The figure shows five light waves. Which light wave is not coherent with the other four?
To determine if light waves are not coherent, we first have to know what it means when light waves are coherent. To do so, we have to compare two different properties of the waves: the frequency and the phase difference. In order for two or more light waves to be coherent, they must have the same frequency and a constant phase difference. If either of these are untrue, then whatever wave we’re looking at will be incoherent.
With this in mind, when we look at this diagram over here, we notice that we’re not given any numbers, just general waveforms. This is because it’s possible to determine if waves have a similar frequency or a constant phase difference just by looking at them. For instance, these two waves here are coherent. They have the same frequency and a constant phase difference. We will get into the specifics for how to determine these properties, but for now let’s just look at them. They look pretty similar, don’t they? The same frequency, a constant phase difference, the only thing that’s different between them is the height of the waves or the amplitude, which luckily does not play a part in whether waves are coherent or not.
But now let’s consider these two waves. Even though they look pretty different, they are still coherent. So while general observation can be helpful, we can’t always rely on it for determining if waves are coherent. We should start by looking at and thinking about these two properties of waves: frequency and phase difference. The frequency of a wave tells us how frequently a wave oscillates over a given period of time. This means that higher-frequency waves will have more complete wave cycles than lower-frequency waves, which allows us to, most of the time, determine which waves have a higher or lower frequency just by looking at them.
But again, this can be confusing when we look at waves with different phase differences. Even though they have the same frequency, they don’t really line up with each other. In these cases, in order to make sure they really do have the same frequency, we should look at the starting points of the waves and the ending points of the waves and then determine how many complete wave cycles are in between those points. This wave at the top has two complete wave cycles with the exact midpoint marking the end of the first wave and start of the second wave.
This wave starts at mid height going upwards and ends at mid height going upwards. So it completes one full wave cycle, just like this third wave, which we see starts at a peak and ends at a peak. So there is also one complete wave cycle here. And when two waves share the same number of wave cycles over the same period in time, it means that they have the same frequency.
With this in mind, let’s now look at the five waves in our diagram and see if any of them have a different frequency. If we just generally observe all of them at first, we may notice that wave (iv) is a little more stretched out than the other waves. But just to be sure, let’s also count the total number of wave cycles that it has, which happens to be eight over this period of time. This differs from the total number of wave cycles in, say, wave (iii), which happens to be about 16.5. And indeed, if we were to measure all of the waves, we would find that waves (i), (ii), (iii), and (v) all have the same number of complete wave cycles, 16.5, which means that wave (iv) has a different frequency than the other waves, which means it can’t meet the same frequency requirement of coherent waves and by extension cannot meet the constant phase difference requirement either.
This is because when waves have different frequencies, their phase relative to each other is always changing. There may be a few points where the phase happens to line up, but coherent waves need a constant phase difference, not one that just happens to line up occasionally. So then wave (iv) has both a different frequency and a nonconstant phase difference compared to the other waves. We only need one property to not match in order for these waves to be incoherent. But in the case of wave (iv), both don’t match.
And in the cases of waves (i), (ii), (iii), and (v), they all have the same frequency and they all have a constant phase difference with each other. We already proved that these waves have the same frequency since they have the same number of complete wave cycles. But just to make sure that these four waves really are coherent, let’s check for a constant phase difference.
The way we can do this for any sets of waves is by lining them up and seeing if they have the same phase at the same points in time. This is to say, do they all have peaks at the same points in time? Do they all have valleys at the same points in time? And what about the midpoints? Do those line up too? If all of our waves have the same phase at all of our chosen points, even the strange ones in between the points we would typically measure, then it means that they have a constant phase difference with each other. When we have multiple waves like this, determining if they are coherent can often be as simple as asking, do the waves look similar except for amplitude? If so, then they are likely coherent.
But it’s always still good to check specifically. And we can’t always rely on just observation when there are only two waves we’re looking at because, for example, these two waves actually are coherent with each other. This is because they not only have the same number of complete wave cycles, two, but also they have a phase difference of 90 degrees that is constant at all points. The bottom wave is always 90 degrees ahead of the top wave at all points, which means that the phase difference is constant and these waves are thus coherent. But this sort of strange scenario only happens when there are two waves.
Since the phase difference must be constant between all three waves, and so while the first two may have a constant phase difference of 90 degrees, and in this case the second and third also having 90 degrees, the first and third have a constant phase difference of zero degrees. Meaning that there is not a constant phase difference amongst all three waves. The only way that they can all have a constant phase difference is if the phase difference between all of them is a constant zero degrees, which is what we see for waves (i), (ii), (iii), and (v), which are all coherent. Wave (iv) is not coherent since it has a different frequency and a nonconstant phase difference.
So out of these five light waves, light wave number (iv) is not coherent with the other four.