c Q * 3z Q g V 3x / \ 3x Q 3z Q y 



+ 3z \3z Q 3x Q 7 3y 



If we pick up terms which are independent of time, we obtain the boundary 

 condition for the steady part of the velocity potential, denoted by U <j> Q 

 as follows. 



8<}> \ 3f 3cf> 3f Q 3(f> 



°^ °+^-^-^=0 (95) 



(»m 



3x / 3x n 3z 3z Q 3y 



Take length n along the outward normal to the hull surface, and designate 

 the direction cosines of the normal as n , n , n . Then the above equation 

 can be written in the form 







3n 



+ n = (96) 



Now let us examine the order of magnitude when the ship is regarded very 



slender. If n is the slope of the hull surface to the longitudinal 



x 

 axis, then its order of magnitude is the slenderness ratio S. We must keep 



in mind the fact that the disturbance velocity potential of a slender body 

 is singular along its longitudinal axis, and the differentiation of it in 

 the direction of the normal changes the order of magnitude by e . This 

 fact can be shown also by adopting so-called strained coordinates which 

 measure lengthwise direction and lateral direction by different scales. 

 This procedure is well known and will not be repeated here. As a conse- 

 quence of this argument, the relation between the order of magnitude of <J> Q 

 and that of 3(J)_/3n is 



3<j) /3n = e" 1 0(<J» ) 



40 



