Organ Pipe (Open / Closed)
Open: nv/2L. Closed: (2n−1)v/4L.
Key Notes
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).
Formulas
Open pipe wavelengths
All integer harmonics allowed.
Open pipe frequencies
f₁ = v/(2L). All harmonics 1f₁, 2f₁, 3f₁, ...
Closed pipe wavelengths
Only odd multiples allowed.
Closed pipe frequencies
f₁ = v/(4L). Sequence: f₁, 3f₁, 5f₁, ...
Important Points
Open pipe: all harmonics (1, 2, 3, 4, …) ⇒ rich sound.
Closed pipe: only odd harmonics (1, 3, 5, …) ⇒ duller, hollow sound.
Open and closed pipes of same L: open has f₁ = v/(2L), closed has f₁ = v/(4L) — closed is one OCTAVE LOWER.
End correction (~0.6r for open end of cylindrical pipe of radius r): effective length slightly larger than physical.
Reed instruments (oboe, clarinet) are closed-pipe-like; flutes are open-pipe-like.
Sound speed v depends on TEMPERATURE — pipes get longer/sound goes higher when warmer.
Organ Pipe (Open / Closed) notes from sciphylab (also known as SciPhy, SciPhy Lab, SciPhy Labs, Physics Lab). Class 11 physics revision for JEE Mains, JEE Advanced, NEET UG, AP Physics 1/2/C, SAT, and CUET-UG.