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- [Voiceover] The frequency
that you'll observe

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when standing next to a speaker

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is determined by the
rate at which wave crests

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strike your location.

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If the speaker moves toward you,

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you'll hear a higher frequency.

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If the speaker moves away from you,

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you'll hear a lower frequency.

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But what will happen if
you run toward the speaker?

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You'll hear a higher frequency
because more wave crests

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will strike you per second.

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If you run away from the speaker,

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you'll hear a lower frequency

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because less wave crests
will strike you per second.

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How do we figure out exactly
what frequency you'll hear?

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To find out, let's zoom
in on what's going on.

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Say a wave crest has just
made it to your location.

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The time it takes until
another wave crest hits you

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will be the period that you'll observe

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since that will be the time you observed

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between wave crests.

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If you're at rest,
you'll just have to wait

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until another wave crest
gets to your location.

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The period you'd observe
would be the actual

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period of the wave emitted by the speaker.

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If you're running towards
the speaker, or wave source,

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you don't have to wait
as long since you'll meet

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the next wave crest somewhere in between.

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If you can figure out how long it takes

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for the next crest to hit you,

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that would be the period that
you'd observe and experience.

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Let's say you're moving
at a constant speed

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that we'll call VOBS, for
the speed of the observer.

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The distance you'll
travel in order to reach

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the next crest will be your speed

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times the time required
for you to get there.

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This time is just going to
be the period you observe

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since it'll be the time you
experience between wave crests.

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We'll write the time as TOBS
for period of the observer.

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Similarly, the distance the
next wave crest will travel

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in meeting you, will be
the speed of the wave VW

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times that same amount of time,

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which is the period you are observing.

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Now what do we do?

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We know that the distance
between crests is the actual

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wavelength of the wave,
not the observed wavelength

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but the actual source wavelength

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emitted by the speaker at rest.

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If we add up the distance that we ran

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plus the distance that the next wave crest

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traveled to meet us,

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they have to equal one
wave length in this case.

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We can now pull out a
common factor of TOBS.

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If we solve this for the
period of the observer,

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we find that it will be equal

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to the wavelength of the source,

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divided by the speed of the wave,

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plus the speed of the observer.

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This is a perfectly fine equation

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for the period experienced
by a moving observer

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but one side's in terms of period

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and the other side's
in terms of wavelength.

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If we want to compare apples to apples

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we can put this wavelength
in terms of period

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by using this formula.

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The velocity of the wave must equal

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the wavelength of the source

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divided by the period of the source.

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Since this wavelength
was the actual wavelength

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emitted by the source or the speaker,

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we have to also use the actual
period emitted by the source

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not the observed period.

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If we solve for the wavelength,

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we'd get that the speed of the wave

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times the period of the source

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has to be equal to the
wavelength of the source.

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We can plug in this
expression for wavelength

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and we get a new equation that says

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that the observed period will be equal to

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the speed of the wave times
the period of the source

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divided by the speed of the wave

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plus the speed of the observer.

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This is a perfectly fine equation to find

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the observed period,

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but physicists and other
people actually prefer

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talking about frequency more than period.

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We can turn this statement
that relates periods

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into a statement that related frequencies

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by just inverting both sides,

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or taking one over both sides.

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We'll get one over the observed period

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equals the speed of the wave
plus the speed of the observer

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divided by the speed of the wave

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times the period of the source.

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But look, one over the observed period

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is just the frequency
experienced by the observer.

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On the right hand side
I'm going to pull out

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a factor of one over the
period of the source,

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which leaves the velocity of the wave

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plus the velocity of the observer

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divided by the velocity of the wave.

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For the final step, we
can put this entirely

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in terms of frequencies by noting

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that one over the period of the source

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is just the frequency of the source.

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Phew, there it is.

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This is the formula to find
the frequency experienced

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by an observer moving
toward a source of sound.

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Note that the faster the observer moves,

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the higher the note or pitch.

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This formula only works
for the case of an observer

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moving toward a source.

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What do we do if the observer

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is moving away from the source?

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Let's start all over
from the very beginning.

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Just kidding.

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Since you're running away from the speaker

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instead of toward it, you can
just stick in a negative sign

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in front of the speed of the observer.

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So here we have it, a single
equation that describes

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the Doppler shift
experienced for an observer

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moving toward or away from a
stationary source of sound.

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Use the plus sign if you're moving

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toward the source of the sound

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and use the negative sound
if you're moving away

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from the source of the sound.