Ogilvie 



be completed by having its reflection added to it, but the reflected half -body 

 must move oppositely to the real body. The same situation obtains for sway 

 and yaw. 



The condition (8a) is the appropriate free surface condition for problems of 

 oscillations at high frequencies. Flows under such conditions are characterized 

 by having no horizontal component of velocity at the undisturbed free surface. 

 A more important property of the ^^ -flows is the fact that they represent the 

 instantaneous fluid response to the motion of the body. If the body is moving and 

 then suddenly stops, the entire fluid motion associated with the sA^ potentials 

 stops. 



As was suggested earlier, the integral terms of the proposed form of solu- 

 tion represent the effects of the free surface. For example, let the ship be at 

 rest until, at t = , it moves impulsively with a large velocity in the k-th mode 

 for a short time. We may idealize this situation by setting 



°-u,(^) = S(t) , the Dirac function. 



For all t >0, 



f 



(l)(x,t) = S(t) ^^(x) + y^(x, t-r) S(T) dr 



= S(t) 4j^(x) + X^^(x,t) . 



This result shows that, for t > 0, v^Cx, t) is just the velocity potential of the 

 motion which results from the impulse of body velocity at t = 0. Furthermore, 

 y^ satisfies the ordinary free smrface condition, (9a), and a homogeneous Neu- 

 mann condition on the body, (9b). Thus v,^(x, t) represents the dispersion of 

 waves caused by the impulse, and this dispersion takes place in the presence of 

 the vinmoving ship hull. 



The potentials for the instantaneous response, v^^Cx), provide initial condi- 

 tions on the potentials which describe the later motion, v^^Cx, t). If we set 

 °^k(^t) - S(t) , the fluid particles which initially made up the free surface, X3 = o, 

 are given a vertical displacement, 



J - 00 ^ 



dt 



x, = 



B0k 

 Bx, 



In the linearized theory of free surface waves, the surface elevation is given by 



g Bt 



and, at t = 0+, this quantity must equal the surface elevation due to the impulse. 

 This, in fact, is the meaning of Eq. (9c). 



28 



