SPEED OF SOUND

Alice Baker

May 7, 2001
Physics II Formal Reports

Introduction:

The speed of sound through a gas can be measured using a speaker that sends a tone into a tube of water with a controllable water level. The sound waves are reflected by the water. When the reflected sound waves are in phase with the incoming sound waves, the sound waves reinforce each other and result in a large amplitude standing wave. The microphone picks up on this large amplitude, and the sine wave seen on the oscilloscope increases in amplitude as well. Immediately after this increase in amplitude, the sine waves cancel each other out, this happens when the phase of the wave reflected by the water cancels out the wave that is emitted by the speaker. The large amplitude, due to the waves being in phase, occurs at water levels that are close to the resonance. Resonance is the point at which the frequency of the wave reflected by the water is equal to the wave emitted through the speaker by the tone generator. These points of resonance in the tube are representative of the nodes in the standing sine waves. By measuring the distances between the nodes on the water tube, we can find the one-half wavelengths. From one node to the next node is a half of a wavelength. We can find the speed of sound through this gas using the equation:

c= f x g Since we have one-half wavelengths then: c = 2f x (g/2) The frequency is known because the tone generator that powers the speaker can be set at certain frequencies. A diagram of the apparatus used to measure the one-half wavelengths is shown below.

This design allows for a very precise measurement of the speed of sound.

Procedure:

Set up the apparatus as shown in fig. 1.
Lower the water level to the bottom of the tube by lowering the jug.
With the tone generator on and producing a tone through the speaker, and the microphone sending a signal to the oscilloscope, slowly raise the water level in the tube by lifting the jug.
At each point where the sound increases followed by a cancellation of sound, mark the tube with a piece of tape. Make sure the point on the tape that marks the node is the same at every resonance.
Measure the distance between the nodes and average them. This average is the one-half wavelength. Multiply the one half wavelength by twice the frequency to find the speed of sound through air.
Compare this experimental speed of sound to the isothermal speed of sound using the (sqrt RT/M) and notice the difference. This difference is do to the fact that sound waves are adiabatic not isothermal. Therefore an adiabatic adjustment is required. The steps taken to find this adjustment is discussed in the Results section.
Repeat all of the above steps for a tube filled with carbon dioxide.

Results:

    With the column filled with air and the frequency of the tone generator at 691 Hz, there were four points on the column where resonance was found. The first one-half wavelength was at 24.3cm from the first resonance point to the second. The second measurement between nodes was at 24.2cm, the third at 24.4cm, and the fourth at 24.3cm. The average distance between the nodes was 24.3cm.
    We propagate uncertainties in our measurements first by finding the uncertainty of the average one-half wavelengths, which is the maximum minus the minimum one-half wavelengths divided by two. This uncertainty is then divided by the average one half wavelength and multiplied by 100% to find the percent uncertainty, which is 0.514%.
    The data is then applied to the formula for the speed of sound; c = 2f x (1/2lavg.). The speed of sound, then, through air is 2 x 691 Hz x 24.3cm, which is 336m/s. The uncertainty is then propagated for this speed of sound by adding and subtracting the percent uncertainty (0.514%) of the 336m/s. This gives a range of 334-338m/s. The isothermal speed of sound through air is around 293m/s. This difference is due to the fact that sound waves are adiabatic, which means they undergo a quick change in temperature with no heat transfer. We can adjust for this factor by multiplying in an adiabatic adjustment called gamma.
    Gamma is equal to the speed of sound at a constant pressure divided by the speed of sound at a constant volume. This gamma is different for different types of molecules. For a monoatomic molecule, gamma= (5/2R) / (3/2R). For a diatomic molecule, gamma = (7/2R) / (5/2R). These differences are due to the differences in degrees of freedom in the molecule. Gamma can also be found by the equation below when you solve for gamma, using the experimental c and the molecular mass of the molecule, the temperature in Kelvin, and the ideal gas constant (R).

The experimental gamma can then be compared to the theoretical gamma (constant pressure/constant volume). Since air is diatomic the gamma should be (7/2R) / (5/2R), or 1.4. The gamma calculated from the equation above with uncertainties is 1.32 +/ - 0.0136. This is not quite 1.4, but the error is quite small; about 6% error which was probably due to a systematic error in the frequency. With this gamma, we now can obtain a speed of sound through air of 336m/s; sqrt (1.32) x 293m/s = 336m/s.
For the speed of sound through carbon dioxide, using the same frequency, the one-half wavelength measurements were: 18.9cm, 18.7cm, 19.1cm, 19.2cm, and 21.5cm. The average one-half wavelength was 19.5cm. The uncertainty of the one-half wavelength was 7.18%. The speed of sound through carbon dioxide was 269m/s plus or minus 7.18%. The range of the speed of sound through carbon dioxide is 250-288m/s. The isothermal speed of sound through carbon dioxide was 238m/s. Since carbon dioxide is diatomic, the gamma would be 1.4, five degrees of freedom, but this molecule could have degrees of freedom for vibration, which would make gamma = (11/2R) / (9/2R), or 1.22, nine degrees of freedom. The calculated gamma from the experimental data is 1.28 plus or minus 0.190, due to the large uncertainty.
Below is a table that lists all of the results for the speed of sound through air and the speed of sound through carbon dioxide.

As you can see there is disagreement in the carbon dioxide results. From these results, the molecule could have five or nine degrees of freedom because both adjusted speed of sound values (263m/s and 282m/s), using the gammas for five and nine degrees of freedom, fall within the range of the speed of sound 250-288m/s. This disagreement is due to the large percent error, which was due to the imprecise measurements of one-half wavelengths.

Conclusions:

    The results for the speed of sound through air of 336m/s, with a gamma of 1.32 plus or minus 0.0136 were fairly accurate. With a gamma of 1.4 for the diatomic structure of air and a 293m/s isothermal speed of sound would give an adiabatic speed of sound of 347m/s. The two values for the speed of sound through air are within experimental error, with a percent error of 3.2%. Even though the gammas are not within experimental error, the results as a whole were quite accurate.
    With a speed of sound through carbon dioxide of 269m/s, and a gamma of 1.28 plus or minus 0.184, we have a large experimental error. This error probably occurred with the differences in the one-half wavelengths, with a maximum one-half wavelength of 21.5cm and a minimum one-half wavelength of 18.7cm. This wide range led to a large percent uncertainty, which led to a wide range of uncertainty in the speed of sound measurements. This error in measurements may have been due to not enough carbon dioxide in the tube, perhaps there was still air at the top of the tube which gave us the one-half wavelength of 21.5cm. A repeat of this portion of the experiment would have led to better results. Also, due to the error in results, I was not able to determine whether carbon dioxide has five degrees of freedom or nine degrees of freedom, from vibration. Both gammas are within the experimental range of the gamma due to the large uncertainty. However, I do believe that carbon dioxide has vibration because we demonstrated a molecule of carbon dioxide motion by throwing a carbon dioxide model into the air. Vibration was clearly seen while the model was in free fall.
    For more information on the speed of sound experiment see the outline for this experiment.
    To see work done by other students in the Physics II class at Warren Wilson College see Peer Formal Reports .
    To learn more about the speed of sound and the adiabatic adjustment see speed of sound .