Toward a simple physical model of double-reed musical instruments : influence of aero-dynamical losses in the embouchure on the coupling between the reed and the bore of the resonator.
Résumé
The air flow model usually considered for physical modeling of wind instruments has to be modified for double-reed instruments. Indeed, it is usually assumed in simple models that air pressure in the reed channel is the same as air pressure at the input of the bore. On the contrary, the first aim of this paper is to explain that, because of the geometrical specificities of the embouchure of double-reed instruments, this assumption can hardly be applied to double reed modeling. A refined (yet still quasi- stationary) model is then proposed, where air pressure at the inlet of the double reed channel and at the input of the bore are linked by a nonlinear relation. This relation is parameterized by a single coefficient which captures all the differences with the usual flow model. This allows an analytical comparison between both models. The second aim of this paper is to study the influence of the flow model on the auto-oscillation of the well known Backus model of reed instruments. Three possible behaviors, qualitatively different, are identified according to the importance of the nonlinear term characterizing the double reed flow model. Two of these behaviors may present an hysteretic character. Analytical predictions are in agreement with numerical simulations.