Smith 



When a = 1000, Simpson's rule fails entirely. Unfortunately, when the oscilla- 

 tions are rapid, many short steps must still be taken to obtain reasonable 

 accuracy. 



A THREE-DIMENSIONAL BODY OSCILLATING IN 

 THE PRESENCE OF A FREE SURFACE 



The previous sections have dealt with work considered as completed. Here 

 we consider some work that is in progress. The problem is that of solving the 

 motion of a true three-dimensional body oscillating with small amplitude in the 

 presence of a free surface. It is the key to the solution of very general prob- 

 lems of motion of a body, according to Ogilvie, Ref. 11, whom we quote: 



"11 we can find velocity potentials for the six problems corre- 

 sponding to the sinusoidal oscillations of a ship in calm water, we 

 can evaluate these potentials far away from the ship (effectively at 

 infinity) and from the resulting simplified functions determine some 

 of the damping coefficients. From the same asymptotic fornas of 

 the potentials we can also find the forces on a ship due to sinusoidal 

 incident waves from any direction, without having to solve the prob- 

 lem of determining the diffracted waves around the ship. In both 

 problems we avoid the necessity of integrating the pressure over 

 the ship hull. It is only necessary to integrate over a simplified 

 mathematical surface far away from the ship. Finally, in any case 

 for which we know the damping coefficients we can find the corre- 

 sponding added-mass coefficients." 



Furthermore, if these six problems can be solved, the force and moment on a 

 ship restrained in incident waves can be computed. ' 



A straightforward and very general attack on this problem is to use the 

 basic method described at the beginning of this paper, but to replace the simple, 

 steady 1/r-type of distribution with an oscillating source distribution that will 

 satisfy the linearized free-surface condition. Wehausen and Laitone present 

 equations for this type of source (9), The general approach is consequently 

 unchanged, except that the Fredholm integral equation acquires a new kernel. 

 The six kinds of motion, rolling, heaving, pitching, surging, etc., are all solv- 

 able by the same method. The only difference is in the boundary conditions on 

 the body, which amounts to no more than different numbers in the column ma- 

 trix of Eq. (7). 



The Oscillating Source Potential 



Let an oscillating point source be located at the point whose Cartesian 

 coordinates are a, b, c. The potential of this source at a field point with co- 

 ordinates X, y, z may be written 



1 



— + 



f" e"^" du 



J„ JR^ + (u+y+b)2 ° ^ ^ 



(29) 



356 



