Q.Obtain an expression for practical determination of end correction –

The end correction is found experimentally by comparing the resonating air-column lengths for two successive resonances (or overtones) with the known wavelength.

1. Pipe closed at one end (resonance tube method): In a resonance tube (closed pipe) experiment, if $L_1$ and $L_2$ are the lengths of the air column for the first and second resonances with a tuning fork of wavelength $\lambda$, then, because of the end correction $e$ at the open end, $L_1 + e = \dfrac{\lambda}{4}, \qquad L_2 + e = \dfrac{3\lambda}{4}$ Subtracting: $L_2 - L_1 = \dfrac{\lambda}{2} \Rightarrow \lambda = 2(L_2-L_1)$ (this itself is used to find the speed of sound). Also, from the two equations, $3(L_1+e) = L_2+e \;\Rightarrow\; 3L_1 + 3e = L_2 + e \;\Rightarrow\; e = \dfrac{L_2-3L_1}{2}$
2. Pipe open at both ends: For an open pipe, there is an end correction $e$ at each open end. If $L$ is the geometrical length and $f_n$ the frequency of the $n^{th}$ harmonic, $f_n = \dfrac{nv}{2(L+2e)}$ …