Waves and wave resistance of thin bodies moving at low speed: The free-surface nonlinear effect

The linearized theory of free-surface gravity flow past submerged or floating bodies is based on a perturbation expansion of the velocity potential in the slenderness parameter e with the Froude number F kept fixed. It is shown that, although the free-wave amplitude and the associated wave resistance tend to zero as F → 0, the linearized solution is not uniform in this limit: the ratio between the second- and first-order terms becomes unbounded as F → 0 with ε fixed. This non-uniformity (called ‘the second Froude number paradox’ in previous work) is related to the nonlinearity of the free-surface condition. Criteria for uniformity of the thin-body expansion, combining ε and F, are derived for two-dimensional flows. These criteria depend on the shape of the leading (and trailing) edge: as the shape becomes finer the linearized solution becomes valid for smaller F. Uniform first-order approximations for two-dimensional flow past submerged bodies are derived with the aid of the method of co-ordinate straining. The straining leads to an apparent displacement of the most singular points of the body contour (the leading and trailing edges for a smooth shape) and, therefore, to an apparent change in the effective Froude number.