# Chapman–Jouguet condition

The Chapman–Jouguet condition holds approximately in detonation waves in high explosives. It states that the detonation propagates at a velocity at which the reacting gases just reach sonic velocity (in the frame of the leading shock wave) as the reaction ceases.[1][2]

David Chapman[3] and Émile Jouguet[4] originally (c. 1900) stated the condition for an infinitesimally thin detonation. A physical interpretation of the condition is usually based on the later modelling (c. 1943) by Yakov Borisovich Zel'dovich,[5] John von Neumann,[6] and Werner Döring[7] (the so-called ZND detonation model).

In more detail (in the ZND model) in the frame of the leading shock of the detonation wave, gases enter at supersonic velocity and are compressed through the shock to a high-density, subsonic flow. This sudden change in pressure initiates the chemical (or sometimes, as in steam explosions, physical) energy release. The energy release re-accelerates the flow back to the local speed of sound. It can be shown fairly simply, from the one-dimensional gas equations for steady flow, that the reaction must cease at the sonic ("CJ") plane, or there would be discontinuously large pressure gradients at that point.

The sonic plane forms a "choke point" that enables the lead shock, and reaction zone, to travel at a constant velocity, undisturbed by the expansion of gases in the rarefaction region beyond the CJ plane.

This simple one-dimensional model is quite successful in explaining detonations. However, observations of the structure of real chemical detonations show a complex three-dimensional structure, with parts of the wave traveling faster than average, and others slower.

## References

1. Cooper, Paul W. (1996), Explosives Engineering, New York: Wiley-VCH, ISBN 0-471-18636-8
2. Fickett, Wildon; Davis, William C. (1979), Detonation, Berkeley: U. Calif. Press, ISBN 0-520-03587-9
3. Chapman, D. L. (1899). "VI.On the rate of explosion in gases". Philosophical Magazine Series 5. 47 (284): 90–104. doi:10.1080/14786449908621243.. Also Archive.org
4. Jouguet, Emile (1905), "Sur la propagation des réactions chimiques dans les gaz" [On the propagation of chemical reactions in gases], Journal de Mathématiques Pures et Appliquées, series 6 (in French), 1: 347–425
Jouguet, Emile (1906), "Sur la propagation des réactions chimiques dans les gaz" [On the propagation of chemical reactions in gases], Journal de Mathématiques Pures et Appliquées, series 6 (in French), 2: 5–85
5. Zel'dovich, Yakov Borissovich (1940). "К теории распространения детонации в газообразных системах" [On the theory of the propagation of detonation in gaseous systems]. Journal of Experimental and Theoretical Physics. 10: 542–568. Translated into English in: National Advisory Committee for Aeronautics Technical Memorandum No. 1261 (1950).
6. See:
• Neumann, John von (1942), Theory of detonation waves, Aberdeen Proving Ground, Maryland: Office of Scientific Research and Development, Report No. 549, Ballistic Research Laboratory File No. X-122
• Progress Report to the National Defense Research Committee, Division B, OSRD-549 (April 1, 1942. PB 31090) 34 pages. (4 May 1942).
• von Neumann, John (1963) [1942], Taub, A. J., ed., John von Neumann, Collected Works, 6, Elmsford, N.Y.: Permagon Press, pp. 178218
7. Döring, Werner (1943). "Über Detonationsvorgang in Gasen" [On the detonation process in gases]. Annalen der Physik. 43: 421–436. Bibcode:1943AnP...435..421D. doi:10.1002/andp.19434350605.