Let’s have a sound wave travelling rightward. Figure (a) shows an imaginary snapshot of the air molecules at one instant of time. You can see that there are regions of high density denoted by C (compression) and regions of low density denoted by R (rarefaction).
Figure (b) is a displacement-position (Δx–x) graph showing the displacement of air molecules (from their equilibrium positions) at different positions. By convention, a positive displacement represents a rightward displacement, and a negative displacement represents a leftward displacement.
Figure (c) shows the pressure at each position along the wave. Note that the pressure at compressions and rarefactions are only slightly above and below the atmospheric pressure. Even for the noise level of a rock concert, we are talking about a pressure variation of a few pascals only.
Now let’s try to make sense of the Δx–x (displacement-position) graphs.
First, if all the air molecules are undisturbed, they will all be sitting at their equilibrium positions and they will be spaced out evenly. No compression, no rarefaction, just constant atmospheric pressure.
Now there is a sound wave passing through. Air molecules are displaced from their equilibrium positions in such a way that alternating regions of compressions and rarefactions are produced.
Compression at position 5
So why is there a compression at position 5? Because the molecule at position 5 has displacement zero, but its neighbor on the left (position 4) has a positive (rightward) displacement, while its neighbor on the right (position 6) has a negative (leftward) displacement. As a result, there is a higher concentration of air molecules at position 5, resulting in high pressure.
Rarefaction at position 9
So why is there a rarefaction at position 9? Because the molecule at position 9 has displacement zero, but its neighbor on the left (position 8) has a negative (leftward) displacement, while its neighbor on the right (position 10) has a positive (rightward) displacement. As a result, there is a lower concentration of air molecules at position 10, resulting in low pressure.
Position 3 and 7
How about at position 3? Notice that even though the molecule at position 3 is displaced rightward, so are its two neighbors on either side (position 2 and 4). The net result is there is no change in the concentration of air molecules at position 3. The pressure at position 3 is still atmospheric pressure.
It’s the same thing at position 7. Even though the molecule at position 7 is displaced leftward, so are its two neighbors on either side (position 6 and 8). The net result is there is no change in the concentration of air molecules and the pressure at position 7 remains at atmospheric pressure.
Cross-over Points and Turning Points
Actually, like so many things in physics, once you get it, it becomes obvious. (1) Compression and rarefaction can only happen at the zero points in the Δx–x graph, because that’s where the displacement is opposite in signs on either side of the zero point, corresponding to air molecules either congregating at or dispersing from these positions. (2) Average pressure occurs at the amplitude points in the Δx–x graph, because at those turning points, the molecules are all displaced in the same direction by the same amount, meaning the spacing between them remain unchanged.