Some Paradoxes of the "cylindrical Saxophone"

The story of the "cylindrical saxophone" started with Benade (1988). The basic idea is that when the length of the missing part of a truncated cone is smaller than the wavelength, and therefore smaller than the length of the truncated cone, the behavior of a conical reed instrument has similarities with that of a cylindrical pipe excited by a reed on its side, at an intermediate location. The shorter part of the cylinder has to be equal to that of the missing part of the cone. This similarity allowed to get caricatures of the pressure waveform inside the mouthpiece, in the form of a Helmholtz (2-state motion), which is well known for bowed string instruments. However some paradoxes remain with this analogy. If the waveform is a Helmholtz motion, the negative pressure episode has a duration corresponding to the resonance frequency of the short length, i.e., a frequency which does not fulfill the condition of the analogy. Furthermore using the simplest approximation deduced from the analogy, the waveform of the radiated sound by the conical instrument should be a Dirac comb. This means that all the Fourier coefficients have the same magnitude, without missing harmonics. After a short review of the literature, these paradoxes will be discussed.