Lesson Video: Interference of Light Waves | Nagwa Lesson Video: Interference of Light Waves | Nagwa

Lesson Video: Interference of Light Waves Physics • Second Year of Secondary School

In this video, we will learn how to define wave interference and describe the constructive and destructive interference of waves that are mutually in or out of phase.

15:52

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.

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