### Video Transcript

Which of the following statements correctly defines what is meant by incoherent light? (A) Two or more light waves are incoherent if they have different frequencies or nonconstant phase differences. (B) Two or more light waves are incoherent if they have the same frequency and constant phase differences. (C) Two or more light waves are incoherent if they have different amplitudes. (D) Two or more light waves are incoherent if they have both different frequencies and different amplitudes. Or (E) two or more light waves are incoherent if they have both different amplitudes and nonconstant phase differences.

In order to make room for some diagrams, we’re going to cut out the first portion of each question, which is the same for all of them. Now then we’re looking for the properties that would make two or more light waves incoherent. The answers we are given talk about these different properties, one of which is frequency, which tells us how frequently a wave oscillates and can also be thought of as the inverse of wavelength, with low-frequency waves having a large wave length and high-frequency waves having a small wavelength.

Two or more light waves are incoherent if their frequencies or wavelengths are different. So these two waves here, for example, are incoherent with each other. However, it is still possible to have waves with the same frequency be incoherent with each other due to a phase difference. To see how this can be, let’s consider these three waves here. They all have the same frequency, which we know because they all consist of one complete wave cycle over the same period in time. Despite this though, these three waves are not coherent with each other due to their phase differences.

A phase difference is a difference in waveform amongst multiple waves at the same point in time. Waves one and two start by going up, while wave three starts by going down. Now if we were to extend the beginning of wave three, we would find the same point that waves one and two start at. The difference between this point and the actual starting point of wave three is approximately one-quarter of a complete wave length. This is the phase difference of wave three. When we compare it to either wave two or to wave one, the phase difference of a wave can also be expressed in degrees. Instead of saying one-quarter of a complete wavelength, we can just say 90 degrees. And if there is no difference in phase, like what we see between waves one and two, then that phase difference is zero degrees.

But even if there is a phase difference between waves, it does not always mean they are automatically incoherent. What actually matters is if there is a constant or nonconstant phase difference. To see what we mean, let’s just look at waves two and three. We know that they have a phase difference of one-quarter of a complete wavelength or 90 degrees with each other. But what matters is not this initial difference in phase but whether that phase difference is held constant over the entirety of both waves, which essentially means if the initial points on the wave have a phase difference of 90 degrees, do all of the other points in time have that same difference as well?

Yes, when you have two waves with a constant frequency with each other, they will also have a constant phase difference at all of the points in time. So this does not hold true for waves that are highly irregular or do not hold a constant frequency over the entirety of the duration we’re measuring them. But waves two and three do have a consistent frequency, meaning that they have a constant phase difference. And when two waves have the same frequency and a constant phase difference, it means they are coherent with each other, which incidentally, since we’re looking for where waves are incoherent, rules out answer (B).

But that is just for two waves. Let’s consider now what a constant phase difference means, when we have three waves. Bringing back wave one here, which remember has the same frequency as waves two and three, we recall that the phase difference between waves one and two is zero degrees, which is a constant phase difference since both of these waves have the same frequency and are consistent. However, we also know that the phase difference between waves two and three is a constant 90 degrees. Zero degrees is not equal to 90 degrees, which means while any two of these waves may have a constant phase difference, all three of them together have a nonconstant phase difference, which means even though they all have the same frequency, they are incoherent with each other.

This is because in order for two or more light waves to be coherent, they must have the same frequency and a constant phase difference. These are the only two requirements. And notably, amplitude does not play a part in whether or not waves are coherent at all. This means that any of the answers involving amplitude cannot be the correct answer. So answers (C), (D), and (E) cannot be correct. So in order for two or more light waves to be incoherent, they must have a different frequency or a nonconstant phase difference.

This “or” is important because as we saw it’s possible for two or more waves to have the same frequency but a nonconstant phase difference between them. So the statement that correctly defines what is meant by incoherent light is (A) two or more light waves are incoherent if they have different frequencies or nonconstant phase differences.