Harmonics of a closed pipe

Problem

A pipe closed at one end supports only odd harmonics. Rank, in increasing order, the first three possible frequencies relative to the fundamental $f_1$. Compare with a pipe open at both ends.

Solution

Principle. In a pipe closed at one end (node at the closed end, antinode at the open end) the allowed frequencies are

$f_n = (2n-1)\,\frac{v}{4L}, \qquad n = 1, 2, 3, \dots$

that is, only the odd multiples of the fundamental: $f_1,\ 3f_1,\ 5f_1, \dots$

First three frequencies.

$\ev{f_1 < 3f_1 < 5f_1}$

Comparison with the open pipe. A pipe open at both ends (antinodes at both ends) has

$f_n = n\,\frac{v}{2L}$

that is, all integer multiples $f_1, 2f_1, 3f_1, \dots$. The open pipe therefore has a complete harmonic series and, for the same length, a fundamental twice that of the closed pipe. The absence of even harmonics in the closed pipe characterises its timbre (e.g. the clarinet).

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