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.
