Full-wave numerical simulation of nonlinear dissipative acoustic standing waves in wind instruments

A finite volume full-wave method is used to simulate nonlinear dissipative acoustic propagation in ducts with a circular cross-section. Thermoviscous dissipative effects, due to bulk viscosity and shear viscosity in the boundary layer adjacent to the duct walls, are also considered. The propagation is assumed to be axisymmetric, and two different geometries are considered: a straight cylindrical tube, and a cylindrical tube joined smoothly to a slowly-flaring bell. Of special interest is the study of the onset of standing waves in the nonlinear regime. The full-wave numerical scheme is particularly well-adapted for this purpose, as it is not necessary to impose boundary conditions at the open end of the duct. A simplified model of excitation is adopted, where the lips are replaced by a spring–mass system which behaves like a pressure valve with a single degree of freedom. The full system behaves as expected, with a feedback cycle established between the pressure valve and the air column. The simulation is validated successfully in the linear regime using a theoretical solution. It is shown that increasing the stiffness of the lips leads to discrete jumps in playing frequency, which is behaviour typical of brass instruments. In the nonlinear regime, shock formation is observed for sufficiently high amplitudes of oscillation, and the radiation of these shock waves by the open end of the ducts can be visualised in the time-domain, along with edge-diffraction effects. The formation and evolution of standing waves in the nonlinear regime, where the effect of these shocks is very noticeable, is also examined.