The first known measurement of the speed of sound was by Gassendi in 1635. In 1698 Newton attempted to derive the value from Boyle's Law; you may have done the Boyle's Law experiment in this laboratory.
This document introduces the Speed of Sound in a Pure Gas experiment. This experiment is very similar to the Standing Waves and Acoustic Resonance experiment. The major difference is that in that experiment acoustic standing waves in air are studied, while here we study acoustic standing waves in Nitrogen.
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We have also prepared a copy of the soundtrack of the video and a summary of the information in this page. For most people, effectively using these will mean having it in hard-copy instead of reading it on-screen. Thus, they are prepared in pdf format. Accessing them will require that you have the Acrobat Reader, which is available free from http://www.adobe.com
In order to do the experiment, somewhat more background information is required than is typical for experiments in this laboratory. This section discusses that needed background.
| A brief non-mathematical introduction to standing waves, prepared for another context, exists. You may access an html version by clicking on the link to the right that says html; it has a file size of 189k. You may access a pdf version by clicking on the link that says pdf; it has a file size of 195k. Either version will appear in a separate window. You should choose the html version of this document if you are intending to read it in a browser; you should choose the pdf version if you intend to print it and read it in hardcopy. |
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A sound wave is a longitudinal wave because the thing that is "waving," the molecules of air, are moving in the same direction as the wave itself. This is different from a transverse wave such as a wave on a string, where the thing that is waving, the string, is moving in a direction that is perpendicular to the direction of the wave's motion.
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The above figure is a slow motion animation of a tuning fork generating a sound wave. The light gray regions represent a low density of air molecules; the dark gray represents high density. The low density regions have a pressure less than atmospheric pressure, while the high density regions have a higher pressure. If you are a Physics student at the University of Toronto you may have studied this relation between density and pressure in the Boyle's Law experiment. |
A graph of the pressure of the air at some moment in time as a function of position will be a sine wave. This sine wave is traveling from left to right in the figure. The pressure at zero amplitude of the wave is just atmospheric pressure. Later it will be important for you to know that microphones measure this pressure wave.
The speed of the sound wave varies with the temperature. The accepted value is:
| vaccepted = 331.4 + 0.61 t (m/s2) | (1) |
where t is the temperature in Celsius.
Just as for all waves, the sound wave travels through the air. The medium of the wave, the air molecules, vibrate back and forth but do not move off to the right with the wave. The molecules' vibration is sinusoidal. Therefore we also can characterise a sound wave in terms of the displacement from equilibrium of the individual air molecules.
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It will be useful to think of a model where the air molecules are connected to each other by springs, as indicated in the above figure. |
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A Flash animation of the oscillations of the air molecules has been prepared. It may be accessed by clicking on the green button to the right. The animation will appear in a separate window, and the size of the animation file is 20k. |
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Some key points in the animation are that:
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Pauses the animation |
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Resumes the animation |
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Step forward one "frame" |
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Step back one "frame" |
In conclusion, where the amplitude of the pressure wave is a maximum or minimum the amplitude of the displacement wave is zero, and vice versa. This is indicated in the following figure.
In most of what follows, we will be analysing the displacement wave. You need to remember that the microphone you will use to measure the wave measures the pressure wave.
Consider a sound wave incident on a barrier that reflects it. The air molecule that is just next to the barrier can not move to the right, so the amplitude of the displacement wave must be exactly zero at that position. The air molecule just to its left collides with the barrier elastically, and rebounds by moving to the left. So in terms of the displacement sound wave, the reflected wave has the opposite orientation to the incident wave: if the incident wave has a positive amplitude the reflected wave has a negative amplitude, and vice versa.
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A Flash animation of an incident displacement sound wave reflecting off a barrier may be accessed by clicking on the red button to the right. The animation will appear in a separate window, and has a file size of 41k. You will see that it has "playback" controls to stop and start the animation. |
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The positions in the animation indicated by the blue lines are called nodes; at these points the sum of the two waves is always exactly zero. The points half-way between the nodes where the amplitude becomes a maximum or minimum value are called antinodes. Of course, a node in the displacement wave shown in the animation is an antinode in the pressure wave.
The distance between the nodes of the sum of the two waves is exactly one-half of the wavelength of the incident and reflected wave. You can see this more clearly in the animation by clicking on the Pause button.
If we have a wave traveling from right to left and reflecting off a barrier, the picture would be just the mirror image of the animation.
Now, imagine we have two barriers, and a sound wave is reflecting back and forth between them. If the left-hand barrier is placed at the position of one of the nodes of the situation shown in the animation we looked at above, then the wave reflected off the left-hand barrier will exactly match the wave traveling from left to right.
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An animation of this may be accessed by clicking on the blue button to the right. It will appear in a separate window, and has a file size of 41k. |
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Thus if the left-hand barrier is at one of the nodes in the total wave produced by the right-hand barrier we form a standing wave. Note that if the left-hand-barrier were not placed at a node, then its reflected wave would not lie on top of the wave incident on the right-hand barrier, so a standing wave would not be formed.
As you can see, if d is the distance between the nodes, it is
related to the wavelength
of the waves that form the standing wave
as:
| = 2 d |
(2) |
Formation of standing waves is sometimes called resonance.
Here is a summary of the material in this sub-section:
Although the animations illustrate the formation of standing waves, to derive that such waves are actually formed involves writing down the mathematical formulation for a traveling wave moving from left to right and the equation for another traveling wave moving from right to left, with the phase relationship between the two waves such that it correctly describes the behavior where the reflection occurs. Then the sum of the two equations may be manipulated into the equation of a standing wave. Such manipulations are not part of this experiment. Instead they are part of theoretical physics, which is the subject of the lecture component of your physics course. The laboratory is teaching the "better half" of physics: experimental physics.
In the previous section we considered the required positions of the barriers so that a standing wave is formed. Here we ask which standing waves can be formed for a fixed position of the barriers. The answer is any wavelength where the nodes of the displacement standing wave correspond to the position of the barriers.
| An animation of the three largest wavelengths that produce standing waves for a fixed position of the barriers may be accessed by clicking on the orange button to the right. It will appear in a separate window, and the file size is 9.2k. |
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| Note that the wavelengths all match the equation: |
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(3) |
where L is the distance between the two barriers.
Although the animation only shows values of n equal to 1, 2, and 3, the equation correctly indicates that there are an infinite number of possible standing waves corresponding to the infinite number of positive integers.
The animation correctly shows that the frequency of different standing waves are different.
| The wavelengths are related to the frequency f of a wave traveling at a speed v according to: |
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(4) |
Equation 4 is true for all waves.
You will measure the wavelength of a sound wave that has set up a standing
wave in the tube by measuring the distance between the nodes of the standing
wave. The frequency f of the sound wave is equal to the frequency
of the voltage from the signal generator. Then you will determine the speed
of sound in the Nitrogen using Equation 4.
The specific heat c of a material is a constant that characterises the amount of heat necessary to raise the temperature of a material. It is defined as:
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(5) |
where Q is the heat necessary to change the temperature of a mass
m by 
T.
Gases expand when their temperature increases. The values of the specific heat depend on the manner in which the gas is constrained. We call the specific heat for a material maintained at constant pressure cp and the specific heat for a material maintained at constant volume cv
The ratio of specific heats 
is related to the specific heat at constant pressure, cp,
and the specific head at constant volume, cV, according
to:
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(6) |
It is, therefore, a dimensionless number whose value depends on the internal structure of the gas molecule. It turns out to be related to the number of internal modes that the molecule has for storing energy; these modes are called degrees of freedom.
The speed of sound v in a pure gas is related to the pressure p
and the density of the gas 
by:
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(7) |
For an ideal gas:
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(8) |
where V is the volume, n the number of moles of the gas, R the gas constant, and T the temperature in Kelvin. Equations 7 and 8 can be combined to give:
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(9) |
where M is the molecular weight of the gas. Re-arranging gives:
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(10) |
From your determination of the speed of sound in Nitrogen, you will use Equation
10 to calculate .
Quantum Mechanics predicts that the values of
are:
| Value of |
Gas type |
|---|---|
| 5/3 |
monatomic gas of point particles |
| 7/5 |
diatomic gas of freely rotating molecules |
| 9/7 |
diatomic gas of rotating and vibrating molecules |
We would expect a real diatomic gas to be capable of rotating and vibrating,
so the value of
should be 9/7.
As you are learning in this laboratory, the real world is often more
complex than theory would have us believe. For example, a graph of The value of If your result for |
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In order to perform the experiment you will need to know how to use an oscilloscope. An introduction to this instrument may be accessed by clicking on the yellow button to the right. It will appear in a separate window, and has a file size of 42k. |
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Earlier we indicated that this experiment is very similar to the Standing Waves and Acoustic Resonance experiment. The major difference is that in that experiment acoustic standing waves in air are studied, while here we study acoustic standing waves in Nitrogen. |
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The blue button to the right will show a diagram of the apparatus for this experiment. You can compare to the apparatus for the Standing Waves experiment by clicking on the red button. Both figures will appear in separate windows. If you wish to print either of these, you will probably need to use landscape mode. You should treat the representations of the Signal Generator and Oscilloscope in the diagrams as generic; the instruments you use may look different from those shown. |
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We have prepared a video of the Standing Waves and Acoustic Resonance apparatus, which you will want to look at for this experiment. The running time of the video is about 6:30 minutes. Links to the video appear below. You will wish to adjust the volume of the speakers on your computer so that you can easily hear the soundtrack.For the higher resolution RealMedia version, you may also wish to increase the size of the video using the controls provided by the player.
| You may download a pdf version of the soundtrack of the video by clicking on the green button to the right. It will appear in a separate window, and has a file size of 76k. | |
The Streaming video will be played by the RealMedia player as it is delivered to your computer by the network. The Download versions of the video will be downloaded to a temporary area on your local hard disc and then shown if your browser is configured to use the appropriate player. You may save these Download versions to a more permanent place that you specify on your computer's discs by right clicking on the link and then saving the Target (Internet Explorer) or Link (Netscape)
 RealMedia
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 Quicktime
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One of the factors that will influence the quality of your data is how well you have adjusted the frequency of the sound wave to achieve a good resonant standing wave. Here is a brief prescription:
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Do not attempt to change the pressure of the gas in the tube! If you suspect problems consult your Demonstrator or the Technologists. |
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Connect the output of the signal generator to the oscilloscope. Generate a voltage of a few hundred Hz and display it on the oscilloscope. In order for the display to be stable, you will want to trigger the oscilloscope on the input voltage.
If you have not used an oscilloscope before this may take a few minutes; this part, then, is your opportunity to begin to learn how to use this instrument.
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Remember the recommendation in the web-document on using an oscilloscope: first take a few moments to locate the major controls on the instrument. Randomly turning knobs and flipping switches is almost certain to not produce the display that you desire. |
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The relationship between the frequency f, in Hz, and the period T, in seconds, is:
f = 1/T |
(11) |
Compare the frequency of the wave as read by the signal generator to the frequency as measured by the oscilloscope.
Connect the output of the signal generator to the loudspeaker. Connect the output of the microphone to the second beam of the oscilloscope and adjust the display so that you can see both the input voltage from the signal generator and the output of the microphone simultaneously.
You should be able to hear the sound wave if you place your ear near the tube.
For some frequency between, say, 200 Hz and 2 kHz, adjust the frequency so that a standing wave exists in the tube.
Question: how does the sound level that you hear with your ear close to the tube correlate with whether or not a standing wave exists in the tube? Can you think of why this is so?
For the lowest frequencies, the maximum amplitude as measured by the microphone does not occur at the position of the loudspeaker, but a noticeable and measurable amount down the tube away from it. This is your first indication that the simple picture of these waves as described above is not quite complete
From the positions of the nodes and/or antinodes, determine the wavelength of the standing wave. Calculate the speed of sound. Repeat for a few different frequencies. Give a final best value for the speed of sound and its error. Compare your result to the accepted value.
Using Equation 10, calculate the ratio of specific heats
and its error. Compare to the theoretical values.
These questions should be answered and turned in to your Demonstrator before beginning the experiment.
As promised at the beginning, we have prepared a summary of the information presented above. It does not attempt to provide a full discussion, but just reviews the "high points" from above.
| You may access the summary by clicking on the red button to the right. The summary is in pdf format, will appear in a separate window, and has a file size of 29k. You may wish to print this document. |
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Earlier versions of this Guide Sheet have been worked on by Joe Vise (1989) and David Harrison (1999 and 2001).
This web version is by David M. Harrison, May, 2003.
This is $Revision: 0.3 $, $Date: 2003/05/27 19:35:37 $ (y/m/d UTC).
