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The present study primarily dwells on the wave propagation in one-dimensional periodic granular dimer (diatomic) chain (ref. Fig. 1, where ε is the mass ratio between beads, α1,2 and μ1,2 are the coefficient of on-site stiffness and damping respectively) mounted on linear elastic foundation (more appropriately called the on-site potential). The interaction between the granular beads is governed by Hertzian interaction law (ref. Eq. (1), where the subscript ‘+’ indicates that only positive values of the terms in the brackets needs to be considered and zero otherwise), which is essentially nonlinear in the absence of applied pre-compression. In the limit of zero pre-compression, the considered system is non-integrable and there are no known analytical methodologies to study wave propagation in such systems. However, recent studies of homogeneous granular chains by Starosvetsky culminated in an interesting analytical method (method of maps) to analyze wave propagation in such systems. In this work we extend the method of maps to investigate the evolution of solitary waves and primary pulses in granular dimers mounted on elastic foundation with and without velocity proportional damping. We propose a methodology based on the multiple time-scale analysis and partition the dynamics of the perturbed dimer chain into slow and fast components. The dynamics of the dimer chain in the limit of large mass mismatch (auxiliary chain) mounted on on-site potential and foundation damping forms the basis for the analysis. A systematic analytical procedure is developed for estimating the slow varying primary pulse response of the beads resulting in a nonlinear map relating the relative displacement amplitudes of two adjacent beads. The methodology is applicable for arbitrary mass ratio between the beads. Several examples (ref. Fig. 2) will be presented to demonstrate the efficacy of the considered method. The amplitude evolution predicted by the described methodology is in good agreement with the numerical simulation of the original system. The present work forms a basis for application of the considered methodology to weakly coupled granular dimers which finds practical significance in designing shock mitigating granular layers.