A new formulation of deterministic and stochastic evolution equations for three-wave interactions involving fully dispersive waves

This paper presents a new and more accurate set of deterministic evolution equations for three-wave interactions involving fully dispersive, weakly nonlinear, irregular, unidirectional waves. The equations are derived directly from the Laplace equation with leading order nonlinearity in the surface boundary conditions. It is demonstrated that previous fully dispersive formulations from the literature have used an inconsistent linear relation between the velocity potential and the surface elevation. As a consequence these formulations 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 formulation, we present improved stochastic evolution equations in terms of the energy spectrum and the bispectrum for unidirectional waves.