Abstract
A single equation governing weakly nonlinear gravity waves traveling on deep water has been derived. This equation, which includes fractional derivatives, can describe any wave motion under the influence of second-order nonlinearity. Deriving dynamic wave behavior is much easier from this equation than using the original set of equations. To obtain a better understanding of wave–wave interaction, several solutions were examined under a periodic boundary condition. The initial value problem for one monochromatic wave and its harmonic shows that the monochromatic wave generates the second harmonic in a certain period which travels with nearly constant velocity. A Stokes-like wavetrain then appears periodically, though not asymptotically, as a result of the periodic generation of the harmonic. A new type of three-wave interaction was seen to exist where two waves of larger wave number interact with each other through a long wave of constant amplitude and constant velocity.
Published Version
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