  The standard analytical approach for studying gravity free-surface waves
generated by a moving body often relies upon a linearization of the physical
geometry, where the body is considered asymptotically small in one or several
of its dimensions. In this paper, a methodology that avoids any such
geometrical simplification is presented for the case of flows at low speeds.
The approach is made possible through a reduction of the water-wave equations
to a complex-valued integral equation that can be studied using the method of
steepest descents. The main result is a theory that establishes a
correspondence between a given physical flow geometry, with the topology of the
Riemann surface formed by the steepest descent paths. Then, when a geometrical
feature of the body is modified, a corresponding change to the Riemann surface
is observed, and the resultant effects to the water waves can be derived. This
visual procedure is demonstrated for the case of two-dimensional free-surface
flow past a surface-piercing ship and over an angled step in a channel.
