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Mathematical problem

Mathematically, the motion of the gas in the tube can be expressed with gas dynamic equations:

\begin{displaymath}\frac{\partial \rho}{\partial t}+\frac{\partial(\rho u)}{\partial x}=0
\end{displaymath} (67)

\begin{displaymath}\frac{\partial m}{\partial t}+
\frac{\partial (\rho u^2+p)}{\partial x}=0
\end{displaymath} (68)

\begin{displaymath}\frac{\partial e}{\partial t}+\frac{\partial (u(e+p))}{\partial x}
\end{displaymath} (69)

where \(e=p/(\gamma-1)+\frac{1}{2}\rho u^2, m=\rho u, \gamma\) is the gas dynamic constant.