Present study concerns the dynamics of a primary pulse propagating through the uncompressed, 2D hexagonally packed granular crystal with spatially varying cross-section and given to onsite perturbation. In this study, the system under consideration is referred to as a fundamental model. We demonstrate that application of the uniform, shock like excitation on the narrow end of granular crystal, leads to the formation of a spatially localized, traveling, primary pulse. At the first stage of propagation, the shape of a spatial distribution of a primary pulse is nearly straight, consequently preserving the main features of incoming pulse. However, along with its propagation, the shape of a spatial, energy distribution slowly deviates from a straight line. Thus, after sufficient number of layers, the moving wave front becomes distributed along the curve, rather than a straight line. We show that a spatial evolution of the strongly localized wave can be efficiently described by a reduced order model comprising the perturbed, purely nonlinear chain of effective particles with the linearly increasing masses and stiffness coefficients. Using the recently developed analytical procedure based on the construction of nonlinear maps and their subsequent homogenization, we derive a closed form, analytical approximation depicting the evolution of the leading edge of the front. Results of the numerical simulations of the reduced order model as well as these of the analytical approximation are found to be in a spectacular agreement with the results of numerical simulations obtained for the full model.
