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

New gravity-capillary waves at low speeds. Part 1: Linear geometries