Physics for the Life Sciences


Experiment 33


Velocity of Sound in Air - Resonance in Air Columns


Purpose

To study resonance in an air column. The velocity of sound will be determined from measurements of the wavelength of standing waves in an air column. The end correction constant for the open end of a closed pipe will be determined.

Apparatus

Resonance tube apparatus, two tuning forks (B4 and C5), meter ruler, rubber mallet, thermometer, Vernier caliper.

Theory

When a tuning fork is set in vibration and held over the open end of an air column, compressions and rarefactions in the air travel down the tube and are reflected at the closed end. If the returning wave is in phase with those produced by the tuning fork as it returns to the fork, a condition of resonance exists and the loudness of the note from the tuning fork is greatly increased. This occurs when the length of the tube is equal to an odd number of quarter wavelengths, that is L = l/4, or 3l/4, or 5l/4 ...



The position of the antinode at the open end of the tube not not exactly at the tubeend but is just outside. This extra distance is called the end correction of the tube and is proportional to the diameter of the tube. Experimentally, it is found that the end correction is approximately 0.4 times the tube diameter. The actual lengths of the resonating air column for the first three resonance conditions are given by,
l/4 = l1 + e, 3l/4 = l2 + e, and 5l/4 = l3 + e,
from which the value of the wavelength may be calculated from
l = 2(l2 - l1), or 2(l3 - l2), or l3 - l1.
The values of the end correction of the tube e can be calculated from
(l2 - 3l1)/2, (3l3 - 5l2)/2, or (l3 - 5l1)/4.
In the experiment, the length of air in a closed pipe is varied by changing the level of the water in the reservoir, as shown in the diagram. The apparatus consists of a glass or plastic tube about a meter long mounted vertically on a tripod with a rubber hose connecting the lower end of the tube to the movable reservoir. A tuning fork is clamped close to the top of the tube with its prongs vertical.
The relation between the velocity of sound in air v, the frequency of the wave f, and its wavelength l, is
v = f x l
The velocity v can then be calculated if the frequency f is known and the wavelength l is measured.
The velocity of sound in air varies with temperature. At 0 C it is 331.7 m/s and increases at the rate of 61 cm/s per degree Celsius rise in temperature, so that
vT = 331.7 + 0.61 T m/s,
where T is the temperature in C.

Procedure

  1. Adjust the water level in the resonance tube by raising the reservoir until the water level is 10 to 12 cm below the top of the tube.
  2. Clamp the tuning fork of higher frequency over the top end of the tube with the prongs about 2 mm above the tube and so that its prongs vibrate vertically.
  3. Strike the fork with the mallet, and quickly lower the level of the water in the resonance tube by lowering the reservoir tank. Record the position when resonance is first observed (the sound will suddenly increase in loudness at this position). Measure the distance l1 from the top end of the tube to the nearest mm.
  4. Repeat procedure (3) three more times and record the data in the table.
  5. Now repeat procedures (3) and (4) for the second position of resonance. This will occur at a distance of a little more than three times the first position.
  6. Again repeat procedures (3) and (4) for the third position of resonance. This will occur at a distance of a little more than five times the first position.
  7. Now repeat procedures (3) through (6) for the second tuning fork and record the data in the table.
  8. Record the temperature of the air in the room in C.
  9. Measure the inside diameter of the pipe to the nearest 0.1 mm using the vernier caliper and record its value.

Data:

Frequency of tuning fork
in Hz
 
First position of resonance length, l1
in cm
Second position of resonance length, l2
in cm
Third position of resonance length, l3
in cm
Wavelength in cm calculated from
2(l2 - l1), 2(l3 - l2) and l3 - l1, using average values of the resonance lengths
Velocity of sound in air in m/s calculated from wavelength in meters x frequency in Hz
Fork 1
1
 
 
 
 
 
2
 
 
 
 
3
 
 
 
 
4
 
 
 
Average
 
 
 
 
 
Fork 2
1
 
 
 
 
 
2
 
 
 
 
3
 
 
 
 
4
 
 
 
Average
 
 
 
 
 


Room temperature = ________ C
Value of velocity of sound in air calculated from 331.7 + 0.61 T = ________ m/s
Average experimental value of the velocity of sound in air = ________ m/s
Percent error ________
Inside diameter of tube, d = ________ cm
Values of end correction ______, ______, ______, ______, ______, ______ cm
Average value of end correction, e = ________ cm
Experimental value of end correction constant k = e/d =________

Calculations

  1. Calculate the average of each set of readings for the first, second, and third resonance positions.

  2. Using the average values of l1, l2, and l3 calculate the wavelength from the relations 2(l2 - l1), 2(l3 - l2) and (l3 - l1) and determine the average values.

  3. Calculate the velocity of sound in air from the relation v = f x l using the values of f given on the tuning forks and the average values of l as determined above. Calculate the average value of v and enter the value in the table.

  4. Calculate the velocity of sound in air at room temperature using the relation given in the theory and the measured value of the air temperature.

  5. Compare the measured and calculated values of the velocity of sound in air and calculate the percent error.

  6. Determine the values of the end correction for each fork using the relations e = (l2 - 3l1)/2, (3l3 - 5l2)/2, and (l3 - 5l1)/4, where l1, l2, and l3 are the average distances.

  7. Calculate the end correction constant k = e/d where d is the inside diameter of the tube.

Questions
  1. What effect would a higher room temperature have on the resonance lengths that you measured in this experiment?




  2. What effect would an atmosphere of helium or carbon dioxide have on the results of this experiment?




  3. Why was it easier to find the first resonance position than the second or third?




  4. How does your value of the end correction constant compare to the accepted value of 0.4? What is the percent difference?





Dr. John Askill, 1999.