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B. L. Weber
and
D. E. Barrick

Abstract

A general hydrodynamic solution is derived for arbitrary gravity-wave fields on the ocean surface by extending Stokes' (1847) original perturbational analysis. The solution to the nonlinear equations of motion is made possible by assuming that the surface height is periodic in both space and time and thus can be described by a Fourier series. The assumption of periodicity does not limit the generality of the result because the series can be made to approach an integral representation by taking arbitrarily large fundamental periods with respect to periods of the dominant ocean waves actually present on the surface. The observation areas and times over which this analysis applies are assumed small, however, compared to the periods required for energy exchange processes; hence an “energy balance” (or steady-state) condition is assumed to exist within the observed space-time intervals. This in turn implies the condition of statistical stationarity of the Fourier height coefficients when one generalizes to a random surface. Part I confines itself to the formulation of a perturbation solution (valid to all orders) for the higher order terms resulting from a two-dimensional arbitrary periodic description of the surface height. The method is demonstrated by deriving (to second order) the height correction to the sea and (to third order) the first nonzero correction to the lowest order gravity-wave dispersion relation.

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D. E. Barrick
and
B. L. Weber

Abstract

In a previous paper (Weber and Barrick, 1977), a generalization of Stokes’ perturbational technique permitted us to obtain solutions to higher orders for gravity-wave parameters for an arbitrary, two-dimensional periodic surface. In particular, the second-order wave-height correction and the third-order dispersion relation correction were derived there. In this paper, we interpret and apply those solutions in a variety of ways. First of all, we interpret the dispersion relation (and its higher order corrections) physically, as they relate to the phase velocity of individual ocean wave trains. Second, the validity of the two results derived previously is established by comparisons in the appropriate limiting cases with classical results available from the literature. It is shown how the solutions—derived for periodic surface profiles—can be generalized to include random wave fields whose average properties are to be specified. Then a number of examples of averaged higher order wave parameters, are given, and in certain cases a Phillips’ one-dimensional wave-height spectral model is employed to yield a quantitative feel for the magnitudes of these higher order effects. Both the derivations and the examples have direct application to the sea echo observed with high-frequency radars, and relationships with the radar observables are established and discussed.

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