Singularity Formation in the Geometry of Perturbed Shocks of General Mach Number

  • W. MostertEmail author
  • D. I. Pullin
  • R. Samtaney
  • V. Wheatley
Conference paper


While planar shock waves are known to be stable to small perturbations in the sense that the perturbation amplitude decays over time, it has also been suggested that plane propagating shocks can develop singularities in some derivative of their geometry (Whitham (1974) Linear and nonlinear waves. Wiley, New York) in a nonlinear, wave reinforcement process. We present a spectral-based analysis of the equations of geometrical shock dynamics that predicts the time to singularity formation in the profile of an initially perturbed planar shock for general shock Mach number. We find that following an initially sinusoidal perturbation, the shock shape remains analytic only up to a finite, critical time that is a monotonically decreasing function of the initial perturbation amplitude. At the critical time, the shock profile ceases to be analytic, corresponding physically to the incipient formation of a “shock-shock” or triple point. We present results for gas-dynamic shocks and discuss the potential for extension to shock dynamics of fast MHD shocks.



This research was supported by the KAUST Office of Sponsored Research under award URF/1/2162-01. V. Wheatley holds an Australian Research Council Early Career Researcher Award (project number DE120102942).


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Copyright information

© Springer International Publishing AG, part of Springer Nature 2019

Authors and Affiliations

  • W. Mostert
    • 1
    Email author
  • D. I. Pullin
    • 1
  • R. Samtaney
    • 2
  • V. Wheatley
    • 3
  1. 1.Graduate Aerospace LaboratoriesCalifornia Institute of TechnologyPasadenaUSA
  2. 2.Mechanical EngineeringPhysical Sciences and Engineering Division, King Abdullah University of Science and TechnologyThuwalSaudi Arabia
  3. 3.Centre for Hypersonics, School of Mechanical and Mining EngineeringThe University of QueenslandSt LuciaAustralia

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