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We focus on solitary waves generated in arrays of lightly contactingspherical elastic granules by shock forces of steep rise and slow decaydurations, and establish a priori: (i) whether the peak value of the resultingsolitary wave would be greater, equal, or less than the peak value of the inputshock force; (ii) the magnitude of the peak value of the solitary waves; (iii)the magnitude of the linear momentum in each solitary wave; (iv) the magnitudeof the linear momentum added to the remaining granules, if the first granule isejected; and (v) a quantitative estimate of the effect of the granules' radius,density and stiffness on force amplification/mitigation. We have supported theanalytical results by direct numerical simulations.

Universal Relations for Solitary Waves in Granular Crystals under Finite Rise-decay Duration Shocks