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## The Doppler effect

# Doppler effect for a moving observer

## Video transcript

- [Voiceover] 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. 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. If you run away from the speaker, you'll hear a lower frequency because less wave crests
will strike you per second. 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 observed between wave crests. If you're at rest,
you'll just have to wait until another wave crest
gets to your location. The period you'd observe
would be the actual period of the wave emitted by the speaker. If you're running towards
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 we'll call VOBS, 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. This time is just going to
be the period you observe since it'll be the time you
experience between wave crests. We'll write the time as TOBS
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. Now what do we do? 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. 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
wave length 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. This is a perfectly fine equation for the period experienced
by a moving observer but one side's in terms of period and the other side's
in terms of wavelength. 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'd get that the speed of the wave times the period of the source has to be equal to the
wavelength of the source. 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. This is a perfectly fine equation to find the observed period, but physicists and other
people actually prefer talking about frequency more than period. We can turn this statement
that relates periods into a statement that related frequencies by just inverting both sides, or taking one over both sides. We'll get one 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, one over the observed period is just the frequency
experienced by the observer. On the right hand side
I'm going to pull out a factor of one 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. For the final step, we
can put this entirely in terms of frequencies by noting that one over the period of the source is just the frequency of the source. Phew, 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. 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? 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 sound
if you're moving away from the source of the sound.