Geodesic
Linear problem of shock wave disturbance analysis. Part~2:~Refraction and reflection of plane waves in the stability case
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 14 (2017), pp. 473-492.

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This part is devoted to the propagation of plane waves in the stability case. First the fact that each post-shock plane wave is accompanied with damped wave, and so the plane waves refraction/reflection quantity characteristics are determined up to the damped waves, is established. The correspondence between angles of incidence and angles of refraction/reflection, i.e. Snell's laws, is obtained. The matrix of generation coefficients as a whole is calculated. Its behaviour for an ideal gas when the pre-shock Mach number tends to infinity, i.e. coefficients' amplification, is investigated. The degree of amplification for different kinds of incident waves is found. Furthermore, numerical calculations of generation coefficients for an ideal gas are performed, in particular, the coefficients' amplification is investigated numerically and the results are found to confirm analytical conclusions. For reflection, all four reflection coefficients are calculated and some of their properties are established. In particular, vanishing of the reflected entropy-vorticity plane waves and mutual suppression of the incident and reflected acoustic plane waves at critical angles of incidence are established. The numerical calculations of reflection coefficients are also performed. A comparison is carried out between the obtained results and the already-known ones. It is found that the known formulas for the refraction/reflection angles and for the generation/reflection coefficients should be specified and corrected. In particular, the existence of so-called abnormal amplification is disproved.
Keywords: shock wave, shock disturbance, entropy-vorticity wave, acoustic wave, incident wave, refraction, transmitted wave, reflection, reflected wave, stability, neutral stability, spontaneous emission
Mots-clés : Fourier transform. 
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     title = {Linear problem of shock wave disturbance analysis. {Part~2:~Refraction} and reflection of plane waves in the stability case},
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E. V. Semenko; T. I. Semenko. Linear problem of shock wave disturbance analysis. Part~2:~Refraction and reflection of plane waves in the stability case. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 14 (2017), pp. 473-492. http://geodesic.mathdoc.fr/item/SEMR_2017_14_a88/

[1] L. D. Landau, E. M. Lifshitz, Course of Theoretical Physics, v. 6, Fluid Mechanics, 2nd edition, Elsevier, 1987 | MR

[2] S. P. D'yakov, “On the stability of shock waves”, Sov. Phys. JETP, 27 (1954), 288–295 | MR | Zbl

[3] S. P. D'yakov, “The interaction of shock waves with small perturbations”, Sov. Phys. JETP, 33 (1957), 948–974 | MR

[4] V. M. Kontorovitch, “Reflection and refraction of sound by shock waves”, Sov. Phys. JETP, 33 (1957), 1527–1528

[5] J. F. McKenzie, K. O. Westphal, “Interaction of linear waves with oblique shock waves”, Phys. Fluids, 11 (1968), 2350–2362 | DOI | Zbl

[6] W. R. Johnson, O. Laporte, “Interaction of Cylindrical Sound Waves with a Stationary Shock Wave”, Phys. Fluids, 1 (1958), 82–94 | DOI | MR | Zbl

[7] G. R. Fowles, G. W. Swan, “Stability of plane shock waves”, Phys. Rev. Lett., 30 (1973), 1023–1025 | DOI

[8] W. K. Van Moorhem, A. R. George, “On the stability of plane shocks”, Journal of Fluid Mechanics, 68 (1975), 97–108 | DOI | Zbl

[9] G. W. Swan, G. R. Fowles, “Shock wave stability”, Phys. Fluids, 18 (1975), 28–35 | DOI | Zbl

[10] G. R. Fowles, “Conditional stability of shock waves — a criterion for detonation”, Phys. Fluids, 19 (1976), 227–238 | DOI | Zbl

[11] G. R. Fowles, “Stimulated and spontaneous emission of acoustic waves from shock fronts”, Phys. Fluids, 24 (1981), 220–227 | DOI | Zbl

[12] M. Mond, I. M. Rutkevich, “Spontaneous acoustic emission from strong ionizing shocks”, Journal of Fluid Mechanics, 275 (1994), 121–146 | DOI | MR | Zbl

[13] J.-C. Robinet, G. Casalis, “Critical interaction of a shock wave with an acoustic wave”, Phys. Fluids, 13 (2001), 1047–1059 | DOI | MR

[14] I. Men'shov, Y. Nakamura, Abnormal amplification of sound waves refracted by an oblique shock wave, JAXA Special Publication, SP-03-002, 2004, 23–28

[15] S. V. Iordanski, “On stability of a plane shock wave”, Journal of Applied Mathematics and Mechanics, 21 (1957), 465–472 | MR

[16] J. J. Erpenbeck, “Stability of Steady-State Equilibrium Detonations”, Phys. Fluids, 5 (1962), 604–614 | DOI | MR

[17] J. J. Erpenbeck, “Stability of Step Shocks”, Phys. Fluids, 5 (1962), 1181–1187 | DOI | MR | Zbl

[18] R. M. Zaidel, “Shock wave from a slightly curved piston”, Journal of Applied Mathematics and Mechanics, 24 (1960), 316–320 | DOI | MR

[19] R. M. Zaidel, “The perturbations propagation in plane shock waves”, Journal of Applied Mechanics and Technical Physics, 4 (1967), 30–39

[20] J. W. Bates, “Initial-value-problem solution for isolated rippled shock fronts in arbitrary fluid media”, Phys. Rev. E, 69 (2004), 056313-1–056313-16 | DOI | MR

[21] J. W. Bates, “Instability of isolated planar shock waves”, Phys. Fluids, 19 (2007), 1–15

[22] A. Tumin, “Initial-value problem for small disturbances in an idealized one-dimensional detonation”, Phys. Fluids, 19 (2007), 106105-1–106105-12 | DOI

[23] G. R. Fowles, A. F. P. Houwing, “Instabilities of shock and detonation waves”, Phys. Fluids, 27 (1984), 1982–1990 | DOI | MR | Zbl

[24] N. M. Kuznetsov, “Contribution to shock-wave stability theory”, Zh. Eksp. Teor. Fiz., 88 (1985), 470–486

[25] N. M. Kuznetsov, “Stability of shock waves”, Usp. Fiz. Nauk, 159 (1989), 493–527 | DOI

[26] J. W. Bates, D. C. Montgomery, “The D'yakov–Kontorovich Instability of Shock Waves in Real Gases”, Phys. Rev. Lett., 84 (2000), 1180–1183 | DOI

[27] A. V. Konyukhov, A. P. Likhachev, V. E. Fortov, S. I. Anisimov, A. M. Oparin, “Stability and ambiguous representation of shock wave discontinuity in thermodynamically nonideal media”, Pis'ma Zh. Eksp. Teor. Fiz., 90 (2009), 28–34

[28] G. F. Carrier, M. Krook, C. E. Pearson, Functions of a complex variable, Hod Books, Ithaka, N. Y., 1983 | MR | Zbl

[29] E. Zauderer, Partial Differential Equations, 3rd edition, Wiley, 2006 | MR | Zbl