closed pipe
Summary
A closed pipe is closed at one end and open at the other, so it has a displacement node at the closed end and an antinode at the open end. Its resonances are only odd harmonics, f = nv/4L with n = 1, 3, 5, …, so its fundamental is half that of an open pipe of equal length.
Detail
Meaning: In MCAT usage, a closed pipe is closed at one end and open at the other. Air cannot move at the closed end, so it is a displacement node (and a pressure antinode); the open end is a displacement antinode. The shortest standing wave that fits is a quarter wavelength, and each next resonance adds half a wavelength, so only odd harmonics occur.
Equation: fn = nv/(4L), n = 1, 3, 5, …, where fn is the resonant frequency (Hz), v is the speed of sound in the air (m/s), L is the pipe length (m), and n is the harmonic number (unitless). Equivalently, λ = 4L/n, with λ in m.
On the MCAT: Passages compare open and closed pipes of equal length, or change L or v and ask how the pitch shifts. A shorter pipe or faster sound speed raises every resonant frequency. The closed-pipe fundamental is half that of an open pipe of the same length (v/4L versus v/2L). A list of resonances in odd ratios (100, 300, 500 Hz) signals a closed pipe.
Common confusion: Students apply λ = 2L/n to every pipe or expect a second harmonic at 2f₁. A closed pipe has no even harmonics, so the next resonance after f₁ is 3f₁. Also keep the node and antinode straight: the closed end is a displacement node but a pressure antinode.
Sources
- Physics (Giancoli)
- Physics (Cutnell and Johnson)
- Fundamentals of Physics (Halliday, Resnick, and Walker)
Written for the MCAT and checked before it was published. For study reference only. Spotted an error? Let us know.