Standing Sound Waves

Standing waves can occur in any wave medium that is enclosed, including the air. For the latter, we can produce standing sound waves, responsible for the rich tones from brass and woodwind instruments, and organs. Of equal importance, standing sound waves within our vocal tract are the basis for producing our voice for others to hear. There are both similarities and differences between standing waves on a string and standing sound waves in a pipe. We will first emphasize the differences, which are most pronounced when we consider a pipe of length L, that is open at one end and closed at the other. Later, we will look at a standing sound waves in a pipe that produces a very similar harmonic series of frequencies as those from a stretched string.

Standing sound waves in a pipe, open at one end and closed at the other end, differ from standing waves on a string. First, there is an obvious difference in the wave medium. For the latter, the leftward going and rightward going traveling waves that are superimposed to form the standing wave pattern are transverse displacements of a string. For standing sound waves, the wave medium is typically air, and the waves themselves are longitudinal, corresponding to regions of compression and rarefaction of the air. A second important difference is how the traveling sound waves are reflected at the two ends of the pipe. At the end of a pipe open to the air, the pressure at the end of the pipe cannot oscillate. Instead, it is fixed at the ambient pressure of the surrounding air. This is completely analogous to the fixed end of a string; in other words, at the open end of the pipe, the standing sound wave pattern must have a node. The situation is different for the closed end of the pipe. Here, the pressure can vary; in fact, for the standing sound wave, the air pressure oscillation at the closed end has its greatest amplitude. In other words, there is an antinode in the standing sound wave pattern at the closed end of the pipe.

For the fundamental standing sound wave, there is only a single node at the open end of the pipe. At the closed end, there is an antinode. The wave pattern looks as drawn below.

The standing sound wave pattern for the fundamental does not even have half a wavelength fitting into the pipe! Instead, the spatial variation of the pressure oscillations starts from a node at the open end and grows to the maximum amplitude oscillation at the closed end. In other words, only one-fourth of a wavelength fits into the pipe for the fundamental. This is expressed in the equation below. Using it and the speed of sound in air, Vs, we get the frequency of the fundamental:

We follow the same rules presented earlier for making standing waves on a string to produce the harmonic series of standing sound waves in a pipe. The next highest frequency is produced by adding a single node to the standing wave pattern for the fundamental. The nodes and antinodes of the resulting pattern are spaced evenly through the tube, starting with a node at the open end and ending with an antinode at the closed end. The resulting standing wave pattern is shown below.

From the shape of the standing wave pattern, we see that three fourths of a wavelength fits into the length, L, of the tube. From the wavelength we can get the frequency of the next standing wave pattern:

Unlike the situation for the string, where the next standing wave mode after the fundamental has a frequency 2f1, for a pipe open at one end and closed at the other, the next standing wave mode after the fundamental has three times higher frequency! This means we have skipped the second harmonic in the harmonic series.

We can repeat the rules discussed earlier to find the next standing wave mode for a pipe. We add a node to the standing wave pattern of the third harmonic, and demand that the nodes and antinodes are evenly spaced along the length of the tube. The resulting pattern looks like:

From the picture, it shouldn't be too hard too see that the three nodes and three antinodes split up the length of the tube into five parts. One full wave cycle in the standing wave pattern is completed in four of these parts, meaning that the wavelength is 4L/5. From this, we can find the frequency of this mode


Harmonic series for organ pipes

From the detailed examples we have done, the pattern of the harmonic series for a standing sound wave in a pipe open at one end and closed at the other should be evident. The series consists of only odd integer multiples of the fundamental frequency. The second, fourth, and all even harmonics are missing! This makes the sound from an organ pipe and our own voices (which to a good approximation can be modeled as a pipe opened at our mouth and closed at the position of the glottis, whose opening is controlled by the vocal cords.) differ from the sound produced by stringed instruments.

We should keep in mind the origin of the difference in the harmonic series between a pipe, open at one end and closed at the other, and a string. The difference arises because at the open end of the pipe there must be a node in the pressure oscillation (just as there is a node at the held ends of the string) but at the closed end of the pipe there must be an antinode, or position where the pressure oscillation has its greatest amplitude.

The harmonic series is different when we talk about a pipe open at both ends. Then, the standing sound wave pattern must have nodes in the pressure oscillations at both ends of the pipe. This means that the harmonic series for such a pipe is very similar to that of a string held at both ends. For a fully opened pipe, there are harmonics occuring at frequencies that are both even and odd multiples of the fundamental.

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Last updated: 11 Oct 1999
Comments: bland@indiana.edu