SLOSHING FREQUENCIES OF LONGITUDINAL MODES FOR A LIQUID CONTAINED IN A TROUGH

The sloshing under gravity is considered for a liquid contained in a h orizontal cylinder of uniform cross-section and symmetric about a vert ical plane parallel to its generators. Much of the published work on t his problem has been concerned with two-dimensional, transverse oscill ations of the fluid. Here, attention is paid to longitudinal modes wit h variation of the fluid motion along the cylinder. There are two know n exact solutions for all modes; these are for cylinders whose cross-s ections are either rectangular or triangular with a vertex semi-angle of 1/4pi. Numerical solutions are possible for an arbitrary geometry b ut few calculations are reported in the open literature. In the presen t work, some general aspects of the solutions for arbitrary geometries are investigated including the behaviour at low and high frequency of longitudinal modes. Further, simple methods are described for obtaini ng upper and lower bounds to the frequencies of both the lowest symmet ric and lowest antisymmetric modes. Comparisons are made with numerica l calculations from a boundary element method.