The diagram shows a longitudinal wave traveling along a spring and four displacement–time graphs. If positive displacement corresponds to compression of the spring, which of the graphs correctly shows the change in displacement with time of the wave on the spring?
Looking at the diagram, the spring is at the bottom, and we know there’s a longitudinal wave traveling along it. Recall that longitudinal waves are defined as having oscillating motion parallel to the direction of propagation. This means that as this spring here carries energy or propagates horizontally, the spring itself oscillates or wiggles back and forth horizontally as well. The directions of propagation and oscillation are parallel because it’s a longitudinal wave. This is in contrast to a transverse wave, like the one shown here in orange. Notice that as this wave propagates horizontally, it oscillates vertically with the medium wiggling up and down as the wave itself travels left or right.
In the diagram, the oscillating motion of the spring is shown by alternating areas of compression and rarefaction. Recall that rarefaction is the opposite of compression as it refers to the regions of a longitudinal wave where the medium is more spread out or less dense. In a spring like we have here, rarefaction is shown where the loops of the spring or more stretched out and farther apart from each other. On the other hand, compression is where the medium is more dense. This can be seen in the diagram where the loops of the spring are all scrunched up and close together.
Now in this question, we want to model the spring’s compression and rarefaction as a wave on a displacement–time graph. So it’s our job to identify which of the graphs A, B, C, or D does this correctly. The graphs show some vertical displacement about some equilibrium position, which is represented by a horizontal black line for each graph. We’ve been told that positive displacement should indicate compression in the spring. So it’ll be helpful to identify the points in the spring that experience the greatest compression. We can see that the spring is most compressed at these three points highlighted in orange. Therefore, the correct graph will show points of maximum positive displacement that line up with these points on the horizontal or time axis.
Notice that both graphs A and B actually show no displacement at these points where the spring’s compression is greatest. This is incorrect. So let’s eliminate these options. Next, let’s look at graph C, specifically at these points on the time axis that we’ve highlighted which show the greatest compression of the spring. C does show displacement at these points, but it’s negative displacement. This goes against what we were told, that positive displacement corresponds to compression of the spring. Therefore, C is incorrect as well.
This leaves us with D, which actually does show positive displacement at the moments when the spring is most compressed. Likewise, the graph properly shows maximum negative displacement at the points when the spring is least compressed, or most rarefied. Therefore, we know graph D correctly shows the change in displacement with time of the longitudinal wave traveling along the spring.