Presets
Playback
Pipe type
Open at the top, or closed by a stopper.
Pipe geometry and air
Length is measured from the upper lip to the top of the pipe.
Wind and voicing (jet)
Wind pressure sets the jet speed; the cut-up sets the jet length.
Display and sound
Animation runs at a slow visual rate; amplitudes are exaggerated.
Layers
Model checks
The science
The pipe as a resonator
Sound in a narrow pipe obeys the 1-D wave equation. A pipe open at both ends (pressure nodes) has modes fn = n c / (2 Leff), n = 1, 2, 3 ... (all harmonics). A stopped pipe, closed at the top (pressure antinode) and open at the mouth, has fn = (2n − 1) c / (4 Leff): odd harmonics only, so the same length sounds an octave lower. The effective length adds an end correction of about 0.6 a at each open end (a = radius): Leff = L + 1.2 a (open) or L + 0.6 a (stopped).
Temperature
The speed of sound is c = 331.3 √(1 + T/273.15) m/s, so pitch rises about 3 cents per kelvin (a cold church plays flat, a full church plays sharp). Reed pipes are far less sensitive because the reed, not the air column, sets the pitch.
The air-jet drive
Wind of pressure p leaves the flue slit at U = √(2p/ρ) (Bernoulli). Disturbances travel along the jet at about 0.4 U, so a jet of length h (the cut-up) has transit time τ = h/(0.4 U). The jet pushes air in and out of the mouth, the pipe's acoustic flow bends the jet, and the loop sustains an oscillation near the resonance whose frequency is closest to the jet's preferred Strouhal frequency f = St·U/h. More wind raises the preferred frequency: the pitch creeps up a little, then the pipe jumps to the next mode (overblowing).
Independent numerical checks
A staggered-grid FDTD solver of pt = −ρc2 ux, ut = −px/ρ with an open (p = 0) or rigid (u = 0) end is excited by a pulse; the spectral peaks of its recorded pressure are compared with the formulas above, and its temperature scaling with √(T2/T1). They run in a background worker.
Honest simplifications
The jet-drive model is a teaching model: the preferred Strouhal number St* (default 0.07, textbook range roughly 0.05-0.15) and the small jet pulling of pitch (up to 3 %) are tuned constants, not measurements of a real pipe. Real voicing also depends on flue width, nicking, languid angle, foot-hole size and the pipe's nonlinear flow losses, which are not modelled. The spectrum is a simple weighted harmonic series (brighter for a lower cut-up/diameter ratio, odd harmonics only for a stopped pipe); the pipe wall is rigid and lossless; the air is dry at 1 atm; the mouth is treated as an open end with the stated end correction.
What to try
- Switch from "8 ft open principal" to "8 ft stopped flute": a pipe about half as long sounds the same note, with only odd harmonics.
- Slide the wind pressure upward on the pitch graph: the pitch creeps sharp, then jumps to the next mode (overblowing).
- Set the temperature to 5 C and then 35 C and watch the pitch drift in cents.
- Widen the cut-up h: the jet-preferred frequency falls and the tone gets softer.
- Turn on Sound and compare the spectra of open and stopped pipes.