Liquid surface waves in parabolic tanks

Surface waves in cylindrical tanks with parabolic cross section formed by two confocal parabolas are studied. Exact general solutions for the inviscid gravity-capillary waves as function of the parabolic curvatures are presented, and the symmetry of the eigensolutions for the velocity potential modes is also investigated. For a complete characterization of the velocity potential and of the amplitude of the liquid surface, their analytic expressions in terms of hypergeometric functions are given. It is shown that for the particular case of two confocal parabolas with the same curvature, the velocity potential is described by the cylindrical functions. The evolution of the nodal structure for gravity-capillary waves for variable excitation, and in terms of the parabolic tank curvature, is analyzed.