On the Secondary Bifurcation of Three-Dimensional Standing Waves

Abstract

This paper contains an analysis of the complex set of periodic solutions that may occur in a fluid filled vessel of rectangular cross-section. A previous analysis by Verma and Kelley [Phys. Fluids, 5 (1962), pp. 52–56] found only simple eigenvalues for the linearized problem. It is shown herein that at critical values of the vessel aspect ratio double eigenvalues occur. Eight nonlinear solution branches are emitted from these double eigenvalues. The solutions along the various branches are derived and the results displayed graphically. It is shown that irregular waves occur along some of these branches.In an interesting development, Bauer, Keller and Reiss [SIAM Rev., 17 (1975), pp. 101–122], in their analysis of shell buckling, showed that the splitting of multiple eigenvalues may lead to secondary bifurcation. This theory is applied to the nonlinear standing wave problem herein, and it is shown that secondary bifurcation does occur in the neighborhood of the double eigenvalue. A perturbation method is u...