The von Neumann Triple Point Paradox

  • Richard Sanders
  • Allen M. Tesdall
Part of the Computational Methods in Applied Sciences book series (COMPUTMETHODS, volume 16)


We describe the problem of weak shock reflection off a wedge and discuss the triple point paradox that arises. When the shock is sufficiently weak and the wedge is thin, Mach reflection appears to be observed but is impossible according to what von Neumann originally showed in 1943. We summarize some recent numerical results for weak shock reflection problems for the unsteady transonic small disturbance equations, the nonlinear wave system, and the Euler equations. Rather than finding a standard but mathematically inadmissible Mach reflection with a shock triple point, the solutions contain a complex structure: there is a sequence of triple points and supersonic patches in a tiny region behind the leading triple point, with an expansion fan originating at each triple point. The sequence of patches may be infinite, and we refer to this structure as Guderley Mach reflection. The presence of the expansion fans at the triple points resolves the paradox. We describe some recent experimental evidence which is consistent with these numerical findings.

Key words

self-similar solutions two-dimensional Riemann problems triple point paradox 


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Copyright information

© Springer Science + Business Media B.V. 2008

Authors and Affiliations

  • Richard Sanders
    • 1
  • Allen M. Tesdall
    • 2
    • 3
  1. 1.Department of MathematicsUniversity of HoustonHoustonUSA
  2. 2.Fields InstituteTorontoCanada
  3. 3.Department of MathematicsUniversity of HoustonHoustonUSA

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