Flutes, organ pipes and clarinets make music with standing sound waves in air columns. The idea is the same as for strings, but the ends work differently. As in Sound: Pitch and Loudness, sound is air moving back and forth along the pipe.
Key idea
Open at both ends (all harmonics): $$f_n = \frac{nv}{2L}, \quad n = 1, 2, 3, \dots$$ Closed at one end (odd harmonics only): $$f_n = \frac{nv}{4L}, \quad n = 1, 3, 5, \dots$$ A closed end is a displacement node; an open end is a displacement antinode.
Why it works
Air can't move through a closed end, so it is a node. At an open end the air moves freely, so it is an antinode. Open–open: antinode to antinode, a whole number of half wavelengths, just like a string. Closed–open: node to antinode, an odd number of quarter wavelengths, $L = n\frac{\lambda}{4}$ with $n$ odd. The even ones would need a node at the open end, which can't happen.
Common mistake
Giving a pipe closed at one end the same harmonics as an open pipe. A closed pipe has only odd multiples of its fundamental, and its fundamental is half that of an open pipe of the same length. Check: closed-pipe frequencies like $150, 250, 350\ \text{Hz}$ step by $2f_1$, and $f_1 = 50\ \text{Hz}$ itself is half the step.
Worked example: Open and closed
Find the first two resonant frequencies of a $0.50\ \text{m}$ pipe ($v = 343\ \text{m/s}$), first open at both ends, then closed at one end.
- Open: $f_1 = \dfrac{343}{2(0.50)} = 343\ \text{Hz}$, then $f_2 = 686\ \text{Hz}$.
- Closed: $f_1 = \dfrac{343}{4(0.50)} = 172\ \text{Hz}$ (171.5), then $f_3 = 3f_1 = 515\ \text{Hz}$.
Answer: Open: $343$ and $686\ \text{Hz}$. Closed: $172$ and $515\ \text{Hz}$.
Worked example: Which kind of pipe?
A pipe resonates at $150$, $250$ and $350\ \text{Hz}$, and at no frequencies between them. Is it open or closed at one end? How long is it? ($v = 343\ \text{m/s}$.)
- Neighbors differ by $100\ \text{Hz}$. If it were open, the frequencies would be multiples of $100$, but $150$ is not.
- They are $3, 5, 7$ times $50\ \text{Hz}$: odd multiples, so the pipe is closed at one end with $f_1 = 50\ \text{Hz}$.
- $L = \dfrac{v}{4f_1} = \dfrac{343}{200} = 1.7\ \text{m}$.
Answer: Closed at one end, about $1.7\ \text{m}$ long