Secondary bifurcation and change of type for three-dimensional standing waves in finite depth

Abstract

The nonlinear periodic free oscillations of irrotational surface waves in a three-dimensional basin with a rectangular cross-section and finite depth are considered. A previous work by Verma & Keller (1962) has analysed the case when the linear natural frequencies are non-commensurate. For particular values of the parameters, however, strong internal resonance occurs (two natural frequencies are equal). Instead of the usual loss of stability and exchange of energy, it is found that the double eigenvalue generates a higher multiplicity of periodic solutions. Eight solution branches are found to be emitted by the double eigenvalues. It is also shown that perturbing the double eigenvalue results in a secondary bifurcation of periodic solutions. The direction of the branches for the multiple and secondary bifurcation changes with the depth. Finally it is shown that the formal solutions obtained are not uniformly valid and an additional expansion in the Boussinesq regime shows that the wave field changes type. One of the solutions in this regime is a field of three-dimensional cnoidal standing waves.