Video Transcript
Which of the following combinations
of motion of a sound wave source and a sound wave observer produces the largest
Doppler shift of the frequency of the wave as measured by the observer? (A) A sound wave source moves at 30
meters per second in the same direction as an observer moving at 20 meters per
second. (B) A sound wave source moves at 15
meters per second toward an observer moving in the opposite direction to the sound
wave source at 20 meters per second. (C) An observer moves toward a
stationary sound wave source at 30 meters per second. (D) A sound wave source moves
toward a stationary observer at 25 meters per second. (E) A sound wave source moves at 20
meters per second away from an observer moving in the opposite direction to the
sound wave source at 20 meters per second.
All right, so of these five motion
combinations, we want to pick which one results in the largest Doppler shift of the
frequency of the wave as measured by the observer. As we get into this, let’s clear a
bit of space at the top of our screen and remind ourselves about what a Doppler
shift is.
Doppler shifts have to do with
motion relative to a source of waves. In all these examples, this source
of waves is producing sound, audible frequencies to our ear. So let’s say that this blue dot
right here is our sound wave source. And these concentric circles
represent wavefronts produced by that source. A wavefront, we can recall, is a
surface that’s being affected by a wave in the same way. In our case, we’ll say that these
wavefronts represent the crests or peaks of a wave.
What we’ve drawn here is one
snapshot in time. But we know that, in general, sound
waves don’t stand still. When they’re emitted by a point
source, then they move out from that point at the speed of sound, roughly 350 meters
per second. Now, in this scenario, we don’t
just have a sound wave source, but we also have an observer of that source. And let’s say that our observer is
this blue dot over here.
As time passes and the source
continues to produce sound waves, these waves, as we mentioned, will travel outward
from the source, and eventually they’ll reach the observer. When they do, depending on the rate
at which these wavefronts pass by the observer, the observer will detect what’s
called a frequency. In general, frequency is equal to
the number of cycles completed by some object or some system every second. In our situation, the wave
frequency, as measured by the observer, would be equal to the number of wavefronts
that pass by the observer every second.
Now, if both our observer as well
as the source of our sound waves were both stationary, then that would mean the
frequency detected by the observer would be the exact same as the frequency emitted
by the source. But if that were not the case, if
instead the source and the observer were in motion relative to one another, then in
that case, there would be what is called a Doppler shift in the wave frequency
observed by the observer.
Here’s how that works. We’ve mentioned that these
wavefronts are moving out at the speed of sound. Now, let’s say our observer was
also moving away from the source of the sound waves. Not as fast as the sound waves
move, which is very fast, but at some speed less than that. So then, let’s say that the arrows
representing the velocity of our sound waves are much bigger than the arrow
representing our observer’s velocity. With our observer moving away from
the source, this means that fewer of these individual wavefronts pass by the
observer every second of time. And that means that the wave
frequency measured by the observer would decrease compared to when the observer was
stationary.
Now, let’s say that our source is
actually producing sound waves at a frequency we’ll call 𝑓 sub s, the frequency of
our source. And then these sound waves reach
our observer, who measures their own frequency — we’ll call 𝑓 sub o. Here’s what we can say about these
two frequencies, the frequency of the source and the observed frequency. Whenever these two frequencies are
equal, that means there is no Doppler shift. But if they’re not equal, that is,
if these frequencies are different, then a Doppler shift has occurred.
Now, getting back to our question,
our question asked, in which of these five scenarios is the Doppler shift the
greatest? In other words, in which of these
scenarios is the difference between the frequency of the source and the observed
frequency greatest? Wherever that difference is
greatest is where the Doppler shift is largest.
So when the source frequency and
the observed frequency are not the same, what is it that makes that difference
bigger or smaller? The answer to that question is the
relative motion of our source and our observer. We saw earlier that when there’s no
relative motion between our source and observer, that is, when they’re both
stationary, then in that case, there’s no Doppler shift. But when they do start moving
relative to one another, say when our observer starts to move away from the source,
then in that case, the observed frequency is no longer equal to the source
frequency. And, and this is the key, the
greater the relative motion between our observer and our source, the greater the
difference between 𝑓 sub o and 𝑓 sub s. In other words, the greater the
Doppler shift will be.
So then, in all these answer
options, the one we’re looking for is the one that shows the greatest difference
between the motion of the source and the observer. Another way of saying that is that
the relative motion between them is largest.
Now, let’s think about relative
motion for a second. Say that I have one object here and
another object here. Now if this object on the left is
moving to the left at a speed of five meters per second. While this object over on the right
is stationary. Then we could say that the relative
speed between these two objects is five meters per second. And if we were to switch things up
so that our object on the left is stationary but our object on the right is now
moving at five meters per second. Then again, the relative speed
between these two objects would be the same. It’s five meters per second.
True to its name, relative motion
between two objects is not concerned with how either one of the objects is moving
individually. But only how the motion between the
two objects is different. And if there is no difference in
their velocity, say both objects were moving to the right at the same speed, then in
that case, the relative speed between them would be zero. And if one of these objects was a
sound wave source and the other was an observer on that source. Then if the relative motion between
them is zero, as it is in this case, then there will be no observed frequency shift
and therefore no Doppler shift.
Now, one last thing about relative
motion. Let’s say that our object on the
right was moving to the right at five meters per second. But our object on the left was
moving to the left at that same speed. In this case, because the objects
are both moving away from one another, the relative speed between them is five plus
five or 10 meters per second. As we evaluate our answer options,
we’ll want to keep in mind this idea of relative speed between the sound wave source
and the sound wave observer.
Knowing all this, let’s now look at
our answer options again, starting with option (A). In this option, our sound wave
source moves at 30 meters per second in the same direction as an observer moving at
20 meters per second. So let’s draw that out. Let’s say that this is our sound
wave source and this is our observer. We’re told that our sound wave
source moves at 30 meters per second in the same direction as our sound wave
observer, with the observer moving at 20 meters per second.
Now, since these two velocities are
in the same direction, the relative speed between the source and the observer is 10
meters per second. That’s because our source is moving
10 meters per second faster than our observer. But they are moving in the same
direction. So let’s record that result. Let’s say that 𝑟 sub s, what we’ll
call the relative speed between our source and observer, in this case is 10 meters
per second. Because that relative speed is
greater than zero, we know that there will be a mismatch between our source and
observed frequencies. And therefore, there will be a
Doppler shift. The question is, will it be greater
than any other shift in the four other answer options? Let’s find out, moving on to option
(B) now.
This option has a sound wave source
moving at 15 meters per second toward an observer moving in the opposite direction
to the sound wave source at 20 meters per second. So here we have our sound wave
source moving at 15 meters per second toward the observer. And then the observer is moving in
the opposite direction of the sound wave source at a speed of 20 meters per
second. Now, since these two speeds are in
opposite directions. The sound wave moving in this case
to the right and the observer in this case to the left. That means that to find the
relative speed between source and observer, we’ll add together these two speeds: 15
meters per second and 20 meters per second. When we do that, the result we find
from this addition is 35 meters per second. So let’s record that value as the
relative speed between the source and observer in option (B).
Now, because this value is greater
than the relative speed identified in option (A), that means that option (B) tells
us about a greater Doppler shift than option (A) does. That means option (A) is off of our
answer choice list. It can’t involve the greatest
Doppler shift. So we clear away option (A) and
move ahead with our current leader, option (B), with a relative speed of 35 meters
per second.
Now, let’s move on to the next
option. Here, we have an observer moving
toward a stationary sound wave source at 30 meters per second. We can draw that like this. Our source is stationary. It’s not in motion. And our observer is moving toward
the source at a speed of 30 meters per second. We can see then that the relative
speed between our source and our observer is 30 meters per second. But then since 30 meters per second
is less than 35 meters per second, that means the scenario described by option (C)
cannot lead to our greatest Doppler shift. Therefore, this option is
eliminated. And we can clear away this choice
too.
On to the next option, this one
says that a sound wave source moves toward a stationary observer at 25 meters per
second. So then our observer isn’t in
motion, but our sound wave source is, toward the observer at a speed of 25 meters
per second. This means the relative speed
between our source and observer is 25 meters per second, which is less than 35
meters per second. So we’ll also cross off this option
from contention.
Our last scenario describes a sound
wave source that moves at 20 meters per second away from an observer moving in the
opposite direction to the sound wave source itself at 20 meters per second. So that could be drawn like
this. Our source is moving at 20 meters
per second to the left, away from the observer, which is moving at 20 meters per
second to the right, away from the source. Because the source and the observer
are moving here in opposite directions, that means to find the relative speed
between them, we’ll add the speed of each one. 20 meters per second plus 20 meters
per second gives a relative speed between source and object of 40 meters per
second. Since this relative speed is
greater than 35 meters per second, that means the situation described in this option
here leads to a greater Doppler shift than the situation described here in option
(B).
So of our answer options, the one
that leads to the greatest Doppler shift in the observed frequency of these sound
waves is our last option. Which describes a sound wave source
moving at 20 meters per second away from an observer moving in the opposite
direction to the sound wave source, also at 20 meters per second.
