Weakly nonlinear gravity three-dimensional unbounded interfacial waves: perturbation method and variational formulation

Abstract

Weakly non-linear behaviour of interfacial short-crested waves with current is presented in this paper. Two approaches are used to determine analytical solutions. First, a perturbation method was applied to determine the fifth-order solutions. The advantage of this method is that it allows for the determination of the harmonic resonance condition which is one of the major short-crested waves characteristics. The second method is Whitham’s Lagrangian approach. From this method, we obtained a quadratic dispersion equation. In the linear case, we have shown that there is a critical cur- rent beyond which steady wave solutions cannot exist. This critical current is associated with the emer- gence of instability. For the non-linear case, the critical current increases with the wave amplitude as in the two-dimensional case.