Current Progress in the Slender Body Theory 



kernel function independent of hull geometry and of the motions, the calculation 

 of which may be carried out once and for all, this being one of the chief objec- 

 tives of the present theory. For sinusoidal oscillations K is a function of the 

 radian frequency w; for general motions, however, K may be interpreted in the 

 usual sense of control theory as a transfer function. 



Using this splitting up of the potential we may now calculate the hydrody- 

 namic forces on the ship, which will be split up in a similar manner. In partic- 

 ular for sinusoidal oscillations we can define in the conventional manner a 

 matrix of frequency dependent damping, added mass, and exciting force coef- 

 ficients, each of which can be decomposed into "wall" and "free surface" por- 

 tions. In this paper we shall focus attention on the latter half of the problem, 

 although in Part II the classical slender body theory is used to find the "wall" 

 forces for oscillations at zero speed. 



n. THE ZERO-SPEED THEORY 



Motions in Oblique Waves 



We shall outline the general analysis for zero forward speed, constructing 

 the velocity potential from Green's theorem in the manner suggested by Vossers 

 (1962). Further details of the present analysis can be fovind in the recent paper 

 by Newman (1964). 



A slender rigid ship is floating with zero mean velocity in the presence of 

 plane progressive incident waves, of amplitude A and angle of incidence /3 rela- 

 tive to the longitudinal x-axis. The resulting fluid velocity vector can be rep- 

 resented by the gradient of a velocity potential 0(x, y, z) e' '■"'^ , including both 

 the known incident wave potential 



pA 

 4)j(x,y,z) = S_ exp [K(z + ix cos /3 + iy sioyS)] (2.1) 



and the unknown disturbance potential ^g due to the presence of the body. Here 

 o) denotes the circular frequency, g the gravitational acceleration, K = co^/g is 

 the wave number, and the z-axis is positive upwards with z = the plane of the 

 undisturbed free surface. It follows from Green's theorem and the boundary 

 conditions of the problem that the disturbance potential satisfies 



1 r r ^^Tt 7)0 



0B(x,y,z) = - ^ J J px.y, z;^,77, O ^^ - 0bC^''^'^) 3i^ 



dS , (2.2) 



where the integral is over the submerged surface s of the ship, the direction of 

 the normal n is out of the ship, and the Green's fvinction is defined (cf. Wehausen 

 and Laitone, 1961) by the expression 



G = G„ + G, , (2.3) 



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