We present an analytical and numerical study of the problem of mechanical impulse propagation through a horizontal alignment of progressively shrinking (tapered) elastic spheres that are placed between two rigid end walls. The studies are confined to cases where initial loading between the spheres is zero (i.e., in the "sonic vacuum" region). The spheres are assumed to interact via the Hertz potential. Force and energy as a function of time for selected grains that comprise the solitary wave are provided and shed light on the system's behavior. Propagation of energy is analytically studied in the hard-sphere approximation and phase diagrams plotting normalized kinetic energy of the smallest grain at the tapered end are developed for various chain lengths and tapering factors. These details are then compared to kinetic energy phase diagrams obtained via extensive dynamical simulations. Our figures indicate that the ratios of the kinetic energies of the smallest to largest grains possess a Gaussian dependence on tapering and an exponential decay when the number of grains increases. The conclusions are independent of system size, thus being applicable to tapered alignments of micron-sized spheres as well as those that are macroscopic and more easily realizable in the laboratory. Results demonstrate the capabililty of these chains to thermalize propagating impulses and thereby act as potential shock absorbing devices. © 2005 The American Physical Society.