When very weak acoustic shock waves (acoustic Mach numbers of order 10−310^{-3} to 10−210^{-2}) strike a rigid wall at grazing incidence, classical mirror (Snell–Descartes) reflection breaks down and a sequence of irregular reflection regimes appears: regular two-shock reflection, von Neumann/Mach reflection, and a 'weak von Neumann' regime with a single curved front. This was demonstrated experimentally for periodic ultrasonic shock waves in water by Marchiano, Coulouvrat, Baskar and Thomas (Phys. Rev. E 76, 056602, 2007) and computed with the paraxial Khokhlov–Zabolotskaya (KZ) equation by the companion work of Baskar, Coulouvrat and Marchiano (J. Fluid Mech. 575, 2007), but it lacks a theory: inviscid three-shock (von Neumann) theory admits no acceptable solution in this regime, the observed stems are curved rather than straight, and no asymptotic analysis predicts the regime boundaries or stem geometry. The problem is to construct a first-principles theoretical account of this acoustical von Neumann paradox — explaining the failure of three-shock theory and the curvature of the observed stems, and quantitatively predicting the boundaries between the Snell–Descartes, regular, von Neumann/Mach and weak von Neumann regimes, including the role of waveform type (step shock, periodic sawtooth, N-wave pulse) and, for unsteady waveforms, the evolution of the reflection pattern with propagation distance — consistent with the existing experimental and numerical observations of these regimes. The 2008–2025 follow-up literature is entirely experimental and computational, while the gas-dynamic von Neumann paradox for self-similar step shocks has been substantially elucidated by Guderley-type multi-wave structures; the transfer of that understanding to the extreme-weak, grazing, unsteady acoustic regime remains open.

Background

When a plane shock wave strikes a rigid wall obliquely, gas dynamics classifies the reflection by how many shock fronts meet. Von Neumann's three-shock theory — an inviscid, locally steady analysis built from the Rankine–Hugoniot jump conditions (the conservation laws relating the flow states on the two sides of a shock front) across incident, reflected and near-wall shocks — predicts either regular reflection (two shocks meeting on the wall) or Mach reflection (a third shock, the Mach stem, joining them at a triple point above the wall). An acoustic shock wave is a steepened high-amplitude sound wave whose strength is measured by the acoustic Mach number MM, the ratio of particle velocity to sound speed, typically 10−310^{-3} to 10−210^{-2} in the experiments discussed here. At grazing incidence (small angle θ\theta between the incident front and the wall), three-shock theory admits no physically acceptable solution for such weak shocks, yet experiments plainly display a three-shock pattern with a triple point. This discrepancy is the von Neumann paradox, classical in gas dynamics and quantified for weak shocks by Colella and Henderson (J. Fluid Mech. 213, 1990). For gas-dynamic step shocks this paradox has been substantially elucidated: Zakharian, Brio, Hunter and Webb (J. Fluid Mech. 422, 2000) resolved a Guderley-type structure — a small supersonic patch and expansion fan adjacent to the triple point, as proposed by Guderley in 1947 — in solutions of the Euler equations; Hunter and Brio (J. Fluid Mech. 410, 2000) derived transonic small-disturbance asymptotics (a reduction of the Euler equations valid for weak, nearly sonic flow perturbations); Tesdall and Hunter (SIAM J. Appl. Math., 2002) computed self-similar solutions containing a sequence of shocks and tiny expansion fans; Skews and Ashworth (J. Fluid Mech., 2005) confirmed the expansion-wave-plus-small-shock structure experimentally; and Vasilev, Elperin and Ben-Dor (Phys. Fluids 20, 046101, 2008) gave an analytical Guderley Mach-reflection account.

In nonlinear acoustics the same question reappears in an accentuated form. Marchiano, Coulouvrat, Baskar and Thomas (Phys. Rev. E 76, 056602, 2007) showed experimentally — using periodic sawtooth ultrasonic shock waves in a water tank at amplitudes about one hundred times smaller than previous air measurements — that mirror reflection fails and that regular, von Neumann/Mach and weak von Neumann regimes occur at grazing incidence. The theoretical companion computation of Baskar, Coulouvrat and Marchiano (J. Fluid Mech. 575, 2007) used the Khokhlov–Zabolotskaya (KZ) equation, the standard paraxial (narrow-angle) evolution equation of nonlinear acoustics combining weak nonlinearity with diffraction, and showed for step shocks a continuous transition governed by the critical parameter sin⁡θ/M\sin\theta/\sqrt{M}, including a weak von Neumann regime in which a single curved shock front replaces the Mach stem; for periodic waveforms the transitions are unsteady, successive shocks of the sawtooth interacting near the wall. These results are numerical descriptions, not explanations: no asymptotic theory of the compressible equations reproduces the regime sequence, predicts where the boundaries lie, or accounts for the curvature of the stem. Later work confirmed and extended the observations without supplying such a theory: schlieren and Mach–Zehnder interferometric experiments with spark-generated N-waves (single-cycle pulses) in air, together with direct numerical simulation of the full Navier–Stokes equations (Karzova et al., JASA 2015–2019; Desjouy et al., JASA and Phys. Fluids 2016), verified Mach-stem formation at genuinely acoustic amplitudes, measured stem growth with propagation distance, and showed that wall roughness shortens the stem; extensions to concave and curved surfaces were made by Gruber and Skews (2013) and Cohen and Skews (2020). Application-driven studies of sonic-boom ground reflection and impulsive jet noise (e.g. Vaughn et al., JASA 2021; Deleu et al., Shock Waves 2025) continue to rely on regime classification because the boundaries are not predictable from theory.

Problem Statement

Construct a first-principles theoretical account of the acoustical von Neumann paradox: the reflection of very weak acoustic shock waves (acoustic Mach number MM of order 10−310^{-3} to 10−210^{-2}) at grazing incidence angle θ\theta on a rigid plane. Starting from the compressible Navier–Stokes (or Euler) equations or a justified asymptotic reduction of them, the answer should: (1) explain why inviscid three-shock (von Neumann) theory admits no acceptable solution in this regime, why the observed near-wall front is a curved stem rather than a straight Mach stem, and which mechanism (nonlinear steepening, diffraction, thermoviscous absorption, and for periodic waveforms the unsteady interaction of successive shocks) sets the stem curvature; and (2) predict quantitatively the boundaries separating the Snell–Descartes (mirror), regular, von Neumann/Mach, and weak von Neumann regimes, for the waveform classes step shock, periodic sawtooth and N-wave pulse. For step shocks the reflection is quasi-steady (pseudo-self-similar), and the boundaries are to be predicted in terms of the critical parameter sin⁡θ/M\sin\theta/\sqrt{M} and MM; for unsteady waveforms the reflection is not self-similar and, at a given value of the critical parameter, all the regimes categorized for the step shock may appear successively as the shock propagates along the wall, so the theory must account for this propagation-distance dependence of the regime sequence. The water-tank periodic-wave experiments of Marchiano et al. (2007) and the KZ computations of Baskar et al. (2007) provide the benchmark phenomenology with which the theory should be consistent.

References

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