On the 1D wave propagation in wind instruments with a smooth profile

Due to the simple properties of plane waves, non lossy straight pipes and their concatenation have been extensively used to compute acoustic transfer functions from bore profiles of wind instruments (input impedance, transmittance, etc). This is also the case for real-time simulations: introducing travelling waves has led to the well-known digital waveguides formalism. Nevertheless, such discontinuous concatenations involve impulse responses composed of pulse trains of Dirac measures, which are structurally unrealistic for smooth bores. Similarly, continuous but non smooth approximations based on conical segments involve discontinuous pulse trains of damped exponentials. This invited paper presents an overview of results that have been elaborated to weaken such artifacts and increase realism, while preserving most of the worthwhile properties of straight pipes. The key steps are based on the use of: (1) a refined 1D wave equation (curvilinear horn equation)based on an isobar map rectification; (2) smooth (C1-regular) junctions of constant-flared acoustic pipes; (3) a radiation model which is compatible with (1); (4) visco-thermal losses. It allows to recover a standard matrix formalism to compute impedances and transmittances of smooth bore parts that yield accurate results. It still make definitions of travelling waves and digital waveguide-like structures possible for the simulation. Finally, by representing smooth bores by very few flared segments (compared to many straight or conical pipes), such descriptions (with a few parameters) are an interesting alternative to optimize wind instrument bores for some criteria (target shape, target impedance, harmonicity, etc)