When a very weak plane shock reflects off a wedge, the region behind the triple point resolves, in inviscid self-similar computations and shock-tube experiments, into Guderley Mach reflection (GMR): a cascade of successively smaller supersonic patches, each terminated by a small reflected shock (shocklet) that returns the flow discontinuously to subsonic speed, with a centered expansion fan issuing from each new triple point. What remains unknown is the quantitative law governing this cascade: how the patch sizes and the strengths of the successive reflected shocks decay from one iterate to the next, whether the cascade terminates after finitely many patches or continues indefinitely in the inviscid limit, and whether the multi-patch structure can be observed experimentally at incident Mach numbers above about 1.1, where the supersonic region is so small that viscous and boundary-layer effects mask it. Centered on the question as left open by Tesdall, Sanders and Popivanov (J. Sci. Comput. 64:721-744, 2015), the present record finds the question open: rigorous analysis has constructed only single local building blocks of the configuration, later high-resolution computations confirm the multi-patch structure and its growth under mesh refinement but yield no asymptotic law, recent asymptotic theory addresses only the leading-order onset regime, and experiments have produced only partial multi-patch evidence at the relevant Mach numbers.

Background

When a plane shock wave meets a wedge at a small angle of incidence, the reflection pattern depends mainly on the shock strength. For sufficiently strong shocks one observes regular reflection (a single reflected shock meeting the incident shock on the wall surface) or Mach reflection, in which the incident and reflected shocks are joined at a triple point off the wall by a third shock, the Mach stem, and by a slip line across which the flow is tangentially discontinuous. Classical three-shock theory due to von Neumann predicts that for very weak incident shocks the flow immediately behind the triple point should be subsonic; yet computations by Colella and Henderson (1990) showed the opposite — a locally supersonic region behind the triple point of weak Mach reflection. This disagreement between the standard three-shock model and both computation and experiment is the von Neumann triple point paradox.

Because the reflection of a plane shock from an infinite wedge involves no length scale, solutions are naturally sought in self-similar form: all flow quantities depend only on the similarity variables x/tx/t and y/ty/t, and the compressible Euler equations reduce to a system of mixed type — elliptic where the self-similar flow is subsonic and hyperbolic where it is supersonic. Guderley (1947), working from the linearized theory of flow at Mach number one, predicted a small supersonic region behind the triple point, terminated by a shock. High-resolution numerical solutions of the self-similar Euler equations by Tesdall, Sanders and Keyfitz (2008) revealed that this supersonic region is not a single patch: it is a cascade of successively smaller supersonic patches, each bounded partly by a sonic line (where the flow speed equals the local speed of sound) and terminated by a small reflected shock, called a shocklet, across which the flow returns discontinuously to subsonic speed, with a centered expansion fan (a family of expansion waves spreading from a single point) issuing from each newly formed triple point. Shock-tube experiments by Skews and co-workers identified the expansion wave at the triple point, the terminating shocklet, and indications of more than one patch. This multi-patch structure is called Guderley Mach reflection (GMR); the simpler pattern with a single supersonic patch and a smooth sonic transition, reported in some computations and experiments, is called Guderley reflection. Tesdall, Sanders and Popivanov (2015) computed more finely resolved self-similar solutions at parameter values where earlier work had found the single-patch structure, found the full cascade in every case, and conjectured that Guderley reflection is really underresolved Guderley Mach reflection. Their computations show that the reflected shocks weaken rapidly from one iterate to the next — the reported density jump across the first three reflected shocks is in the approximate ratio 6.7:1.7:1 — and that the whole supersonic region is minute, its height of order a few tenths of a percent of the Mach stem height. What the computations do not provide is any theoretical prediction of this decay, any determination of whether the cascade stops after finitely many patches, and experiments resolving the individual patches above incident Mach numbers of about 1.1.

Problem Statement

Determine the asymptotic structure of the supersonic patch cascade in Guderley Mach reflection, as left open by Tesdall, Sanders and Popivanov (2015). The core theoretical question is: starting from the inviscid, self-similar compressible Euler equations (or an asymptotic reduction with controlled error), predict the law by which the successive supersonic patches shrink and the successive reflected shocks weaken across iterates — in particular whether the decay of patch sizes and of reflected-shock strengths is geometric in the iterate index, with what ratio, and on what flow quantities that ratio depends — and determine whether the sequence of supersonic patches and shock/expansion-wave pairs terminates after finitely many iterates or continues indefinitely in the inviscid limit, including the accumulation behavior of the cascade at the triple point. The experimental status of the same structure — multi-patch Guderley Mach reflection established in the very weak regime, but the individual expansion waves and terminating shocklets not yet definitively resolved at incident Mach numbers above about 1.1, where viscous and boundary-layer effects mask the tiny supersonic region — is background context for this theoretical question, not a separate deliverable.

References

  1. Tesdall, Allen M.; Sanders, Richard; Popivanov, Nedyu (2015). 'Further Results on Guderley Mach Reflection and the Triple Point Paradox', Journal of Scientific Computing 64(3):721-744. DOI 10.1007/s10915-015-0028-1, https://doi.org/10.1007/s10915-015-0028-1
  2. Tesdall, Allen M.; Sanders, Richard; Keyfitz, Barbara L. (2008). 'Self-Similar Solutions for the Triple Point Paradox in Gasdynamics', SIAM Journal on Applied Mathematics 68(5):1360-1377. DOI 10.1137/070698567, https://doi.org/10.1137/070698567
  3. Colella, P.; Henderson, L. F. (1990). 'The von Neumann paradox for the diffraction of weak shock waves', Journal of Fluid Mechanics 213:71-94. DOI 10.1017/S0022112090002221, https://doi.org/10.1017/S0022112090002221
  4. Skews, B. W.; Ashworth, J. T. (2005). 'The physical nature of weak shock wave reflection', Journal of Fluid Mechanics 542:105-114. DOI 10.1017/S0022112005006543, https://doi.org/10.1017/S0022112005006543
  5. Skews, Beric William; Li, Gavin; Paton, Randall (2009). 'Experiments on Guderley Mach reflection', Shock Waves 19(2):95-102. DOI 10.1007/s00193-009-0193-y, https://doi.org/10.1007/s00193-009-0193-y
  6. Cachucho, A.; Skews, B. W. (2011). 'Guderley reflection for higher Mach numbers in a standard shock tube', Shock Waves 22:141-149. DOI 10.1007/s00193-011-0349-4, https://doi.org/10.1007/s00193-011-0349-4
  7. Lai, Geng; Sheng, Wancheng (2015). 'Centered wave bubbles with sonic boundary of pseudosteady Guderley Mach reflection configurations in gas dynamics', Journal de Mathématiques Pures et Appliquées 104:179-206. DOI 10.1016/j.matpur.2015.02.005, https://doi.org/10.1016/j.matpur.2015.02.005
  8. Lai, Geng; Sheng, Wancheng (2016). 'Two-dimensional centered wave flow patches to the Guderley Mach reflection configurations for steady flows in gas dynamics', Journal of Hyperbolic Differential Equations 13:107-128. DOI 10.1142/S0219891616500028, https://doi.org/10.1142/S0219891616500028
  9. Kochi, S. R. Siva Prasad; Ramakrishna, M. (2024). 'Numerical Investigation of the Von Neumann Paradox for Weak Shocks Using a Higher Order Discontinuous Galerkin Method and Overset Grids', in Advances in Shock Interactions (ISSW33), Springer Proceedings in Physics, pp. 21-30. DOI 10.1007/978-981-97-6099-2_3, https://doi.org/10.1007/978-981-97-6099-2_3
  10. Hew, Justin Kin Jun (2026). 'Distinguished Scaling and UTSD Structure in Weak Shock Reflection at Nearly Glancing Incidence', arXiv:2606.18618 (journal version in Transport Phenomena, DOI 10.1515/tp-2026-0071). https://arxiv.org/abs/2606.18618
  11. Hew, Justin Kin Jun (2026). 'Finite-Strength Sensitivity and Euler-UTSD Correspondence for Guderley-Mach Reflection', arXiv:2608.15704. https://arxiv.org/abs/2608.15704
  12. Guderley, K. G. (1947). 'Considerations of the structure of mixed subsonic-supersonic flow patterns', Technical Report F-TR-2168-ND, U.S. Air Materiel Command, Wright Field, Ohio. Identifier: report no. F-TR-2168-ND; no DOI or stable public URL exists for this 1947 report — it is cited in the reference list of Tesdall, Sanders and Popivanov (2015), Journal of Scientific Computing 64(3):721-744, DOI 10.1007/s10915-015-0028-1