### Video Transcript

The following five functions can be used to model five light waves. Which of the five waves is not coherent with the other four? (A) π¦ equals sin π₯, (B) π¦ equals two sin π₯, (C) π¦ equals eight sin π₯, (D) π¦ equals 0.5 sin π₯, and (E) 0.75 sin π₯ over two minus π over two.

Here weβve drawn out each of the light waves modeling the five functions, and we can clearly see that they are different from each other. But not all of these differences will matter when weβre looking at whether the light waves are coherent with each other, since the only properties that coherent light waves have to share are having the same frequency and a constant phase difference. Notably, amplitude is absent from this list. It is not a factor at all for determining whether waves are coherent or not. So then to determine the frequency and phase difference for all five of these functions, we can use this equation which mathematically describes a wave: π¦ is equal to π΄ sin ππ₯, where π΄ is the amplitude of the wave and π represents the frequency.

Looking back at the five functions, we see that all of them have a different value of π΄, meaning that they all have a different amplitude, with function (A) having a value of one since one times sin π₯ is just sin π₯. But amplitude doesnβt matter when looking at whether waves are coherent. What really matters is the value of π which represents the frequency. For functions (A), (B), (C), and (D), we see that there is no value in front of π₯, meaning that the π-value for each of these is just equal to one, since one times π₯ is of course just π₯. The only exception for the π-value is with function (E) with an π₯ over two, meaning that the π value is equal to one-half. So because function (E) has a different frequency from all of the other functions, it must mean that function (E) is not coherent.

But just to really make sure, letβs also look at the phase difference that it has. To determine if there is a phase difference when looking at this wave equation, we want to look for any additions or subtractions inside of the brackets of the sine function, which we are representing as the Greek letter π. This π is the phase difference. And since waves are only coherent if they have a constant phase difference between all of them, then it must mean that this value of π has to be the same for any coherent waves. Functions (A), (B), (C), and (D), we already know, have the same value of π, frequency. But since there is also no addition or subtraction going on between the brackets of the sine function, it means that they all have a phase difference of zero, since π₯ plus zero is of course just π₯.

But when we look at function (E), we see that it has a phase difference of negative π over two, which is not constant with the other phase differences of the other functions. So function (E) not only has a different frequency, but also a nonconstant phase difference. Either one of these properties by themselves would be good enough to say that it is not coherent with the other four waves, but together it is unambiguous. Function (E) is not coherent with the other four.