SLENDER BODY THEORY FOR AN 

 OSCILLATING SHIP AT FORWARD SPEED 



W. P. A. Joosen 



Netherlands Ship Model Basin 

 Wageningen, Netherlands 



ABSTRACT 



A linearized theory is developed for an oscillating slender body which 

 is moving along a straight line on the free surface of an ideal fluid. 

 Green's function is used to formulate the velocity potential. Some as- 

 sumptions are made about the order of magnitude of the Froude number 

 and the frequency with respect to the slenderness parameter. 



The first order term of the potential is derived by asymptotic expansion. 



INTRODUCTION 



During the past few years several papers have been published on the subject 

 of slender body theory for surface ships. In the slender body theory the beam- 

 length ratio t is supposed to be small with respect to unity and of the same 

 order as the draft-length ratio. This is in contrast with the thin ship theory, 

 where only the beam-length ratio is assumed to be small. The principal task of 

 the theory is to provide the expansion of the velocity potential in terms of the 

 slenderness parameter e. 



Ursell [l] has solved the problem of an oscillating slender body of revolu- 

 tion at zero forward speed for the case of small and moderate frequency param- 

 eter as well as for the case of large frequency. He derived two terms in the 

 series expansion. 



Newman [2] followed another approach, suggested by Vossers [3] starting 

 from Green's theorem. He treated the problem of an oscillating slender body of 

 arbitrary shape at small or moderate frequency in the presence of incoming 

 waves. He derived the first order terms of the velocity potential and of the 

 forces and moments. 



A difficulty arises in the equation of motion for pitch and heave, because it 

 appears that the force due to hydrostatic pressure and the Froude-Krylov force 

 is of lower order than the hydrodynamic forces (added mass and damping). A 

 similar result was obtained already by Peters and Stoker [4] in the thin ship 

 theory. 



167 



