On the Nonlinear Theory for Gravity Waves on the Ocean's Surface. Part II: Interpretation and Applications

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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.

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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