Fundamental of a Closed Pipe

A pipe sealed at one end must have a node (no air motion) at the closed end and an antinode (maximum motion) at the open mouth. The longest wave satisfying both fits just a quarter of a wavelength inside, so the fundamental is v/4L — an octave lower than an open pipe of the same length. Blow across a 20 cm test tube and you get roughly 343/(4 × 0.20) ≈ 429 Hz, close to concert A.

This quarter-wave trick is why a clarinet plays nearly an octave deeper than a flute of the same length: the reed end acts closed while the flute is open at both ends. It also explains the closed pipe's hollow timbre — the boundary conditions only permit odd harmonics (f, 3f, 5f...), stripping out every even overtone. Your own ear canal is a closed pipe about 2.5 cm deep, resonating near 3400 Hz, which is precisely where human hearing is most sensitive — and where audiologists find the most noise damage.