The classification of pseudo-steady shock-wave reflection over a straight wedge proposed by Semenov, Berezkina and Krassovskaya (Shock Waves, 2012) maps the existence domains of regular and irregular reflection types onto the plane of wedge angle and incident-shock strength for a single gas, carbon dioxide (specific-heat ratio γ\gamma = 1.29), and leaves open the physical structure inside the irregular non-Mach reflection domain — the von Neumann paradox region where the classical three-shock theory admits no solution. The source models the reflected disturbance there as a Prandtl–Meyer compression fan (von Neumann reflection, vNR), while Guderley-type four-wave patterns with centered expansion fans and local supersonic patches have since received substantial numerical, experimental, and analytical support, without a conclusive domain-wide determination. This problem asks for the physically and mathematically established wave configuration in that domain — including which assumption of three-shock theory fails there — together with the γ\gamma-dependence of the reflection-domain map and the unresolved near-wall structures (slipstream behavior under the compression pulse, terminal double Mach reflection). The core of this problem remains open: no rigorous Euler-level selection theory exists, the single-patch versus multi-patch question persists, the strong-shock part of the domain is untouched, and the γ\gamma-dependence of the map is unexplored.

Background

When a plane shock wave strikes an inclined plane wall (a wedge), the reflected wave system grows, to leading order, without changing shape when measured in self-similar coordinates; such flows are called pseudo-steady. Two inviscid ideal-gas patterns are classical: regular reflection (RR), in which incident and reflected shock waves meet on the wall, and Mach reflection (MR), in which the two shocks are joined above the wall by a third, nearly normal shock — the Mach stem — with all three meeting at a triple point from which a contact discontinuity (slipstream) separates flows of equal pressure and different velocity. Since von Neumann's work in the 1940s, the expected pattern has been read off algebraic theories: the two-shock theory for RR and the three-shock theory for MR, which require the states behind each shock to satisfy the Rankine–Hugoniot jump conditions (conservation of mass, momentum, and energy across an oblique shock) and are conveniently visualized with shock polars — curves of attainable post-shock states in the pressure/flow-deflection plane. Standard monographs (Ben-Dor, 'Shock Wave Reflection Phenomena', 2007) organize pseudo-steady reflection by wedge angle β\beta, incident-shock strength (the density ratio ζ\zeta across the incident shock, or the shock Mach number MsM_s), and the specific-heat ratio γ\gamma of the gas, into single, transitional, and double Mach reflection (SMR, TMR, DMR), according to the number of triple points and the shape of the reflected wave; a terminal double Mach reflection (TDMR), a variant of double Mach reflection expected for very strong shocks in gases with low γ\gamma, is the least understood of these.

Semenov, Berezkina, and Krassovskaya (2012) proposed a finer classification built on the interplay of two processes: the reflection of the incident wavefront at the wedge surface and the deflection of the post-shock flow around the wedge leading edge. Distinguishing the shapes of the reflected wave, the Mach stem, and the slipstream, they introduced subtypes named after early investigators of these configurations (single Mach–Smith, weak single Mach–Smith, single Mach–White, single Mach–Cabannes reflection, and their double and triple variants) and computed a map of existence domains in the (β\beta, ζ\zeta) plane for a perfect gas with γ\gamma = 1.29 (carbon dioxide), bounded by the two-shock regular-reflection limit, the three-shock (Cabannes) boundaries, and sonic conditions controlling whether compression disturbances from the leading edge can reach the triple point.

The map has a hole. Below the lowest three-shock boundary lies a domain in which the three-shock theory admits no solution at all, yet an irregular, Mach-like reflection is observed — the von Neumann paradox, known since the 1940s from weak-shock diffraction studies. Colella and Henderson (1990) named the observed pattern von Neumann reflection (vNR): the reflected 'shock' degenerates near the triple point into a continuous compressive disturbance — modeled as a Prandtl–Meyer compression fan, a centered wave of isentropic compression — that merges smoothly into the Mach stem, so there is no triple point in the strict sense. Guderley (1957) had proposed a different local structure: a four-wave configuration in which a centered expansion fan issues from the triple-point region and encloses a small patch of supersonic flow (relative to the triple point) behind the reflected wave, adjacent to the subsonic flow required by the three-shock relations. Later work distinguishes a single-patch pattern (Guderley reflection, GR) from a multi-patch pattern with a sequence of tiny supersonic patches, each terminated by a small shock (a shocklet) — Guderley Mach reflection (GMR). Semenov et al. (2012) adopt the compression-fan (vNR) model and, relying on their group's earlier simulations, extend the vNR domain to strong incident shocks on thin wedges, not only weak shocks; they also present evidence that the slipstream under the leading-edge compression pulse does not roll up into a spiral vortex. They explicitly flag as unresolved: which local structure is actually realized in the irregular non-Mach domain, which assumption of the three-shock theory fails there (in particular, whether the reflected disturbance can be a continuous compression wave rather than a shock), how the whole map changes with γ\gamma, the near-wall slipstream behavior, and the structure of TDMR.

Problem Statement

Consider a plane shock wave of strength ζ\zeta (the density ratio across the incident shock, equivalently the incident-shock Mach number MsM_s) reflecting from a straight wedge of angle β\beta in an inviscid perfect gas with specific-heat ratio γ\gamma, in the pseudo-steady (self-similar) regime.

The 2012 classification of Semenov, Berezkina, and Krassovskaya provides existence domains of the reflection types in the (β\beta, ζ\zeta) plane for a single gas, γ\gamma = 1.29 (CO₂), and models the reflection in the irregular non-Mach domain — the region below the lowest three-shock boundary where the classical three-shock theory admits no solution, i.e. the von Neumann paradox region — as a von Neumann reflection whose reflected disturbance is a Prandtl–Meyer compression fan, including its extension to strong incident shocks on thin wedges.

The problem is to complete this classification physically and, where possible, mathematically:

(1) Determine the wave configuration actually realized in the irregular non-Mach reflection domain, across its full extent in (β\beta, ζ\zeta, γ\gamma): whether the reflected disturbance emanating from the triple-point region is a continuous compression fan (von Neumann reflection), a four-wave Guderley-type configuration with a centered expansion fan and local supersonic patch(es) (Guderley reflection or Guderley Mach reflection), or a configuration that varies within the domain — and identify which specific assumption of the three-shock theory (in particular, that the reflected wave is a shock rather than a continuous compression wave) is responsible for its failure there.

(2) Establish how the boundaries between the existence domains of the reflection types — including the irregular non-Mach domain itself and the sub-type boundaries derived from sonic and disturbance-reachability conditions — depend on the specific-heat ratio γ\gamma (the 2012 map was computed only for γ\gamma = 1.29).

(3) Resolve the near-wall structure questions that the source could not settle and that enter the subtype assignments: whether the slipstream under the leading-edge compression pulse rolls up into a spiral vortex or folds without circulation (as the source's single simulation suggested), and the near-wall structure of the terminal double Mach reflection (TDMR).

These deliverables serve one objective — a physically validated, γ\gamma-parameterized classification of pseudo-steady shock reflection: the subtype map of (2) is meaningful only once the local structure of (1) is settled, and the near-wall behavior of (3) is one of the three shape criteria on which the subtype assignments rest. The community-terminology component of the source's discussion (acceptance of the new subtype names) is a matter of scientific convention rather than a research question and is not part of this problem.

References

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