  An analog to the equations of compressible flow that is based on the inviscid
Burgers equation is utilized to investigate the effect of spatial discreteness
of energy release on the propagation of a detonation wave. While the
traditional Chapman-Jouguet (CJ) treatment of a detonation wave assumes that
the energy release of the medium is homogeneous through space, the system
examined here consists of sources represented by $\delta$-functions embedded in
an otherwise inert medium. The sources are triggered by the passage of the
leading shock wave following a delay that is either of fixed period or randomly
generated. The solution for wave propagation through a large array
($10^3$-$10^4$) of sources in one dimension can be constructed without the use
of a finite difference approximation by tracking the interaction of
sawtooth-profiled waves for which an analytic solution is available. A
detonation-like wave results from the interaction of the shock and rarefaction
waves generated by the sources. The measurement of the average velocity of the
leading shock front for systems of both regular, fixed-period and randomized
sources is found to be in close agreement with the velocity of the equivalent
CJ detonation in a uniform medium wherein the sources have been spatially
homogenized. This result may have implications for the applicability of the CJ
criterion to detonations in highly heterogeneous media (e.g., polycrystalline,
solid explosives) and unstable detonations with a transient and
multidimensional structure (e.g., gaseous detonation waves).
