
Something about sound
The harmonic seriesThe sound waves going up and down the instrument add up to give a standing wave, a vibration pattern of the air in the instrument. Several different such patterns are possible. On a flute, with all keys down, you can play about seven or eight different notes. Their pitches (approximate) are given below. The frequencies of these sounds are whole number multiples of the frequency of the lowest (f_{1}). We call them the harmonic series. Try playing the series on any instrument, without changing the fingering. You will notice the halfsharp on the 7th. (For more detail, see Flute acoustics and Clarinet acoustics
Harmonics and the different instrument boresWhy can the air in the flute vibrate in these different ways? Well, the tube is open to the air at both ends, so the pressure is pretty close to atmospheric, but the air is free to move in and out. Inside the tube the pressure can be higher or lower, but the air is less free to move. The diagram on the left shows the different vibration patterns or modes that satisfy the condition of the flute: zero pressure and maximum vibration at both ends. The top graph is the pattern of a wave whose length is twice that of the flute (2L, say), the second has wavelength 2L/2, the third 2L/3, and so on. The frequency is the speed of sound divided by this wavelength, and that gives the harmonic series f_{1}, 2f_{1}, 3f_{1} etc. (This is a slight simplification: the pressure node is a little distance outside the pipe, and so L, the effective length of the tube that should be used in such calculations, is a little longer than the physical length of the tube. The end effect is about 0.6 times the radius at an open end.)
These graphs show the wave patterns in the three simplest air columns: open cylinder, closed cylinder and cone. The red line represents sound pressure and the blue line represents the amount of air vibration. These pipes all have the same lowest note or fundamental. Note that the longest wavelength is twice the length of the open cyclinder (eg flute), twice the length of the cone (eg oboe), but four times the open length of the closed cylinder (eg clarinet). Thus a flutist or oboist plays C4 using (almost) the whole length of the instrument, whereas a clarinetist can play approximately C4 (written D4) using only half the instrument (ie removing the lower joint and bell). Important: in all three diagrams, the frequency and wavelength are the same for the figures in each row. When you look at the diagrams for the cone, this may seem surprising, because the shapes look rather different. This distortion of the simple sinusoidal shape is due to the variation in cross section along the tube. See Pipes and harmonics, where this point is discussed in detail. There is a more detailed discussion of standing waves in pipes in the introduction to flute acoustics, introduction to clarinet acoustics and introduction to saxophone acoustics, which also have a discussion of the use of register holes to produce harmonics. The effects of different bores are discussed in more detail in Pipes and harmonics. Flutes vs reed InstrumentsReed instruments are different: the end in the player's mouth is not open to the outside air, so the air is not maximally free to move in and out. The pressure is not fixed at atmospheric  in fact it can have its maximum value at this closed end. Consider the clarinet: it is mainly cylindrical and is open to the outside air at the bell end, but closed at the end in the mouth.The vibration patterns that the clarinet can play are shown in the diagram in the middle. The lowest wave is four times as long as the tube (4L'), the next is 4L'/3, the next 4L'/5 etc. So it only produces the odd members of the harmonic series (see above). Two consequences: first, that a clarinet can play nearly an octave lower (twice the wavelength) than a flute of the same length. Second, it "overblows a twelfth"  you have to go up 12 scale steps (3 times the frequency) before you can restart the same fingering. This is explained in more detail in the introduction to clarinet acoustics.
The bores of woodwind instruments. The diameters are exaggerated. The flute (top) and clarinet (middle) are nearly cylinders. The oboe, saxophone and bassoon are nearly conical (right). (See also Pipes and harmonics and Flutes vs clarinets.) Conical bores: oboes, bassoons and saxophonesWhat about oboes, bassoons and saxophones? Like the clarinet, they are closed at one end and open at the other, but the difference is that their air columns are in the shape of a cone. The resulting pressure and air motion vibrations are shown in the right hand diagram. When these waves get out into the outside world, they have the same frequencies as those from an open pipe of the same length. So an oboe, which is about the same length as the flute or the clarinet, has a lowest note close to that of the flute and, like the flute, it plays all of the harmonic series. To say more requires mathematics. Flute players can control which vibration pattern or mode they produce by the way they blow. In reed instruments, there is an octave hole or register hole which helps obtain the higher notes. Its purpose is to open up the tube to the outside air at or near one of the points where the air pressure should be atmospheric for the high vibrations.FlutesTo set the mood, listen to flutist Geoffrey Collins play some Debussy.I expect that most of us have played a note by blowing over the top of a bottle. The air in the bottle is springy and can vibrate, rather like a spring with a mass on it. When you blow across the top of the bottle, the stream of air from your lips can be deflected up or down by the expanding and contracting air in the bottle. When the stream is deflected down, some of it goes into the bottle, increasing the vibration. Thus the power in the stream of air can sustain the vibration in the bottle. (For an analysis of the sound made blowing across the top of a bottle, see Helmholtz Resonance). The mouthpiece of the flute (diagram below) works on the same principle  a jet of air passes a volume of air (the air in the tube of the instrument) which can vibrate. This is an oversimplified account, so go follow this link for a more detailed introduction to flute acoustics. Air jet or reed excites the vibration
The ClarinetTo set the mood, listen to Catherine McCorkill play a couple of phrases from the Mozart concerto.The clarinet has a single reed which swings in and out, cutting off and opening the stream of air as the pressure in the tube goes up and down, so in principle the operation is much like that of the double reeds. Clarinets come in a range of sizes, from sopranos that are 3/4 the size of the normal one, to contrabass clarinets which look like a plumber's nightmare. We saw above that the clarinet has only the odd numbered members of the harmonic series, so the gap between the first register and the second is a frequency ratio of three (a musical twelfth, or 19 semitones). All other woodwind players can play a scale of one octave and then use (nearly) the same fingerings again for the next register. A clarinetist must ascend twelve scale steps to repeat the fingerings. Because this exceeds the number of fingers on standard players, clarinets have four or five keys for the little fingers and extra keys for the knuckles of the index fingers. (See also the Introduction to clarinet acoustics.) The SaxophoneTo set the mood, listen to a movement of a quartet for saxophone, flute, bassoon and cello.The saxophone has a mouthpiece and reed much like that of a clarinet, but it is approximately a conical tube (like the oboe and bassoon) rather than a cylinder (like the clarinet). So it plays all the harmonics and has an octave between first and second registers. See the Introduction to saxophone acoustics. (See also Pipes and harmonics for some explanations about the importance of the conical bore and how it changes the harmonics.) The saxophone has a larger bore angle (and so a wider diameter at the bell) than any of the other woodwinds and this makes it possible to play rather louder. Like clarinets, saxophones come in a large family from tiny sopraninos to huge contrabasses. We have just posted a data base on saxophone acoustics. See this French saxophone site for a great series of pics on the fabrication of saxophones. Double reeds: Oboes and bassoonsIn the oboe and bassoon the sound is produced by a double reed (see the diagram and photographs above). The bassoon is the bass of the woodwind family  a long, folded conical tube reamed and mandrilled into four pieces of maple. Both have conical bores, like the saxophone, but their smaller angle makes them less loud than the saxophone. Qualitatively, the double reed shares many of the properties of a single reed. However, the geometry is more complicated. Further several effects that we have neglected here become important in double reeds. See this Introduction to double reed acoustics.
More about woodwinds

This photo shows the common orchestral woodwinds in ascending order of pitch. (Figure from The acoustics of woodwind musical instruments. Click to enlarge.)
