Name: _______________________________________
Lab Section: ___________
Lab Partners: ___________________________________________________ Grade: ____________
Lab 3: Resonance in an Open-Closed Tube
Pre-Lab
Introduction One resonating system we all carry with us where ever we go, the vocal tract, can be modeled as a system of
resonating tubes. For the vocal tract, as is true of the resonances of other tube systems like brass and woodwind
instruments, many frequencies are inputted into the tube, but only those that meet the resonance condition can be heard. For
wind instruments, often the tube is lengthened or shortened during playing in order to change the pitch. That is, the
condition for resonance has purposely been changed to select a different resonant frequency. In this lab, you will be
investigating a open-closed resonating tube in very different way. You will be inputting just one frequency, and changing
the length of tube in order to keep this single frequency in resonance with the system. For any traveling wave of just one
frequency, the speed is given by
vf
(1)
where f is the frequency of oscillation and is the wavelength. So, you will be holding the frequency constant, and the
velocity of the wave, which is the speed of sound, will also be constant.
1) Why will the velocity of sound be constant for this experiment? What are the only ways to change it?
2) So if the velocity and frequency are constant throughout the experiment, what does that tell you about the
wavelength throughout the experiment?
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Visualization Case 1: Let’s assume a tube of length L is entirely filled with air. Let’s also assume that this length meets the
resonance condition where you have three nodes along the tube length. So the resonant length for this case is L. Draw a
diagram of the tube with the resonating wave. Near your diagram, write down how many wavelengths are present for this
mode, and solve for the wavelength in terms of L. Case 2: Imagine you have filled your tube with water, thereby shortening
the length, such that you will hear resonance again. Draw this. Near your diagram, write down the new length (as a fraction
of L) required for this next resonance. Again, write down how many wavelengths are shown in you diagram and solve for
the wavelength in terms of L. Case 3: Repeat what you did for Case 1 and 2 for the next shorter length that gives you
resonance.
Case 1:
Case 2:
Case 3:
3) What do you notice about your analytical solutions for wavelength for each case? Is this what you expected? Can
you get this information from the picture without doing any math? Explain.
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4) From the pattern you see, what is the next shorter length that gives resonance? Note that there might not be one, at
some point you run out of tube.
The cases you drew on the previous page correspond to lengths of the tube that will allow for resonances at a fixed input
frequency.
5) From your pictures and calculations, by how much of a wavelength do you need to change your tube in order to
get the next resonance?
6) Conceptual Picture: If you aren’t at one of these resonant lengths, what happens? Basically, why don’t all lengths
work? What is going on in terms of the transmitted waves from the tuning fork and reflected waves off the ends of
the tube. If it helps, contrast this with what happens at one of the resonant lengths. Draw some pictures below.
Think sine waves and superposition. Just describe in words and pictures.
7) Conceptual Picture: So what happens to a tube where you introduce many frequencies (you try to drive it at
many at once; this is what happens when you blow into it). Why does it pick out ones that it “likes”? Again, just
words and pictures.
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Name:_______________________________________
Lab Section: ___________
Lab Partners: ___________________________________________________ Grade:____________
Lab 3: Resonance in an Open-Closed Tube
Introduction One resonating system we all carry with us where ever we go, the vocal tract, can be modeled as a system of
resonating tubes. For the vocal tract, as is true of the resonances of other tube systems like brass and woodwind
instruments, many frequencies are inputted into the tube, but only those that meet the resonance condition can be heard. For
wind instruments, often the tube is lengthened or shortened during playing in order to change the pitch. That is, the
condition for resonance has purposely been changed to select a different resonant frequency. In this lab, you will be
investigating a open-closed resonating tube in very different way. You will be inputting just one frequency, and changing
the length of tube in order to keep this single frequency in resonance with the system. For any traveling wave of just one
frequency, the speed is given by
vf
(1)
Experiment: Calculating the Speed of Sound
To experimentally measure the speed of sound, assume the frequency value stamped on the tuning fork to be correct and devise
a way to measure the wavelength from finding the lengths of tube that meet the resonance condition. You can then use equation
(1) to calculate speed of sound from your measured value of wavelength.
1) How do you intend to use information about the different resonant lengths to calculate a value of wavelength?
Note that in reality there is a correction factor on any open end of a resonant tube. It’s as if the open end is really a
bit longer than the physical end by 0.61r, where r is the radius of the tube. However, depending on how you decide
to find wavelength, you may not need the absolute length of the tube and therefore don’t need to worry about this
factor. Use your drawings to help.
In order to locate the lengths for which resonance occurs, raise the water level (by raising the can) until it is near the top of the
tube. Strike the tuning fork with the rubber mallet (never use a hard object). Lower the water level, listening for a sudden
increase in the intensity of the sound indicating a resonance. Use a string to mark the position of the water level at which the
sound intensity is at its highest. Slowly move the water level past the string a few times to improve the accuracy of the
measurement. Continue lowering the water level until you have found the other resonances. Record the position of the
resonances and the uncertainty of the position. Repeat the process for the three more tuning forks (four forks total). Measure
the temperature of the air in the room; also record the uncertainty in that measurement.
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Analysis:
2) Method 1: Make a graph of your data points such that velocity is the slope of the line. Write down the calculated
velocity and uncertainty here as well as on your graph and spreadsheet. Be sure to include error bars on your
graph.
3) Method 2: From your many measurements of wavelengths corresponding to different tuning forks, find many
values of velocity using equation (1). Take an average value and a standard deviation. The standard deviation,  ,
is a conservative estimate of your error. You can do better if the uncertainty is random and Gaussian- the standard
deviation of the mean (SDOM), SDOM   N . If you take many measurements of a quantity x, it is typical
quote the final answer as x  SDOM . In addition to your spreadsheet, write your average value of velocity,
standard deviation, and standard deviation of the mean here.
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4) Method 3: Take just one of the values of velocity that you calculated from wavelength. Find the uncertainty in
this value by propagating error. Write it down here as well as on the spreadsheet. The uncertainty from
propagation should be approximately equal to the standard deviation using method 2. Is it? If one is greater, which
one is correct? O.k.,we are both men/women of action and lies do not become us: rather which one is a more
conservative estimate of the uncertainty?
After you have calculated the velocity of sound three ways compare it to the accepted value
v  331.5 m s  0.61
m
s C
T
(2)
where T is the temperature of the room in Celsius. Be sure to use the error in T to find the error in the velocity.
5) Write down all of your values of velocity from your three methods with corresponding uncertainty. Write down
the accepted value. Does the accepted value lie within your measured value  the respective uncertainties from
your three methods. This will tell you how well you know the parameters of your experiment.
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Post-lab Questions
1) Would it be possible to have 3/2 of a wavelength in your tube? Draw a picture that shows why or why not.
2) What could you experimentally do to your setup in order to produce a different value of wavelength but maintain
the same frequency?
3) Which one is the better method of calculating velocity and why? Did you overestimate or underestimate the error?
Why?
4) Method 2 shows us how taking many measurement rather than just one is better. However, don’t get too excited,
the reduction of uncertainty by taking many measurements grows very slowly. It is always in your interest to make
 small by careful measurement rather than taking boat-loads of crappy data. Too see this, calculate how many
more measurements you would have needed to take in order to reduce the uncertainty by a factor of 100 (darn
you, N !).
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