Stellar wind bow shocks have been seen in association with a wide variety of stellar objects, from pulsars to young stars. A new solution method is presented for bow shocks in the thin-shell limit, stressing the importance of the conserved momentum within the shell. This method leads to exact analytic solutions to the classical problem of Baranov, Krasnobaev, & Kulikovskii. Simple formulae are given for the shell's shape, mass column density, and velocity of shocked gas at all points in the shell. These solutions will facilitate detailed comparison between observed sources and bow shock models.