Abstract

The speed of sound in air was measured in a thin column of air at 351.0 K and heights ranging from 20.0 cm to 65.8 cm.  I found that that speed of sound was 332.8 +/- 6.09 m/s, which did not compare well with the theoretical value, for reasons explained later in this paper.  The theoretical value for the speed of a mechanical wave can be given as the square root of the ratio of the specific heats at constant pressure and volume times the pressure of the air and divided by the density of the air.

Introduction

This experiment consisted of comparing the theoretical value of the speed of sound obtained by measuring atmospheric conditions, to that measured based on resonances of sound as follows.  A speaker was used to project sound waves into a long pipe whose length was much greater than its width thus allowing me to assume that the waves were essentially one dimensional in character.  The end opposite the speaker was closed and the end with the speaker was left open.  In addition a microphone was placed at the end with the speaker and used to look for resonances.  In order to determine the speed of sound I measured the frequencies of multiple harmonics of the standing wave for different column heights.  This data was then fit to the expression L = (v_s/f_n)(n/2-1/4), where L is the column length, v_s is the speed of sound in air, f_n is the n^th harmonic frequency and n is the number of the harmonic.  A value of vs was extracted from the numerical fits to the above expression.  This was then compared to the speed of a mechanical wave given by Ö(gp/r) where g is the ratio of the specific heats of the gas at constant pressure and volume, p is the air pressure and r is the density of air.

Experiment

The experimental apparatus was as follows : a 90.3 cm pipe arranged vertically on a stand with the top end open with a speaker immediately above it and a microphone placed level to the rim.  The base of the pipe was connected to a rubber tube which allowed, by means of a container of variable height attached to the stand, water to enter into the pipe and thus control the height of the air space in which the waves were transmitted.  The microphone was attached to an oscilloscope so as to allow one to detect resonances frequencies by the changes in the amplitudes of the waves.  The speaker was connected to a basic function generator omitting a sine wave pulse.  The height of the water level in the pipe was read from a scale placed on the pipe itself.  Thus the length of the air space could be found as the difference between the water level (measured by the meniscus) and the top of the pipe.  The frequency on the generator was varied until one visually obtained a maximum on the oscilloscope.  Then 4 readings were taken with the oscilloscopes measure function so as to determine the best approximation of the actual maximal resonance frequency.  When no more resonances were detectable at a given height within the frequency range available on the generator the height was adjusted.  In total resonances were found for six different values of the air column height.  This data was then used to find experimental values for the speed of sound.

Results/Analysis

The results from the resonance frequencies clearly fall on well defined lines corresponding to the nth order harmonics.

Note: the X and Y error bars too negligible to be well defined on this graph, with Length error of 1 part in a 1000 (accuracy limit of equipment) and 1/Freq error in the range of 2 * 10^-5 or smaller (based on std deviation).

Using this one can easily assign resonance modes to all the data except the three points marked above in Red, it is believed that this data represents a natural resonance mode of some piece of equipment other than the air column, since it is clearly independent of the column height.  For the remainder of this analysis those three points will be discounted.  Also as there is exactly one point representative of the n=8 mode this will also be ignored in the analysis of data relative to constant n.

Since the ultimate goal is to determine the speed of sound we utilize the equation :
v_s = (4*L*f_n)/(2*n-1), where v_s is the speed of sound (m/s), L is the Length of the column (m), f_n is the n^th resonance frequency (Hz), and n is the resonance number.  Two useful results can be derived from our data, the first by holding L constant and looking at the ratio of f_n to (2n-1), and the second by holding n constant and looking at the product of L*f_n.  These were examining by plotting points on the graphs of f_n vs (2n-1) and and f_n vs 1/L and then finding the slopes as presented below.
 
 

Value of L (cm +/- 0.1 50% confidence)	Slope of fn vs (2n-1)	Error of Slope (66% confidence interval)	Value for vs (m/s)	Error in vs based on std dev and analysis
20.0	405.508	6.46	324.4	5.17
29.5	278.101	18.42	328.2	21.73
38.4	218.766	7.31	336.1	11.23
48.2	174.591	14.43	336.6	27.83
55.8	152.242	11.09	339.8	24.76
65.8	129.768	13.56	341.5	35.7

Mean Value v_s : 334.4  Error of Mean : 9.56
 

Value of n	Slope of fn vs 1/L	Error of Slope (based on scatter residual)	Value of vs (m/s)	Error in vs based on error of slope
1	80.6	1.5	322.5	2.97
2	247.3	16.0	329.7	21.33
3	411.5	45.48	329.3	36.38
4	579.5	40.03	331.2	22.91
5	747.2	45.62	332.1	20.27
6	920.1	24.5	334.6	8.91
7	1101.9	0.3*	339.0	0.1*

*: based on limited data error estimate biased down.
Mean Value v_s : 331.2  Error of Mean : 7.56

For a combined mean of : 332.8 +/- 6.09 m/s.

Thus there is obviously close agreement between the two methods of computing the velocity of sound.

Additionally the theoretical value for the speed of sound was calculated from the following atmospheric data :
 

Temperature (deg C) :	78.0 (+/- 0.5 instrumental limit)
Barometer mmHg :	656 (+/- 1 instrument limit)
Relative Humidity :	89%
Vapor Pressure H20 mmHg:	251.8
Gamma (Ratio of Specific Heat at Constant Pressure to Specific Heat at Constant Volume for air - Calculated from composition of atmosphere information)	1.404 (+/- 0.03 standard dev)

With this information we can calculate the first the air pressure in Pascals as p = d*g*h, where p is pressure in Pascals, d is density of mercury (kg/m^3), g is the gravity at the surface of the earth (m/s^2) and h is the height of mercury (m).  From that we may arrive at the density of air by using r = 1.2929 (273.13 / T) (p-0.3783v) / 760, where r is the density of air (kg/m^3), T is the temperature in K, p is the pressure (mmHG), and v is the vapor pressure of H2O (mmHG).

Thus we find p = 8.708 * 10 ^ 4 +/- 1.3 * 10^3 Pascals and r = 0.7422 +/- 0.00169 kg/m^3

From this we may use v_s = Ö(gp/r), where the variables are as listed in the introduction.  Therefore we compute the value of v_s to be 405.9 +/- 5.309 m/s.

Now this presents a problem as 405.9 is greatly different from the experimental values found earlier.  I believe that the problem may lie with the value taken for the relative humidity on that day.  While it does come directly from the measurements taken in the room one that day, many people myself included felt at the time that 89% humidity did not correlate to our personal perception of the humidity at the time.  However we lacked any way to check this.  Having an improperly high humidity level would have the perceived effect of forcing up the speed of sound calculations.  Therefore this is my primary candidate for being the red herring in the calculations.

Conclusion

It this experiment I measured the velocity of sound by means of its resonant properties and compared this to a theoretical value of the speed of mechanical waves.  The experimental evidence seemed to give a well defined value of for the speed of sound, however that value was in conflict with the theoretical value as measured by atmospheric data at the time.  It is my opinion that the theoretical answer is wrong due to inaccuracy in one of the tools available to take humidity data, and the lack of any available means to obtain an alternative measurement.  Therefore it is my conclusion that the speed of sound is most closely that given in experiment on that day.