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Development of perturbations in plane shock waves

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Investigation of the stability of plane shock waves as regards nonuniform perturbations was first performed by D'yakov [1]. He obtained criteria for stability, and showed that perturbations grow exponentially with time in the case of instability. Iordanskii [2] has shown that in the case of stability, the perturbations are attenuated according to a power law. However, the stability criteria of [2] do not agree with the results of [1], Kontorovich [3] has explained the cause of the apparent discrepancies, and asserts the correctness of the criteria of [2]. A power law for the attenuation of perturbations has also been obtained in [4,5] under a somewhat different formulation of the boundary conditions.

The Cauchy problem with perturbations is examined in §1 of this paper, results are obtained for cases of practical interest, and the asymptotic behavior is investigated.

In §2 the effect of a low viscosity on the development of perturbations is examined. It is shown that when t→∞ the amplitude of perturbations is attenuated mainly as exp(-αt), where α>0 does not depend on the form of the boundary conditions at the shock wave front. The results of §2 were used in processing the experimental data of [6], which made it possible to determine the viscosity of a number of substances at high pressure.

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  1. 1.

    S. P. D'yakov, “A note on stability of shock waves,” ZhETF, 27, no. 3 (9), 1954.

  2. 2.

    S. V. Iordanskii, “A note on the stability of a plane stationary Shock wave,” PMM, 21, no. 4, 1957.

  3. 3.

    V. M. Kontorovich, “Anote on stability of shock waves,” ZhETF, 33, no. 6(12), 1957.

  4. 4.

    N. C. Freeman, “Theory of stability of plane Shock waves,” Proc. Roy. Soc., A. vol. 228, p. 341, 1955.

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    R. M. Zaidel, “The Shock wave from a slightly distorted piston,” PMM, 24, no. 2, 1960.

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    A. D. Sakharov, R. M. Zaidel, V. N. Mineev, and A. G. Oleinik, “Experimental investigation of the stability of shock waves and of the mechanical properties of a material at high pressures and temperatures,” DAN SSSR, 159, no. 5, 1964.

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    M. A. Lavrent'ev and B. V. Shabat, Methods in the Theory of Functions of a Complex Variable, 2-nd ed. [in Russian], 1958.

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    A. G. Istratov and V. B. Librovich, “The influence of transfer processes on the stability of a plane flame front,” PMM, 30, no. 3, 1966.

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    Ya. B. Zel'dovich, S. B. Kormer, M. V. Sinitsyn, and K. B. Yusko, “Investigation of the optical properties of transparent materials at ultra-high pressure,” DAN SSSR, 138, no. 6, 1961.

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    L. D. Landau and E. M. Lifshits, Mechanics of Continuous Media [in Russian], 1953.

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In conclusion, the author expresses his gratitude to A. D. Sakharov for valuable advice, and to A. G. Oleinik and V. N. Mincer for useful discussions. The author also thanks G. I. Barenblatt, L. A. Galin, and others who took part in a seminar at the Institute for Problems in Mechanics, for their interesting discussion and valuable comments.

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Zaidel, R.M. Development of perturbations in plane shock waves. J Appl Mech Tech Phys 8, 20–25 (1967).

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  • Viscosity
  • Boundary Condition
  • Experimental Data
  • Mathematical Modeling
  • Attenuation