Vertical motion of a floating sphere in a sine-wave sea

Abstract

The vertical motion (heave) of a freely floating sphere is studied under the action of incident sine waves. Forced heave is defined as the vertical motion of sphere in still water, while free heave is defined as the vertical motion of sphere in the sine-wave sea. The total velocity potential describing the motion is decomposed into three terms: incident wave potential, diffracted wave potential (potential as if the sphere were fixed in sine-wave sea), and forced heave potential (potential as if the sphere were in forced vertical motion in still water). The forced heave and diffraction problems are solved separately. The linearized equation of motion is then used to effect the synthesis of the free motion via two unknowns: amplitude of vertical displacement and phase difference.Both the radiation and diffraction problems are solved by expansions of non-orthogonal functions (wave-free potentials). These functions are trivial solutions of the Sommerfeld radiation condition in the sense that they attenuate faster than O(r−½) and it is necessary to add multipole terms which have the proper behaviour at infinity. Infinite systems of linear equations are obtained for the unknown expansion coefficients and the unknown source strengths of the multipole. The added mass, damping coefficient and wave-making coefficient in forced heave are studied as well as the force on the fixed sphere. In addition the added mass and damping coefficient in free heave are obtained.The horizontal motion (surge) of the freely floating sphere can be obtained by methods similar to those employed for the vertical motion. The author is presently working on this problem. Surge and heave are not independent motions but can be treated separately because of the symmetry of the sphere.