Coupon Accepted Successfully!


Resonating Cavities

In the last chapter we looked at the sound produced by a plucked guitar string or a struck piano string. Now we will look at resonating pipes, like organ pipes. While the resonating cavity of a soft drink bottle or of an oboe are more complicated than the pipes in this section, the principle behind all these pipes is the same. Standing waves are set up in the cavities, and these produce sound of a particular pitch and timbre.

A closed pipe is a pipe closed at one end and open at the other. If we excite the air column, the air in the pipe vibrates longitudinally. The variable x gives the location along the length of the pipe, and Δx gives the tiny displacement an air particle can have. (See Figure 12-6.) The double arrow shows the air particle moving back and forth. Since its equilibrium point is in the middle, the distance from one side to the other side of the displacement is 2Δx.

..\art 12 jpg\figure 12-tf.jpg

Figure 12-6

At the closed end, the air cannot move back and forth, while it is completely free to do so at the open end. Thus the closed end is a displacement node, and the open end is an antinode. Compare this with the vibrations in Section 11.G, which had nodes at both ends. Any graph we draw for a closed pipe must have a node on one end and an antinode on the other. The fundamental has no nodes in the middle of the pipe away from the ends (Figure 12-7).

..\art 12 jpg\figure 12-tg.jpg

Figure 12-7


What is shown ends up being one fourth of a wave. A full wave looks like this:

..\art 12 jpg\figure 12-tq.jpg


The graph shows the first fourth: from zero point to maximum. Thus the full wave is four times the length of the pipe.


A boy blows across the top of a bullet casing (a cylinder closed at one end, open at the other) which is 0.03 m long. What is the frequency of the note he hears (the fundamental)? (The speed of sound is 343 m/s.)



The fundamental mode is shown in Figure 12-7, and the wavelength is λ = 4 (0.03 m) = 0.12 m. Thus f = v/λ = 2860 Hz. (At a different temperature the sound speed will be different.)

Some hints for drawing these diagrams appear at the end of Section 11.G. For closed pipes, each successive harmonic has one additional node.

Now try drawing the second harmonic without looking at Figure 12-8.

For the second harmonic we have drawn three fourths of a wave, so L = 3/4 λ, and λ = 4/3 L. The next harmonic is shown in Figure 12-9, but try to draw it also without looking. What is λ? (Did you get 4/5 L?)

Let's go back and look at the fundamental. Note this peculiar fact: If we are thinking in terms of displacement of air particles, then the node is at the closed end and the antinode is at the open end. If we are thinking in terms of pressure variation (see the beginning of the chapter), then the closed end is the antinode and the open end is the node. Figure 12-10 shows the fundamental in both cases. The frequency we calculate comes to the same, of course. You should check this point.

..\art 12 jpg\figure 12-th.jpg ..\art 12 jpg\figure 12-ti.jpg
Figure 12-8 Figure 12-9
..\art 12 jpg\figure 12-tj.jpg ..\art 12 jpg\figure 12-tk.jpg
Figure 12-10 Figure 12-11


An open pipe is open at both ends, like an organ pipe (Figure 12-11). If we consider displacement of air particles, then both ends are antinodes. We find that we cannot draw a mode with no nodes in the middle of the pipe, since such a mode would make no sound. Thus the fundamental has one node (Figure 12-12).

..\art 12 jpg\figure 12-tl.jpg
Figure 12-12

If we consider pressure variation, then both ends are nodes, and we can draw a mode with no nodes in the middle. Try drawing these graphs yourself without looking. They represent the fundamental and the second and third harmonics. Draw also the graphs for the fourth harmonic.


Test Your Skills Now!
Take a Quiz now
Reviewer Name