The Memory Trick
๐ก Open: All Harmonics ยท Closed: Odd Only
Air columns inside pipes form standing waves, just like strings, but the boundary conditions differ based on whether the pipe's ends are open or closed. An open pipe (open at both ends) has a fundamental frequency fโ = v/2L and supports all harmonics. A closed pipe (closed at one end) has a fundamental frequency fโ = v/4L and supports only odd harmonics (1st, 3rd, 5th...).
Why It Works
An open end must be a displacement antinode (air is free to move there), while a closed end must be a displacement node (air can't move against a solid boundary). These different boundary requirements at each end are what force the different fundamental frequency formulas and harmonic patterns.
Step by Step
Comparing Open and Closed Pipes
1
Open pipe โ resonates at L = nฮป/2
With both ends open (both antinodes), the pipe supports a standing wave pattern where the fundamental has half a wavelength fitting the pipe length, and every integer harmonic above that also fits.
A flute is an open pipe (open at both the mouth hole and the far end) โ it produces all harmonics, giving it a rich, full tone.
2
Closed pipe โ resonates at L = nฮป/4 (odd n only)
With one end closed (a node) and one open (an antinode), only odd-numbered harmonics fit the boundary conditions โ even harmonics are physically impossible in this configuration.
A clarinet is a closed pipe (closed at the reed end) โ this is why it has a notably different, more hollow-sounding timbre than an open pipe instrument like a flute, since it's missing all even harmonics.
3
The missing harmonics change the timbre
Even though two instruments might play the 'same note' (same fundamental frequency), whether they're open or closed pipes changes which additional harmonics are present, which is a major reason different instruments sound different playing the identical pitch.
A flute and clarinet playing the same fundamental note sound distinctly different because the flute includes even harmonics the clarinet physically cannot produce.
๐ฅ Worked Example
An open pipe is 0.5 m long. Using the speed of sound in air (v โ 343 m/s), what is its fundamental frequency?
1
Apply the open pipe formula: fโ = v/2L.
2
Plug in values: fโ = 343 / (2 ร 0.5) = 343/1 = 343 Hz.
3
Compare to a closed pipe of the same length: using fโ = v/4L instead would give 343/2 = 171.5 Hz โ exactly half the open pipe's fundamental, illustrating why closed pipes sound an octave lower than open pipes of the same physical length.
โ ๏ธ Most Common Resonance in Pipes Mistakes
The most common trap is using the same fundamental frequency formula for both open and closed pipes โ they're different (v/2L vs v/4L) because the boundary conditions at a closed end (a node) differ fundamentally from an open end (an antinode).
โ Quick Self-Test
1) Write the fundamental frequency formula for an open pipe (open at both ends). fโ = v/2L. 2) Write the fundamental frequency formula for a closed pipe (closed at one end). fโ = v/4L. 3) Which harmonics does a closed pipe support โ all, or only odd? Only odd harmonics (1st, 3rd, 5th...). 4) Is a flute an open or closed pipe instrument? Open pipe. 5) For the same physical length, is a closed pipe's fundamental frequency higher or lower than an open pipe's? Lower โ exactly half, since f = v/4L is half of f = v/2L for the same L.