The motion of a polytropic compressible fluid or gas in a non-uniform channel with elastic walls and a specified cross-section $A(x,p)$ depending on a given point $x$ and the pressure $p$ is examined. The form of a cross-section $A=k{x}^{\alpha }{p}^{\beta }$ is considered in this paper. Particular cases arising from the analysis that lead to an exact solution of the equations of the hydraulic longitudinal waves are given. For the case $\beta \gamma +1=0$ the general solution is obtained. In addition, we study the solutions with a linear profile of velocity for the case $\beta \gamma +1\ne 0$. An illustration of the behavior of the solution when the initial velocity is varied is given.

中文翻译：

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水力纵波方程的Sedov型解

多向可压缩流体或气体在具有弹性壁和指定横截面的非均匀通道中的运动 $\mathrm{一个}（X，p）$ 取决于给定的点 $X$ 和压力 $p$被检查。横截面形式$\mathrm{一个}=ķX^{\alpha }{p}^{\beta }$本文考虑。给出了由分析引起的，导致水力纵波方程精确解的特殊情况。对于这种情况$\beta \gamma +\mathrm{1个}=0$获得了一般的解决方案。此外，我们研究了情况下速度线性分布的解$\beta \gamma +\mathrm{1个}\ne 0$。给出了当初始速度变化时溶液行为的图示。