Steady radiating gravity waves: an exponential asymptotics approach

The radiation of steady gravity surface waves by a uniform stream over locally confined smooth topography is analyzed based on potential flow theory. The linear solution to this classical problem is readily found by Fourier transforms, and the nonlinear response has been studied extensively by numerical methods. Here an asymptotic analysis is made for subcritical flow in the low Froude number limit, where the wavelength of the radiating waves is much shorter than the horizontal extent of the topography. In this regime, the radiating wave amplitude, although formally exponentially small with respect to the Froude number, is determined by a fully nonlinear mechanism even for small topography amplitude. Based on comparisons of asymptotic results with direct numerical computations, it is argued that this mechanism controls the wave response for a broad range of flow conditions, beyond those assumed by the asymptotic analysis, in contrast to linear theory which has very limited validity.