Video Transcript
In the figure shown, which of the resultants corresponds to the resultant of the interfering waves?
Here, we see two waves, one in red and one in blue, that interfere with one another. Below these are three possible resultants of that interference. When two waves interfere with one another, their displacements combine at every point in space to yield a resultant wave. One way we can figure out which of these three resultants is correct is to look at the locations of their peaks.
Note that the peaks of the green wave occur here, those of the purple wave at these locations, and these are the peaks of the black wave. These peaks we see are all at different locations along the length of the wave. Therefore, if we can figure out where the peaks of these two interfering waves will be generated, we’ll be able to identify which of the three resultants actually represents that resulting wave.
To get oriented, let’s draw a line that represents the displacement of zero on our two waves that interfere. Any wave value above this line has a positive displacement, and any wave value below it has a negative displacement. If we were to mark out the peaks of our blue wave, those peaks would be here, here, and here. Similarly, those of the red wave are at these spots.
The resultant wave, the wave that results from adding together the displacements of the red and blue wave at each point in space, will have peaks that are right in the middle of each pair of blue and red dots. Note that these dots are not separated by a significant fraction of the wavelength of these waves. These waves, we say, are interfering nearly completely constructively.
In between each pair of blue and red dots, the red and blue waves cross over one another. That is, they have an intersection point, here, here, and here. At these points, the displacements of each wave are the same. Therefore, to find the resulting displacement at these three points, we take the distance between each one of these points and our displacement equals zero axis and we double it. That will be the resulting wave displacement.
These three green dots represent the resulting displacement of the wave after combining the red and blue wave at these points along the vertical gray dashed lines. Because these points are midway between the peaks of the respective waves we’re adding together, they represent the maxima, the peaks, of the resulting wave. Half of the wave resulting from the red and blue waves interfering would look like this.
And looking along the vertical gray dashed lines, we see that the peaks of our resultant wave line up with the peaks of resultant wave (A). Therefore, this will be our choice for the way that corresponds to the resultant of the red and blue interfering waves. Note that we would have found the same answer if, instead of looking at the peaks, we have looked at the values or troughs of these waves. Either approach confirms that it’s resultant (A) that corresponds to the resultant of the interfering waves.
