Q.What is meant by end correction in resonance air column apparatus?

The resonance air column apparatus is one of the simplest laboratory techniques for measuring the speed of sound in air at room temperature, and it also lets an unknown tuning-fork frequency be checked against a known one. It consists of a vertical cylindrical glass tube about one metre long, open at the top end A, with its lower end B connected via a rubber tube to a water reservoir R whose height can be raised or lowered to adjust the level of water standing inside the glass tube; the water surface acts as a closed reflecting end (forcing a node there), while the open top acts as an open end (forming an antinode), so the apparatus behaves exactly like a closed organ pipe of adjustable length. A vibrating tuning fork of known frequency $f$ is held just above the open top; as the water level (and hence the length of the air column above it) is slowly varied, the tube resonates -- produces a sudden, loud sound -- whenever the length of the air column matches an odd multiple of a quarter wavelength of the tuning fork's note. Because the antinode is not formed exactly at the open rim itself but a little distance above it, a small end correction $e$ must be included: the first resonance occurs at length $L_1$ where $\lambda/4 = L_1+e$, and continuing to raise the air column length, the second (next) resonance occurs at $L_2$ where $3\lambda/4 = L_2+e$. Subtracting the first equation from the second eliminates the unknown end correction entirely, $\lambda/2 = L_2-L_1 = \Delta L$, giving the wavelength directly from tw …