管乐器中的声行波作为非线性齐次气体动力学方程的解
Sound travellingwaves in wind instruments as solutions to non linear homogeneous gas dynamics equations
摘要：
声传播通常用线性齐次波动方程来描述，而管道中的空气流动则用气体动力学方程来描述，采用变截面，这对应于非线性非齐次系统。本文的目的是展示一个共同的周期性的解决方案，这两个模型，与几个自由的参数，如频率或振幅，能够代表任何声音。通过考虑与管道材料（例如木材或黄铜）有关的摩擦项，当截面满足与管乐器的一般形状完全对应的某种条件时，就有可能建立这样的解析解。得出了在风力发电设备中，形状带来了线性的结论。
英文摘要：
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.
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