Obtain an expression for practical determination of end correction – (i) for a pipe open at both ends and (ii) for a pipe closed at one end.

Answer: (i) Open pipe: \(e = \dfrac{n_2l_2 - n_1l_1}{2(n_1 - n_2)}\); (ii) closed pipe: \(e = \dfrac{n_2l_2 - n_1l_1}{n_1 - n_2}\).

The antinode at an open end is formed slightly outside the pipe, at a distance \(e\) called the end correction (\(e \approx 0.3d\), \(d\) = inner diameter). It is found experimentally using two tuning forks of known frequencies \(n_1\) and \(n_2\) (\(n_1 \gt n_2\)). Let \(v\) be the speed of sound in air.

(i) Pipe open at both ends (an end correction at each end, effective length \(l + 2e\)). The pipe length is adjusted so that it resonates in its fundamental mode with fork \(n_1\) at length \(l_1\) and with fork \(n_2\) at length \(l_2\). For the fundamental mode \(\lambda = 2(l + 2e)\), so

(ii) Pipe closed at one end (one end correction, effective length \(l + e\)). The resonating air column lengths for the fundamental mode with forks \(n_1\) and \(n_2\) are \(l_1\) and \(l_2\). For the fundamental mode \(\lambda = 4(l + e)\), so

(With a single fork in a resonance tube, the first and second resonance lengths \(l_1\), \(l_2\) give \(l_1 + e = \lambda/4\) and \(l_2 + e = 3\lambda/4\), so \(e = \dfrac{l_2 - 3l_1}{2}\).)