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We present the exact small-amplitude linear Laplace-transform theory describing the propagation of an initially planar detonation front through a gaseous mixture with nonuniform density perturbations, complementing earlier normal-mode results for nonuniform velocity perturbations. The investigation considers the fast-reaction limit in which the detonation thickness is much smaller than the size of the density perturbations, so that the detonation can be treated as an infinitesimally thin front with associated jump conditions given by the Rankine-Hugoniot equations. The analytical development gives the exact transient evolution of the detonation front and the associated disturbance patterns generated behind for a single-mode density field, including explicit expressions for the distributions of density, pressure, and velocity. The results are then used in a Fourier analysis of the detonation interaction with two-dimensional and three-dimensional isotropic density fields to provide integral formulas for the kinetic energy, enstrophy, and density amplification. Dependencies of the solution on the heat-release parameter and propagation Mach number are discussed, along with differences and similarities with results of previous analyses for non-reacting shock waves. (C) 2013 AIP Publishing LLC.