Q.An air column in a pipe which is closed at one end, will be in resonance with the vibrating body of frequency 83 Hz. Then the length of the air column is:

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 …