Relation between length of air column and wavelength
Consider the first position of resonance and suppose the lower fork prong is in its extreme upper position. As the prong moves downward through half a vibration it sends a pulse of compression down the tube, which is reflected from the water surface and returns to the mouth of the tube (Fig. 29.7 (a)). If, at the moment the
compression reaches the top of the tube, the prong has just reached its lowest position and is about to move upwards it will begin to send a rarefaction down the tube.
This rarefaction will, in turn, travel down the tube and be reflected back to the top in the time it takes the prong to complete the vibration (Fig. 29.7 (b)). At this stage the fork has made one vibration and has sent out one complete wavelength of sound. Since, as we have seen, the sound has twice traveled up and down the tube, it follows that the length of the air column is one-quarter of a wavelength.
Similarly, it can be shown that, in the second position of resonance, the fork
makes three complete vibrations in the time it takes the sound to travel four times the length of the tube. In this case the length of the air column is equal to three quarters of a wavelength.