Both laser and electron beam-based additive manufacturing of Ti-6Al-4V are under consideration for application to aerospace components, and offer significant increases in efficiency and flexibility compared to conventional manufacturing methods. A critical concern for these processes is the ability to obtain a consistent and desirable microstructure and corresponding mechanical properties of the deposit. To this end, recent work has focused on the development of simulation-based process maps for solidification cooling rate and thermal gradient (the key parameters controlling microstructure) as a function of deposition process variables (beam power and velocity). Process map results have been further plotted on solidification maps to predict trends in grain size and morphology for Ti-6Al-4V. However, these results have been limited to semi-infinite geometries, where a steady-state melt pool exits away from free-edges. The extent to which transient changes in melt pool geometry near free-edges also affect solidification microstructure is still unclear, and is the focus of the current study. Much of the authors' prior work has been based on the steady-state Rosenthal solution for a moving point heat source. In the current study, the Rosenthal solution is modified to include the effects of free-edges. This is accomplished by superposition of two point heat sources symmetrically approaching one another, with the line of symmetry representing the free-edge. The result is an exact solution for the case of temperature-independent properties, which supplements nonlinear finite element modeling currently ongoing. Dimensionless results for melt pool geometry, solidification cooling rate and thermal gradient are determined numerically with MATLAB, and plotted as a function of distance from the free-edge. Results are further plotted on solidification maps to predict trends in grain size and morphology for Ti-6Al-4V. Results suggest that melt pool geometry is more sensitive to free-edges than solidification microstructure, which is an important result for process developers.