Stationary waves in a closed pipe (odd harmonics only)
A pipe closed at one end has a displacement node at the closed end and a displacement antinode at the open end, so L = (2n − 1)λ/4. This gives λ = 4L/(2n − 1) and f = (2n − 1)v/4L, so only the odd harmonics exist (f₁, 3f₁, 5f₁ …). The fundamental is λ = 4L, exactly one octave below an open pipe of the same length.
λ = 4L/(2n − 1) · f = (2n − 1) v/(4L) · odd harmonics onlyWhat this actually means
The odd-only result is not a rule to memorise, it is forced by the geometry. You need a node at one end and an antinode at the other, and the shortest such pattern is a quarter of a wavelength. Every extra half wavelength you add keeps the pattern legal, hence quarter, three-quarters, five-quarters and so on.
So there is no second harmonic and no fourth harmonic in a closed pipe. If a question offers you a frequency of exactly 2f₁ for a closed pipe, it is wrong.
Careful with overtone numbering: the first overtone of a closed pipe is the third harmonic, at 3f₁, not 2f₁. This is the single most common closed-pipe mistake.
A closed pipe of length L sounds an octave lower than an open pipe of the same length, because 4L is twice 2L. That is why closing off a pipe on a wind instrument drops the pitch so dramatically.
For sound, the closed end is a displacement node but a pressure antinode, which is why the pressure variation is largest exactly where the air is not moving.
Calling the first overtone of a closed pipe the second harmonic at 2f₁, when it is the third harmonic at 3f₁.
Prove it — watch it be true
- Select the 3D closed pipe and set the mode selector to n = 1, confirming a node at the closed end and an antinode at the open end.
- Try to select the even harmonics and confirm the mode selector skips them as impossible.
- Step to the next available mode and confirm it is the third harmonic at 3f₁, with λ = 4L/3.