I have played the saxophone for a very long time and really enjoy it. Although I have played it for so long, I have never learned the physics behind how blowing on a little piece of wood generates sound. In making a sound on the saxophone, one blows air at a high pressure through the mouthpiece. The reed controls the air flow through the instrument and acts like an oscillating valve. The reed, in cooperation with the resonances in the air in the instrument, produces an oscillating component of both flow and pressure. Once the air vibrates, some of the energy is radiated as sound out of the bell and any open holes. A much greater amount of energy is lost as a sort of friction with the wall. The column of air in the saxophone vibrates much more easily at some frequencies than at others. These resonances largely determine the playing frequency and thus the pitch, and the player in effect chooses the desired resonances by suitable combinations of keys. Also, the saxophone acts as a closed end resonator, and, more simply, a conical pipe. The natural vibrations in the saxophone that cause it to play notes are standing waves. The standing waves in a cone of length L have wavelengths of 2L, L, 2L/3, L/2, 2L/5... in other words 2L/n, where n is a whole number. The wave with wavelength 2L is the fundamental, that with 2L/2 is called the second harmonic, and that with 2L/n the nth harmonic. The frequency equals the wave speed divided by the wavelength, so this longest wave corresponds to the lowest note on the instrument: Ab on a Bb saxophone, Db on an Eb saxophone.
For a more complete overview, visit the University of South Wales website on acoustics: https://newt.phys.unsw.edu.au/jw/saxacoustics.html#overview