Video Transcript
The figure shows five light
waves. Which light wave is not coherent
with the other four?
When we are looking for light waves
that are coherent with each other, it means that we’re looking for light waves that
have the same frequency and a constant phase difference. Now, determining if waves have the
same frequency can be done through observation. Do they look the same or different
when passing the same fixed point over the same period of time? These three waves all have
different frequencies, since they are all different from each other. But sometimes a difference in phase
can make it difficult to tell if two waves really do have the same frequency. These waves on the left share the
same frequency with their counterparts on the right.
And the way that we can determine
that this is correct is by looking at how many complete wave cycles these waves have
and comparing them to each other. This first wave starts at a
midpoint going up. So a complete wave cycle start to
finish would be about here, meaning it has one and what looks like one-half of a
complete wave cycle. Its partner to the right has a wave
start at the peak of a wave, meaning that the wave ends at the next peak in the
wave, for one and a half of a complete wave cycle. Whenever waves passing the same
fixed point over the same period in time cover the same number of complete wave
cycles, it must mean that they have the same frequency, provided of course that they
have a consistent frequency, not one that changes as the wave moves.
With this in mind, looking at the
five light waves in the diagram, we see that waves i, iii, iv, and v all start at
the bottom of a valley, meaning that the start and endpoints of each of these wave
cycles would be at the bottom of these valleys. For wave number ii, we see that the
wave starts at a midpoint moving down, meaning that is the area that we would take
for determining the start and end of a complete wave cycle. And we find that for all five of
these waves, they all have the same number of complete wave cycles, four.
So now we just have to determine if
these waves have the same phase difference. We certainly know that the second
wave has a different phase, since the starting and ending points to determine its
wave cycles are different than the other four. As a reminder, phase is typically
measured in degrees or radians and refers to specific points along the path of a
typical sine wave. For simplicity, we’re just going to
show the values in degrees. The beginning of a wave is zero
degrees, peaks are 90 degrees, the midpoint of a wave going down is 180 degrees,
valleys are 270, and ends of waves are 360. Or since the end of a wave is the
beginning of a new wave, they can also be considered zero degrees.
Using these phase measurements, we
find that since the first wave in the diagram starts at the bottom of a valley, we
can say that it starts at a phase of 270 degrees. And the same is true for waves iii,
iv, and v. Wave ii, however, looks like it’s
starting at about halfway on a wave going down, meaning that it’s starting with a
phase of 180 degrees. Now, at first, it may seem like the
second wave, since it has a different starting phase than the other waves, must not
be coherent. But we’re not looking for a
constant phase. We’re looking for a constant phase
difference, which just means that the difference in phase between two waves has to
be constant.
And using the starting phases for
wave i and wave ii, we find this difference to be 90 degrees. And since these two waves are
consistent in their waveforms, and especially because we know they already share the
same frequency, we can assume that the phase difference is the same for all of the
points on the waves. But just to be sure, let’s measure
the phase difference between these two points. This point at the top of the wave
is 90 degrees, and this point on wave ii is 360 or zero degrees.
When we take the difference between
these two points, we want to get an answer that is not negative. So, rather than doing 90 minus 360,
which would give us negative 270 degrees, we will instead treat it as zero degrees,
meaning that the phase difference between these two points is 90 degrees, which
matches with the original starting points, meaning that there is definitely a
constant phase difference, which means that wave i and wave ii are coherent with
each other. And since waves iii, iv, and v are
just the same as wave i, except for a difference in amplitude, and remember
amplitude does not affect coherency, this means that wave ii is coherent with wave
iii, iv, and v, too, but only independently.
While wave ii and any other wave
will have a constant phase difference of 90 degrees, the other waves, excluding ii,
will have a constant phase difference of zero degrees with each other, which means
that while there may be a constant phase difference between any two of these waves,
there won’t be a constant phase difference between three or more waves when they
include wave number ii.
So waves i, iii, iv, and v are all
coherent with each other at the same time, since they share the same frequency and
have a constant phase difference of zero degrees, which means when determining the
coherency of all five of these light waves at once, wave ii is the light wave that
is not coherent with the other four.
