Abstract

Brass instruments sound brassy due to nonlinearity inside the instrument (not, as the name would suggest, because they look like they are made of brass). This work aims to answer the question “can you hear the shape of a trumpet?”, by considering weakly nonlinear sound in a cylindrical duct with varying diameter and curved centreline. In particular, a rather neat tensorial construction (due to James McTavish, as is almost all of this work) gives a relatively straightforward way to generalize any linear quantity to include weakly nonlinear effects. For example, matrices relating wave transmission, reflection, or impedance, can all be generalized to weak nonlinearity. This includes, we believe for the first time, the weakly nonlinear version of the boundary condition at an open end of a duct (c.f. the linear Wiener-Hopf solution by Munt). We can, therefore, ask questions such as whether the end correction at a duct exit varies with the amplitude of the sound. Some preliminary musical results for a simplified trombone will also be included.