Linearized Potential Flow Theory for ACVs in a Seaway 



(* - v) r +4>r -*$"+£■ =o 



y y z z t 



x x 



also on z = f . 



Eliminating f as before, we derive 



(_ |t * V |^ )2<l> " g *z + 2V4> - V( *t " V *x ) +4 V *' V [ (V4>)2 ] 



t[« 



1 1 <|t ■ V |i ) P s (x ' y) + *x P s (X ' Y) + Vs (X,Y 



x y 



This is the exact free surface condition on z = f and the 

 approximate condition on z = is obtained as before by a Taylor 

 expansion of 4>, giving the final result 



4> -2 V$ + V * -g<*>+ 2V<|>. V(<t> - V$ ) 

 tt xt xx z t x 



1 P s d 2 



+ _ (* . V$ + — ) 3- (* - 2 V* . + V $ - g< 

 gt xPdz x tt xt xx° 



1 



(37 - V3-) p + * p + $ p 



dt dx s x s v s 



x ' y 



+-^— Vp -|_(* "- y* ) + (* 3 ) = 



p g *s dz t x 7 ' 



X 



(4-7) 



IV. 3 Conditions on the Hull Surfaces 



Setting w= in (1-9) the kinematic condition applicable to the 

 wetted hull surfaces is 



(* - V) H +<S> H +4>'H +H =0 

 x x y y z z t 



i.e. V*.VH+(v-Vv-)H = 

 .dt dx 



where the hull surface is given in the moving co-ordinate system by 

 an equation of the form 



H (x, y, z;t) = 



The above condition stipulates that the normal velocity of the hull at 

 each point is equal to that of the contiguous fluid particle, i. e. that 



127 



