The reflection of a weak shock wave is considered using a shock polar. We present a sufficient condition under which the von Neumann paradox appears for the Euler equations. In an attempt to resolve the von Neumann paradox for the Euler equations, two new types of reflection configuration, one called the von Neumann reflection (vNR) and the other called the Guderley reflection (GR), are observed in numerical calculations. Finally, we obtain that GR is a reasonable configuration and vNR is an unreasonable configuration to resolve the von Neumann paradox.

中文翻译：

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欧拉方程的冯·诺依曼悖论

可以使用激波极性来考虑弱激波的反射。我们提出了一个足以满足欧拉方程的冯·诺依曼悖论的条件。为了解决欧拉方程的冯·诺依曼悖论，在数值计算中观察到两种新型的反射配置，一种称为冯·诺依曼反射（vNR），另一种称为Guderley反射（GR）。最后，我们认为GR是合理的配置，而vNR是解决冯·诺依曼悖论的不合理配置。