Linearized Potential Flow Theory for ACVs in a Seaway 



we may now linearize this by setting 



2 

 sin0 = and cos = 1 + 0(0 ) 



bearing in mind that p is of 0(/3), f is of 0(5, /3) and 2 is of 

 0(5 2 ,5/3 ,/3 ,5a ,/3«,a 2 ) . The linearization will certainly be valid for the 

 moments of-<the order we are considering and therefore 



M p = // p s | - { y + ( f- *) r y } y + { x - i - h Q » + ( r - £>r x 1 1 



.jjjfp. [-{y+U-i)r y }*+{« 



+ |(x - x - h Q 0) fy-y M k dxdy 



As the motion is confined to the longitudinal plane we may conclude 

 as before that for this symmetrical motion p , f and f x are even 

 functions of y whilst f v is an odd function so that the integrals of 

 all the terms containing f may be set equal to zero and for the 

 same reason 



// 



p y dxdy = 



and ^o 



p f y dx dy = 



// 



A S °A 



The i and k integrals therefore vanish and we are left only with the 



j component or the pitching moment 



■//>.[ 



m p c : II p s |x-x-h G » + (r-z) r s 



dxdy 



This result is obvious as there cannot be a rolling moment or yawing 

 moment in symmetrical motion in the longitudinal plane normal to the 

 crests of the waves. 



Substituting for p s and for f in terms of 4> and carrying 

 out the correction implied in replacing S by S Q as before, the 

 final result is : 



181 



