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An air displacement wave is also an air pressure wave

The nodes of the displacement wave, where the air is not rushing back-and-forth but is doing the most piling-up-and-spreading-out, are the antinodes of the pressure wave. The antinodes of the displacement wave, where the air is rushing back-and-forth the most, but is not piling up or spreading out at all, are the nodes of the pressure wave. Both waves must have exactly the same frequency, of course; they are actually just two aspects of the same sound wave.

At an open end of the tube, there is nothing to stop the air rushing in and out, and so it does. What the air cannot do at the open end is build up any pressure; there is nothing for the air to build up against, and any drop in pressure will just bring air rushing in from outside the tube. So the air pressure at an open end must remain the same as the air pressure of the room. In other words, that end must have a pressure node (where the air pressure doesn't change) and (therefore) a displacement antinode.

Since being exposed to the air pressure outside the instrument is what is important, the "open end" of a wind instrument, as far as the sound waves are concerned, is the first place that they can escape - the first open hole. This is how woodwinds change the length of the wave, and the pitch of the note. For more on this, please see Wind Instruments – Some Basics .

Harmonic series in tubes

As explained in the previous section , the standing waves in a tube must have a (displacement) node at a closed end and an antinode at an open end. In an open-open tube, this leads to a harmonic series very similar to a harmonic series produced on a string that's held at both ends. The fundamental , the lowest note possible in the tube, is the note with a wavelength twice the length of the tube (or string). The next possible note has twice the frequency (half the wavelength) of the fundamental, the next three times the frequency, the next four times, and so on.

Allowed waves in an open tube

These are the first four harmonics allowed in an open tube. Any standing wave with a displacement antinode at both ends is allowed, but the lower harmonics are usually the easiest to play and the strongest harmonics in the timbre .

But things are a little different for the tube that is closed at one end and open at the other. The lowest note that you might be able to get on such a tube (a fundamental that is unplayable on many instruments) has a wavelength four times the length of the tube. (You may notice that this means that a stopped tube will get a note half the frequency - an octave lower - than an open tube of the same length.) The next note that is possible on the half-closed tube has three times the frequency of the fundamental, the next five times,and so on. In other words, a stopped tube can only play the odd-numbered harmonics.

Allowed wavelengths in a stopped tube

Again, these are the lowest (lowest pitch and lowest frequency) four harmonics allowed. Any wave with a displacement node at the closed end and antinode at the open end is allowed. Note that this means only the odd-numbered harmonics "fit".

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Source:  OpenStax, Sound, physics and music. OpenStax CNX. Jan 06, 2005 Download for free at http://cnx.org/content/col10261/1.1
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