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# Using Waves to Measure the Speed of Sound (page 2)

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Source:
Author: Jerry Silver

### Expected Results

As before, the speed of sound at 20 degrees centigrade is 343 meters per second.

The speed of sound (in meters per second) as a function of temperature (in degrees centigrade) is v = 331 + 0.6T.

Using a 440 Hz tone, the distance separating the microphones to get a 343 meter per second value for the speed of sound is 0.78 meters (78 centimeters).

A 1000 Hz tone would require a 0.343 meter separation to result in the expected value for the speed of sound.

### Why It Works

Two waves traveling in the same direction add together to form a new wave. If the crests of the two waves rise at the same time and place, the waves are said to be in phase and reinforce each other to produce a louder sound. This is called constructive interference. This occurs when the two sources of the sound are separated by exactly one full wavelength. (One wave gets a onewavelength head start, but both are in phase at the detector.) If we know the wavelength and the frequency of the sound, we can easily determine its velocity according to the relationship v = f.

Destructive interference occurs when one wave crests while the trough of a second wave is passing. This happens when the source of the two waves is separated by half a wavelength.

### Two microphones/one speaker

If you can set up two microphones to your oscilloscope, there is another way to do this that shows the process of interference more clearly. In this case, you follow basically the same procedure as the previous one, except you have one speaker and two microphones. You move the microphones until you observe destructive interference. This occurs when the crest of one wave is at the same place as the trough of the other wave, as shown in Figure 77-3. This method does not work using a sound intensity meter or by listening carefully, as did the previous method.

Following this method, you can use the capability that many oscilloscopes have to detect the point at which the waves are separated by one-half a wavelength. This involves plotting one signal versus the other on the display. When this x versus y plot is a straight line with a slope equal to a negative one, as shown in Figure 77-4, your signals are 180 degrees out of phase and separated by one-half wavelength.

### Interference along a line/double slit analogy

Another configuration that can be used to find positions of a constructive and destructive interference is shown in Figure 77-5. This method is analogous to the double slit technique used by Thomas Young with light and is explored in Project 83. The wavelength is given by:

= dsin

where d is the separation between the speakers and is the angle between the midpoint between the speakers and the point where constructive interference is identified.

### Speed of light

Using a similar principle, the speed of light can also be measured in the lab. An apparatus is commercially available that determines the wavelength of a known frequency of light by measuring the distance between positions of constructive interference. The measurement is much trickier than the one in this experiment because the speed of light is so much greater than the speed of sound. The same basic approach, however, can be applied to either sound or light.

### The Point

The speed of sound can be determined if the wavelength and frequency of the wave are known. The wavelength for a given frequency can be determined by finding the distance at which constructive interference occurs.

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