Nonlinear Wave Propagation in Brass Instruments

Abstract

The study of wave production and propagation is a common phenomenon seen within a variety of math and physics problems. This thesis in particular will investigate the production and propagation of sound waves through musical instruments. Although this field of work has been examined since the late 1800s, approaching these types of problems can be very difficult. With the exception of the last fifty years, we have only been able to approach such problems by linearizing the necessary equations of gas dynamics. Without the use of a computer, one can only get so far in studying nonlinear acoustic problems. In addition, the numerical theory for nonlinear problems is incomplete. Proving stability is challenging and there are a variety of open problems within this field.

This thesis will be examining the propagation of sound waves specifically through brass instruments. However, we will not be able to fully examine this problem in a master’s thesis because of the complexity. Instead, the objective is to provide a foundation and global picture of this problem by merge the fields of nonlinear acoustics as well as computational and analytical gas dynamics.

To study the general behaviour of nonlinear wave propagation (and to verify previous findings), experiments have been carried on a trumpet. The purpose of these experiments is take measurements of the sound pressure waves at various locations along the instrument in order to understand the evolution of the wave propagation. In particular, we want to establish if the nonlinear distortion is strong enough to have musical consequences; and if there are such outcomes, what prerequisites are required for the observable behaviour. Additionally, by using the discontinuous Galerkin numerical method, a model of the system will be presented in this thesis. It will then be compared with the experimental data to verify how well we were able to describe the nonlinear wave motion within a trumpet.