Shock Waves

, Volume 1, Issue 4, pp 251–273 | Cite as

A generalisation of the theory of geometrical shock dynamics

  • J. P. Best


The study of the propagation of a shock down a tube of slowly varying cross sectional area has proved to be most valuable in understanding the dynamics of shocks. A particular culmination of this work has been the theory of geometrical shock dynamics due to Whitham (1957, 1959). In this theory the motion of a shock may be approximately computed independently of a determination of the flow field behind the shock. In this paper the propagation of a shock down such a tube is reconsidered. It is found that the motion of the shock is described by an infinite sequence of ordinary differential equations. Each equation is coupled to all of its predecessors but only to its immediate successor, a feature which allows the system to be closed by truncation. Of particular relevance is the demonstration that truncation at the first equation in the sequence yields the A-M relation that is the basis for Whitham's highly successful theory. Truncation at the second equation yields the next level of approximation. The equations so obtained are investigated with analytic solutions being found in the strong shock limit for the propagation of cylindrical and spherical shock waves. Implementation of the theory in the numerical scheme of geometrical shock dynamics allows the computation of shock motion in more general geometries. In particular, investigation of shock diffraction by convex corners of large angular deviation successfully yields the observed inflection point in the shock shape near the wall. The theory developed allows account to be taken of non-uniform flow conditions behind the shock. This feature is of particular interest in consideration of underwater blast waves in which case the flow behind the shock decays approximately exponentially. Application of the ideas developed here provides an excellent description of this phenomenon.

Key words

Geometrical shock dynamics A-M relation Shock diffraction Underwater blast 


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  1. Arons AB (1954) Underwater explosion shock wave parameters at large distances from the charge. J Acoust Soc Am 26:343–345Google Scholar
  2. Cole RH (1948) Underwater Explosions. Princeton University PressGoogle Scholar
  3. Guderley G (1942) Starke kugelige und zylindrische Verdichtungsstösse in der Nähe des Kugelmittelpunktes bzw der Zylinderachse. Luftfahrtforschung 19:302–312Google Scholar
  4. Henshaw WD, Smyth NF, Schwendeman DW (1986) Numerical shock propagation using geometrical shock dynamics. J Fluid Mech 171:519–545Google Scholar
  5. Kirkwood JG, Bethe HA (1942) Progress report on the pressure wave produced by an underwater explosion I. OSRD Rept 588Google Scholar
  6. Kucera A (1991) A boundary integral method applied to the growth and collapse of bubbles near a rigid boundary. J Comp Phys (To appear)Google Scholar
  7. Lighthill J (1949) The diffraction of blast II. Proc Roy Soc London A 200:554–565Google Scholar
  8. Maslov VP (1980) Propagation of shock waves in the isentropic nonviscous gas. J Sov Math 13:119–163Google Scholar
  9. Osborne MFM, Taylor AH (1946) Non-linear propagation of underwater shock waves. Phys Rev 70:322–328Google Scholar
  10. Poché LB (1972) Underwater shock-wave pressures from small detonators. J Acoust Soc Am 51:1733–1737Google Scholar
  11. Prasad P, Srinivasan R (1988) On methods of calculating successive positions of a shock front. Acta Mechanica 74:81–93Google Scholar
  12. Rogers PH (1977) Weak-shock solution for underwater explosive shock waves. J Acoust Soc Am 62:1412–1419Google Scholar
  13. Skews BW (1967a) The shape of a diffracting shock wave. J Fluid Mech 29:297–304Google Scholar
  14. Skews BW (1967b) The perturbed region behind a diffracting shock wave. J Fluid Mech 29:705–719Google Scholar
  15. Srinivasan R, Prasad P (1985) On the propagation of a multi-dimensional shock of arbitrary strength. Proc Indian Acad Sci (Math Sci) 94:27–42Google Scholar
  16. Whitham GB (1957) A new approach to problems of shock dynamics. Part I, Two dimensional problems. J Fluid Mech 2:146–171Google Scholar
  17. Whitham GB (1958) On the propagation of shock waves through regions of non-uniform area or flow. J Fluid Mech 4:337–360Google Scholar
  18. Whitham GB (1959) A new approach to problems of shock dynamics. Part II, Three-dimensional problems. J Fluid Mech 5:369–386Google Scholar
  19. Whitham GB (1974) Linear and nonlinear waves. Wiley, New YorkGoogle Scholar
  20. Whitham GB (1987) On shock dynamics. Proc Indian Acad Sci (Math Sci) 96(2):71–73Google Scholar

Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • J. P. Best
    • 1
  1. 1.Materials Research Laboratory (MRL), DSTOMelbourneAustralia

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