Examples of trapped modes in the presence of freely floating structures

A freely floating motion-trapping structure can be defined as one or more rigid bodies floating on the surface of a fluid which extends to infinity in at least one direction, whose free motion under its natural hydrostatic restoring force is coupled to that of the surrounding fluid in such a way that no waves are radiated to infinity. The resulting local time-harmonic oscillation of the structure and the surrounding fluid is called a motion-trapped mode. Such a structure would, if displaced slightly from its equilibrium position and released, ultimately oscillate indefinitely at the trapped-mode frequency. Previous examples of motion-trapping structures have been devised using an inverse approach in which the shape of pairs of such structures is determined implicitly by sketching certain streamlines. In this paper an alternative direct approach to the construction of motion-trapping structures in the form of a pair of identical floating cylinders of rectangular cross-section in two dimensions is presented. It is also shown that a thick-walled axisymmetric heaving circular cylinder can act as a motion-trapping structure.