Sloshing frequencies of longitudinal modes for a liquid contained in a trough

The sloshing under gravity is considered for a liquid contained in a horizontal cylinder of uniform cross-section and symmetric about a vertical plane parallel to its generators. Much of the published work on this problem has been concerned with twodimensional, transverse oscillations of the fluid. Here, attention is paid to longitudinal modes with variation of the fluid motion along the cylinder. There are two known exact solutions for all modes ; these are for cylinders whose cross-sections are either rectangular or triangular with a vertex semi-angle of in. Numerical solutions are possible for an arbitrary geometry but few calculations are reported in the open literature. In the present 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 obtaining upper and lower bounds to the frequencies of both the lowest symmetric and lowest antisymmetric modes. Comparisons are made with numerical calculations from a boundary element method.