Deterministic and stochastic evolution equations for fully dispersive and weakly nonlinear waves

This paper presents a new and more accurate set of deterministic evolution equations for the propagation of fully dispersive, weakly nonlinear, irregu lar, multidirectional waves. The equations are derived directly from the La place equation with leading order nonlinearity in the surface boundary cond itions. It is demonstrated that previous fully dispersive formulations from the literature have used an inconsistent linear relation between the veloc ity potential and the surface elevation. As a consequence these formulation s are accurate only in shallow water, while nonlinear transfer of energy is significantly underestimated for larger wave numbers. In the present work we correct this inconsistency. In addition to the improved deterministic fo rmulation, we present improved stochastic evolution equations in terms of t he energy spectrum and the bispectrum for multidirectional waves. The deter ministic and stochastic formulations are solved numerically for the case of cross shore motion of unidirectional waves and the results are verified ag ainst laboratory data for wave propagation over submerged bars and over a p lane slope. Outside the surf zone the two model predictions are generally i n good agreement with the measurements, and it is found that the accuracy o f e.g., the energy spectrum and of the third-order statistics is considerab ly improved by the new formulations, particularly outside the shallow-water range. (C) 1999 Elsevier Science B.V. All rights reserved.