Standing Waves in Air

Just like a string fixed at both ends, an air column inside a hollow cylinder can vibrate only at certain frequencies. That is the physics principle behind all musical wind instruments. A flute, for example, will produce a sound if you blow across its mouthpiece. The mouthpiece takes over the function of the vibrator in the previous investigation. If you want to change the pitch of the note, you must change the effective length of the flute by covering or uncovering holes along its body. You can also excite certain modes by changing the way you blow across the mouthpiece. However, to do this reliably takes a considerable amount of skill.

In this investigation, you will excite a standing sound wave in the air column inside a plastic tube with an adjustable plug. The adjustable plug has magnets embedded in it and can be moved from outside with a slider that also has magnets. Moving the plug changes the length $L$ of the air column inside the tube. The excitation of the air column is made by a speaker placed at the opening of the tube and a signal generator program to create a single tone with a known frequency.

The sound wave in the air column has boundary conditions different from those of the fixed string, which has nodes at both ends. The end of the tube is open to the ambient air, so the pressure at that point must always be constant and equal to the ambient pressure; in other words, the end of the tube is a pressure node. On the other side, the air column ends at the plug. At this point, the air has nowhere to go, i.e. large pressure can build up. This means that this point constitutes a pressure antinode. Therefore, the two boundary conditions are a node at the end of the tube and an antinode at the adjustable plug.

Figure 1: Standing waves in an air column (pressure view). The first three modes ( $n=1,3,5$ ) are shown.

This is illustrated in Fig. 1, where the first three standing pressure waves in the air column are shown. The relations between the length of the air column and the wavelengths of the three waves are also shown. It is clear from this figure that the relation between length of air column $L$ and wavelength $\lambda$ is given by

$\displaystyle \lambda = \frac{4 L_{n/4}}{n} \qquad (n=1,3,5,7,\ldots )$

where $n$ is an odd integer.

A warning is in order here. While the idea of a node at the end of the tube and an antinode at the adjustable plug in the tube results in the “pretty” theoretical picture shown in Fig. 1, the reality of the situation is a bit more murky. Exactly where at the mouth of the plastic tube does the pressure node form? Is there an effect of the tube diameter on node formation? These are among serious open experimental questions that may cause a difference between the measured and the true $L$ . Step 3 of the procedure will show a practical way to try to get around these difficulties.

Procedure

Make sure the speaker is attached to the audio cable on your bench. Place the speaker near one of the two ends of the tube. The speaker should be close to the tube but should not touch the tube. Bring the adjustable plug as close as possible to the end of the tube where the speaker is located. Pick a random frequency value in the 400 Hz – 500 Hz range and start the Signal Generator applet at the bottom of this page by clicking on the Play button. You should hear the buzzing sound from the speaker. Start moving the plug very slowly away from the speaker. This will change the length of the air column inside the tube. When the air column attains just the right length, there will be a rather sudden increase in the sound intensity, indicating a resonance (standing wave). When you move the adjustable plug further away from the speaker, the sound volume will just as quickly decrease again. Record the position $L_{1/4}$ at which the sound intensity is a maximum by using a meter stick. Move the adjustable plug a bit more away from the speaker, then spot the position of the maximum again, this time by moving the plug towards the speaker. Measure $L_{1/4}$ a total of three times and calculate the average location of the maximum and its error.
Move the adjustable plug farther away from the speaker again very slowly and search for the ( $n=3$ ) resonance (see Fig. 1). Make three measurements of the location $L_{3/4}$ of the 2nd sound maximum using the same method as in step 1. Repeat the same procedure to find the ( $n=5$ ) resonance.
Find the distance between the first and second resonance, and determine the wavelength $\lambda$ by subtraction of the two measurements:
$\lambda=(L_{3/4}-L_{1/4})\times 2$
This subtraction procedure should yield a more accurate determination of $\lambda$ than the direct calculation, for instance $\lambda=L_{1/4}\times 4$ , since it eliminates part of the systematic measurement error, namely the uncertainty in the exact shape and position of the pressure node at the open end of the air column. Make an estimate of the error $\delta\lambda$ in the wavelength measurement.
Repeat steps 1 through 3 for two other randomly picked frequencies in the 600 Hz – 900 Hz range and in the 1000 Hz – 1200 Hz range respectively.
Plot $\lambda$ (vertically) vs. $1/f$ (horizontally) with error bars showing $\delta\lambda$ . Obtain the slope and determine its error. Show mathematically that this slope is the speed of sound in air. Compare your result with the value of the speed of sound given in the introduction to this experiment. Do the values agree within the calculated uncertainty?

IPL Signal Generator

400 Hz

1200 Hz

Frequency: 400 Hz