  When traditional linearised theory is used to study gravity-capillary waves
produced by flow past an obstruction, the geometry of the object is assumed to
be small in one or several of its dimensions. In order to preserve the
nonlinear nature of the obstruction, asymptotic expansions in the low-Froude or
low-Bond number limits can be derived, but here, the solutions invariably
predict a waveless surface at every order. This is because the waves are in
fact, exponentially small, and thus beyond-all-orders of regular asymptotics;
their formation is a consequence of the divergence of the asymptotic series and
the associated Stokes Phenomenon.
  By applying techniques in exponential asymptotics to this problem, we have
discovered the existence of new classes of gravity-capillary waves, from which
the usual linear solutions form but a special case. In this paper, we present
the initial theory for deriving these waves through a study of
gravity-capillary flow over a linearised step; this will be done using two
approaches: in the first, we derive the surface waves using the the standard
method of Fourier transforms; in the second, we derive the same result using
exponential asymptotics. Ultimately, these two methods give the same result,
but conceptually, they offer different insights into the study of the
low-Froude, low-Bond number problem.
