Gravity-capillary waves in reduced models for wave-structure interactions

This paper is concerned with steady-state subcritical gravity-capillary waves that are produced by potential flow past a wave-making body. Such flows are characterised by two non-dimensional parameters: the Froude number, and the inverse Bond number,. When the size of the wave-making body is formally small, there are two qualitatively different flow regimes and thus a single bifurcation curve in the plane. If, however, the size of the obstruction is of order one, then, in the limit, Trinh &Chapman (J. Fluid Mech., vol. 724, 2013, pp. 392-424) have shown that the bifurcation curve widens into a band, within which there are four new flow regimes. Here, we use results from exponential asymptotics to show how, in this low-speed limit, the water-wave equations can be asymptotically reduced to a single differential equation, which we solve numerically to confirm one of the new classes of waves. The issue of numerically solving the full set of gravity-capillary equations for potential flow is also discussed.