### Video Transcript

Which of the waves shown corresponds to the resultant of the other two waves?

Here, we have three waves: wave C, wave B, and wave A. Two of these waves can be combined so that the resultant is the remaining wave. In other words, we want to test whether we could add together waves C and B and get wave A or add together waves B and A and get wave C or combine wave C with wave A and get wave B. Those are the three possibilities.

To get started, let’s note that the peaks of all three of these waves line up vertically. That is, a vertical line that passes through the peaks of any one of these waves passes through a peak of the other two. Not only do these waves have the same wavelength then, they are also in phase. We do see, however, that there is a difference between these waves. If we were to draw horizontal lines representing a displacement of zero for each of our three waves, note that the amplitude of wave A, the wave in black, is just about equal to the amplitude of wave B, the wave in red, but that the amplitude of wave C is significantly larger. In fact, the amplitude of wave C looks like it’s about double that of waves B and A. This is a clue to our answer.

We know that when two waves combine to form a resultant wave, the displacement of that resultant wave equals the combined displacements of the two interfering waves. So, if along one of these vertical orange lines, say this one here, we measure the displacement of the black wave and the displacement of the red wave, if we add those two displacements together, we would just about get the displacement of the blue wave at that same point. We could try other points along the wavelength to confirm what we’re finding. Say that we look, for example, along this vertical dashed line. Along this line, the displacement of the black wave is zero. That of the red wave is also zero, and the blue wave displacement is zero as well. So far then, it’s looking like if we add waves B and A together, then we get wave C.

Just to make sure this is the case, let’s test one more point. Now, let’s look along this vertical dashed line, one that passes through minima of these curves. Here, the displacement of the black curve has this value. The red wave has a displacement of this value. And if we add those two negative displacements together, we would get about this negative displacement depicted on wave C. We have our answer then. Of these three waves, wave C corresponds to the resultant of the other two waves, wave B and wave A.