Understanding and Prediction of Ship Motions 



cannot solve this equation, but we simplify it for the special case of the slender 

 body, and the resulting equation can be solved. It turns out that the integral 

 equation which must be solved relates to a set of two-dimensional problems 

 again, but this time we obtain an additional part of the solution which explicitly 

 represents interaction effects between sections. 



It seems to be desirable at this point to restrict ourselves to the case of 

 zero forward speed, since it has been worked out in detail and we can be rea- 

 sonably confident of the results. In detail, the approach which follows is that of 

 Newman (1964); the general concept is still Vossers'. 



In order to simplify matters, let us assume immediately that all disturb- 

 ances are sinusoidal in time. For the potential we write Re (g^x) e'^'^*} , and we 

 have similar expressions for all other variables. This is not necessary and 

 perhaps not desirable, but it is certainly convenient. We also stipulate that 0(x) 

 represents the potential for motion -induced diffraction waves, but not for inci- 

 dent waves. 



By Green's theorem, we can write an expression for the potential at any 

 point in the interior of the fluid: 



1 r f ^0(^) 3G(x,cf)l 



<^(-) = -^ l{''('^'i'^^ -^(l^—^]^''- (25) 



G(x,^), a Green's function, is any function which satisfies Laplace's Equation (in 

 three dimensions) except at x = f , where it has the behavior: 



G(x,a = (l/|x-||) + 



The domain of integration, I, must be a closed surface with x in its interior, 

 and ^ is the dummy varialale which ranges over 1. Under these conditions, (25) 

 is a very general equation, and its usefulness for us depends on our selecting 

 G(x,^) in a meaningful way. 



We choose G(x,|) as the potential function of a pulsating source located at 

 ^, the potential satisfying the linearized free surface condition on X3 = and an 

 appropriate radiation condition at infinity. Also, we define the closed surface 

 2 as s^ + Sf + s^, where s^ is the wetted surface of the ship, s^ is the mean free 

 surface, that is, the plane X3 = outside the ship, and s^ is a closing surface 

 far away from the ship, at "infinity." If now we assume that 0(x) satisfies the 

 linearized free surface condition and a radiation condition, the integrals over s^ 

 and s^ vanish, leaving only the integral over S^, in (25). 



We have left a gap in our logic by assuming that we can use the linearized 

 free surface conditions. However, we get away with it in the case of zero for- 

 ward speed, for we actually have two means of supporting the linearization: If 

 there are incident waves, we may assume them small, so that the ship motions 

 will also be small (even if it is a "fat ship"), or we can concentrate on the as- 

 sumption of slenderness of the ship, in which case even finite amplitude mo- 

 tions will produce small amplitude disturbances. It is evident that the linearized 



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