Everybody has created a stationary resonant harmonic sound wave by whistling or blowing over a beer bottle or by swinging a garden hose or by playing the organ. In this section we will see how to compute the harmonics of a given (simple) pipe geometry for an imaginary organ pipe that is open or closed at one or both ends.
The way we proceed is straightforward. Air cannot penetrate a closed pipe end. The air molecules at the very end are therefore ``fixed'' - they cannot displace into the closed end. The closed end of the pipe is thus a displacement node. In order not to displace air the closed pipe end has to exert a force on the molecules by means of pressure, so that the closed end is a pressure antinode.
At an open pipe end the argument is inverted. The pipe is open to the air (at fixed background/equilibrium pressure) so that there must be a pressure node at the open end. Pressure and displacement are out of phase, so that the open end is also a displacement antinode.
Actually, the air pressure in the standing wave doesn't instantly equalize with the background pressure at an open end - it sort of ``bulges'' out of the pipe a bit. The displacement antinode is therefore just outside the pipe end, not at the pipe end. You may still draw a displacement antinode (or pressure node) as if they occur at the open pipe end; just remember that the distance from the open end to the first displacement node is not a very accurate measure of a quarter wavelength and that open organ pipes are a bit ``longer'' than they appear from the point of view of computing their resonant harmonics.
Once we understand the boundary conditions at the ends of the pipes, it is pretty easy to write down expressions for the standing waves and to deduce their harmonic frequencies.