Structure of the instability associated with harmonic resonance of short-crested waves

Harmonic resonance of a short-crested water wave field, one of the three-dimensional water waves, is due to the multiple-like structure of the solutions. In a linear description, the harmonic resonance is due to the resonance between the fundamental of the wave and one of its harmonics propagating at the same phase speed for particular wave parameter regions depending on the depth of the fluid, the wave steepness, and the degree of three-dimensionality. Former studies showed that (i) an Mth-order harmonic resonance is associated with a super-harmonic instability of class I involving a (2M + 2)-mode interaction and (ii) the instability corresponds to one sporadic "bubble" of instability. However, these first-step studies dealt with nonbifurcated solutions and thus incomplete solutions. In the present study, the complete set of short-crested wave solutions was computed numerically. It is found that the structure of the solutions is composed of three branches and a turning point that matches two of them. Then, the stability of the bifurcating solutions along their branches was computed. Of interest, another bubble of instability was discovered that is very close to the turning point of the solutions, and the instabilities are no longer sporadic. Harmonic resonances of short-crested water waves are thus associated with two bubbles of instability; the first one is located in a branch in which the turning point is present, and the second one is located in another branch that is continuous in the vicinity of the bifurcation point.