Oblique wave groups in deep water

Abstract

Oblique wave groups consist of waves whose straight parallel lines of constant phase are oblique to the straight parallel lines of constant phase of the group. Numerical solutions for periodic oblique wave groups with envelopes of permanent shape are calculated from the equations for irrotational three dimensional deep water motion with nonlinear upper free surface conditions. It is shown that some analytical solutions for oblique wave groups calculated from the two (horizontal) dimensional nonlinear Schrodinger equation are in error because they ignore the resonant forcing of certain harmonics in two dimensions. The numerical method casts doubts also on the physical relevance of solutions of the nonlinear Schrodinger equation for which the envelope of the group passes through zero. Particular attention is given to oblique wave groups whose group to wave angle is in the neighbourhood of the critical angle tan⁻¹ (1/√2), corresponding to waves on the boundary wedge of the Kelvin ship-wave pattern.