Among musicians little is known about the physics of the saxophone. An ignorance which makes stories are being told that do not really match what's actually going on inside the instrument. However, a saxophone is not a mystery, a saxophone obeys the laws of nature.
Sound waves exist in two distinctly different forms: the standing and the travelling wave. Although a complete discussion of the physics of waves is far beyond the scope of this site, yet the following: inside the instrument we find a standing wave which owes its character to the fact that on both ends it is reflected upon itself. In this way a vibration inside the instrument comes into being that is going up and down and not one that is going in a forward movement only and which, as is sometimes mistakenly stated, starts at the reed and travels down through the instrument. To be able to exist this way, the mirror image must accurately fit the original one. Given a certain sound velocity, on a certain (instrument) length this mirror image fits the original only at a particular pitch. The result is a wave motion which is linked in a controllable way to the length of the instrument and thus with a controllable pitch. We should keep in mind that both the mouthpiece end and the open bottom end (whether it be the bell or the first open tone hole) act as a mirror for this standing sound wave. What's reflected is the wave motion itself; the separate particles stay more or less in place.
animated cartoon with thanks to Dan Russell, Kettering University
The animation shows us a standing wave. There is both an open and a closed end. It is clear that we find a wave which sets off from nowhere and which goes nowhere: the wave just stands still in a corresponding length of tube. In our case this will be the closed length (if only by keys) of the instrument.
In red we see a graph of the pressure as it builds up and diminshes along the tube. So, in a standing wave, there are spots with pressure differences and without any pressure differences. In places without pressure differences air particles move up and down and in places without any movement there are pressure differences (particles moving in opposite directions from both sides). So, there are two types of extremes: spots with pressure diffrences only (we call them pressure–antinodes) and spots with movement only (we call them displacement–antinodes) and in between all the possible transitional stages.
At the reed side of the instrument, which functions (more or less) as a closed end, there will be clear pressure differences. The reed acts as a controlable lock to the instrument. When pressure on the reed side is high, the reed will open and new energy can be added into the tube. When pressure on the reed side is low, the reed will close.
Further down the tube all different stages of the standing wave will occur.
At the bell side of the instrument pressure will always be atmospheric; here pressure differences are impossible, but movement is. Here we find a to–and–fro going movement of air particles which sets the outside air in motion. It will generate a travelling wave departing from the instrument to reach our ears. As far as the instrument itself is concerned, this travelling wave is a loss of energy which, in order to keep the standing wave inside going, needs to be compensated by supplying new energy to the top of the instrument: so we blow! (Here I do not take into account all the other losses such as friction and thermal losses. In fact, these other losses taken together are a lot bigger than the loss of energy by sound emission. The sound pressure inside the instrument is a great lot bigger than what we will ever get out of it...)
And finally, you see that the wave can be reflected on two different types of endings: an open and a closed one. Here, the mirror image each time has an inverse character: at the open end movement swings to–and–fro whereas at the closed end pressure pulsates up and down. Yet in both cases, the mirror end is situated right at a maximum of these aspects of the standing wave!