hear a higher frequency and if the speaker moves away from you you'll hear a lower frequency but what will happen if you run toward the speaker you'll hear a higher frequency because more wave crests will strike you per second and if you run away from the speaker you'll hear a lower frequency because less wave crests will strike you per second but how do we figure out exactly what frequency you'll hear to find out let's zoom in on what's going on say a wave crest has just made it to your location the time it takes until another wave crest hits you will be the period that you'll observe since that will be the time you observe between wave crests if you're at rest you'll just have to wait until another wave crest gets to your location and the period you'd observe would be the actual period of the wave emitted by the speaker but if you're running toward the speaker or wave source you don't have to wait as long since you'll meet the next wave crest somewhere in between if you can figure out how long it takes for the next crest to hit you that would be the period that you'd observe and experience let's say you're moving at a constant speed that will call vo BS for the speed of the observer the distance you'll travel in order to reach the next crest will be your speed times the time required for you to get there but this time is just going to be the period you observe since it will be the time you experience between wave crests so we'll write the time as T OBS for period of the observer similarly the distance the next wave crest will travel in meeting you will be the speed of the wave VW times that same amount of time which is the period you are observing but now what do we do well we know that the distance between crests is the actual wavelength of the wave not the observed wavelength but the actual source wavelength emitted by the speaker at rest so if we add up the distance that we ran plus the distance that the next wave crest traveled to meet us they have to equal one wavelength in this case we can now pull out a common factor of Tobs if we solve this for the period of the observer we find that it will be equal to the wavelength of the source divided by the speed of the wave plus the speed of the observer so this is a perfectly fine equation for the period experienced by a moving observer but one sides in terms of period and the other sides in terms of wavelength so if we want to compare apples to apples we can put this wavelength in terms of period by using this formula the velocity of the wave must equal the wavelength of the source divided by the period of the source since this wavelength was the actual wavelength emitted by the source or the speaker we have to also use the actual period emitted by the source not the observed period if we solve for the wavelength we get that the speed of the wave times the period of the source has to be equal to the wavelength of the source so we can plug in this expression for wavelength and we get a new equation that says that the observed period will be equal to the speed of the wave times the period of the source divided by the speed of the wave plus the speed of the observer so this is a perfectly fine equation to find the observed period but physicists and other people actually prefer talking about frequency more than period so we can turn this statement that relates periods into a statement that relates frequencies by just inverting both sides or taking 1 over both sides and we'll get 1 over the observed period equals the speed of the wave plus the speed of the observer divided by the speed of the wave times the period of the source but look 1 over the observed period is just the frequency experienced by the observer and on the right hand side I'm going to pull out a factor of 1 over the period of the source which leaves the velocity of the wave plus the velocity of the observer divided by the velocity of the wave and for the final step we can put this entirely in terms of frequencies by noting that 1 over the period of the source is just the frequency of the source so whoo there it is this is the formula to find the frequency experienced by an observer moving toward a source of sound note that the faster the observer moves the higher the note or pitch but this formula only works for the case of an observer moving toward a source what do we do if the observer is moving away from the source okay well let's start all over from the very beginning just kidding since you're running away from the speaker instead of toward it you can just stick in a negative sign in front of the speed of the observer so here we have it a single equation that describes the Doppler shift experienced for an observer moving toward or away from a stationary source of sound use the plus sign if you're moving toward the source of the sound and use the negative sign if you're moving away from the source of the sound