AP® Physics 2: Algebra-Based · Unit 14 · Topic 14.6
Standing Waves in Air Columns
A standing wave forms in an air column when reflected sound waves interfere, with open ends forcing antinodes and closed ends forcing nodes. The pipe's length and which ends are open decide which frequencies resonate.
An air column resonates when a standing wave fits neatly between its boundaries. Air is free to move at an open end, so pressure stays near atmospheric there, which forces a displacement antinode. Air can't move at a closed end, so that spot is always a displacement node. The pipe's length, combined with these fixed boundary conditions, picks out which wavelengths can stand still inside it. Only those wavelengths resonate; everything else cancels itself out through interference on the way back.
A pipe open at both ends needs an antinode at each end, so it fits a half wavelength at its fundamental, and every integer multiple after that: harmonics 1, 2, 3, 4, all present, with fn = nv/(2L). A pipe closed at one end needs a node at the closed end and an antinode at the open end, so the shortest wave that fits is a quarter wavelength, not a half. Stretching further only adds odd numbers of quarter wavelengths, so only odd harmonics show up: 1, 3, 5, with fn = nv/(4L) for odd n. That's why a closed pipe sounds different from an open one even at the same length: it's missing every even harmonic.
This connects directly to the diffraction and interference ideas nearby in the unit. A standing wave is really two traveling waves of the same frequency moving in opposite directions, interfering constructively at antinodes and destructively at nodes. Blow across a bottle and the air column acts as a closed-at-bottom, open-at-top pipe; pouring out water lengthens the resonating column, lowers f1 = v/(4L), and drops the pitch.
Key terms
- Node
- A point in a standing wave where the air barely moves, forced at any closed end of a pipe.
- Antinode
- A point in a standing wave with maximum displacement, forced at any open end of a pipe.
- Fundamental frequency
- The lowest frequency that fits the pipe's boundary conditions, labeled f1.
- Harmonic series
- The set of resonant frequencies a pipe supports, all integer multiples of f1 for open pipes but only odd multiples for pipes closed at one end.
Watch it explained
- How it worksHow does an open end force an antinode while a closed end forces a node?
- How it worksCompare the harmonic series of a pipe open at both ends with one closed at one end.
- Worked exampleFind the fundamental frequency of a 0.50 m pipe closed at one end with sound at 343 m/s.
- Common mistakeWhy isn't the third harmonic of a closed pipe three times as many half wavelengths long?
Check yourself
End of topic.