In each of the following diagrams, five light waves are shown. Which of the diagrams shows incoherent light?
Whether waves are coherent or incoherent depends on two factors: the frequency of the waves and the phase difference of the waves. Specifically, for two or more light waves to be coherent, they must have the same frequency and a constant phase difference. Other properties of the waves, like height or amplitude, do not matter for determining whether or not they are coherent.
So then we want to look at each of the five waves in these five diagrams to determine which set of waves is incoherent, which is to say “Which set of waves do not have the same frequency or a constant phase difference?” We can sometimes look at waves and determine these properties just through observation. For instance, these two waves are pretty similar-looking, aside from their amplitude. And it’s because they have the same frequency and a constant phase difference, meaning that they are coherent with each other.
But simple observation isn’t enough. Very tiny changes in frequency or phase difference can mean the difference between whether waves are coherent or not. So don’t rely on just observation. In order to accurately compare the frequency of waves when we’re just given the waves with no numbers, we look at how many complete wave cycles there are over the same period of time and compare them to each other. In this case, both of these waves consist of one complete wave cycle over the same period of time, meaning that they must have the same frequency.
Now let’s look at finding phase difference, which can be a little bit trickier. This is because, while phase differences can be pretty obvious sometimes, they can also be subtle. A good way to determine if multiple waves have a constant phase difference between all of them is to choose some point on the wave usually peaks, because they’re easy to spot, and then determine if those parts of the waves occur at the same point in time, which they don’t in this case for these three waves. The third wave has a slight phase offset, meaning that there is not a constant phase difference amongst all three of these waves. So these three together are incoherent.
With all of this in mind, let’s start looking at the light waves in the diagrams, starting with diagram (A). When first looking at all of these waves, we may notice that they have different amplitudes. But again, this does not matter for whether or not waves are coherent, just frequency and phase difference. So to determine the frequency for these waves, let’s count each wave’s number of complete wave cycles. We’ll find, when we finish, that each wave has eight complete wave cycles, meaning that these waves all have the same frequency.
Now for determining whether they have a constant phase difference between them, let’s look at the peaks. At the same point in time, all of the peaks line up, and not just the first peaks, but all of the other peaks as well, which indicates a constant phase difference. It’s important to look at more than one point in time when determining whether waves have a phase difference or not, since even waves with a phase difference can just happen to line up sometimes. But this is not the case here; all of these waves are coherent, meaning that diagram (A) does not show incoherent light.
So let’s look at (B) now. Diagram (B) is a case where just observation can actually be enough. We don’t have to look very close to determine that this is just the same wave repeated five times, consisting of 12 complete wave cycles, and of course lining up at any points that we choose. The waves in diagram (B) are all coherent.
So let’s look at (C). All of these waves consist of approximately 11 and one-quarter complete wave cycles, meaning that they have the same frequency. And all of the peaks lining up indicates a constant phase difference, meaning the waves in diagram (C) are coherent.
So we’re going to have to now look at diagram (D). The waves in diagram (D) all consist of the same number of complete wave cycles, about 3 and a half. And any points that we choose all line up at the same point in time, meaning a constant phase difference. So just like the last three diagrams, the light in diagram (D) is coherent.
So we must now turn our attention to diagram (E). When we first look over all of these waves individually, we may notice that the third wave has a tiny discrepancy in where it starts. The other four waves all start at about midheight of the wave going down, while the third wave starts up just a little bit higher and also ends a little bit higher too. And when we measure out the number of complete wave cycles for these waves, we find that this third wave has 12, while all of the other waves have 15. This third wave does not have the same frequency as the other four. This is why it’s important to count the number of complete wave cycles because otherwise that would have been tough to spot.
This diagram also demonstrates why it’s important to look at more than one point in time when determining if there is a phase difference, since the peaks of the other four waves very nearly line up with the third one here. But over here, they clearly don’t. Even though it is just the third wave that has a different frequency and nonconstant phase difference from the others, all it takes is one wave to make an entire set of waves incoherent. So because of this third wave having a different frequency and a nonconstant phase difference, diagram (E) is the one that shows incoherent light.