Air columns closed at both ends are of limited interest unless, as in Example 17.4, you are inside the column. Columns of air that emit sound are open at one or both ends. Many musical instruments fit this description. For example, a flute is a tube of air open at both ends. The flutist blows across one end to create a standing wave inside the tube, and a note of that frequency is emitted from both ends of the flute. (The blown end of a flute is open on the side, rather than across the tube. That is necessary for practical reasons of how flutes are played, but from a physics perspective this is the “end” of the tube because it opens the tube to the atmosphere.) A trumpet, however, is open at the bell end but is closed by the player’s lips at the other end.

You saw earlier that a wave is partially transmitted and partially reflected at a discontinuity. When a sound wave traveling through a tube of air reaches an open end, some of the wave’s energy is transmitted out of the tube to become the sound that you hear and some portion of the wave is reflected back into the tube. These reflections, analogous to the reflection of a string wave from a boundary, allow standing sound waves to exist in a tube of air that is open at one or both ends.

Not surprisingly, the boundary condition at the open end of a column of air is not the same as the boundary condition at a closed end. The air pressure at the open end of a tube is constrained to match the atmospheric pressure of the surrounding air. Consequently, the open end of a tube must be a pressure node. Because pressure nodes and antinodes are interchanged with those of the displacement wave, an open end of an air column is required to be a displacement antinode. (A careful analysis shows that the antinode is actually just outside the open end, but for our purposes we’ll assume the antinode is exactly at the open end.)

FIGURE 17.15 shows displacement and pressure graphs of the first three standing-wave modes of a tube closed at both ends (a closed-closed tube), a tube open at both ends (an open-open tube), and a tube open at one end but closed at the other (an open-closed tube), all with the same length L. Notice the pressure and displacement boundary conditions. The standing wave in the open-open tube looks like the closed-closed tube except that the positions of the nodes and antinodes are interchanged. In both cases there are m half-wavelength segments between the ends; thus the wavelengths and frequencies of an open-open tube and a closed-closed tube are the same as those of a string tied at both ends:

FIGURE 17.15 The first three standing sound wave modes in columns of air with different boundary conditions.

Full Alternative Text

Full Alternative Text

The open-closed tube is different. The fundamental mode has only one-quarter of a wavelength in a tube of length L; hence the $m=1$ wavelength is ${\text{λ}}_1=4L$. This is twice the ${\text{λ}}_1$ wavelength of an open-open or a closed-closed tube. Consequently, the fundamental frequency of an open-closed tube is half that of an open-open or a closed-closed tube of the same length. It will be left as a homework problem for you to show that the possible wavelengths and frequencies of an open-closed tube of length L are

Notice that m in Equations 17.18 takes on only odd values.