The variation of Tropical cyclone azimuthal wind speed (V) with distance from storm center (r) is a fundamental aspect of storm structure that has important implications for risk and damages. The theoretical model of Emanuel (2004), which applies well outside the rainy core of the storm, matches radiatively-driven subsidence and Ekman suction rates at the top of the boundary layer to obtain a nonlinear differential equation for dV/dr. This model is particularly appealing because of its strong physical foundation, but has no known analytic solution for V(r). In this paper, I obtain an analytic solution to V(r) for the Emanuel (2004) outer wind model. Following previous work, I then use this solution to explore properties of merged wind models that combine the outer model with an inner model that applies to the rainy core of a storm.