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The tapered-slot (Vivaldi) endfire element is a promising candidate for broadband phased arrays. These arrays often are designed by using estimates from infinite arrays and ignoring edge effects arising from truncation of the infinite periodic array. However, the strong mutual coupling that contributes to wide bandwidth performance often leads to severe truncation effects within the finite Vivaldi array. This study presents a time-domain approach to coupling and truncation phenomena with the help of; (1) full-wave numerical simulation and (2) wave-based physical modeling. First, the full-wave solution provides the terminal currents for all elements in the finite array. Next, the coupling and truncation effects are modeled by waves propagating across the aperture. Time gating of the computed terminal currents enables decomposition of the wave events. The dispersion characteristics of the waves propagating across the aperture can be quantified and edge reflection coefficients can be estimated. This process provides physical insight to better understand the behavior of finite Vivaldi arrays.