A long, narrow column of air, such as the air in a tube or pipe, can support a longitudinal standing sound wave. Longitudinal waves are somewhat trickier than string waves because a graph—showing displacement parallel to the tube—is not a picture of the wave.

To illustrate the ideas, FIGURE 17.13 is a series of three graphs and pictures that show the $m=2$ standing wave inside a column of air closed at both ends. We call this a closed-closed tube. The air at the closed ends cannot oscillate because the air molecules are pressed up against the wall, unable to move; hence a closed end of a column of air must be a displacement node. Thus the boundary conditions—nodes at the ends—are the same as for a standing wave on a string.

FIGURE 17.13 The $m=2$ standing sound wave in a closed-closed tube of air.

Although the graph looks familiar, it is now a graph of longitudinal displacement. At $t=0$, positive displacements in the left half and negative displacements in the right half cause all the air molecules to converge at the center of the tube. The density and pressure rise at the center and fall at the ends—a compression and rarefaction in the terminology of Chapter 16. A half cycle later, the molecules have rushed to the ends of the tube. Now the pressure is maximum at the ends, minimum in the center. Try to visualize the air molecules sloshing back and forth this way.

FIGURE 17.14 combines these illustrations into a single picture showing where the molecules are oscillating (antinodes) and where they’re not (nodes). A graph of the displacement $\text{Δ}x$ looks just like the $m=2$ graph of a standing wave on a string. Because the boundary conditions are the same, the possible wavelengths and frequencies of standing waves in a closed-closed tube are the same as for a string of the same length.

FIGURE 17.14 The $m=2$ longitudinal standing wave can be represented as a displacement wave or as a pressure wave.

It is often useful to think of sound as a pressure wave rather than a displacement wave, and the bottom graph in Figure 17.14 shows the $m=2$ pressure standing wave in a closed-closed tube. Notice that the pressure is oscillating around $p_{\text{atmos}}$, its equilibrium value.

We showed in 《 SECTION 16.6 that the pressure in a sound wave is minimum or maximum at points where the displacement is zero, and vice versa. Consequently, the nodes and antinodes of the pressure wave are interchanged with those of the displacement wave. You can see in Figure 17.13 that the gas molecules are alternately pushed up against the ends of the tube, then pulled away, causing the pressure at the closed ends to oscillate with maximum amplitude—an antinode—at a point where the displacement is a node.