Fundamental of an Open Pipe

Open both ends of a tube and the air is free to move at each mouth, so both ends must be antinodes. The longest standing wave that fits between two antinodes is half a wavelength, giving λ = 2L and a fundamental of v/2L — one full octave above a pipe of the same length stopped at one end. A flute is about 60 cm of open pipe and sounds near 343/(2 × 0.60) ≈ 286 Hz at its lowest; a clarinet of much the same length plays nearly an octave lower, because its reed end behaves as closed.

The bigger difference is in the timbre rather than the pitch. An open pipe supports every harmonic — f, 2f, 3f, 4f — while the closed pipe permits only the odd ones, which is why a flute sounds bright and round and a clarinet hollow and woody. It also changes what happens when a player overblows: a flute jumps a clean octave to its second harmonic, a clarinet has no second harmonic to jump to and leaps a twelfth to the third instead, which is why the two instruments have entirely different fingering logic above the break. One practical warning sits on top of the arithmetic: the air just outside each mouth moves along with the column, so the acoustic length is longer than the tube. The usual allowance is about 0.6 of the pipe radius at each open end — roughly 30 mm on a 25 mm bore — which is why organ builders and pipe makers cut long and tune down.