Linearized Potential Flow Theory for ACVs in a Seaway 



As the position of the free surface denoted by z = f is un- 

 known we will endeavour to eliminate the derivatives of J" from 

 (4-1) by using (4-2) in order to derive an explicit equation for <t> . 

 Denoting the terms on the right-hand side of (4-2) by R, for the time 

 being, and noting that R is a function of z and t and that its value 

 is to be taken when z = f , the derivatives of f are evaluated and 

 substituted in (4-1) giving the condition 



(^T-V-l-) R + w(yR -xR )+V<t».VR-<i) = 

 dt dx' x y z 



where R is, of course, 



(4-3) 



R = — 



g 



1 2 



<|> - V4> + w (y<j> - x<i> ) + — (V4>) 

 t x x y 2 



and eliminating the derivatives of R, we derive finally, 



(4- . V-l-) 2 *+ 2V$.V(* - V$ ) +^-V<i>. V 

 ^t dx t x 2 



(V*)' 



- g* z = s 



(4-4) 



where S is an algebraic expression of a complicated nature each 

 term of which has, however, o> or o> as a factor. This condition 

 can therefore be simplified when u is set equal to zero and in the 

 case of our coplaner motion in a straight course, 



(It - v ^) 2 * - g* + 2V*. v(* - V4> ) +4-v<J>.v[ (v$) 2 1 = 



dt dx z t x c l J 



(4-5) 



This is the exact free surface condition. We have made no 

 approximations so far, but the condition is only applicable on the 

 actual free surface z = f . Although we have elimanated X from the 

 equation itself, we are still in some difficulty as we do not know the 

 position of the free surface for the application of the condition the- 

 reon. We will therefore attempt to derive a condition, even if it is an 

 approximate one, which can be imposed on the known surface z = 0. 

 This, in effect, is the first stage in the linearization of the problem. 



Assuming that the potential may be continued analytically from 

 the actual free surface z = f to the plane z = 0, we may expand it 

 in the form of Taylor's series 



<|>(x, y, z;t) = 4>(x, y,o;t) + f^— 



dz 



+ 

 z = 



125 



