Hidden symmetries of parametrically forced waves

Abstract

The dynamics of parametrically excited surface waves in square containers reveal the effects of symmetry at several levels. In addition to the expected square symmetry D₄ admitted by the fluid equations and the boundary conditions, there are hidden translational and rotational symmetries that further constrain the linear and nonlinear behaviour of the fluid. As a result one finds unexpected degeneracies among the linear wave frequencies and unexpected branches of nonlinear solutions in the bifurcation equations for the surface waves. These additional symmetries are not obvious since they are not symmetries of the square container and consequently do not preserve the boundary conditions of the problem. The author can include them in a theoretical analysis by extending the fluid equations of the original problem to larger domains with greater symmetry; in this enlarged problem the previously hidden symmetries now enter in the usual way. Among other prerequisites, this extension depends on the square container having straight sidewalls.