Question

Show that the fundamental frequency of vibration of the air column in a pipe open at both ends is double that of a pipe of the same length and closed at one end.

Answer 1

Let L_O and L_C be the lengths of a pipe open at both ends and a pipe closed at one end, respectively. Let n_O and n_C be their corresponding fundamental frequencies. Then, ignoring the end corrections,

n_O = \(\frac v{2L_O}\) and n_C = \(\frac v{4L_C}\)

where v is the speed of the sound in air.

Given that L_O = L_C = L (say),

n_O = \(\frac v{2L}\) and n_O = \(\frac v{4L}\)

∴ n_O = 2 \((\frac v{4L})\) = 2n_C