Quarter-Wavelength Resonance

A tube closed at one end naturally rings loudest at the pitch whose sound wave is four times the tube's length.

A pipe sealed at one end traps a standing wave that fits just a quarter of itself inside, ringing at f = c / (4 x L).

What it is

A tube closed at one end rings loudest at the pitch whose wave is four times the tube's length.

Key facts

Speed of sound in air = 343 m/s at 20 degrees C (rises ~0.6 m/s per +1 degree C)
Closed-tube (quarter-wave) formula: f = c / (4 x L). c = speed of sound (m/s), L = tube length (m), f = frequency (Hz)
Resonant wavelength = 4 x L: the tube fits exactly 1/4 of the wave (wavelength = c / f)
1 m closed tube: f = 343 / (4 x 1) = 85.75 Hz. 0.5 m = 171.5 Hz. 2 m = 42.9 Hz. Half L = +1 octave
Closed tube makes ODD harmonics only: f, 3f, 5f, 7f (1 m tube hums at ~86, 257, 429 Hz...)
OPEN-both-ends tube is HALF-wave: f = c / (2 x L), all harmonics; an octave HIGHER than a same-length closed tube
Closed end = pressure ANTINODE / velocity NODE; open mouth = pressure NODE / velocity antinode
End correction: add ~0.6 x radius to L at an open end, so real f sits slightly LOWER than the formula
+6 dB = 4x sound pressure; doubling pressure = +6 dB; doubling power = +3 dB; -3 dB = half-power point (sets bandwidth Q)
Quarter-wave bass trap depth: L = c / (4 x f_target). Kill 100 Hz needs 343/400 = 0.86 m deep

How it works

Sound enters the tube and reflects off the sealed end.
The sealed end forces air velocity to zero (a node) and pressure to peak (antinode).
The open mouth lets air move freely (velocity antinode, pressure node).
Only waves that fit exactly a quarter-wavelength between those two ends reinforce.
That trapped standing wave rings loudest at f = c / (4 x L).
Longer tube means a longer wave fits, so a lower note.

Real examples

Blowing across a test tube or a beer bottle: the air column rings at f = c / (4 x L)
Didgeridoo: a long closed-ish pipe gives a deep drone around 60-75 Hz
Organ 'stopped' pipes: a capped pipe sounds an octave lower than an open pipe of the same length
A long HVAC duct or backstage corridor humming at one stubborn bass pitch
A quarter-wave bass trap stood in a room corner to soak up one room mode

How it helps in live sound

Got a venue duct/gap droning? Measure its length, calc f = 343/(4xL), then notch that Hz with a tight EQ band
Build a quarter-wave trap to L = 343/(4 x f_problem): a 90 Hz boom needs ~0.95 m depth
Use RTA/REW to find the offending peak, then confirm it matches a tube length you can see (duct, riser cavity, doorway gap)
Stuff long open cavities with rockwool to broaden and tame the resonance
Stack traps in room corners (pressure maxima) where bass modes are strongest for most absorption
Remember it is ODD harmonics only: a closed cavity humming at 86 Hz also rings at ~257 Hz and ~429 Hz

Everyday analogy

Like a child on a swing: push at exactly the right rhythm (the tube's natural note) and it builds huge, push off-rhythm and nothing happens.

Watch out

Myth: a longer tube gives a higher note. Wrong, it is inverse: longer L = LOWER f, because f = c / (4 x L).

Fun fact

A capped organ pipe sounds a full octave LOWER than an open pipe of identical length, so builders use stopped pipes to get deep bass in half the space.

Key takeaways

Closed-one-end tube resonates at f = c / (4 x L).
The resonant wavelength is exactly 4x the tube length.
Longer tube = lower note (inverse relationship).
Closed tubes ring on ODD harmonics only (f, 3f, 5f).
Open-both-ends tube is half-wave: f = c / (2 x L), all harmonics.
Quarter-wave traps are sized to kill one target frequency.