Video Transcript
In this lesson, we are going to
discuss the interference of light waves. Although we’ll be focusing on
light, most of the concepts that we’ll cover will actually be relevant to all kinds
of different waves. So let’s get started by defining
interference. Interference is the result of two
waves overlapping in space so that at each position in space, their displacements
add. We’ll be particularly interested in
the interference that result in two special cases, when the displacement of the two
waves have the same sign at each position in space and when the displacement of the
two waves have opposite signs at each position in space.
Before we discuss those cases,
let’s see what it means to add the displacements of two waves. We’ll illustrate this by
considering two pulses that are initially apart and then move towards each other and
begin to overlap. To help us keep track of everything
in this discussion, we’ve labeled a set of axes on the side over here. The horizontal axis is
position. So the horizontal location of the
wave on the screen actually represents its position in space. The vertical axis is
displacement. So the vertical location of a point
of the wave on the screen represents the displacement of the wave at that position
in space.
Lastly, we’ve drawn the dashed line
to represent the point where the displacement is zero. So all of the displacements above
this line are positive. And all of the displacements below
this line are negative. Let’s now see what happens when we
allow these two pulses to come together and overlap.
Here, we’ve drawn the two pulses at
an instant in time where their troughs completely overlap. Recall that the troughs of a wave
are the portions near where the displacement reaches a minimum. These are the portions of the wave
that look like troughs or valleys. The corresponding parts of the wave
that look like hills are called crests. Let’s now return to our two pulses’
overlap and try to find the resulting displacement at this position in space. To do this, we must add the
displacements of the two pulses. If we knew the numerical value of
these displacements, we could add them like we add any two numbers.
Since we don’t know the numerical
value of these displacements, let’s do the addition graphically. First, we’ll draw an arrow
representing the displacement of each of the pulses at the position we’re interested
in. We put the tail of our arrow at the
line of zero displacement and the head of our arrow at the displacement of the
pulse. The arrow we just drew represents
the blue pulse’s displacement at the position we’re interested in. The arrow for the orange pulse’s
displacement at this position would be the same, since the displacement itself is
the same. When we draw the arrows this way,
the size of the arrow represents the magnitude of the displacement, while the
direction of the arrow represents the sign. Where arrows pointing downward
represent negative displacements and arrows pointing upward represent positive
displacements.
To add these two arrows, we then
simply move one of them so that the tail aligns with the head of the other
arrow. Since we already have the orange
arrow drawn, let’s add the blue arrow. We start with the tail of the blue
arrow at the head of the orange arrow and then simply draw the blue arrow. The head of the blue arrow now
points to the resulting displacement of the position we’re interested in. The resulting displacement has a
larger magnitude than the two initial displacements because the two initial
displacements have the same sign, so their magnitude’s added.
We’ve now found the resulting
magnitude at a single point. We can find the resulting magnitude
everywhere there’s an overlap by applying the same procedure. The parts of the pulses that
overlap have the same displacement at each position. So the total displacement wherever
they overlap will just be twice the displacement of one of the pulses. In the two regions where there is
no overlap, the resulting pulse will just look like the initial pulses. We have thus found the total shape
from adding the displacements of two overlapping pulses at a particular instant in
time. Now that we know how to add
displacements to get a resultant wave, let’s turn to our special cases. First, let’s see what happens when
we overlap two pulses where the displacements of each pulse have the same sign at
each position in space.
The easiest way to do this is with
two identical pulses. So let’s overlap another pulse on
top of the pulse that we’ve already drawn. Now, let’s start adding
displacements like we did before. Just like our previous example, the
orange arrows representing the displacement at each position will be exactly the
same as the blue displacement arrows at the same positions. Shifting each orange arrow so that
its tail matches up with the head of the corresponding blue arrow, we find the
resulting displacement from this overlap at several positions in space. When we apply this method to the
entire pulse, we find that we get a resulting pulse with a shape similar to that of
the initial pulses but with an enhanced amplitude.
This enhancement in amplitude
occurs because the displacements of the two pulses have the same sign at each
position in space. In other words, the crests align
with crests, and the troughs align with troughs. Note that although the wave now has
a larger amplitude, there are still portions of the wave that have zero
displacement. The reason for this is that at the
points with zero displacement, the initial pulses had zero displacement. And zero plus zero is still
zero. We call this kind of interference,
where crests align with crests and troughs align with troughs resulting in an
amplitude that is enhanced, constructive interference.
Now, let’s consider a second
special case of interference, where the crests of one wave align with the troughs of
another wave. Here, again, we have a blue
pulse. This time, though, we’ll overlap an
orange pulse so that the crests of the blue pulse align with the troughs of the
orange pulse and vice versa. Let’s see what happens when we now
try to add the displacements. This time, though, at every
position where the orange displacement arrow points downward, the blue displacement
arrow points upward with the same magnitude. And everywhere where the blue
displacement arrow points downward, the orange displacement arrow points upward with
the same magnitude.
This means that when we start
drawing the arrows with the tails of the blue arrows aligned to the heads of the
orange arrows, we’ll find that whenever we draw an orange arrow pointing upwards,
the blue arrow will point straight back down to zero. And whenever we draw an orange
arrow pointing downwards, the blue arrow will point straight back up to zero. In fact, this will be true at every
position along the two waves, and the resulting displacement at every position in
space will be zero.
This happens because the
displacements of each wave have opposite signs at each position in space. In other words, the crests of one
wave align with the troughs of the other and vice versa. This results in a wave with
diminished amplitude relative to the two interfering waves. In our specific case, the
diminished amplitude is zero. In other words, the waves have
totally canceled out. We call this kind of interference,
where crests align with troughs and the amplitude is diminished, destructive
interference. If the crests of one wave do not
fully align with the crests of another wave, then the interference between them is
neither constructive nor destructive. Let’s now examine another condition
called the phase difference that can characterize whether the interference between
two waves is constructive or destructive.
To help us understand this
quantity, we’ve drawn two zero displacement lines and put a blue wave on one and an
orange wave on the other. These two waves are really
overlapping. We’ve just drawn them separately
for the sake of visual clarity. Recall that the distance associated
with one complete cycle of the wave’s displacement is called the wavelength and is
usually given the symbol 𝜆. As we can see, the wavelength is
also the distance between successive crests of the wave.
If we look at the crests of the
second wave, we can see that they’re the same distance apart as the crests of the
first wave. So the second wave and the first
wave have the same wavelength. Because our main focus is light
waves. And anyway, because for two waves
to interfere, they must be the same type of wave and be located at the same position
in space. Because these two waves have the
same wavelength, they also must have the same frequency. We can see, however, that even
though the waves have the same wavelength, their crests are not aligned. In our diagram, this misalignment
is one-quarter of one wavelength and is the same for every orange peak and the blue
peak immediately behind it.
We call this distance the phase
difference. And we define it as the distance
between the crests or troughs or really any two corresponding points of two
waves. We could also define a phase
difference in terms of angles if we assign a phase to each point of the two waves,
for example, by letting all of the crests have zero degrees. And then that phase cycles through
360 degrees one full cycle around a circle by the time we get to the next crest. To find the phase difference, we
would then simply subtract the phases of the two waves at a particular position in
space. In this case, that would give us 90
degrees because 90 degrees is one-quarter of 360. So 90 degrees and one-quarter of a
wavelength are both one-quarter of a full cycle of the wave.
At any rate, the only reason we
have a well-defined phase difference in our diagram at all is because the two waves
have the same wavelength and the same frequency. If the waves had different
wavelengths, the phase difference that we calculated from one pair of crests
wouldn’t match the phase difference we calculated from a different pair of
crests. However, in such a situation, the
interference between the two waves would also neither be constructive nor
destructive. As a result, it is worthwhile to
try to match phase difference to either constructive or destructive
interference. Because situations of purely
constructive and purely destructive interference should also have well-defined phase
differences.
Let’s now redraw our diagram to
show waves that would interfere constructively and try to find the associated phase
difference. We’ve now redrawn the orange wave
so that if the blue wave and orange wave overlapped, they would interfere
constructively. And we can see this is the case
because the crests would align with crests and the troughs would align with
troughs. Let’s take a look at the associated
phase difference. The distance between the positions
where the orange wave peaks and where the blue wave peaks is zero because those
peaks occur at the same positions in space. So the phase difference between
these two waves is zero.
The Greek letter 𝜙 is often used
when discussing phases. So symbolically, we can write 𝜙
two minus 𝜙 one, the phase of the second wave minus the phase of the first wave, to
represent the phase difference. Using this, we can write the
condition for constructive interference between two waves as the phase difference 𝜙
two minus 𝜙 one equals zero. Let’s now find the phase difference
associated with destructive interference. This time, instead of crests that
align with crests and troughs that align with troughs, we have crests than align
with troughs and troughs that align with crests. Let’s take a look at the phase
difference.
For these two waves, the phase
difference is one-half of one wavelength. And we can also see that, in this
particular case, it doesn’t matter if we measure from the orange peak to the blue
peak immediately before or immediately after it, since in either direction the
distance is half a wavelength. We can thus write as the condition
for destructive interference that the phase difference between the two waves is half
a wavelength. Any other phase difference between
zero and a full wavelength that is neither zero nor half a wavelength would result
in interference that is neither wholly constructive nor wholly destructive.
For phase differences one
wavelength or greater, the fact that the wave repeats its shape every wavelength
means that we can subtract whole numbers of wavelengths from the phase difference
without affecting the relative alignment of the crests of the two waves. So the only phase differences we
need to concern ourselves with are those that are between zero and a full
wavelength, since all other phase differences are equivalent to one of those. Let’s now apply all of these ideas
to an example.
Two waves with the same wavelength
and frequency move in the same direction, with one wave leading the other by one
whole wavelength. Is the interference between the
waves constructive, destructive, or neither constructive nor destructive?
Let’s start by recalling the
definition of interference, specifically constructive and destructive
interference. Interference results when waves
overlap so that their displacements add. If crests align with crests and
troughs align with troughs, the interference is called constructive and the
resulting amplitude is enhanced. On the other hand, if the crests
align with troughs and the troughs align with crests, the interference is called
destructive and the resulting amplitude is diminished compared to the interfering
waves.
Let’s draw a picture of our two
waves to see if they correspond to constructive or destructive interference or
neither. Here, we’ve drawn one of our
waves. The dashed line represents zero
displacement. We’ve also labeled the wavelength
of this wave, that is, the distance associated with one full cycle of
displacements. Let’s now draw our second wave on
this dotted line so that the two waves have the same wavelength and frequency and
also one of the waves leads the other by one whole wavelength. Note that the fact that the waves
move in the same direction just tells us that whatever condition we find at one
particular instant in time will persist for all instants in time.
To draw a wave with the same
wavelength and frequency as the one we’ve already drawn, we simply need to draw a
wave with the same shape. To draw it so that one wave leads
the other by a whole wavelength, we simply need to start drawing our wave one
wavelength after we started drawing our first wave. So our first wave starts along this
vertical line. And we’ll start our second wave
along this vertical line, one whole wavelength later. So here’s our second wave. It has the same shape as the first
wave, but it is delayed in starting by one wavelength. Let’s now check on the alignment of
the crests and troughs to determine what kind of interference is present.
As we can clearly see, the crests
of the first wave align with the crests of the second wave. And the troughs of the first wave
align with the troughs of the second wave. And this is true at all of the
corresponding crests and troughs that we’ve drawn in this picture. Crests aligning with crests and
troughs aligning with troughs is the condition for constructive interference. So the interference between these
two waves is totally constructive. This occurs because waves repeat
their shape every wavelength.
Therefore, when we shifted our
second wave over by one whole wavelength, we didn’t actually change the relative
position of crests and troughs from what would have been had we not shifted the wave
at all as represented by this dotted curve. But had we not shifted the wave at
all, the waves would have been identical and clearly interfered constructively. Which confirms that, indeed, the
interference between these two waves is constructive.
Now that we’ve seen an example,
let’s summarize what we’ve learned in this lesson. In this lesson, we spoke about
interference, which occurs when two waves overlap and their displacements add at
each position in space. To help us characterize the
interference between two waves, we talked about the phase difference between
them. The phase difference is the
distance between corresponding points on the two waves, usually measured between
crests. We can also use angles to represent
the phase difference, where one wavelength corresponds to 360 degrees and some
fraction of a wavelength corresponds to the same fraction times 360 degrees.
In either case, it’s only
meaningful to define a constant phase difference between two waves when they have
the same frequency and wavelength. In such a situation, we can also
talk about two different kinds of interference. There’s constructive interference,
where the crests of one wave align with the crests of the other, as do the
troughs. This results in a wave with an
enhanced amplitude relative to the amplitudes of the original waves. Because the displacements have the
same sign at each position in space. We can also state this condition as
the phase difference between the waves being zero. Since this means that the waves
align exactly and so the crests align with crests and the troughs align of
troughs.
The other type of interference that
we can find for waves of the same wavelength and frequency is destructive
interference, which results when the crests of each wave align with the troughs of
the other wave. This gives the resulting wave a
diminished amplitude compared to the amplitudes of the interfering waves. Because the displacements of the
two waves have opposite signs at every position in space. And finally, the phase difference
corresponding to destructive interference is one-half of a wavelength, which causes
the crests to be aligned with troughs and the troughs to be aligned with crests.
