Determination of the Velocity of Sound in Air

If a vibrating tuning-fork be held immediately over the opening of a tube which is open at one end and closed at the other, and of suitable length, the column of air in the tube will vibrate in unison with the fork, and thus act as a resonator and reinforce its vibrations. The proper length of the tube may be determined experimentally.

If we regard the motion of the air in the tube as a succession of plane wave pulses sent from the fork and reflected at the closed end, we see that the condition for resonance is that the reflected pulse must reach the fork again at a moment when the direction of its motion is the opposite of what it was when the pulse started. This will always be the case, and the resonance will in consequence be most powerful, if the time the pulse takes to travel to the end of the tube and back to the fork is exactly half the periodic time of the fork.

Now the pulse travels along the tube with the constant velocity of sound in air; the length of the tube must be, therefore, such that sound would travel twice that distance in a time equal to one half of the periodic time of the fork.

If n be the vibration frequency of the fork, 1/n is the time of a period, and if l be the required length of the resonance tube and v the velocity of sound, then

or

In words, the velocity of sound is equal to four times the product of the vibration frequency of a fork and the length of the resonance column corresponding to the fork.

This formula (1) is approximately but not strictly accurate. A correction is necessary for the open end of the pipe; this correction has been calculated theoretically, and shown to be nearly equivalent to increasing the observed length of the resonance column by an amount equal to one half of its diameter.⁽¹⁾

Introducing this correction, formula (1) becomes

where r is the radius of the resonance tube.

This furnishes a practical method of determining v. It remains to describe how the length of the resonance tube may be adjusted and measured. The necessary capability of adjustment is best secured by two glass tubes as A, B, in fig. 17, fixed, with two paper millimetre scales behind them, to two boards arranged to slide vertically up and down in a wooden frame; the tubes are drawn out at the bottom and connected by india-rubber tubing. The bottoms of the tubes and the india-rubber connection contain water, so that the length of the column available for resonance is determined by adjusting the height of the water. This is done by sliding the tubes up or down.

The position to be selected is the position of maximum resonance, that is, when the note of the fork is most strongly reinforced. The length of the column can then be read off on the paper scales. The mean of a large number of observations must be taken, for it will be noticed, on making the experiment, that as the length of the tube is continuously increased the resonance increases gradually to its maximum, and then gradually dies away. The exact position of maximum resonance is therefore rather difficult of determination, and can be best arrived at from a number of observations, some on either side of the true positioa

From the explanation of the cause of the resonance of a tube which was given at the outset, it is easily seen that the note will be similarly reinforced if the fork has executed a complete vibration and a hal£ or in fact any odd number of half-vibrations instead of only one half-vibration. Thus, if the limits of adjustment of the level of the water in the tube be wide enough, a series of positions of maximum resonance may be found. The relation between the velocity of sound, the length of the tube, and the vibration frequency of the fork, is given by

where x is some integer.

This gives a series of lengths of the resonance tube, any two consecutive ones differing by v/2n.

Now v/n is the wave-length in air of the note of the fork. So that with a tube of sufficient length, a series of positions of maximum resonance can be determined, the difference between successive positions being half the wavelength in air of the note of the fork.

Introducing the correction for the open end of the pipe, the formula (3) for determining the velocity of sound becomes

[The most suitable diameter of the tube for a 256 fork is about 5 centimetres; for higher forks the diameter should be less.]

Experiment. - Determine the lengths of the columns of air corresponding to successive positions of maximum resonance for the given fork and deduce the velocity of sound in air.

Enter results thus : -

Vibration frequency of fork, 256 per sec.
Lengths of resonance columns :
  (1) Mean of twelve observations, 31 cm.
  (2) Mean of twelve observations, 97 cm
Radius of tube, 2.5 cm.
Velocity of sound, from (1) 343.40 m/s
                   from (2) 340.00 m/s

(1)	See Lord Rayleigh's Sound, vol. ii. §307 and Appendix A.