- [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.