A cylindrical pipe exhibits normal modes that are harmonically related. If the pipe is open at both ends then there is a pressure node at each end of the tube and the tube must hold an integral number of half-wavelengths. Thus the modes have the form and .
If the pipe is closed at one end and open at the other then there
is a pressure
node at the open end and a pressure anti-node at the closed end. In
the tube must hold an odd integral number of quarter-wavelengths.
modes have the form and .
Note that the lowest note from a half-closed tube is 1 octave lower than that from an open tube of the same length. Note also that all the even harmonics are missing from a half-closed tube.
A jet of air blown across a sharp edge produces an edgetone, a clear single frequency tone, in addition to a certain amount of noise. The frequency of the tone depends on the speed of the jet of air and on the distance between the jet and the edge approximately as .
Organ flue pipes, flutes, whistles, and recorders all operate by coupling a jet of air flowing over and edge to a resonator. In this case the frequency of oscillation is controlled by the transit time for a pressure pulse to go down the tube and back and so by the length of the tube. The air jet is forced into and out of the pipe by the resonance and the instrument sounds at a resonant frequency of the pipe, usually very much lower than the edge tone.
Drilling holes in the side of a blown pipe allows the effective length of the pipe to be shortened. A pipe with a hole in it will sound the same note as an entire pipe a little longer than the distance from the fixed end of the pipe to the hole. The sound wave continues some way past the hole. The bigger the hole is, the higher the note, the lounder the sound, and the less high frequency content the sound has.