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20 February 2004

Sound Velocity from Interference Observations

by John W. Dooley, Physics Department
Millersville University

This month's experiment requires that the left and right sound sources be separated by distances as large as 1 meter. It is best to use small loudspeakers instead of earphones to get enough sound intensity.

The experiment is best understood by thinking of sound as being a wave. The feature of waves that is important for understanding this experiment is called superposition. You can observe superposition in waves on the surface of a liquid in a carry-out coffee cup. (I used a cup made of expanded, rigid polystyrene plastic.) Tap the cup on the side, near the top, with a pencil as shown below and watch the ripples that form in surface of the liquid. Among the patterns that form is one in which ripples start out very small near the walls of the cup. They travel to the center of the cup and erupt upward in a spike.

The eruption is caused by smaller waves traveling from the edge to the center and piling up on top of each other. The height of combined waves is the sum of the heights of each wave.

The opposite can also happen. A depression (or trough) in the liquid surface travels just as fast as a bump (or crest). If the two meet, they tend to cancel. It is as though the trough has negative height. When the depth of the trough is added to the height of the crest, the sum will be zero or nearly so. We will see that pairs of sound signals can combine in a similar way.

We will use the computer to produce an identical sound from both stereo speakers, and observe how the sounds combine. It is important that both speakers emit exactly the same wave. When one emits a crest so does the other. The troughs must also match.

The figure below shows the wave idea, with waves emitted together and detected by a box that represents your ear. The speaker pushes forward and back. When it pushes forward, a high-pressure crest forms in front of it and travels toward the detector. When the speaker pulls back, it creates a low-pressure trough that also travels toward the detector. In the figure, a crest is just emerging from each speaker. The next crest to the right was emitted by the previous push of the speaker.

If the frequency of the sound is, for example, 500Hz, the speakers emit 500 crests per second. Each crest is emitted 1/500 of a second (.002 seconds) after the other. The time between emission of successive crests is called the period of the sound. Because all the crests travel at the same speed (the speed of sound), we can calculate the distance by which one crest leads the next. That distance is called the wavelength. The wavelength is equal to the speed of sound (in meters per second) multiplied by the period (in seconds).

In the figure above, the speakers are next to each other, and the waves arrive together, crest for crest and trough for trough. When the speakers are next to each other, we hear a loud tone. In the figure below, one speaker has been moved so that its crest occurs in the same place as a trough of the other speaker. The sum of these two waves is zero - they cancel. When this is done, we little or no sound.

The figure below shows the speakers separated by such a large distance that the wave crests match and we hear loud again. In this figure, the two speakers are separated by one wavelength. By measuring the distance between the two speakers in this figure, we can determine the sound wavelength. Since the wavelength is the sound speed times the sound period, we can calculate the sound speed from the measured wavelength and the known period: The sound speed is the wavelength divided by the period.

The photograph below shows the experiment, with two speakers and a meter stick. In the figure, GoldWave is playing a 500 Hz tone in continuous loop mode, so the tone never stops playing. Both speakers emit exactly the same tone. The listener is positioned facing the two speakers, at the same level as the speakers. In the setup shown, the speakers are separated so that the listener hears nothing, because the two sound waves cancel.

A simple test of the wave nature of sound is possible in this configuration. With the speakers set so that you "hear quiet," we assert that the two waves are canceling each other. To test this, cover one of the speakers with your hand. Now you should hear sound from only one speaker. Since there is no second sound wave to cancel it, it should be louder. Making the sound louder by covering one speaker is one of those pleasant counter-intuitive examples which convinces us that there must be something to the wave theory of sound.

The speakers were moved to this "quiet" separation starting from where both speakers were side-by-side. While a speaker is being moved, the sound loudness gradually decreased, and the movement was stopped when the sound intensity was heard to be a minimum.

This first minimum occurs when the speakers are separated by half of a wavelength. In the photograph, the speakers are separated by about 35 cm, so the wavelength of sound at 500Hz is about 0.7 meter. At 500Hz, the period is 1/500 second, and we calculate a speed of sound of approximately 350 m/sec. This agrees roughly with the speed of sound calculated for sound pulses in an earlier article ( http://www.sas.org/E-Bulletin/2003-11-14/features/index.html ).

To make a fair comparison, we need to estimate the uncertainty in our value for the velocity of sound. The best way to make the estimate is to repeat the measurement with different parameters and see how much variation we get in our numbers from one trial to the next. For example, we can increase the speaker separation in the photograph until the sound is loud again. Then the separation of the two speakers is one wavelength. We measure the speaker separation and calculate the speed again.

The following three files are tones of 500Hz, 1000Hz, and 1500 Hz, each an octave higher than the previous. Each is almost 2Mb in size, so if you have dial-up connection, you may prefer to make your own.

500hz.wav

1000hz.wav

1500hz.wav

With these, you can make more independent measurements of the sound speed. When the frequency doubles, we expect that the wavelength will be cut in half, since doubling the frequency cuts the period in half. A wave crest has only half the head-start on the crest that follows it. (The implicit assumption is that the speed of sound is the same for all frequencies. Like most assumptions, this is not exactly true.)

If you want to go further with this experiment, you can try to see the effect of temperature on the speed of sound. Simple theory (see http://www.grc.nasa.gov/WWW/K-12/airplane/sound.html for example) holds that the speed of sound should vary as the square root of the absolute temperature (which we measure in degrees Kelvin, or K). Room temperature is on the order of 300K on the absolute scale. A moderately cold winter day is about 30K colder than that. By moving the experiment from indoors to outdoors on a winter day, we change the temperature by about 10%. It turns out that this implies a change of about 5% in the speed of sound. Can you measure the speed with enough accuracy to detect this change?

Finally, we should note that you can do this experiment with much simpler equipment. The simple experiment is described at http://muweb.millersville.edu/~physics/exp.of.the.month/24/