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The two-dimensional sloshing of a fluid in a horizontal circular cylindrical container and the three-dimensional sloshing of a fluid in a spherical container are considered. The linearized theory of water waves is used to determine the frequencies of free oscillations under gravity of an arbitrary amount of fluid in such tanks. Special coordinate systems are used and the problems are formulated in terms of integral equations which are solved numerically for the eigenvalues. Detailed tables of the sloshing frequencies are presented for a range of fill-depths of the containers.

The numerical method of Longuet-Higgins & Cokelet (1976), for waves on deep water, is extended to account for a horizontal bottom contour, and used to investigate breaking waves in water of finite depth. It is demonstrated that a variety of overturning motions may be generated, ranging from the projection of a small-scale jet at the wave crest (of the type that might initiate a spilling breaker) to large-scale plunging breakers involving a significant portion of the wave. Although there seems to be a continuous transition between these wave types, a remarkable similarity is noticed in the overturning regions of many of the waves.Three high-resolution computations are also discussed. The results are presented in the form of interrelated space-, velocity- and acceleration-plane plots which enable the time evolution of individual fluid particles to be followed. These computations should be found useful for the testing of analytical theories, and may also be applied, for example, to studies of slamming forces on shipping and coastal structures.

Trapped modes in the linearized water-wave problem are free oscillations of an unbounded fluid with a free surface that have finite energy; it has been known for some time that such modes are supported by certain structures when held fixed. This paper investigates the problem of a freelyfloating structure that is able to move in response to the hydrodynamic forces acting upon it and it is shown that trapped modes also exist in this problem. For a freely-floating structure a trapped mode is a coupled oscillation of the fluid and the structure.