A horn model for the open end of brass instruments

Over the years, brass wind instruments have been improved empirically and manufactured using molds over which brass sheets are hammered without recourse to any analytical model. Although there have been some attempts to describe analytically the geometry of horns of various wind instruments using the linear acoustic approximation, these models are based on mathematical rather than physical arguments. We present a family of curves that strongly resemble horn profiles of some brass instruments, such as trumpets and trombones, and use them to study experimentally boundary layer separation and vortex formation in these geometries. The curves are solutions of an ordinary differential equation. Scaling and balance of inertial and centripetal effects provide the basis for derivation of the equation, and suggests a dimensionless parameter that governs separation. Using particle image velocimetry techniques, vortex formation inside the horns can be observed, allowing for the determination of the values of the dimensionless parameter. Our experimental setup includes a blue diode laser sheet and a high-speed camera, operating at an acquisition rate of 5,760 frames per second, which allows for adequate analysis of a 96 Hz waveform. Vortex formation at the open end of the different horns is reduced by the specific geometries employed, but for very high sound pressure levels (140 dB) is still observed.