Video Transcript
In the figure shown, which of the
resultants corresponds to the resultant of the interfering waves?
At the top of our screen, we see
these two waves, one in red and one in blue, that interfere. This means the waves combine in
such a way that each wave’s displacement at every point in space is added to the
other wave’s displacement at every point. For each wave, this dashed orange
line represents a displacement of zero. When these waves combine, we want
to know which of these three candidate resultants A, B, and C is the actual
resultant of the interference of these waves. Notice that the peaks, the maxima,
of waves A and B and C are all at different locations. That is, the peaks of resultant A
here, here, and here do not line up with the peaks of resultant B here, here, and
here or those of resultant C. All these peaks are at different
locations in space.
One way to figure out which of
these resultants is correct is to see where the peaks of these two interfering waves
are created. We might think that those peaks are
located, say, at the peaks of one of the two waves that are combined together. In fact, the peaks of the resultant
wave will occur in between these pairs of blue and red dots. To see this more clearly, let’s
focus in on one of these such pairs. We see that the peak of the red
wave in this box occurs along the gray vertical dashed line. At the same point in space, the
value of the blue wave is down here. That means when we add these two
wave values together, the displacements of the two waves at this point, we would get
a value very nearly equal simply to the peak of the red wave at that point. The displacement of the blue wave
is nearly zero, so it contributes almost nothing. And in fact, over at the peak of
the blue wave, something similar happens. The displacement of the red wave at
this point is nearly zero. So, the combined displacement of
the two waves is just about the displacement of the blue wave.
Now, let’s look at the values of
these waves in between these two green dots. Specifically, we’ll look at where
the waves cross one another. At that point, both waves have the
same displacement. If we add those displacements
together, we get a value of about this much. This displacement is the distance
between our axis and our two dots doubled. We see that this peak is clearly
higher than the peaks of either of our individual waves on either side. And we could expect our resultant
wave to follow a shape like this in between these points.
We’ve identified the location of
one of the peaks of our resultant wave. Between the other pairs of red and
blue dots, the peaks would be in a similar location. If we draw vertical lines down
through these peaks, we see that they align with those of resultant A, but not with
those of resultant B or resultant C. Therefore, we know which resultant
to pick as the actual resultant of our two interfering waves. It’s resultant A.
