Gravitational oscillations in a rotating paraboloidal basin: A classical problem revisited

T.S. Murty, M.I. El-Sabh


The effect of ellipticity of the free surface on the frequencies of axisymmetric normal gravity modes in a rotating shallow layer of liquid contained in a circular cylinder with paraboloidal bottom is investigated. The classical shallow-water theory due to Lord Kelvin for the flat bottom case neglects the curvature of the free surface due to rotation. In 1962 Fultz and Platzman treated the flat-bottom case, taking into account the effect of this ellipticity. The present work is similar to that of Fultz and Platzman, but is more general and treats the flat bottom as a special case. Suitable nondimensional parameters are defined, namely a depth parameter, a rotation parameter, and a frequency parameter. The problem is formulated in terms of cylindrical polar coordinates. It is a Sturm-Liouville problem and the differential operator is self-adjoint. The problem is solved by the Galerkin method, but the solution thus obtained is not in a closed form. The problem is also solved in a closed form by making use of the Legendre functions, but since these functions are not tabulated very extensively the solution involving the Galerkin method is used for computational purposes. A discussion of these Legendre function solutions is given and the results are compared with those of the Galerkin method whenever possible. Poincare treated the gravity modes in a parabola without a cylinder and not taking the ellipticity into account. The writers treated this case including the effect of ellipticity obtained results that differ radically from those in which ellipticity is neglected. In particular, according to the present theory, rotation has not effect on the frequency of the fundamental mode


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