INTRODUCTION A loud speaker and microphone are placed near the end of a glass tube. The tube has an adjustable water level which the student can control through flexible tubing and an open jug. See the diagram. The speaker generates a continuous tone sine wave, which is reflected from the water level. The microphone detects the waves reflected from the water level as well as direct waves from the speaker. At certain water levels (which determine the length of the air column) the reflections from the tube are in phase with the speaker vibration - setting up a large-amplitude standing wave. The increase in amplitude of the sound vibration is detectable on the oscilloscope. At water levels very close to the resonance there is a cancellation of the sound - the phase of the reflected wave is such that the reflected wave cancels the wave from the speaker. This gives a very precise length of the air column for the standing wave. By detecting as many standing waves as possible in the length of the glass tube, the average wavelength is determined. From the relation between wavelength l, frequency, f, and speed c,
 

(1)

1. Set-up the apparatus, choosing a convenient frequency so that the oscilloscope displays the waveforms and resonances may be detected. Try to direct the microphone as much as possible to the top of the glass tube. If the waveform on the oscilloscope is not a pure sine wave, the amplitude is too large. Clean-up any water spills.
2. Find the resonance positions. Adjust the water level so that a best position of resonance is obtained. Pieces of tape may be used to mark the water levels. Repeat for as many consecutive levels as possible (no more than 10). Determine the average distance between resonances - this is half a wavelength.
3. Estimate the variation in this distance by calculating half the range of values of the separation. For those familiar with statistics, the standard deviation of all the separations may be used. Since the wavelength is twice the separation of resonances, the uncertainty in wavelength is twice the uncertainty in resonance separations. Calculate the percentage uncertainty in wavelength:

 

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4. Calculate the speed of sound and the uncertainty in speed of sound. From the frequency of the audio oscillator and Eq. 1 above calculate the speed of sound in air. The relative uncertainty of the speed of sound is the sum of the relative uncertainty in each of the terms making up a product or quotient. In Eq. 1 where we have:

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the relative uncertainty in c is:
	(4)

Since  is assumed 0,  is the only contribution to the relative uncertainty in c. In general if a quantity contains products and

quotients of terms such as
	(5)
then
	(6)

Relative uncertainties of products and quotients add. Having found the relative uncertainty, the absolute uncertainty is found by multiplying the best value by the relative uncertainty.
 

Example:

	(7)
where 5% is	(8)
	(9)

5. Repeat the experiment for CO₂. CO₂ is more massive than air, so with the water all the way in the bottom of the tube, the tube may be filled with CO₂ from a gas cylinder. When finding resonances, you can move the water level only one direction. If the water level is lowered again, air enters the tube. You may be required to perform the CO₂ experiment more than once to get satisfactory data.

Comparison with theory. From the theoretical expression for the speed of sound discussed in class, the speed of sound depends on the ideal gas constant , the absolute temperature , the molecular mass , and a dimensionless constant, , which is determined by the structure of the molecule. Calculate  for both air and CO₂ using the experimental speed of sound, the molecular mass and the absolute temperature. Also calculate the relative uncertainty of the experimental  and compare the experimental values with the model values discussed in class.

PROBLEM SOLVING - Due Tuesday, but graded separately from lab.
 textbook: Probs: pg. 493: 22; pg 497: 36; pg 500: 81, 82, 83, 117, 120, 129, 134