In this paper, Maruo's formulas of wave drift forces acting on a freely floating body in waves are derived without the momentum theory. Firstly a method to estimate the surface integral of freely floating body is proposed. Then Kashiwagi's 1st and 2nd reciprocity, that is the energy conservation law and directivity of outgoing wave amplitude at far field respectively as for freely floating body in waves proven by Kashiwagi recently, is shown in three dimensions without Green's second identity. Finally reciprocity with relation to wave drift forces is shown.

INTRODUCTION

Some relations as for radiation waves and hydrodynamic forces acting on a floating body in waves are known (Mei, 1989). For example, Haskind-Hanaoka theorem, symmetry of radiation forces between direction and mode, and Bessho-Newman relations (Bessho, 1965; Newman, 1975) etc. They are derived by using Green's second identity systematically. To apply Green's second identity to a pair of functions means to consider the selected function to order of its amplitude square. On the other hand wave drift forces which are of order of wave amplitude square are derived from the momentum theory (Maruo, 1960). By the way, recently Kashiwagi (2006) have proven two relations between Kochin functions of freely floating body in waves in two dimensions by using Green's second identity and equation of motion. The 1st Kashiwagi's reciprocity represents the energy conservation law among two waves and contains the energy theorem used in Maruo's drift force theory. The 2nd Kashiwagi's reciprocity states the directivity of outgoing waves and contains case of well known diffraction problem perfectly. These facts imply that Mauro's drift force formulas and Newman's drift moment formula (1967) may be obtained from applying Green's second identity to a pair of appropriate functions. To consider this, at first a method to evaluate the surface integral of freely floating body which wetted surface oscillates with small amplitudes is proposed, where the time variation of volume integral is approximated by the surface integral at static equilibrium position as well as the transport theorem (Newman, 1977).

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