18 June 2004

Inside a standing wave

John W. Dooley, Physics Department
Millersville University

In a previous installment (http://www.sas.org/E-Bulletin/2004-04-02/features/body.html) we learned that that resonance in a tube can be understood in terms of the superposition of waves that bounce back and forth between the ends of the tube. This installment lets us stand inside the tube and experience the superposition all along its length. As with the previous installment, Dr. Conrad Miziumski showed me this experiment.

This experiment is best done outdoors using a pair of speakers to eliminate unwanted reflections. Two such speakers are illustrated in the figure above. (Pay no attention to the flower pot.) It will probably be most interesting to stop reading at this point and set up the experiment. You can drive the speakers with the 1500 Hz wave downloaded here, or you can create your own tone, in stereo, so that both speakers are driven with the same signal.

Cover one ear, and place your head between the speakers. Move your head back and forth along the line between the speakers. You will hear the sound become loud and soft, depending on the location of your uncovered ear. Measure the separation of the "loud spots," and compare that length to the sound wavelength (about .23 meters for 1500 Hz sound in air). The results might seem mysterious.

For reasons that we will see, this phenomenon is called a "standing wave" resonance. There is a connection between this experiment with two traveling waves and the resonance in a tube.

Previously, we focused upon what happened at the ends of a resonant tube. With multiple reflections, there were many waves propagating, but there is a way to simplify our thinking. We imagine the superposition of all the left-traveling waves into one single wave, traveling left. Then we do the same for all the right traveling waves to get a single wave, traveling right. Now we can think about the superposition of just two waves, one going left and one going right.

The animated .gif file at right illustrates the idea. The blue and red waves have the same frequency and wavelength, but travel in opposite directions. The yellow wave is the sum of the two counter-propagating waves. The yellow wave represents the sound that you hear when you place your head between the speakers.

Although the yellow wave is composed of two traveling waves, it seems to "stand still" and simply oscillate up and down. For this reason it is called a standing wave.

As the animation suggests, there are places in space at which the sum is always zero. These locations are called nodes. In between the nodes, the sum oscillates up and down. Think a moment about the frequency of the sum wave. Imagine placing your ear in the middle of the pattern with only one (e.g., the left-traveling) wave present.

You would hear the same tone that the speaker emits. Now, with both waves present, you hear the sum. As the animation suggests, the sum is always at a positive maximum when the two waves "agree" on creating a maximum. The sum sounds the same pitch as the individual waves. The sum is loud at locations where the yellow line makes large excursions. Close examination of the yellow animated line shows that the separation between the loud spots is exactly 1/2 of a sound wavelength. If we call the wavelength L, the separation of loud zones is L/2. Does this assertion agree with your measurements?

We have connected the two-wave experiment to the resonance experiment. It turns out that we can also connect it to the Doppler shift experiment . (In fact, I regard this as the whole business of physics: To analyze the connection between seemingly diverse experiments.) To see the connection, try this if you can: Move from the left speaker toward the right speaker at a constant velocity, v. You will hear a sequence of loud sounds. Call the time between loud sound occurrences T. During the time between hearing loud sounds, you travel a distance vT = L/2 .

According to the Doppler arguments, when we move to the right at velocity v, we detect a higher pitch for the left-traveling wave. The increase in frequency is given (to good accuracy) by (1500Hz)*(v/c), where c is the speed of sound in air. (c varies with temperature and humidity, but I use 345 meters per second, because it is easy to remember.) Similarly, we detect a lower pitch for the right-traveling wave, the one we are "running away from."

As you walk, you now hear two slightly different tones, separated in frequency by 2*(1500Hz)*(v/c).

Since frequency (f) is related to the speed of sound and the wavelength by c=fL , this frequency difference can also be written as 2*v/L .

As mentioned in a previous article , and discussed in more detail here , the superposition of slightly different tones produces a "flutter" in loudness, or a "beat." The frequency of the beat is exactly equal to the difference in frequency between the two tones.

Since the period is the inverse of the frequency , the Doppler effect predicts a time T = L/(2v) , exactly the same result that we got by thinking simply about walking from one loud spot to another.

Once again, I wish to thank Dr. Miziumski for this example of the unity of physics.