# Preprint 2008-010

# Sound traveling-waves in wind instruments as solutions to non linear non homogeneous gas dynamics equations

## Alain-Yves LeRoux

**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 homogeneous 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 wind instruments.
The conclusion is that in a wind instrument,
the shape yields linearity.