Video: Understanding the Conditions of Constructive and Destructive Interference

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. 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?

06:18

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.

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