Organ Pipe (Open / Closed)

Organ pipe: a tube with sound standing waves. Two main types: OPEN at both ends, or CLOSED at one end.

OPEN pipe: both ends antinodes (air can move freely). λ_n = 2L/n. ALL harmonics: f₁, 2f₁, 3f₁, …

CLOSED pipe: closed end node (air can't move), open end antinode. λ_n = 4L/(2n−1). Only ODD harmonics: f₁, 3f₁, 5f₁, …

Open pipe fundamental: f₁ = v/(2L). Closed pipe fundamental: f₁ = v/(4L) — HALF the open pipe.

Same length L: open pipe sounds an OCTAVE higher than closed pipe.

Pipe organs use both types for different sounds; reed instruments (clarinet) behave like closed pipes; flute behaves like open pipe.

End correction: actual antinode is slightly OUTSIDE the open end — small correction to L.

Sound speed in pipe: same as in surrounding air (v depends only on temperature, not pipe geometry).