Sound travellingwaves in wind instruments as solutions to non linear homogeneous gas dynamics equations

TL;DR

This paper demonstrates that the shape of wind instruments allows nonlinear gas dynamics equations to be approximated by linear wave equations, providing a common periodic solution that models sound propagation.

Contribution

It introduces an analytic solution linking nonlinear gas dynamics with linear wave models, considering friction and shape conditions of wind instruments.

Findings

Shape of wind instruments induces linearity in sound propagation

Analytic periodic solutions can represent any sound in wind instruments

Friction effects are incorporated into the model

Abstract

The sound propagation is usually described by a linear homogeneous wave equation, though the air flow in a duct is described by the gas dynamics equations, using a variable cross section, which corresponds to a non linear non homogneous system. The aim of this paper is to exhibit a common periodic solution to both models, with several free parameters such as frequency or amplitude, able to represent any sound. By taking in account a friction term linked to the material (wood or brass for instance) of the duct, it is possible to build an analytic such solution when the cross section fullfills some condition which corresponds exactly to the general shape of the wind instruments. The conclusion is that in the wind intruments, the shape brings the linearity.