Video: Sketching the Resulting Wave due to the Interference of Two Waves of Different Wavelengths

Two waves of different wavelengths move toward each other, as shown in the diagram. Which of the other diagrams, (a), (b), (c), and (d), shows the result of the two waves interfering?

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Video Transcript

Two waves of different wavelengths move toward each other as shown in the diagram. Which of the other diagrams, a, b, c, and d, shows the results of the two waves interfering?

Okay, so in this question, we’ve got wave number one and wave number two moving toward each other. And we need to find which one of these four diagrams, a to d, shows the result of the first two waves interfering. Now, let’s imagine what happens when these two waves move toward each other. We can see that the first one is moving to the right and the second one is moving to the left. And then sometime later, they’re at the point where they’re about to overlap. Here’s the first wave and here’s the second.

Now, what happens when waves overlap and interfere with each other is that the vertical displacement of one wave gets added to the vertical displacement of the other. In other words, if we think about an imaginary horizontal axis along which the waves are moving, then the vertical displacement at each point of each wave is simply the distance away that the wave is from the horizontal axis.

And so coming back to the diagram we were given in the question and drawing a horizontal line, we can see that at, for example, this point, the first wave’s vertical displacement is positive but relatively small. Whereas, at this point, the first wave’s vertical displacement is the maximum possible displacement that it will have. And it’s also positive. And we can think about the same thing for the second wave. We can see that, for example, this point has a relatively small but negative vertical displacement, whereas this point has the maximum possible vertical displacement in the negative direction. And of course, this is under the implicit assumption that this direction is positive and this direction is negative.

We could even assign these points’ vertical displacement values, if we were to draw in a vertical axis. And we were to put a little arrow on the horizontal axis and call it, let’s say, the 𝑥-axis. The vertical axis could then be the 𝑦-axis. And at that point, it’s very easy to see how the vertical displacement at every point would have a certain 𝑦-value. And the 𝑦-value would represent the vertical displacement away from the horizontal axis.

So we can now imagine what would happen if this centre point of the first wave and this centre point of the second wave move together until they were overlaying each other. So this centre point moved here and was at this point now. And this centre point moved this way and also was in the same position. So here is the centre point of the first wave. And let’s draw the second wave overlapping that but in pink. So here’s the second wave where its centre point is now at exactly the same position as the centre point of the first wave.

And now, what we have to do to find the result of these two waves’ interference is to simply add the vertical displacements at every single point along the line. So if we say that this now is the horizontal axis, the dotted blue line, then from here all the way to here, the vertical displacements of both waves is zero, because they both lie on the horizontal axis. And the same is true from here all the way to here.

The interesting stuff is going to happen in this region. So let’s zoom in slightly to that region. Now we can see that actually the first wave stays at the same zero displacement for quite a while longer, whereas at all of these points, the pink wave has some negative displacement. So if we were to find a resultant displacement, then, at all of these points, it would simply be zero plus whatever the negative displacement here is.

So the resultant wave is actually going to follow the pink wave. It’s going to look exactly like the pink wave, until we get to this point here because at that point now, what we have is some negative value due to the pink wave. But now, to these values we’re adding the positive vertical displacement of the orange wave. And so the resultant wave is going to look something like this. And having realised this, we then get to a point where the vertical displacement of the orange wave becomes zero once again. And so the resultant wave simply becomes a sum of zero plus whatever the vertical displacement of the pink wave is.

In other words, it follows the pink wave. And so the resultant wave, which we’ve drawn in blue, looks something like this. And remember, that’s assuming that the centre points of the two waves are exactly overlapping. But the point is that, out of the diagrams a, b, c, and d, only diagram c shows us what we’d expect to see when these two waves interfere with each other. Therefore, that’s our final answer to this question.

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