The two waves shown in the diagram have the same frequency and wavelength as each other, but with different initial displacements. If the two waves interfere, which of the other diagrams — A, B, C, and D — best shows how the resultant wave compared to the two waves?
Okay, now in this question, we’ve been given two waves: the orange one and the blue one. And we’re told that they have the same frequency and wavelength as each other. However, they have different initial displacements. All this means is that at any given point in space — let’s say at this point here — there’re very different points of their cycle. For example, this one is very near its trough, whereas this one is very near its peak. That’s all the different initial displacements bit says.
Now, if the two waves interfere, which of the other diagrams — A, B, C, and D — best shows how the resultant wave compares to the two waves? So we haven’t yet put the other diagrams A, B, C, and D on screen. But that’s because we don’t actually need them. We can work out for ourselves what the resultant wave is going to look like. Then, once we’ve done that, we can compare what we’ve got to the diagrams A, B, C, and D.
So let’s go about finding out what this resultant wave is going to look like. Now, it helps to be told that the two waves that we’ve got — the orange one and the blue one — have the same frequency and wavelength as each other because that is a prerequisite in order for us to be able to have interference. For two waves to interfere, they must have the same frequency and wavelength. So that makes life easier for us.
Now, the other thing is in this diagram they’ve been drawn separately. The orange wave has been drawn about here and the blue one has been drawn about here. This is just for the sake of clarity. However, if these two waves are to interfere, then they must overlap. In other words, they must look a little bit like this. They both occupy the same space. They’re not just oscillating in phase with each other. In fact, these two waves are specifically travelling out of phase with each other because whenever we’re at the maximum of one wave, we’re at the minimum of another and vice versa.
This suggests out-of-phase behavior. And what we’d expect to see as a resultant wave is complete destructive interference. What do we mean by that? Well, let’s draw a set of axes so we can work out what the resultant wave looks like. Now, we draw the axis through the centres of these two waves so that we can see that the amplitude of the orange wave for example is the distance between zero displacement which is here and maximum displacement which is here. And the same applies for the blue wave. This is its amplitude.
Now, the diagram shows that the amplitude of the two waves is exactly the same. And since these two waves are out of phase with each other, they should completely cancel each other out. In other words like we said earlier, we’ll get complete destructive interference. So how does that work? Well, what we can do to work out what the resultant wave looks like is to simply add up the waves at each point along this axis.
So let’s take this point here. Well, at that point, the blue wave is at its maximum value, which along the vertical axis. Let’s call this capital 𝐴 for amplitude. And at the same point in space, the orange wave is at its minimum value. Let’s call this negative 𝐴. Because remember the amplitudes of the two waves is the same, so the minimum and maximum value in each case must also be the same, except the minimum value is negative and the maximum value is positive.
So what we do to find out the value of the resultant wave at this point in the centre is simply add up positive 𝐴 and negative 𝐴. Positive 𝐴 plus negative 𝐴 is zero. So at this point, the resultant wave is going to be here. Let’s move on to another point. Let’s say this point here. Well, at that point, the orange wave is equally as high as the blue wave is low down. Let’s call these two values positive 𝐵 and negative 𝐵. And the reason for this is because remember these two waves have the same amplitude, same frequency, and same wavelength and they’re just out of phase with each other.
So we add these two values up to give us a resultant wave of 𝐵 plus negative 𝐵, which once again is zero. And we can repeat this process along the entirety of the horizontal axis. We will find the resultant wave looks something like this. It’s zero constantly. Now of course this is a badly drawn diagram. So there are going to be some errors in it. For example, in this region here, in reality, the blue line and the orange line should cross each other over upon the axis and so the resultant there as well is zero. So once again, the total resultant wave is just going to be a flat line along the horizontal axis.
And so let’s draw that a little bit more clearly. Here’s the orange wave, here’s the blue wave, and here’s the resultant wave. At this point, what we can do is to look at diagrams A, B, C, and D and see which one shows us a resultant of simply a flat line. So let’s bring up those diagrams on screen now and here they are.
Now as we said earlier, we want the resultant wave to be a flat line. Now, these diagrams are showing us the first wave the orange one, the second one the blue one, and what a possible resultant wave could look like, the black one.
So in this diagram, diagram A, this is not a flat line. In diagram B, it is a flat line. In diagram C, it’s not a flat line and same thing with diagram D. So it looks like we’ve got the answer to our question. Diagram B best shows us how the resultant wave compares to the two waves.