In this paper we examine the history of the studying the dynamic compatibility conditions for gas-dynamic discontinuities, which determine the ratio between values of the gas-dynamic variables before the discontinuity and right behind him. The concepts of a shock wave, shock and the shock polar are introduced. The formation of ideas about the shock waves as a narrow region with abrupt changes in gas-dynamic parameters is shown with a staged scientific studies as an example. The relationship between the physical nature of gas-dynamic discontinuities and the appearance of singularities in solutions of the Euler equations for an ideal gas is shown. Burgers equation, which allows to simulate the shock waves is discussed. The article can serve as a brief introduction to the theory of gasdynamic discontinuities. It proposes the modern idea of gasdynamic discontinuities as the features arising in the solution of hyperbolic partial differential equations.

• Arnold, A. I. (1992) Ordinary differential equations. Berlin: Springer-Verlag. 336p.
• Arnold, V. I. (1988) Geometrical methods in the theory of ordinary differential equations. Berlin: Springer-Verlag. 250p.
• Earnshaw, S. (1858) On the mathematical theory of sound. Proceedings of Royal Society of London. Direct access: https://ia601609.us.archive.org/26/items/jstor-111545/111545.pdf
• Earnshaw, S. (1860) On the mathematical theory of sound. Philosophical Transactions of the Royal Society of London, 150(8), 133–48. 
• Hugoniot, H. (1889) Propagation du mouvement dans les corps. Chapitre V. Sur les discontinuités qui se manifestent dans la propagation du mouvement. Journal de l’École Polytechnique, LVIII, 68–125.
• Karman, T. & Burgers, I. (1939) General aerodynamic theory-perfect fluids. Springer, 173 p.
• Lagrange, J. L. (1788) Mecanique analytique. Paris, PGP, 522 p.
• Poisson, S. D. (1808) Memoire sur la theorie du son. Journal de l’Ecole Polytechnique, 7(14), 319–392.
• Rankine, W. J. (1869) On the thermodynamic theory of waves of finite longitudinal disturbance. Proceedings of the Royal Society of London, 18(15), 80–83.
• Rankine, W. J. (1870) On the thermodynamic theory of waves of finite longitudinal disturbance. Philosophical Magazine, 39(4), 306–309.
• Riemann, B. (1860) Uber die Fortpflanzung ebener Luftwellen von endlicher Schwingweite. Abhandlungen der koniglichen Gesellschaft der Wissenschaften zu Gottingen, 8, 43.
• Stodola, A. (1903) Beitrag zur Stromung von Gasen und D ä mpfen durch Rohre mit veranderlichem Querschnitt. Zeitschrift des Vereins deutcher Ingenieure, 47, 1787–88.
• Stokes, G. G. (1848) On a difficulty in the theory of sound. Philosophical Magazine, 3(33), 349-356
• Uskov, V. N. (1980) Shock waves and their interaction. Leningrad: Leningrad Mechanical Institute press. 88 p.
• Uskov, V. N. (1983) Interference of stationary gasdynamic discontinuities. Supersonic gas jets, 22, 46-53.
• Uskov, V. N. (2000) Optimal one-dimensional shock waves running on gas flow. Proceedings of XV Session of the International School on Models of Continuum Mechanics, 3, 63–78.
• Uskov, V. N. & Chernyshov, M. V. (2014) Extreme shockwave systems in problems of external supersonic aerodynamics. Thermophysics and Aeromechanics, 21(1), 15–30.
• Uskov, V. N. & Mostovykh, P. S. (2006) Extreme properties of an oblique shock wave traveling along the gas flow. Proceedings of Fourth Polyakhov reading: Selected Papers of the International Scientific Conference on Mechanics, 444–354.
• Uskov, V. N. & Mostovykh, P. S. (2010) Interference of stationary and non-stationary shock waves. Shock Waves, 20(2), 119–29.
• Uskov, V. N., Tao, G., Omelchenko, A. V. (2002) On the behavior of gas-dynamic variable at oblique shock. In V.N. Uskov (Eds.), Collection of articles, 179–191.
• Vieille, P. (1899) Sur les discontinuiti é s produites par la det é nte brusque de gaz comprim é s. Comptes Rendus, 27, 1228–1230.
• Zeldovich, Ya. B. (1970) Gravitational instability: An approximate theory for large density perturbations. Astronomy and Astrophysics, 5(1), 84–89.

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