A weakly nonlinear model for sound propagation in brass instruments

We present a summary of work on a new weakly nonlinear model of sound in curved 2D or 3D waveguides without mean flow. Modelling begins by consid- ering a general duct geometry, allowing for curvature and width variation, and also in 3D for torsion. Equations are projected onto a Frenet-Serret frame with arclength s, then expanded at each s in terms of the modes of an equiradial straight duct. This is the multi-modal method. We introduce a matricial ad- mittance that encodes the radiative properties of the duct; this allows for the prescription of a radiation condition at the duct outlet. Previous modelling has taken this to be the characteristic admittance: while mathematically tractable, this is highly physically idealised and corresponds to a duct opening into a room of equal radius. We instead derive a radiation condition consisting of a greatly enlarged duct with the original duct at the centre, which tells us more about the passage of sound from the outlet. This allows for more direct physical compar- isons with the model, e.g. the end correction for organ pipes or the harmonic series of a trumpet.