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On nonlinear interactions in a spectrum of inviscid gravity–capillary surface waves

Using the Fourier components of the velocity potential at the surface in an iterative perturbation theory, the author derives expressions for the modal frequency shifts and modal decay rates in a spectrum of in viscid gravity–capillary surface waves. The expressions differ from those calculated by the perturbation methods of Hasselmann (1962) and Valenzuela & Laing (1972), which are based on expanding the velocity potential about the equilibrium fluid level z = 0. This difference is shown to be due to the failure of the z = 0 perturbation method to converge rapidly enough to produce lowest-order corrections that are smaller than unperturbed quantities. A physical explanation for this failure is given.