How to Determine the Velocity of Sound
The aim of this experiment is to determine the Velocity of sound in air by measuring the resonant frequency of air columns of variable length.
The variables being used in the experiment are as follows:
L = Length of air column (meters)
f = resonant frequency (Hertz)
c = Velocity of sound (meters/second)
t = air temperature (Celsius)
x = 1/f, the reciprocal of the measured resonant frequency (period) (seconds)
y = Length of the air column (meters)
m = slope of the linear regression (meters/second)
e = y intercept
Experimental Method
A speaker and oscillator was used to produce and record standing waves within a plastic cylinder, closed at one end, in order to determine the velocity of sound in air.
Results
Data Analysis
The data was stored in a data array consisting of two columns, x and y. X is a list of the reciprocals of the resonant frequencies measured for each of the different lengths of the air column used in the experiment. y is a list of different lengths of the air column.
The data was linear, as the resonant frequency had been transformed by the reciprocal (1/f) function, so the standard linear equation, y=mx+e, could be used to determine the slope for the data.
The following set of equations were used to obtain the slope for each part of the experiment.
meanx = sum(x)/len(x)
meany = sum(y)/len(y)
meanxx = sum(x*x)/len(x)
meanxy = sum(x*y)/len(y)
m = ((meanx*meany)-meanxy)/((meanx**2)-meanxx)
e = meany-m*meanx
Determining the velocity of sound in air:
Using the slope to determine the re it was simply a matter of multiplying the slope of the linear regression by 4. This resulted in:
c = 4m = 341.9624
As there are 7 sets of data and the data is linear, the middle values correspond to the average of the overall data. The relationship of the resonant frequency to the velocity of sound can be used to confirm that the answer fits with the linear regression:
The slope should equal 341.9624/f
The velocity of sound through air should equal 4*341.9624/400
Length of air column (Meters) = 0.2
Measured resonant frequency (Hertz) (f) = 400
m = c/f (velocity over frequency) = 85.4906
c = 4*(c/f) = 4*m = 341.9624
c/f*10^2 = m
341.9624/400*10**2 = 85.4906
4*(341.9624/400*10**2) = 341.9624
Experimental Discussion
Given the measurment of the resonant frequency peaks was done with my own ears, the uncertainty on the slope is quite large. There were several measurements taken in which I wasn't sure of the optimal point at which to record the frequency. By electronically tabulating the peaks in the volume of the resonant frequency, a higher level of accuracy might be able to be determined.
Another way of determining the peaks in resonant frequency might be to precisely measure the movement of the water or the slight shaking of the cylinder with a high framerate camera, a measurement of the wave period could be manually extracted from this data as well, considering it would be possible to visualise the peaks and troughs of the waves.
Conclusion
The coefficient of determination of the Length of air column as a function of the period of the resonant frequency was 0.9962, revealing that the data matched the linear model with a high level of accuracy. Much of the variance involved in the model was likely due to the precision of the individual measurements taken, as explained above.
The final value of the velocity of sound in air that was calculated was close to the expected value and fell within the uncertainty range. A more precise experimental method would likely lead to a value that closer reflects the measured estimated value according to the accepted velocity of sound in air.
The variables being used in the experiment are as follows:
L = Length of air column (meters)
f = resonant frequency (Hertz)
c = Velocity of sound (meters/second)
t = air temperature (Celsius)
x = 1/f, the reciprocal of the measured resonant frequency (period) (seconds)
y = Length of the air column (meters)
m = slope of the linear regression (meters/second)
e = y intercept
Experimental Method
A speaker and oscillator was used to produce and record standing waves within a plastic cylinder, closed at one end, in order to determine the velocity of sound in air.
- The oscillator was first adjusted to a reasonable volume at 250 Hz so that the experimenters could hear the tone produced by the speaker. The cylinder was also emptied of any liquid that may have been inside.
- Given the accepted value of the velocity of sound in air at 0֯ Celsius is c = 331.5, and the temperature in the room was, t = 23.5 (Celsius), an estimate of the velocity of sound in the room was determined using the equation, cest = 331.5+0.607*23.4
- The cylinder was measured in length.
- An estimate of the resonant frequency was determined with the equation fest = cest/4L, with L being the length of the part of the cylinder filled with air.
- The oscillator was adjusted until the resonant frequency was determined by measuring the frequency when the volume peaks before suddenly dropping.
- The length of the air cylinder was reduced by 0.05 meters by filling it with water.
- Steps 3, 4, 5, and 6 were repeated until 7 rows of data were recorded.
Results
Data Analysis
The data was stored in a data array consisting of two columns, x and y. X is a list of the reciprocals of the resonant frequencies measured for each of the different lengths of the air column used in the experiment. y is a list of different lengths of the air column.
The data was linear, as the resonant frequency had been transformed by the reciprocal (1/f) function, so the standard linear equation, y=mx+e, could be used to determine the slope for the data.
The following set of equations were used to obtain the slope for each part of the experiment.
meanx = sum(x)/len(x)
meany = sum(y)/len(y)
meanxx = sum(x*x)/len(x)
meanxy = sum(x*y)/len(y)
m = ((meanx*meany)-meanxy)/((meanx**2)-meanxx)
e = meany-m*meanx
Determining the velocity of sound in air:
Using the slope to determine the re it was simply a matter of multiplying the slope of the linear regression by 4. This resulted in:
c = 4m = 341.9624
As there are 7 sets of data and the data is linear, the middle values correspond to the average of the overall data. The relationship of the resonant frequency to the velocity of sound can be used to confirm that the answer fits with the linear regression:
The slope should equal 341.9624/f
The velocity of sound through air should equal 4*341.9624/400
Length of air column (Meters) = 0.2
Measured resonant frequency (Hertz) (f) = 400
m = c/f (velocity over frequency) = 85.4906
c = 4*(c/f) = 4*m = 341.9624
c/f*10^2 = m
341.9624/400*10**2 = 85.4906
4*(341.9624/400*10**2) = 341.9624
Experimental Discussion
Given the measurment of the resonant frequency peaks was done with my own ears, the uncertainty on the slope is quite large. There were several measurements taken in which I wasn't sure of the optimal point at which to record the frequency. By electronically tabulating the peaks in the volume of the resonant frequency, a higher level of accuracy might be able to be determined.
Another way of determining the peaks in resonant frequency might be to precisely measure the movement of the water or the slight shaking of the cylinder with a high framerate camera, a measurement of the wave period could be manually extracted from this data as well, considering it would be possible to visualise the peaks and troughs of the waves.
Conclusion
The coefficient of determination of the Length of air column as a function of the period of the resonant frequency was 0.9962, revealing that the data matched the linear model with a high level of accuracy. Much of the variance involved in the model was likely due to the precision of the individual measurements taken, as explained above.
The final value of the velocity of sound in air that was calculated was close to the expected value and fell within the uncertainty range. A more precise experimental method would likely lead to a value that closer reflects the measured estimated value according to the accepted velocity of sound in air.
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