Detonation wave is a shock wave in a reactive medium that is sustained by energy released in chemical reactions that the shock itself triggers and sustains.
Schematically, the detonation wave is shown in the figure here. In the frame of the wave, the fresh mixture enters the shock at speed D, starts burning in the reaction zone until completely converted to products.
It was discovered and experimentally investigated by Abel (1869), Berthelot (1881), Berthelot and Vieille (1883), and Mallard and Le Chatelier (1883).
The first theory of detonations was proposed by Chapman in 1899 (England), Jouguet in 1905 (France), and by Mikhelson in 1890 (Russia). In this theory, it is assumed that chemical reactions take place instantaneously inside the shock - in the schematics above, the reaction-zone length would shrink to zero. The reaction products are assumed to flow at a locally sonic speed relative to the shock, which is called the Chapman-Jouguet condition.
Modern theories of detonation originate from the works of Zel'dovich, von Neumann, and Doering in 1940s. In the ZND theory, chemical reactions behind the shock proceed at a finite rate that depends on the local chemical composition and thermodynamic state. The schematics above corresponds to the ZND model.
It was discovered experimentally in 1926 (spinning detonation) and from the late 1950s to 1960s (pulsating and cellular detonations) that gaseous detonations are unstable and posess a multi-dimensional structure. Theoretical explanations of these complex phenomena rest on the understanding that the underlying reactive Euler equations predict instability of the shock-reaction-zone structure. Pioneering works are those of Zaidel (1961) and most prominently of Erpenbeck (1960s).
My interest in detonation is in building simplified rational models capable of predicting the observed dynamical features of detonation waves. A list of my works on detonation theory and numerical simulation is (for more details, see Theory and Simulation subpages):
D. S. Stewart and A. R. Kasimov, State of Detonation Stability Theory and Its Application to Propulsion, Journal of Propulsion and Power, 22, No. 6, 1230-1244, 2006. (PDF)
D. S. Stewart and A. R. Kasimov, Theory of detonation with an embedded sonic locus, SIAM Journal of Applied Mathematics, 66, No. 2, 384-407, 2005. (PDF)
A. R. Kasimov and D. S. Stewart, Asymptotic theory of evolution and failure of self-sustained detonations, Journal of Fluid Mechanics, 525, 161-192, 2005. (PDF)
A. R. Kasimov and D. S. Stewart, On the dynamics of self-sustained detonations: A numerical study in the shock-attached frame, Physics of Fluids, 16(10), 3566-3578, 2004. (PDF)
A. R. Kasimov and D. S. Stewart, Spinning instability of gaseous detonations, Journal of Fluid Mechanics, 466, 179-203, 2002. (PDF)
A. R. Kasimov and D. S. Stewart, Theory of detonation initiation and comparison with experiment, Report #1035, Theoretical & Applied Mechanics, University of Illinois at Urbana-Champaign, 2004.Champaign, 2004. (PDF)
Good general references on detonation are: