  In the study of low-speed or low-Froude flows of a potential gravity-driven
fluid past a wave-generating object, the traditional asymptotic expansion in
powers of the Froude number predicts a waveless free-surface at every order.
This is due to the fact that the waves are, in fact, exponentially small and
beyond-all-orders of the naive expansion. The theory of exponential asymptotics
indicates that such exponentially-small water waves are switched-on across
so-called Stokes lines -- these curves partition the fluid-domain into
wave-free regions and regions with waves. In prior studies, Stokes lines are
associated with singularities in the flow field such as stagnation points, or
corners in submerged objects or rough beds. In this work, we present a smoothed
geometry that was recently highlighted by Pethiyagoda et al. [Int. J. Numer.
Meth. Fluids. 2018; 86:607--624] as capable of producing waves, yet
paradoxically exhibiting no obvious Stokes line insofar as conventional
exponential asymptotics theory. In this work, we demonstrate that the Stokes
line for this smooth geometry originates from an essential singularity at
infinity in the analytic continuation of free-surface quantities. We discuss
some of the difficulties in extending the typical methodology of exponential
asymptotics to general wave-structure interaction problems with smooth
geometries.
