Lineavized Potential Flow Theory for ACVs in a Seaway 



in the domain occupied by water, i. e. for all Z > in the case of 

 deep water of infinite extent or in the domain 



d (X, Y) > Z > 



in the case of shallow water of infinite extent. 



In the moving co-ordinate system the velocity potential may 

 be written 



<£(X, Y, Z : t) = $(X + xcos a - ysin a , Y + zsina + ycos a , z;t) 



= $(x, y, z;t), say. 



It is then easy to derive from (1-1) the following equations for the 

 transformation of various derivatives between the two systems : 



2 2 



<f) = 4> cos a - <i> sina , 0„, = <t> cos a - 2$ sin cos a + <i> sin a 

 ^X x y XX xx xy yy 



2 2 



<I) = 4> sin a + $ cos a , d) = <j> sin a + 2<f> sina cosa + 4> cos a 

 x y x y YY xx xy yy 



^Z z ^ZZ zz 



(1-3) 



so that 



[v$ (X, Y, Z;t)] 2 = [v<l>(x,y,z;t)] 2 (l -4) 



and 



V 2 $ (X, Y, Z;t) = V 2 *(x,y, z;t) (1-5) 



It follows that <t> is a harmonic function in the (x, y, z) system in 

 the same way as $ is un the (X, Y, Z) system. 



II. 3 Bernoulli's Equation 



The compressibility of water may be ignored even at the high 

 speeds attained by ACVs at present and we may write Bernoulli's 

 law as 



111 



