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The classical Rankine-Hugoniot jump conditions, an important cornerstone of modern shock wave physics: ideal assumptions vs. reality

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Abstract

The purpose of this paper is to discuss from a historical point of view the classical Rankine-Hugoniot (RH) relations in more detail than usually done in standard textbooks. Particularly focusing on the last seventy years, this paper (i) reviews their validity and limitations as interpreted by numerous users; (ii) summarizes their enormous extension also to other branches of science and engineering; and (iii) discusses the nontrivial problem of error estimation. Originally, the RH relations were derived for a plane-parallel steadily propagating aerial shock with a step wave profile; i.e., a wave with zero rise-time and constant thermodynamic as well as kinematic parameter values behind the shock front. But real shock waves are in most cases three-dimensional, have finite rise-times, and in almost all cases are unsteady waves. These real properties must produce a systematic error when applying the RH relations and the cardinal question arises how large this error will be in comparison with a random error caused by the applied high-speed diagnostics. However, numerical procedures of studying systematic errors of unsteady wave propagation are difficult to carry out because of various reasons and still pending.

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References

  1. Rankine-Hugoniot jump conditions. Wikipedia; http://en.wikipedia.org/wiki/Rankine%E2%80%93Hugoniot_conditions.

  2. Multivariable calculus − Rankine-Hugoniot jump conditions derivation. Mathematics Stack Exchange; http://math.stackexchange.com/questions/865142/rankine-hugoniot-jump-condition-derivation.

  3. P.O.K. Krehl, Shock wave physics and detonation physics − a stimulus for the emergence of numerous new branches in science and engineering, Eur. Phys. J. H. 36, 85–152 (2011).

    ADS  Google Scholar 

  4. A. Mazzia, Numerical Methods for the Solution of Hyperbolic Conservation Laws. Rapporto Tecnico No. 68, Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate, Università de Padova, Italia, 1998.

  5. W.C. Griffith, W. Bleakney, Shock waves in gases, Am. J. Phys. 22, 597–612 (1954).

    ADS  Google Scholar 

  6. G. Ben-Dor, Shock Wave Reflection Phenomena (Springer, New York, 1992).

  7. C. Chalons, P. Goatin, Transport-equilibrium schemes for computing contact discontinuities in traffic flow modeling, Commun. Math. Sci. 5, 533–551 (2007).

    MATH  MathSciNet  Google Scholar 

  8. D.F. Cioffi, C.F. McKee, E. Bertschinger, Dynamics of radiative supernova remnants, Astrophys. J. (Part 1) 334, 252–265 (1988).

    ADS  Google Scholar 

  9. G.W. Swan, D.E. Duvall, C.K. Thornhill, On steady wave profiles in solids, J. Mech. Phys. Solids 21, 215–227 (1973).

    MATH  ADS  Google Scholar 

  10. Y.J. Horie, Classification of steady-profile shocks in liquids, J. Appl. Phys. 45, 759–764 (1974).

    ADS  Google Scholar 

  11. L. Euler, Continuation des recherches sur la théorie du mouvement des fluides, Hist. Acad. Roy. Sci. Belles Lettres (Berlin) 11, 316–361 (1757).

  12. G.G. Stokes, On a difficulty in the theory of sound, Phil. Mag. 33 [III], 349–356 (1848).

    Google Scholar 

  13. G.F.B. Riemann, Über die Fortpflanzung ebener Luftwellen von endlicher Schwingungsweite. Abhandl. Königl. Gesell. Wiss. Gött. 8 [Math. Physik. Kl.], 243–265 (1860).

    Google Scholar 

  14. W.J.M. Rankine, On the thermodynamic theory of waves of finite longitudinal disturbance [read Dec. 16, 1869], Phil. Trans. Roy. Soc. Lond. 160, 277–286 (1870); Supplement. Ibid. pp. 287–288 (1870).

    Google Scholar 

  15. P.H. Hugoniot, Mémoire sur la propagation du mouvement dans les corps et plus spécialement dans les gaz parfaits. 1e Partie, J. Ecole Polytech. (Paris) 57, 3–97 (1887); Mémoire sur la propagation du mouvement dans les corps et plus spécialement dans les gaz parfaits. 2e Partie, J. Ecole Polytech. (Paris) 58, 1–125 (1889).

    Google Scholar 

  16. R.J.E. Clausius, Über verschiedene für die Anwendung bequeme Formen der Hauptgleichungen der mechanischen Wärmetheorie, Ann. Phys. 125 [II], 353–400 (1865).

    Google Scholar 

  17. P.O.K. Krehl, History of Shock Waves, Explosions, and Impact – a Chronological and Biographical Reference (Springer-Verlag, Berlin, 2009), pp. 340–341, 387–389, 1075–1077, 1094–1095, 1148–1149, 1175–1176.

  18. R. Chéret, The life and work of Pierre-Henri Hugoniot, Shock Waves 2, 1–4 (1992).

    ADS  Google Scholar 

  19. J.N. Johnson, R. Chéret, Shock waves in solids: an evolutionary perspective, Shock Waves 9, 193–200 (1999).

    MATH  ADS  Google Scholar 

  20. M. Salas, The curious events leading to the theory of shock waves, Shock Waves 16, 477–487 (2007); A Shock-Fitting Primer (CRC Press, Boca Raton, FL, 2010).

    MATH  ADS  Google Scholar 

  21. R.H. Cole, Underwater Explosions (Princeton University Press, Princeton, NJ, 1948)

  22. C.E. Needham, Blast Waves (Springer-Verlag Berlin, Heidelberg, 2010), pp. 11–15.

  23. R. Courant, K.O. Friedrichs, Supersonic Flow and Shock Waves. Springer-Verlag, New York (corrected 5th printing 1999), pp. 121–126, 336–338.

  24. D.H. Weinberg, Astronomy 825: Radiative Gas Dynamics, Chap. 7: Shocks (Winter 2003). Dept. of Astronomy, Mathematical and Physical Sciences, Ohio State University; http://www.astronomy.ohio-state.edu/_dhw/A825/notes7.pdf.

  25. A. Bressan, Hyperbolic conservation laws – an illustrated tutorial (Dept. of Mathematics, Penn State University, University Park, PA (2009); http://www.math.psu.edu/bressan/PSPDF/clawtut09.pdf.

  26. J.W. Forbes, Shock Wave Compression of Condensed Matter – a Primer (Springer-Verlag, Berlin, 2012), pp. 179–185.

  27. G.W. Sutton, A. Sherman, Engineering Magnetohydrodynamics (McGraw-Hill Book Co., New York, 1965).

  28. J. von Neumann, The point source solution. In Blast Wave, edited by K. Fuchs, J.O. Hirschfelder, J.L. Magee, R. Peierls, J. von Neumann (Rept. La-2000, LASL, 1947), pp. 27–55.

  29. L.M. Barker, R.E. Hollenbach, Shock-wave studies of PMMA, fused silica, and sapphire, J. Appl. Phys. 41, 4208–4226 (1970).

    ADS  Google Scholar 

  30. C.E. Ragan III, M.G. Silbert, B.C. Diven, Shock compression of molybdenum to 2.0 TPa by means of a nuclear explosion, J. Appl. Phys. 48, 2860–2870 (1977).

    ADS  Google Scholar 

  31. B. Hayes, Particle-velocity gauges system for nanosecond sampling rate of shock and detonation waves, Rev. Sci. Instrum. 52, 594–603 (1981).

    ADS  Google Scholar 

  32. S. Minshall, Properties of elastic and plastic waves determined by pin contactors and crystals, J. Appl. Phys. 26, 463–469 (1955).

    ADS  Google Scholar 

  33. S.R. BrinkleyJr., J.G. Kirkwood, Theory of the propagation of shock waves, Phys. Rev. 71, 606–611 (1947).

    MATH  MathSciNet  ADS  Google Scholar 

  34. D. Benson, An efficient, accurate, simple ALE method for nonlinear finite element programs, Computat. Meth. Appl. Mech. Eng. 72, 305–350 (1989).

    MATH  ADS  Google Scholar 

  35. Arbitrary Lagrangian Eulerian and Fluid-Structure Interaction: Numerical Simulation, edited by M. Souli, D.J. Benson (Wiley, London, UK and Hoboken, NJ, 2010).

  36. W. Bleakney, D.K. Weimer, C.H. Fletcher, The shock tube: a facility for investigations in fluid dynamics, Rev. Sci. Instrum. 20, 807–815 (1949).

    ADS  Google Scholar 

  37. R. LeVeque, Numerical Methods for Conservation Laws (Birkhäuser, Berlin, 1992), Chap. 2.1: Integral and differential forms, pp. 14–16.

  38. R.K. Tsou, Conservation equations, in Dynamic Response of Materials to Intense Impulsive Loading, edited by P.C. Chou, A.K. Hopkins. Rept. AD-768-416, Air Force Materials Laboratory, Wright Patterson Air Force Base, OH (1972), Chap. 2, pp. 7–42.

  39. V.N. Kukudzhanov, Numerical Continuum Mechanics (De Gruyter GmbH, Berlin/Boston, 2013), pp. 9–13.

  40. Ames Research Staff, Equations, Tables, and Charts for Compressible Flow, Rept. NACA 1135 (1953); Normal shock waves, pp. 6–8.

  41. Rankine-Hugoniot spreadsheets. Dewey McMillin & Associates Ltd., Victoria, BC V8N 2A4, Canada; see www.blastanalysis.com.

  42. J.M. Walsh, M.H. Rice, R.G. McQueen, F.L. Yarger, Shock-wave compressions of twenty-seven metals. Equations of state of metals, Phys. Rev. 108 [II], 196–216 (1957).

    Google Scholar 

  43. J.M. Walsh, M.H. Rice, Dynamic compression of liquids from measurements on strong shock waves, J. Chem. Phys. 26, 815–823 (1957).

    ADS  Google Scholar 

  44. M. van Thiel, A.S. Kusubov, A.C. Mitchell, Compendium of Shock Wave Data. Rept. UCRL-50108, Lawrence Radiation Laboratory (LRL), Livermore, CA (1977).

  45. LASL Shock Hugoniot Data Bank, edited by S.P. Marsh (University of California Press, Berkeley, 1980).

  46. SESAME: The Los Alamos National Laboratory Equations-of-State Database., edited by S.P. Lyon, J.D. Johnson. Rept. LA-UR-92-3407 (1992).

  47. J.D. Johnson, G.I. Kerley, G.T. Rood, Recent Developments in the SESAME Equation-of-State Library, edited by B.I. Bennett. Rept. LA-7130, LASL, Los Alamos, NM (1978); Equations of state – theoretical formalism. Los Alamos Science No. 26, 192 (2000).

  48. I.V. Lomonosov, K.V. Khishchenko, P.R. Levashov, D.V. Minakov, A.S. Zakharenkov, J.B. Aidun, International shock-wave data base. IPCP and JIHT, RAS and SNL, USA; see http://www.ihed.ras.ru/elbrus12/program/restore.php?id=820.

  49. P.R. Levashov, K.V. Khishchenko, I.V. Lomonosov, V.E. Fortov, Database on shock-wave experiments and equations of state available via Internet, in APS Topical Conference on Shock Compression of Condensed Matter − 2003, edited by M.D. Furnish, Y.M. Gupta, J.W. Forbes. AIP Conf. Proc. 706, 87–90 (2004).

    ADS  Google Scholar 

  50. J.M. Dewey, The Rankine-Hugoniot equations: their extensions and inversions related to blast waves, in Proc. 19th International Symposium on Military Aspects of Blast and Shock (MABS) [Calgary, Alberta, Canada; Oct. 1−6, 2006]. CD-ROM, publ. by Defence R&D, Suffield, Alberta, Canada.

  51. E. Jouguet, La théorie thermodynamique de la propagation des explosions, in Verhandlungen des 2. Int. Kongresses für Technische Mechanik, edited by E. Meissner [Zurich, Switzerland; Sept. 12−17, 1926]. Füssli, Zurich (1927), pp. 12–22.

  52. R. Becker, Stoßwelle und Detonation, Z. Phys. 8, 321–362 (1922).

    ADS  Google Scholar 

  53. Encyclopaedic Dictionary of Physics, edited by J. Thewlis (Pergamon Press, Oxford, 1962), p. 718.

  54. S.D. Poisson, Sur la chaleur des gaz et des vapeurs, Ann. Chem. Phys. 23 [II], 337–353 (1823).

    Google Scholar 

  55. A.F. Viñas, J.D. Scudder, Fast and optimal solution to the Rankine-Hugoniot problem. NASA Memorandum 86214 (May 1985).

  56. S.K. Chakrabarti, Theory of Transonic Astrophysical Flows (World Scientific Publishing Co., Singapore, 1990), pp. 48–53.

  57. A. Siegenthaler, J. Madhani, Outline of a theory of non-Rankine-Hugoniot shock wave in weak Mach reflection. 14th Australasian Fluid Mechanics Conference (Adelaide Univ., Adelaide, Austr.; 10–14 Dec. 2001).

  58. A.H. Bepp, Underwater explosion measurements from small charges at short ranges, Phil. Trans. Roy. Soc. Lond. 244, 153–175 (1951).

    ADS  Google Scholar 

  59. J.M. Walsh, R.H. Christian, Equation of state of metals from shock wave measurements, Phys. Rev. 97, 1544–1556 (1955).

    ADS  Google Scholar 

  60. M.H. Rice, R.G. McQueen, J.M. Walsh, Compression of solids by strong shock waves, in Solid State Physics. Advances in Research and Applications, edited by F. Seitz, D. Turnbull (Academic Press, New York and London, 1958), Vol. 6, pp. 1–63.

  61. N. Curle, Rankine-Hugoniot law, in Encyclopaedic Dictionary of Physics, edited by J. Thewlis (Pergamon Press, Oxford, 1962), pp. 194–195.

  62. The Effects of Nuclear Weapons, edited by S. Glasstone. Prepared by the US Dept. of Defense, published by the US Atomic Energy Commission (Feb. 1964), Chap. 3.72.

  63. J.M. Dewey, The properties of a blast wave obtained from an analysis of the particle trajectories, Proc. Roy. Soc. Lond. A 324, 275–299 (1971).

    ADS  Google Scholar 

  64. Prof. John M. Dewey, private communication on June 8, 2012.

  65. B.K. Godwal, S.K. Sikka, R. Chidambaram, Equation of state theories of condensed matter up to about 10 TPa, Phys. Rep. (Rev. Ser. Phys. Lett.) 102, 121–197 (1983).

    ADS  Google Scholar 

  66. E. Murr, K.P. Staudhammer, Shock wave fundamentals: effects on the structure and behavior of engineering materials, in Shock Waves for Industrial Applications, edited by E. Murr (Noyes Publs., Park Ridge, NJ, 1988), pp. 13–15.

  67. M.A. Barrios, D.G. Hicks, T.R. Boehly, D.E. Fratanduono, D.D. Meyerhofer, J.H. Eggert, P.M. Celliers, G.W. Collins, High-precision measurements of the EOS of hydrocarbons at 1−10 Mbar using laser-driven shock waves. LLE (Laboratory for Laser Energetics) Rev. 121, 6–21 (2009).

    Google Scholar 

  68. J.W. Forbes, Shock Wave Compression of Condensed Matter – a Primer (Springer-Verlag, Berlin, 2012), pp. 31–57.

  69. Dr. Charles E. Needham, private communication on Dec. 17, 2013.

  70. J. von Neumann, R.D. Richtmyer, A method for the numerical calculation of hydrodynamic shocks, J. Appl. Phys. 21, 232–237 (1950).

    MATH  MathSciNet  ADS  Google Scholar 

  71. H.L. Brode, Blast wave from a spherical charge, Phys. Fluids 2, 217–229 (1959).

    MATH  ADS  Google Scholar 

  72. W. Band, G.E. Duvall, Physical nature of shock propagation, Am. J. Phys. 29, 780–785 (1961).

    MATH  ADS  Google Scholar 

  73. Lord Rayleigh, J.W. Strutt, Aerial plane waves of finite amplitude, Proc. Roy. Soc. Lond. A 84, 247–284 (1910).

    MATH  ADS  Google Scholar 

  74. W. Band, Introduction to Mathematical Physics (Van Nostrand Company, Inc., Princeton, NJ, 1959).

  75. G.E. Duvall, Shock waves in condensed media. Int. School of Physics Enrico Fermi (Lago di Como, Italy; July 14–26, 1969). Proc. publ. in Physics of High Energy Density, edited by P. Caldirola (Academic Press, New York, 1971), Vol.48, pp. 7–50.

  76. G.R. Fowles, R.F. Williams, Plane stress wave propagation in solids, J. Appl. Phys. 41, 360–363 (1970).

    ADS  Google Scholar 

  77. R. Fowles, Determination of constitutive relations from plane wave experiments (US Defense Technical Information Center, Fort Belvoir, VA, 1970).

  78. G.R. Fowles, Conservation relations for spherical and cylindrical stress waves, J. Appl. Phys. 41, 2740–2741 (1970).

    ADS  Google Scholar 

  79. M. Cowperthwaite, R.F. Williams, Determination of constitutive relationships with multiple gauges in non-divergent waves, J. Appl. Phys. 42, 456–462 (1971).

    ADS  Google Scholar 

  80. L. Seaman, Lagrangian analysis for multiple stress or velocity gages in alternating waves, J. Appl. Phys. 45, 4303–4314 (1974).

    ADS  Google Scholar 

  81. J.B. Aidun, Y.M. Gupta, Analysis of Lagrangian gauge measurements of simple and nonsimple plane waves, J. Appl. Phys. 69, 6998–7014 (1991).

    ADS  Google Scholar 

  82. W.J. Murri, D.R. Curran, C.F. Peterson, R.C. Crewdson, Response of Solids to Shock Waves. Tech. Rept. No. 001-71, Poulter Laboratory of SRI, Menlo Park, CA (1971). Later published in Advances in High Pressure Research, edited by R.H. Wentorf Jr. (Academic Press, London and New York, 1974), Vol. 4, pp. 1–163.

  83. G.I. Taylor, The conditions necessary for discontinuous motion in gases, Proc. Roy. Soc. Lond. A 84, 371–377 (1910).

    MATH  ADS  Google Scholar 

  84. J.M. Kelly, P.P. Gillis, Shock thickness in viscoplastic solids, J. Appl. Mech. 37, 163–170 (1970).

    ADS  Google Scholar 

  85. R.T. Walsh, Finite difference methods, in Dynamic Response of Materials to Intense Impulsive Loading, edited by P.C. Chou, A.K. Hopkins. Rept. AD-768-416, Air Force Materials Laboratory, Wright Patterson Air Force Base, OH (Aug. 1972), Chap. 7, pp. 363–403.

  86. G.E. Duvall, R.A. Graham, Phase transitions under shock wave loading, Rev. Mod. Phys. 49, 523–579 (1977).

    ADS  Google Scholar 

  87. L. Barker, α-phase Hugoniot of iron, J. Appl. Phys. 46, 2544–2546 (1975).

    ADS  Google Scholar 

  88. R.A. Graham, Measurement of wave profiles in shock-loaded solids, in High-Pressure Science and Technology, edited by K.D. Timmerhaus, M.S. Barber (Plenum Publ. Corp., New York, 1979), Vol. 2, pp. 854–869.

  89. D.C. Wallace, Equation of state from weak shocks in solids, Phys. Rev. B20, 1495–1502 (1980).

    ADS  Google Scholar 

  90. R. Blandford, D. Eichler, Particle acceleration at astrophysical shocks: a theory of cosmic ray origin, Phys. Rep. 154, 1–75 (1987).

    ADS  Google Scholar 

  91. T.J. Ahrens, Equation of state, in High Pressure Shock Compression of Solids, edited by J.R. Asay, M. Shahinpoor (Springer-Verlag, New York, 1993), pp. 75–114.

  92. Y. Sano, Shock jump equations for unsteady wave fronts, J. Appl. Phys. 82, 5382–5390 (1997).

    ADS  Google Scholar 

  93. Y. Sano, Shock jump equations for unsteady wave fronts of finite rise time, J. Appl. Phys. 84, 6606–6613 (1998).

    ADS  Google Scholar 

  94. Y. Sano, I. Miyamoto, Generalized smooth and weak-discontinuous unsteady waves, J. Math. Phys. 41, 6233–6247 (2000).

    MATH  MathSciNet  ADS  Google Scholar 

  95. Y. Sano, T. Sano, Unsteady state Rankine-Hugoniot jump conditions, in 15th APS Topical Conference on Shock Compression of Condensed Matter – 2007, edited by M. Elert, M.D. Furnish, R. Chau, N.C. Holmes, J. Nguyen, AIP Conf. Proc. 955, 267–270 (2007).

  96. Y. Sano, T. Sano, Jump across an outgoing spherical shock wave front, in 15th APS Topical Conference on Shock Compression of Condensed Matter – 2007, edited by M. Elert, M.D. Furnish, R. Chau, N.C. Holmes, J. Nguyen, AIP Conf. Proc. 955, 271–274 (2007).

  97. W.W. Anderson, Jump conditions for nonsteady waves, in 14th APS Topical Conference on Shock Compression of Condensed Matter – 2005, edited by M.D. Furnish, M.L. Elert, T.P. Russell, C.T. White, AIP Conf. Proc. 845, 1303–1306 (2006).

  98. A. Balogh, R.A. Treumann, Physics of Collisionless Shocks (Springer-Verlag, New York, 2013), p. 30.

  99. P.L. Sachdev, Shock Waves and Explosions. Monographs and Surveys in Pure & Applied Mathematics, No. 132 (Chapman & Hall/CRC, Boca Ration, FL, 2004), pp. 38–39.

  100. F. de Hoffmann, E. Teller, Magneto-hydrodynamic shocks, Phys. Rev. 80, 692–703 (1950).

    MATH  ADS  Google Scholar 

  101. Shocks and discontinuities (magnetohydrodynamics). Wikipedia;http://en.wikipedia.org/wiki/Shocks_and_discontinuities_(magnetohydrodynamics).

  102. A.H. Taub, Relativistic Rankine-Hugoniot equations, Phys. Rev. 74 [II], 328–334 (1948).

    MATH  MathSciNet  ADS  Google Scholar 

  103. S.M. Carioli, Solutions of the Rankine-Hugoniot relations in relativistic magnetohydrodynamics, Phys. Fluids 29, 672–675 (1986).

    MATH  MathSciNet  ADS  Google Scholar 

  104. X.-B. Lin, Generalized Rankine-Hugoniot condition and shock solutions for quasi-linear hyperbolic systems. Dept. of Mathematics, North Carolina State University, Raleigh, NC (June 2, 2000); see http://www4.ncsu.edu/xblin/preprint/shock.pdf.

  105. V.M. Shelkovich, Delta-shocks in the Navier-Stokes system of granular hydrodynamics. Poster presented at the 14th Int. Conference on Hyperbolic Problems: Their Theory, Numerics, Applications (HYP 2012) [Università di Padova, Italy; June 25–29, 2012].

  106. V.M. Shelkovich, Concept of delta-shock type solutions to systems of conservation laws and the Rankine-Hugoniot conditions, Operator Theory: Advances & Applications 231, 297–305 (2013).

    MathSciNet  Google Scholar 

  107. S.K. Lele, Shock-jump relations in a turbulent flow, Phys. Fluids A 4, 2900–2905 (1992).

    MATH  MathSciNet  ADS  Google Scholar 

  108. M.A. Liberman, Introduction to Physics and Chemistry of Combustion: Explosion, Flame, Detonation (Springer-Verlag, Berlin, 2008).

  109. C.M. Tarver, Chemical energy release in one-dimensional detonation waves in gaseous explosives, Combust. Flame 46, 111–133 (1982).

    Google Scholar 

  110. H. Nieuwenhuijzen, C. de Jager, M. Cuntz, A. Lobel, L. Achmad, A generalized version of the Rankine-Hugoniot relations including ionization, dissociation, radiation and related phenomena, A&A 280, 195–200 (1993).

    ADS  Google Scholar 

  111. H. Ockendon, J.R. Ockendon, Waves and Compressible Flow (Springer-Verlag, New York, 2004).

  112. Y. He, X. Hu, Y. Hu, Z. Jiang, J. Lü, Rankine-Hugoniot relations of an axial shock in cylindrical non-neutral plasma, Phys. Plasmas 13, 092116 (2006).

    MathSciNet  ADS  Google Scholar 

  113. Y. Zheng, Systems of Conservation Laws: Two-Dimensional Riemann Problems (Birkhäuser, Boston, 2001), pp. 86–88.

  114. K.C. Hall, A linearized Euler analysis of unsteady flows in turbomachinery. Ph.D. thesis, Dept. of Aeronautics & Astronautics, MIT, Cambridge, USA (May 1987); later partly publ. with W.S. Clark and C.B. Lorence in J. Turbomach. 116, 477–488 (1994).

  115. R.F. Chisnell, The normal motion of a shock wave through a non-uniform one dimensional medium, Proc. Roy. Soc. Lond. A232, 350–370 (1955).

    MathSciNet  ADS  Google Scholar 

  116. R.F. Chisnell, The motion of a shock wave in a channel, with applications to cylindrical and spherical shock waves, J. Fluid Mech. 2, 286–298 (1957).

    MATH  MathSciNet  ADS  Google Scholar 

  117. G.B. Whitham, On the propagation of shock waves through regions of non-uniform area or flow, J. Fluid Mech. 4, 337–368 (1958).

    MATH  MathSciNet  ADS  Google Scholar 

  118. G.P. Zank, Y. Zhou, W.H. Matthaeus, W.K.M. Rice, The interaction of turbulence with shock waves: a basic model, Phys. Fluids 14, 3766–3774 (2002).

    MathSciNet  ADS  Google Scholar 

  119. Yu.P. Raizer, Heating of a gas by a powerful light pulse, Sov. Phys. J. Exp. Theor. Phys. 21, 1009–1017 (1965).

    ADS  Google Scholar 

  120. E. Daniel, J. Massoni, Jump relations across shock waves in condensed multiphase flows: a comparison between numerical and analytical solutions. 18ème Congrès Français de Mécanique (Grenoble, France; August 27–31, 2007).

  121. W. Fickett, W.C. Davis, Detonation (University of California Press, Berkeley and Los Angeles, 1979), pp. 16–20, 98–102.

  122. S.L. Gavrilyuk, R. Saurel, Rankine-Hugoniot relations for shocks in heterogeneous mixtures, J. Fluid Mech. 575, 495–507 (2007).

    MATH  MathSciNet  ADS  Google Scholar 

  123. O. Thual, Modeling rollers for shallow water flows, J. Fluid Mech. 728, 1–4 (2013).

    MATH  ADS  Google Scholar 

  124. Y.C. Huang, F.G. Hammitt, T.M. Mitchell, Note on shock wave velocity in high-speed liquid-solid impact, J. Appl. Phys. 44, 1868–1869 (1973).

    ADS  Google Scholar 

  125. R. Ghoshal, N. Mitra, Non-contact near-field underwater explosion induced shock-loading of submerged rigid structures: nonlinear compressibility effects in fluid structure interaction, J. Appl. Phys. 112, 024911 (2012).

    ADS  Google Scholar 

  126. H.G. David, S.D. Hamann, Some properties of compressional waves in Lennard-Jones-and-Devonshire liquids, Austral. J. Chem. 14, 372–386 (1961).

    Google Scholar 

  127. A. Satoh, Rankine-Hugoniot relations for Lennard-Jones liquid, J. Fluid Eng. 116, 625–630 (1994).

    Google Scholar 

  128. R.J. Seeger, H. Polachek, On shock-wave phenomena: waterlike substances, J. Appl. Phys. 22, 640–654 (1951).

    MATH  MathSciNet  ADS  Google Scholar 

  129. L. Davison, R.A. Graham, Shock compression of solids, Phys. Rep. 55, 255–379 (1979).

    ADS  Google Scholar 

  130. L. Davison, Shockwave structure in porous solids, J. Appl. Phys. 42, 5503–5512 (1971).

    ADS  Google Scholar 

  131. L.G. Bolkhovitinov, Yu.B. Khvostov, The Rankine-Hugoniot relation for shock waves in very porous media, Nature 274, 882–883 (1978).

    ADS  Google Scholar 

  132. P.P. Gillis, Elastic precursor decay in tantalum, J. Appl. Phys. 42, 2145–2146 (1971).

    ADS  Google Scholar 

  133. G.E. Duvall, Shock waves and equations of state, in Dynamic Response of Materials to Intense Impulsive Loading, edited by P.C. Chou, A.K. Hopkins. Rept. AD-768-416, Air Force Materials Laboratory, Wright Patterson Air Force Base, OH (1972), Chap. 4, pp. 89–118.

  134. J.K. Chao, B. Goldstein, Modification of the Rankine-Hugoniot relations for shocks in space, J. Geophys. Res. 77, 5455–5466 (1972).

    ADS  Google Scholar 

  135. J.J. Sanderson, R.A. Uhrig Jr., Extended Rankine-Hugoniot relations for collisionless shocks, J. Geophys. Res.: Space Phys. 83, 1395–1400 (1978).

    ADS  Google Scholar 

  136. D. Winterhalter, M.G. Kivelson, R.J. Walker, C.T. Russell, The MHD Rankine-Hugoniot jump conditions and the terrestrial bow shock: a statistical comparison, Adv. Space Res. 4, 287–292 (1984).

    ADS  Google Scholar 

  137. E.C. Roelof, S.M. Krimigis, D.G. Mitchell, R.B. Decker, J.D. Richardson, M. Gruntsman, H. Funsten, D. McComas, Implications of generalized Rankine-Hugoniot conditions for the PUI population at the Voyager 2 termination shock, in Proc. 9th Annual International Astrophysics Conference, edited by J. Le Roux, G.P. Zank, A.J. Coates, V. Florinski. AIP Conf. Proc. 1302, 133–141 (2010).

  138. R.E. Lee, S.C. Chapman, R.O. Dendy, Numerical simulations of local shock reformation and ion acceleration in supernova remnants. 31st EPS Conference on Plasma Physics [London, UK; June 26–July, 2, 2004]. ECA 28G, Paper O-4.15 (2004).

  139. G. Pallocchia, A.A. Samsonov, M.B. Bavassano Cattaneo, M.F. Marcucci, H. Rème, C.M. Carr, J.B. Cao, Interplanetary shock transmitted into the Earth’s magnetosheath: cluster and double star observations, Ann. Geophys. 28, 1141–1156 (2010).

    ADS  Google Scholar 

  140. R.D. Blandford, C.F. McKee, Fluid dynamics of relativistic blast waves, Phys. Fluids 19, 1130–1138 (1976).

    MATH  ADS  Google Scholar 

  141. Y. Gao, C.K. Law, Rankine-Hugoniot relations in relativistic combustion waves. arXiv:1210.3455 [astro-ph.CO] (2012).

  142. P. Jenny, B. Müller, Rankine-Hugoniot-Riemann solver with considering source terms and multi-dimensional effects, J. Comput. Phys. 145, 575–610 (1997).

    ADS  Google Scholar 

  143. S. Jaisankar, S.V.R. Rao, A central Rankine-Hugoniot solver for hyperbolic conservation laws, J. Comput. Phys. 228, 770–798 (2009).

    MATH  MathSciNet  ADS  Google Scholar 

  144. A. Konyukhov, A. Likhachev, V. Fortov, S. Anisimov, Nonlinear analysis of stability of plane shock waves in media with arbitrary thermodynamic properties, in 28th Int. Symposium on Shock Waves, edited by K. Kontis (Springer-Verlag, Heidelberg & Berlin, 2012), Vol. 2, pp. 531–536.

  145. R.F. Chisnell, The motion of a shock wave through a nonuniform one-dimensional medium, Proc. Roy. Soc. Lond. A 232, 350–370 (1957).

    MathSciNet  ADS  Google Scholar 

  146. L. Crussard, Ondes de choc et onde explosive, Bull. Soc. Industrie Minérale 6 [IV], 257–364 (1907); Propriété de l’onde explosive, C. R. Acad. Sci. Paris 144, 417–420 (1907).

    Google Scholar 

  147. G.E. Duvall, Semiannual report, 1 February 1973 to 31 July 1973. Contract No.DAAG-46-C-0104, AMMRC, Watertown, MA (1973).

  148. S.D. Hamann, Effects of intense shock waves, in Advances in High Pressure Research, edited by R.S. Bradley (Academic Press, London & New York, 1966), Vol. 1, pp. 85–141.

  149. D.P. Dandekar, Behavior of porous tungsten under shock compression at room temperature, J. Appl. Phys. 48, 2871–2879 (1977).

    ADS  Google Scholar 

  150. C.F. Petersen, W.J. Murri, M. Cowperthwaite, Hugoniot and release-adiabat measurements for selected geological materials, J. Geophys. Res. 75, 2063–2072 (1970).

    ADS  Google Scholar 

  151. H. Eyring, R.E. Powell, G.H. Duffey, R.B. Parlin, The stability of detonation, Chem. Rev. 45, 69–181 (1949), Appendix A.

    Google Scholar 

  152. Dr. Jerry W. Forbes, private communications on Nov. 22, 2014.

  153. M. Müller, Energy dissipated at the shock wave during its propagation in sea water. Colloquium Fluid Dynamics, Institute of Fluid Dynamics, Prague (2007).

  154. A.J. Eggers Jr., One-Dimensional Flows of an Imperfect Diatomic Gas. NACA Rept. 959 (1950).

  155. F. Marconi, M. Salas, L. Yaeger, Development of computer code for calculating the steady super/hypersonic inviscid flow around real configurations. Vol. 1: Computational technique. Rept. NASA CR-2675 (1976).

  156. G.R. Fowles, Shock wave compression of hardened and annealed 2024 aluminum, J. Appl. Phys. 32, 1475–1487 (1961).

    ADS  Google Scholar 

  157. D.C. Pack, Shock wave phenomena, in Research Frontiers in Fluid Dynamics, edited by R.J. Seeger, G. Temple (Interscience Publications, New York, 1965), Chap. 8, pp. 212–249.

  158. G.E. Duvall, Concepts of shock wave propagation, Bull. Seismology Soc. Am. 52, No. 4, 869–893 (Oct. 1962).

  159. T.-P. Liu, Hyperbolic conservation laws with relaxation, Commun. Math. Phys. 108, 153–175 (1987).

    MATH  ADS  Google Scholar 

  160. J.W. Taylor, M.H. Rice, Elastic-plastic properties of iron, J. Appl. Phys. 34, 364–371 (1963).

    ADS  Google Scholar 

  161. M. Sichel, Structure of weak non-Hugoniot shocks, Phys. Fluids 6, 653–662 (1963).

    MATH  MathSciNet  ADS  Google Scholar 

  162. J.M. Dewey, Spherical shock waves, in Handbook of Shock Waves, edited by G. Ben-Dor, O. Igra, T. Elperin (Academic Press, San Diego, 2001), Vol. 2, pp. 441–481.

  163. H. Grad, The profile of a steady plane shock wave, Commun. Pure Appl. Math. 5, 257–300 (1952).

    MATH  MathSciNet  Google Scholar 

  164. Y.B. Zel’dovich, On the possibility of rarefaction shock waves, Zh. Eksp. Teor. Fiz. 16, 363–364 (1946).

    Google Scholar 

  165. W.E. Drummond, Multiple shock production, J. Appl. Phys. 28, 998–1001 (1957).

    MathSciNet  ADS  Google Scholar 

  166. S.S. Kutateladze, Al. A. Borisov, A.A. Borisov, V.E. Nakoryakov, Experimental detection of a rarefaction shock wave near a liquid-vapor critical point, Sov. Phys. Dokl. 25, 392–393 (1980).

    ADS  Google Scholar 

  167. G.I. Taylor, The dynamics of the combustion products behind plane and spherical detonation fronts in explosives, Proc. Roy. Soc. Lond. A 200, 235–247 (1949/1950).

    ADS  Google Scholar 

  168. Y.M. Gupta, Shock waves in condensed media. In McGraw-Hill Encyclopedia of Science & Technology, 9th edn. (McGraw-Hill Book Co., New York, 2005), pp. 438–439.

  169. W. Band, G.E. Duvall, Physical nature of shock propagation, Am. J. Phys. 29, 780–785 (1961).

    MATH  ADS  Google Scholar 

  170. L. Barker, L. Hollenbach, Shock wave study of the alpha-epsilon phase transition in iron, J. Appl. Phys. 45, 4872–4887 (1974).

    ADS  Google Scholar 

  171. H.A. Bethe, E. Teller, Deviations from Thermal Equilibrium in Shock Waves. BL Rept. X-117, BRL, Aberdeen Proving Ground, MD (1941).

  172. G.E. Duvall, G.R. Fowles, Shock waves, in High Pressure Physics and Chemistry, edited by R.S. Bradley (Academic Press, New York, 1963), Vol. 2, p. 212.

  173. G.A. Lyzenga, T.J. Ahrens, W.J. Nellis, A.C. Mitchell, The temperature of shock-compressed water, J. Chem. Phys. 76, 6282–6286 (1982).

    ADS  Google Scholar 

  174. L.V. Al’tshuler, K.K. Krupnikov, B.B. Lebedev, V.I. Zhuchikin, M.I. Brazhnik, Dynamic compressibility and equation of state of iron under high pressure, Sov. Phys. 7, 606–614 (1958).

    Google Scholar 

  175. A.C. Mitchell, W.J. Nellis, Shock compression of aluminum, copper, and tantalum, J. Appl. Phys. 52, 3363–3374 (1981).

    ADS  Google Scholar 

  176. R.G. Shreffler, W.E. Deal, Free surface properties of explosive-driven metal plates, J. Appl. Phys. 24, 44–48 (1953).

    ADS  Google Scholar 

  177. W.J. Carter, S.P. Marsh, J.N. Fritz, R.G. McQueen, The equation of state of selected materials for high-pressure references, in Accurate Characterization of the High-Pressure Environment, edited by E.C. Lloyd. NBS Special Publication No. 326, US Government Printing Office, Washington, DC (1971), pp. 147–158.

  178. F.G. Friedlander, The diffraction of sound pulses. I. Diffraction by a semi-infinite plate, Proc. Roy. Soc. Lond. A 186, 322–344 (1946).

    MATH  ADS  Google Scholar 

  179. W.E. Baker, Explosions in Air (University of Texas Press, Austin, 1973).

  180. M. Larcher, Pressure-time functions for the description of air blast waves. JRC Technical Note 46829, Joint Research Centre, Ispra, Italy (2008).

  181. H. Honma, I.I. Glass, C.H. Wong, O. Hoist-Jensen, D. Xu, Experimental and numerical studies of weak blast waves in air, Shock Waves 1, 111–119 (1991).

    ADS  Google Scholar 

  182. G.R. Fowles, Experimental techniques and instrumentation, in Dynamic response of materials to intense impulsive loading, edited by P.C. Chou, A.K. Hopkins. Rept. AD-768-416, Air Force Materials Laboratory, Wright Patterson Air Force Base, OH (1972), Chap. 8, pp. 405–480.

  183. T.J. Ahrens, Shock wave techniques for geophysics and planetary physics, Meth. Exp. Phys. 24, 185–210 (1987).

    Google Scholar 

  184. J.W. Forbes, Shock Wave Compression of Condensed Matter – a Primer (Springer-Verlag, Berlin, 2012); pp. 68–79.

  185. L.M. Barker, M. Shahinpoor, L.C. Chhabildas, Experimental and diagnostic techniques, in High-pressure shock compression of solids, edited by J.R. Asay, M. Shahinpoor (Springer-Verlag, New York, 1993), pp. 43–73.

  186. Y. Beers, Introduction to the Theory of Error (Addison-Wesley, London, 1957), p. 4.

  187. R.W. Goranson, D. Bancroft, B.L. Burton, T. Blechar, E.E. Houston, E.F. Gittings, S.A. Landeen, Dynamic determination of the compressibility of metals, J. Appl. Phys. 26, 1472–1479 (1955).

    ADS  Google Scholar 

  188. R.G. McQueen, S.P. Marsh, Equation of state for nineteen metallic elements from shock-wave measurements to two megabars, J. Appl. Phys. 31, 1253–1269 (1960).

    ADS  Google Scholar 

  189. F.S. Minshall, The dynamic response of iron and iron alloys to shock waves, in Response of Metals to High-Velocity Deformation, edited by V.F. Zackay, P.G. Shewmon (Interscience, New York, 1961), pp. 249–272.

  190. R.E. Duff, E. Houston, Measurement of the Chapman-Jouguet pressure and reaction zone length in a detonating high explosive, J. Chem. Phys. 23, 1268–1273 (1955).

    ADS  Google Scholar 

  191. N.L. Coleburn, J.W. Forbes, Irreversible transformation of hexagonal boron nitride by shock compression, J. Chem. Phys. 48, 555–559 (1968).

    ADS  Google Scholar 

  192. R. Schall, G. Thomer, Flash Radiographic Measurement of the Shock Compressibility of Magnesium Alloy, Lucite, and Polyethylene. Rept. AFSWC-TDR-62-134, Air Force Systems Command, Kirtland Air Force Base, NM (1962).

  193. R. Schall, Die Zustandsgleichung des Wassers bei hohen Drucken nach Röntgenblitzaufnahmen intensiver Stoßwellen, Z. Angew. Phys. 2, 252–254 (1950).

    Google Scholar 

  194. R. Schall, G. Thomer, Röntgenblitzaufnahmen von Stoßwellen in festen, flüssigen und gasförmigen Medien, Z. Angew. Phys. 3, 41–44 (1951).

    Google Scholar 

  195. F. Jamet, G. Thomer, Flash Radiography (Elsevier, Amsterdam, 1976), pp. 120–122.

  196. G.E. Duvall, Problems in shock wave research [invited paper], in Conference on Metallurgical Effects at High Strain Rates [Albuquerque, NM; Feb. 5−8, 1973], edited by R.W. Rohde, B.M. Butcher, J.R. Holland, C.H. Karnes (Plenum Press, New York, 1973), pp. 1–13.

  197. M. Ross, W. Nellis, A. Mitchell, Shock-wave compression of liquid argon to 910 kbar, Chem. Phys. Lett. 68, 532–535 (1979).

    ADS  Google Scholar 

  198. A.H. Jones, W.H. Isbell, C.J. Maiden, Measurements of the very-high-pressure properties of materials using a light-gas gun, J. Appl. Phys. 37, 3493–3499 (1966).

    ADS  Google Scholar 

  199. C.E. Morris, Shock-wave equation-of-state studies, Shock Waves 1, 213–222 (1991).

    ADS  Google Scholar 

  200. G.V. oriskov et al., Shock compression of liquid deuterium up to 109 GPa, Phys. Rev. B 71, 092104 (2005).

    ADS  Google Scholar 

  201. G. Chabrier et al., Hydrogen and helium at high density and astrophysical implications, in High Energy Density Laboratory Astrophysics, edited by S.V. Lebedev (Springer-Verlag, Dordrecht, The Netherlands, 2007), pp. 257–261.

  202. J.W. Forbes, Shock Wave Compression of Condensed Matter – a Primer (Springer-Verlag, Berlin, 2012), pp. 82–90.

  203. R.F. Smith, J.H. Eggert, A. Jankowski, P.M. Celliers, M.J. Edwards, Y.M. Gupta, J.R. Asay, G.W. Collins, Stiff response of aluminum under shockless compression to 110 GPa, Phys. Rev. Lett. 98, 065701 (2007).

    ADS  Google Scholar 

  204. P. Krehl, Measurement of low shock pressures with piezoresistive carbon gauges, Rev. Sci. Instrum. 49, 1477–1484 (1978).

    ADS  Google Scholar 

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Dr. Krehl is a retired staff scientist from the Fraunhofer-Institute for High-Speed Dynamics, Ernst-Mach-Institut (FhG-EMI) at Freiburg/Breisgau, Germany.

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Krehl, P.O.K. The classical Rankine-Hugoniot jump conditions, an important cornerstone of modern shock wave physics: ideal assumptions vs. reality. EPJ H 40, 159–204 (2015). https://doi.org/10.1140/epjh/e2015-50010-4

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