Video Transcript
The diagram shows the wave fronts
of two waves that have been diffracted through equally narrow gaps. Both waves have the same speed,
wavelength, frequency, and initial displacement as each other.
Before we get to our questions,
let’s take a look at this diagram. We see in this diagram the left gap
as well as the right gap which are openings in his barrier for light waves to travel
through. Now even though these two different
light waves have different colours, we’re told that they have the same speed,
wavelength, frequency, and initial displacement as one another. This means that for all intents and
purposes, the light travelling to the left gap is identical to the light travelling
to the right gap. After the light makes it through
these gaps, it diffracts; it spreads out. And we see that it starts to mix,
the light from the left gap with the light from the right gap.
In this diagram, a particular
notation is used to represent wave fronts of this diffracted light. Each wave front is represented
using a line. And the curvature of this line
shows where the wave is travelling. Along with all this, we have these
four points 𝐴, 𝐵, 𝐶, and 𝐷 marked out on the sketch. In this exercise, we’re going to
answer questions about each one of these points starting out with point 𝐴. Regarding this point, we want to
know how many wavelengths of this light is the left-hand gap from point 𝐴. How many wavelengths of this light
is the right-hand gap from point 𝐴? Is the interference between the two
light waves at point 𝐴 constructive or destructive?
As we start answering these
questions, let’s clear some space on screen. Now in this first question, we’re
asked how many wavelengths of this light is the left-hand gap from point 𝐴. An important point to realise here
is that the light that goes through the left gap and the light that goes though the
right gap is the same light. So again, even though these waves
are coloured differently, they have the same wavelength, frequency, phase, and so
on. So when the question asks about
this light, it’s referring simply to the light in the diagram which is all the same
light.
Knowing that, we want to know how
many wavelengths of this light is the left-hand gap from the point marked out as
𝐴. Looking at our diagram, we see
where this point is. It’s located on a wave front for
the waves coming from the left gap and the waves coming from the right gap. And it’s at this point that we can
remember an important fact about wave fronts. And that is that the distance
between adjacent wave fronts that is two wave fronts that are next to each other is
equal to a wavelength.
Going back to our diagram then,
that means that point 𝐴 which is on the second wave front of the light that went
through the left gap is a distance of one wavelength and then two wavelengths from
that left-hand gap. That’s because we have two complete
wave front cycles between that point and the gap. So that’s our answer to this first
question. There are two wavelengths of light
from the left-hand gap to point 𝐴 in the diagram.
Moving on to the next question,
this one asks how many wavelengths of this light is the right-hand gap from point
𝐴. To figure this out, we’ll do
something similar. We see that point 𝐴 is on the
second wave front of the light that came through the right-hand gap. And that means for this light as
well, there are one and then two complete wavelengths between that gap and point
𝐴. Once again then, our answer is
two. That’s the number of wavelengths of
this light the right-hand gap is from point 𝐴.
And then we have this last question
about point 𝐴: is the interference that occurs there constructive or
destructive? To answer this question, we can
recall something helpful about wave interference. We can recall that when the crest
of one wave overlaps with the crest of another wave, then that wave interference is
constructive. On the other hand, when the crest
of one wave overlaps with the trough of another wave, then that is destructive
interference. So looking again at point 𝐴, we
see that this point lies along the wave crests of both the light from the left-hand
and the light from the right-hand gaps. So we have crest overlapping with
crest. By our definition, that’s
constructive interference.
Now that we’ve answered these
questions about point 𝐴 in the diagram, let’s answer the same questions but this
time about point 𝐵 rather than point 𝐴. So looking at point 𝐵 on our
diagram, we first want to know how many wavelengths of the light the left-hand gap
is from this point. To figure this out, we can again
count wave fronts. Starting from the left-hand gap, we
have one, two, three, four wave fronts, which means there are four wavelengths of
this light from that gap to point 𝐵.
Next, we want to answer the same
question, except now our start point is the right-hand gap. Once again, we’ll count the wave
fronts starting from this gap. We count one, two, three, four wave
fronts. And that tells us that there are
four wavelengths of the light from the right-hand gap to point 𝐵.
Next, we want to know whether the
interference at point 𝐵 is constructive or destructive. We see this point is at the overlap
of two wave crests. Therefore, the interference at this
point is constructive. That finishes point 𝐵. So now let’s answer these questions
about point 𝐶.
Locating this point on the diagram,
we see this it’s on a crest of a wave coming from the left gap. But it’s in between crests of the
wave coming from the right gap. So answering this first question,
how many wavelengths of light is the left-hand gap from this point, if we count wave
fronts starting from that gap, we count one, two, three. But then moving on to the question
about how many wavelengths this point is from the right-hand gap, if we count wave
fronts once more, we get one, two, three, and then three and a half to point 𝐶.
The fact that point 𝐶 is three and
a half wavelength from the right-hand gap tells us that for the wave coming from
this gap, this point, point 𝐶, is at a low point, a trough of that wave. This means when we consider the
last question is the interference at this point constructive or destructive, we see
that point 𝐶 lies along a crest of the wave coming from the left-hand gap. But it lies along a trough of the
wave coming from the right-hand gap. In other words, we have wave crest
overlapping with wave trough. This is destructive
interference.
And finally, we’ll do this all once
more with point 𝐷. Starting at the left-hand gap, if
we count wave fronts, we count one, two, three, four to point 𝐷. That’s the number of wavelengths of
light from that gap to the point. And then from the right-hand gap,
we count one, two, three wave fronts. Notice though that point 𝐷 lies
along a wave crest of both waves. Therefore, we have crest
overlapping with crest and the interference is constructive.
