Shock wave propagation in a variable cross-section channel is a recurrent issue in the literature. Seminal work regarding this flow configuration has been proposed by Whitham (1958) through the derivation of a one-dimensional approach connecting the shock Mach number and the area channel: the A-M relation. It is based on strong theoretical restrictions: (i) shock equations applied on a $C^{+}$, (ii) omission of the postshock influence, and (iii) initial conditions at rest. It has been the focus of many studies aimed at generalizing it. However, very little attention has been paid to the study of the shock motion outside the varying cross-sectional region since Russell (1967). The objective of the current work is to describe and to explain the shock wave behavior in a constant area channel behind a convergent or divergent channel. It is found that the shock propagation in the downstream uniform area region is influenced by the postshock flow unsteadinesses. Therefore, the Whitham model is not suited to the shock motion study in a constant area region downstream of a convergent or divergent area region. A detailed flow description is provided and a quasisteady model for determining the waves intensity at large times is proposed. This model gives accurate results without any assumptions on the shock strength, area variation or the shock upstream state. Finally, this study points out the limits of the use of Whitham's theory in a variable area channel with increasing section variation rate (where $\left|dA/A(x+dx)\right|\ge \left|dA/A\left(x\right)\right|$).

中文翻译：

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横截面通道内的冲击运动及其对下游流量的影响

冲击波在变截面通道中的传播是文献中经常出现的问题。Whitham（1958）通过推导连接冲击马赫数和区域通道的一维方法提出了关于这种流动配置的开创性工作：AM关系。它基于强大的理论限制：（i）将冲击方程应用于$C^{+}$，（ii）省略了震后的影响，以及（iii）静止时的初始条件。它一直是许多旨在对其进行概括的研究的重点。然而，自从拉塞尔（1967）以来，很少有人关注变化截面区域之外的冲击运动。当前工作的目的是描述和解释在收敛或发散通道后面的恒定面积通道中的冲击波行为。已经发现，冲击后在下游均匀区域区域内的冲击传播受到震后流动不稳定性的影响。因此，Whitham模型不适合在收敛或发散区域下游的恒定区域中进行冲击运动研究。提供了详细的流描述，并提出了用于确定大时间波强度的准稳态模型。该模型给出了准确的结果，而没有任何关于冲击强度，面积变化或冲击上游状态的假设。最后，本研究指出了在截面变化率增大的情况下，在可变区域通道中使用Whitham理论的局限性（其中$\left|d\mathrm{一种}/\mathrm{一种}（X+dX）\right|\ge \left|d\mathrm{一种}/\mathrm{一种}（X）\right|$）。