Converging Shocks

  • Nicholas Apazidis
  • Veronica Eliasson
Part of the Shock Wave and High Pressure Phenomena book series (SHOCKWAVE)


In the beginning of this chapter, we give on overview of early experimental work on generation of converging shocks by various methods ranging from annular shock tubes to cylindrical as well as spherical explosion chambers. These early experimental results along with Guderley’s solution raise important questions of self-similarity and stability of converging shocks. Experimental results showing the dependence of the power-law exponent on the adiabatic exponent for various gases are presented and discussed. We then give an overview of theoretical and numerical results on the stability of converging shocks based on the theory of geometrical shock dynamics. A number of experimental results on shock convergence show that converging shock experiences tendency toward planarity, e.g., generation of plane sides and sharp corners in initially cylindrical shock front. In this respect several sections of this chapter are devoted to experimental as well as numerical work on convergence of polygonal shocks and their ability to preserve symmetry and thus enhance the final energy density. Production of cylindrical and spherical converging shocks by a gradual change in the shock tube cross-section has been proposed by several researchers. We discuss the basic theoretical and numerical results as well as their experimental realization leading to extreme conditions at the focal area with gas temperatures in excess of 30,000 K. The end of this chapter is devoted to shock generation and focusing in water by means of exploding wire techniques. Experimental findings showing extreme states of matter at the focal area of a converging shock in water generated by a moderate input of initial energy are discussed.


  1. 1.
    Ahlborn, B., Fong, K.: Stability criteria for converging shock waves. Can. J. Phys. 56(5), 1292–1296 (1978)CrossRefGoogle Scholar
  2. 2.
    Aki, T., Higashino, F.: A numerical study on implosion of polygonally interacting shocks and consecutive explosion in a box. In: Current Topics in Shock Waves: 17th Proceedings of the International Symposium on Shock Waves and Shock Tubes, Bethlehem, PA, 17–21 July (A91-40576 17-34), pp. 167–172. American Institute of Physics, New York (1989)Google Scholar
  3. 3.
    Antonov, O., Efimov, S., Yanuka, D., Kozlov, M., Gurovich, V.T., Krasik, Y.E.: Generation of extreme state of water by spherical wire array underwater electrical explosion. Phys. Plasmas 19, 102702 (2012)CrossRefGoogle Scholar
  4. 4.
    Antonov, O., Efimov, S., Yanuka, D., Kozlov, M., Gurovich, V.T., Krasik, Y.E.: Generation of converging strong shock wave formed by microsecond timescale underwater electrical explosion of spherical wire array. Appl. Phys. Lett. 102, 124104 (2013)CrossRefGoogle Scholar
  5. 5.
    Antonov, O., Efimov, S., Yanuka, D., Kozlov, M., Gurovich, V.T., Krasik, Y.E.: Diagnostics of a converging strong shock wave generated by underwater explosion of a spherical wire array. J. Appl. Phys. 115, 223303 (2014)CrossRefGoogle Scholar
  6. 6.
    Apazidis, N.: Focusing of weak shock waves in confined axisymmetric chambers. Shock Waves 3, 201–212 (1994)zbMATHCrossRefGoogle Scholar
  7. 7.
    Apazidis, N.: Numerical investigation of shock induced bubble collapse in water. Phys. Fluids 28, 046101 (2016)CrossRefGoogle Scholar
  8. 8.
    Apazidis, N., Lesser, M.B.: On generation and convergence of polygonal-shaped shock waves. J. Fluid Mech. 309, 301–319 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Apazidis, N., Lesser, M.B., Tillmark, N. Johansson, B.: An experimental and theoretical study of converging shock waves. Shock Waves 12, 39–58 (2002)zbMATHCrossRefGoogle Scholar
  10. 10.
    Apazidis, N., Kjellander, M., Tillmark, N.: High energy concentration by symmetric shock focusing. Shock Waves 23, 361–368 (2013)CrossRefGoogle Scholar
  11. 11.
    Balasubramanian, K., Eliasson, V.: Numerical investigations of the porosity effect on the shock focusing process. Shock Waves 23(6), 583–594 (2013)CrossRefGoogle Scholar
  12. 12.
    Barbry, H., Mounier, C., Saillard, Y.: Transformation d’un choc plan uniforme en choc cylindrique ou spherique uniforme Classical and quantum mechanics, general physics (A1110), Report CEA-N–2516, France (1986)Google Scholar
  13. 13.
    Baronets, P.: Imploding shock waves in a pulsed induction discharge. Fluid Dyn. 19, 503–508 (1984)CrossRefGoogle Scholar
  14. 14.
    Betelu, S.I., Aronson, D.G.: Focusing of noncircular self-similar shock waves. Phys. Rev. Lett. 87(7), 074501 (2001)Google Scholar
  15. 15.
    Book, D., Löhner, R.: Simulation and theory of the quatrefoil instability of a converging cylindrical shock. In: Current Topics in Shock Waves: 17th Proceedings of the International Symposium on Shock Waves and Shock Tubes, Bethlehem, PA, 17–21 July (A91-40576 17-34), pp. 149–154. American Institute of Physics, New York (1989)Google Scholar
  16. 16.
    Bond, C., Hill, D.J., Meiron, D.I., Dimotakis, P.E.: Shock focusing in a planar convergent geometry: experiment and simulation. J. Fluid Mech. 641, 297–333 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Brode, H.L.: Quick estimates of peak overpressure from two simultaneous blast waves. Tech. rep., Tech. Rep. DNA4503T, Defense Nuclear Agency, Aberdeen Proving Ground, MD (1977)Google Scholar
  18. 18.
    Butler, D.: Converging spherical and cylindrical shocks. Report No. 54/54, Burgess Hill, New York (1954)Google Scholar
  19. 19.
    Cass, A.S.: Comparison of first generation (Dornier HM3) and second generation (Medstone STS) lithotriptors: treatment results with 13,864 renal and ureteral calculi. J. Urology. Am. Urological Ass. 153, 588–592 (1995)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Cates, J., Sturtevant, B.: Shock wave focusing using geometrical shock dynamics. Phys. Fluids 9(10), 3058–3068 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Chaudhuri, A., Hadjadj, A., Sadot, O., Ben-Dor, G.: Numerical study of shock-wave mitigation through matrices of solid obstacles. Shock Waves 23, 91–101 (2013)CrossRefGoogle Scholar
  22. 22.
    Chessire, G., Henshaw, W.D.: Composite overlapping meshes for solution of partial differential equations. J. Comput. Phys. 1, 1 (1990)Google Scholar
  23. 23.
    Chester W.: The propagation of shock waves in a channel of non-uniform width. Quart. J. Mech. Appl. Math. 6(4), 440–452 (1953)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Chester, W.: The quasi-cylindrical shock tube. Philos. Mag. 45, 1239–1301 (1954)Google Scholar
  25. 25.
    Chisnell, R.F.: The normal motion of shock wave through a non-uniform one-dimensional medium. Proc. R. Soc. A 232, 350–370 (1955)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Chisnell R.F.: The motion of a shock wave in a channel, with applications to cylindrical and spherical shock waves. J. Fluid Mech. 2(3), 286–298 (1957)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Chisnell, R.F.: An analytic description of converging shock waves. J. Fluid Mech. 354, 357–375 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Christopher, T.: Modeling the Dornier HM3 lithotripter. J. Acoust. Soc. Am. 96(5), 3088–3095 (1994)CrossRefGoogle Scholar
  29. 29.
    Christopher, P.T., Parker, K.J.: New approaches to nonlinear diffractive field propagation. J. Acoust. Soc. Am. 90(5), 488–499 (1991)CrossRefGoogle Scholar
  30. 30.
    Cocchi, J.P.,Saurel, R.,Loraud, J.C.: Treatment of interface problems with Godunov-type schemes. Shock Waves 65, 347–357 (1996)zbMATHCrossRefGoogle Scholar
  31. 31.
    Coleman, A.J., Saunders, J.E.: A survey of the acoustic output of commercial extracorporeal shockwave lithotripters. Ultasound Med. Biol. 15, 213–227 (1989)CrossRefGoogle Scholar
  32. 32.
    Davitt, K., Arvengas, A., Caupin, F.: Water at the cavitation limit: density of the metastable liquid and size of the critical bubble. Europhys. Lett. 90, 16002 (2010)CrossRefGoogle Scholar
  33. 33.
    De Neef, T., Hechtman, C.: Numerical study of the flow due to a cylindrical implosion. Comput. Fluids 6, 185–202 (1978)Google Scholar
  34. 34.
    Demmig, F., Hemmsoth, H.H.: Model computation of converging cylindrical shock waves – initial configurations, propagation, and reflection. In: Current Topics in Shock Waves: 17th Proceedings of the International Symposium on Shock Waves and Shock Tubes, Bethlehem, PA, 17–21 July (A91-40576 17-34), pp. 155–160. American Institute of Physics, New York (1989)Google Scholar
  35. 35.
    Dennen, R.S., Wilson, L.N.: Electrical generation of imploding shock waves. In: Exploding Wires, pp. 145–157. Plenum Press, New York (1962)CrossRefGoogle Scholar
  36. 36.
    Dimotakis, P.E., Samtaney, R.: Planar shock cylindrical focusing by a perfect-gas lens. Phys. Fluids 18, 031705 (2006)CrossRefGoogle Scholar
  37. 37.
    Dumitrescu, L.Z.: On efficient shock-focusing configurations. In: Proceedings, 11th Australian Fluid Mechanics Conference, University of Tasmania, Hobart, Australia (1992)Google Scholar
  38. 38.
    Eliasson, V., Gross, J.: Experimental investigation of shock wave amplification using multiple munitions. In: Ben-Dor, G., et al. (eds.) 30th International Symposium on Shock Waves 2, pp. 1017–1021 (2017)CrossRefGoogle Scholar
  39. 39.
    Eliasson, V., Apazidis, N., Tillmark, N., Lesser, M.B.: Focusing of strong shocks in an annular shock tube. Shock Waves 15, 205–217 (2006)CrossRefGoogle Scholar
  40. 40.
    Eliasson, V., Apazidis, N., Tillmark, N.: Controlling the form of strong converging shocks by means of disturbances. Shock Waves 17, 29–42 (2007)CrossRefGoogle Scholar
  41. 41.
    Eliasson, V., Tillmark, N., Szeri, A.J., Apazidis, N.: Light emission during shock focusing in air and argon. Phys. Fluids 19, 106106 (2007)Google Scholar
  42. 42.
    Eliasson, V., Kjellander, M., Apazidis, N.: Regular versus Mach reflection for converging polygonal shocks. Shock Waves 17, 43–50 (2007)CrossRefGoogle Scholar
  43. 43.
    Eliasson, V., Mello, M., Rosakis, A.J., Dimotakis, P.E.: Experimental investigation of converging shocks in water with various confinement materials. Shock Waves 20, 395–408 (2010)CrossRefGoogle Scholar
  44. 44.
    El Mekki-Azouzi, M., Ramboz, C., Lenain, J.-F., Caupin, F.: A coherent picture of water at extreme negative pressure. Nat. Phys. 9, 38–41 (2013)CrossRefGoogle Scholar
  45. 45.
    Evans, A.K.: Instability of converging shock waves and sonoluminescence. Phys. Fluids 22(3), 5004–5011 (1996)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Fisher, J.C.: The fracture of liquids. J. Appl. Phys. 19, 1062–1067 (1948)CrossRefGoogle Scholar
  47. 47.
    Fong, K., Ahlborn, B.: Stability of converging shock waves. Phys. Fluids 22(3), 416–421 (1979)CrossRefGoogle Scholar
  48. 48.
    Fujumoto, Y., Mishkin, E.: Analysis of spherically imploding shocks. Phys. Fluids 21, 1933 (1978)zbMATHCrossRefGoogle Scholar
  49. 49.
    Gardner, G.H., Book, D.L., Bernstein I.B.: Stability of imploding shocks in the CCW approximation. J. Fluid Mech. 114, 41–58 (1982)zbMATHCrossRefGoogle Scholar
  50. 50.
    Glass, I.I.: Shock Waves and Man. University of Toronto Institute for Aerospace Studies, Toronto (1974)Google Scholar
  51. 51.
    Godunov, S.K.: A difference scheme for numerical solution of discontinuous solution of hydrodynamic equations. Math. Sbornik 47, 271–306 (1959)Google Scholar
  52. 52.
    Guderley, G.: Starke kugelige und zylindrische Verdichtungsstöße in der Nähe des Kugelmittelpunktes bzw. der Zylinderachse. Luftfahrt Forsch. 19, 302–312 (1942)Google Scholar
  53. 53.
    Gustafsson, G.: Focusing of weak shock waves in a slightly elliptical cavity. J. Sound Vib. 116(1), 137–148 (1987)zbMATHCrossRefGoogle Scholar
  54. 54.
    Hafner, P.: Strong converging shock waves near the center of convergence: a power series solution. J. Appl. Math. 48, 1244 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  55. 55.
    Hamilton, M.F.: Transient axial solution for the reflection of a spherical wave from a concave ellipsoidal mirror. J. Acoust. Soc. Am. 93(3), 1256–1266 (1993)CrossRefGoogle Scholar
  56. 56.
    Henshaw, W.D., Smyth, N.F., Schwendeman, D.W.: Numerical shock propagation using geometrical shock dynamics. J. Fluid Mech. 171, 519–545 (1986)zbMATHCrossRefGoogle Scholar
  57. 57.
    Hikida, S., Needham, C.E.: Low amplitude multiple burst (lamb) model. Tech. rep., S-cubed Final Report, S-CUBED-R-81-5067 (1981)Google Scholar
  58. 58.
    Hornung, H.G., Pullin, D.I., Ponchaut, N.F.: On the question of universality of imploding shock waves. Acta Mech. 201, 31–35 (2008)zbMATHCrossRefGoogle Scholar
  59. 59.
    Hosseini, S.H.R., Takayama, K.: Implosion from a spherical shock wave reflected from a spherical wall. J. Fluid Mech. 530, 223–239 (2005)zbMATHCrossRefGoogle Scholar
  60. 60.
    Johansson, B., Apazidis, N., Lesser M.B.: On shock waves in a confined reflector. Wear 233–235, 79–85 (1999)CrossRefGoogle Scholar
  61. 61.
    Johnsen, E., Colonius, T.: Shock-induced collapse of a gas bubble in shockwave lithotripsy. J. Acoust. Soc. Am. 124(4), 2011–2020 (2008)CrossRefGoogle Scholar
  62. 62.
    Johnsen, E., Colonius, T.: Numerical simulations of non-spherical bubble collapse. J. Fluid Mech. 629, 231–262 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  63. 63.
    Kandula, M., Freeman, R.: On the interaction and coalescence of spherical blast waves. Shock Waves 18, 21–33 (2008)zbMATHCrossRefGoogle Scholar
  64. 64.
    Keefer, J.H., Reisler, R.E.: Simultaneous and non-simultaneous multiple detonations. In: Proceeding of the 14th International Symposium on Shock Waves and Shock Tubes, New South Wales, Australia, pp. 543–552 (1984)Google Scholar
  65. 65.
    Kjellander, M., Tillmark, N., Apazidis, N.: Thermal radiation from a converging shock implosion. Phys. Fluids 22, 046102 (2010)zbMATHCrossRefGoogle Scholar
  66. 66.
    Kjellander, M., Tillmark, N., Apazidis, N.: Shock dynamics of strong imploding cylindrical and spherical shock waves with real gas effects. Phys. Fluids 22, 116102 (2010)zbMATHCrossRefGoogle Scholar
  67. 67.
    Kjellander, M., Tillmark, N., Apazidis, N.: Experimental determination of self-similarity constant for converging cylindrical shocks. Phys. Fluids 23(11), 116103 (2011)CrossRefGoogle Scholar
  68. 68.
    Kjellander, M., Tillmark, N., Apazidis, N.: Energy concentration by spherical converging shocks generated in a shock tube. Phys. Fluids 24, 126103 (2012)CrossRefGoogle Scholar
  69. 69.
    Kleine, H.: Time resolved shadowgraphs of focusing cylindrical shock waves. Study treatise at the Stoßenwellenlabor, RWTH Achen, FRG (1985)Google Scholar
  70. 70.
    Knystautas, R., Lee, B., Lee, J.: Diagnostic experiments on converging detonations. Phys. Fluids. Suppl. 1, 165–168 (1969)Google Scholar
  71. 71.
    Kozlov, M., Gurovich, V.T., Krasik, Y.E.: Stability of imploding shocks generated by underwater electrical explosion of cylindrical wire array. Phys. Plasmas 20, 112107 (2013)CrossRefGoogle Scholar
  72. 72.
    Lazarus, R.: Self-similar solutions for converging shocks and collapsing cavities. SIAM J. Numer. Anal. 18, 316 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  73. 73.
    Lazarus, R., Richtmyer, R.: Similarity Solutions for Converging Shocks. Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM (1977)Google Scholar
  74. 74.
    Liverts, M., Apazidis, N.: Limiting temperatures of spherical shock wave implosion. Phys. Rev. Lett. 116, 014501 (2016)Google Scholar
  75. 75.
    Matsuo, H., Nakamura, Y.: Experiments on cylindrically converging blast waves. J. Appl. Phys. 51, 3126–3129 (1980)Google Scholar
  76. 76.
    Matsuo, H., Nakamura, Y.: Cylindrically converging blast waves in air. J. Appl. Phys. 52, 4503–4507 (1981)CrossRefGoogle Scholar
  77. 77.
    Matsuo, M., Ebihara, K., Ohya, Y.: Spectroscopic study of cylindrically converging shock waves. J. Appl. Phys. 58(7), 2487–2491 (1985)CrossRefGoogle Scholar
  78. 78.
    McMillen, J.H.: Shock wave pressures in water produced by impact of small spheres. Phys. Rev. 68(9,10),198–210 (1945)CrossRefGoogle Scholar
  79. 79.
    Mishkin, E.A., Fujimoto, Y.: Analysis of a cylindrical imploding shock wave. J. Fluid Mech. 89(1), 61–78 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  80. 80.
    Müller, M.: Comparison of Dornier lithotripters: measurement of shock wave fields and fragmentation effectiveness. Biomed. Tech. 35, 250–262 (1990)Google Scholar
  81. 81.
    Nakamura, Y.: Analysis of self-similar problems of imploding shock waves by method of characteristics. Phys. Fluids 26, 1234 (1983)zbMATHCrossRefGoogle Scholar
  82. 82.
    Neemeh, R.A., Ahmad, Z.: Stability and collapsing mechanism of strong and weak converging cylindrical shock waves subjected to external perturbation. In: Proceeding of the 14th International Symposium on Shock Waves and Shock Tubes, Berkeley, CA, 28 July–2 Aug, pp. 423–430. Stanford University Press, Stanford (1986)Google Scholar
  83. 83.
    Norris, A.N.: Flexural waves on narrow plates. J. Acoust. Soc. Am. 113, 2647–2658 (2003)CrossRefGoogle Scholar
  84. 84.
    Perry, R.W., Kantrowitz, A.: The production and stability of converging shock waves. J. Appl. Phys. 22(7), 878–886 (1951)CrossRefGoogle Scholar
  85. 85.
    Ponchaut, N., Hornung, H.G., Mouton, D.I.: On imploding cylindrical and spherical shock waves in a perfect gas. J. Fluid Mech. 560, 103 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  86. 86.
    Qiu, S., Eliasson, V.: Interaction and coalescence of multiple simultaneous and non-simultaneous blast waves. Shock Waves 26(3), 287–297 (2016)CrossRefGoogle Scholar
  87. 87.
    Qiu, S., Liu, K., Eliasson, V.: Parallel implementation of geometrical shock dynamics for two-dimensional converging shock waves. Comput. Phys. Commun. 207, 186–192 (2016)zbMATHCrossRefGoogle Scholar
  88. 88.
    Ramsey S.D., Kammb J.R., Bolstad J.H.: The Guderley problem revisited. Int. J. Comput. Fluid Dyn. 26(2), 79–99 (2012)MathSciNetCrossRefGoogle Scholar
  89. 89.
    Roberts, D.E., Glass, I.I.: Spectroscopic investigation of combustion-driven spherical implosion waves. Phys. Fluids 14, 1662–1670 1971CrossRefGoogle Scholar
  90. 90.
    Roig, R.A., Glass, I.I.: Spectroscopic study of combustion-driven implosions. Phys. Fluids 20, 1651–1656 (1977)CrossRefGoogle Scholar
  91. 91.
    Saillard, Y., Barbry, H., Mounier, C.: Transformation of a plane uniform shock into cylindrical or spherical uniform shock by wall shaping. In: Proceedings of the XV-th International Symposium on Shack Tubes and Waves. Stanford University Press, Stanford (1985)Google Scholar
  92. 92.
    Saito, T., Glass, I.: Temperature measurements at an implosion focus. Proc. R. Soc. Lond. A 384, 217–231 (1982)CrossRefGoogle Scholar
  93. 93.
    Sankin, G.N., Zhou, Y., Zhong, P.: Focusing of shock waves induced by optical breakdown in water. J. Acoust. Soc. Am. 123(6), 4071–4081 (2008)CrossRefGoogle Scholar
  94. 94.
    Schwendeman, D.W., Whitham, G.B.: On converging shock waves. Proc. R. Soc. Lond. A 413, 297–311 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  95. 95.
    Schwendeman, D.W.: On converging shock waves of spherical and polyhedral form. J. Fluid Mech. 454, 365–386 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  96. 96.
    Sembian, S., Liverts, M., Tillmark, N., Apazidis, N.: Plane shock wave interaction with a cylindrical column. Phys. Fluids 28, 056102 (2016)Google Scholar
  97. 97.
    Sommerfeld, M., Müller, H.M.: Experimental and numerical studies of shock wave focusing in water. Exp. Fluids 6, 209–216 (1988)CrossRefGoogle Scholar
  98. 98.
    Stan, C.A., Willmont, P.R., Stone, H.A., Koglin, J.E., Mengling, L., Aquila, A.L., Robinson, J.S., Gumerlock, K.L., Blaj, G., Sierra, R.G., Boulet, S., Guillet, S.A.H., Curtis, R.H., Vetter, S.L., Loos, H., Turner, J.L., Decker, F.-J.: Negative pressures and spallation in water drops subjected to nanosecond shock waves. Phys. Chem. Lett. 7, 2055–2062 (2016)CrossRefGoogle Scholar
  99. 99.
    Stanyukovich, K.: Unsteady Motion of Continuous Media. Pergamon, Oxford (1960)CrossRefGoogle Scholar
  100. 100.
    Stanyukovich, K.P.: Unsteady Motion of Continuous Media. Pergamon Press, Oxford (1960)CrossRefGoogle Scholar
  101. 101.
    Starkenberg, J.K., Benjamin, K.J.: Predicting coalescence of blast waves from sequentially exploding ammunition stacks. Tech. rep., Army Research Lab Report ARL-TR-645 (1994)Google Scholar
  102. 102.
    Sturtevant, B., Kulkarny, V.A.: The focusing of weak shock waves. J. Fluid Mech. 73(04), 651–671 (1976)CrossRefGoogle Scholar
  103. 103.
    Sun, M., Takayama, K.: An artificially upstream flux vector splitting scheme for the Euler equations. J. Comput. Phys. 189(1), 305–329 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  104. 104.
    Takayama, K., Onodera, O., Hoshizawa, Y.: Experiments on the Stability of Converging Cylindrical Shock Waves. Shock Waves Marseille IV, pp. 117–127. Springer, Berlin (1984)Google Scholar
  105. 105.
    Takayama, K., Kleine, H., Grönig, H.: An experimental investigation of the stability of converging cylindrical shock waves in air. Exp. Fluids 5, 315–322 (1987)Google Scholar
  106. 106.
    Taylor, G.: The formation of a blast wave by a very intense explosion. I. Theoretical discussion. Proc. R. Soc. Lond. A Math. Phys. Sci. 201, 159–174 (1950)zbMATHCrossRefGoogle Scholar
  107. 107.
    Trevena, D.H.: Cavitation an generation tension in liquid. J. Phys. D: Appl. Phys. 17, 2139–2164 (1984)CrossRefGoogle Scholar
  108. 108.
    Van Dyke, M., Guttman, A.: The converging shock wave from a spherical or cylindrical piston. J. Fluid Mech. 120, 451 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  109. 109.
    Wan, Q., Eliasson, V.: Numerical study of shock wave attenuation in two-dimensional ducts using solid obstacles – How to utilize shock focusing techniques to attenuate shock waves. Aerospace 2, 203–221 (2015)CrossRefGoogle Scholar
  110. 110.
    Wang, C., Eliasson, V.: Shock wave focusing in water inside convergent structures. Int. J. Multiphys. 6, 267–282 (2012)CrossRefGoogle Scholar
  111. 111.
    Wang, C., Qiu, S., Eliasson, V.: Quantitative pressure measurement of shock waves in water using a schlieren-based visualization technique. Exp. Tech. (2013).
  112. 112.
    Wang, C., Qiu, S., Eliasson, V.: Investigation of shock wave focusing in water in a logarithmic spiral duct, part 1: Weak coupling. Ocean Eng. 102, 174–184 (2014). CrossRefGoogle Scholar
  113. 113.
    Wang, C., Grunenfelder, L., Patwardhan, R., Qiu, S., Eliasson, V.: Investigation of shock wave focusing in water in a logarithmic spiral duct, part 2: strong coupling. Ocean Eng. 102, 185–196 (2015)CrossRefGoogle Scholar
  114. 114.
    Watanabe, M., Takayama, K.: Stability of converging cylindrical shock waves. Shock Waves 1, 149–160 (1991)CrossRefGoogle Scholar
  115. 115.
    Watanabe, M., Onodera, O., Takayama, K.: Shock wave focusing in a vertical annular shock tube. Theor. Appl. Mech. 32, 99–104 (1995)Google Scholar
  116. 116.
    Welsh, R.L.: Imploding shocks and detonations. J. Fluid Mech. 29, 61–79 (1967)zbMATHCrossRefGoogle Scholar
  117. 117.
    Whitham, G.B.: A new approach to problems of shock dynamics Part I Two-dimensional problems. J. Fluid Mech. 2, 145–171 (1957)MathSciNetzbMATHCrossRefGoogle Scholar
  118. 118.
    Whitham, G.B.: A new approach to problems of shock dynamics Part II Two-dimensional problems. J. Fluid Mech. 5, 369–386 (1957)MathSciNetzbMATHCrossRefGoogle Scholar
  119. 119.
    Whitham, G.B.: A note on shock dynamics relative to a moving frame. J. Fluid Mech. 31, 449–453 (1968)zbMATHCrossRefGoogle Scholar
  120. 120.
    Whitham, G.: Linear and Nonlinear Waves. Wiley, New York (1974)zbMATHGoogle Scholar
  121. 121.
    Wilson, D.A., Hoyt, J.W., McKune, J.W.: Measurement of tensile strength of liquids by an explosion technique. Nature 253, 723–725 (1975)CrossRefGoogle Scholar
  122. 122.
    Wu, J., Neemeh, R., Ostrowski, P.: Experiments on the stability of converging cylindrical shock waves. AIAA J. 19, 257–258 (1981)CrossRefGoogle Scholar
  123. 123.
    Zel’dovich, Y.B., Raizer, Y.P.: Physics of shock waves and high-temperature hydrodynamic phenomena. Dover Publications, New York (1966)Google Scholar
  124. 124.
    Zhai, Z., Liu, C., Qin, F., Yang, J., Luo, X.: Generation of cylindrical converging shock waves based on shock dynamics theory. Phys. Fluids 22, 041701 (2010)zbMATHCrossRefGoogle Scholar
  125. 125.
    Zheng, Q., Durben, D.J., Wolf, G.H., Angel, C.A.: Liquids at large negative pressures: water at the homogeneous nucleation limit. Science 254, 829–832 (1991)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2019

Authors and Affiliations

  • Nicholas Apazidis
    • 1
  • Veronica Eliasson
    • 2
  1. 1.MechanicsKTH-Royal Institute of TechnologyStockholmSweden
  2. 2.University of California, San DiegoLa JollaUSA

Personalised recommendations