Developments in Theory of Bulbous Ships 



3 3 



z=0 



d^drj 



(50) 



where (' is the intersection of the ship surface and the z = o plane. Since the 

 ship beam length ratio B/L = e is considered to be small, in general, (50) is 

 omitted in the first order theory. 



Wehausen (1962) considered a systematic, formal, yet thorough estimation 

 of the order of magnitude in the Green's formula with the exact boundary condi- 

 tions of the potential. For a ship with the draft H as small as the beam b, he 

 estimated I in (50) is O(e^) while the main integral around the ship hull in (48) 

 is 0( c2) , In fact it has been known that the effect of the draft behaves like 

 exp C-CF) where C is a function of Froude number and even for the case K/B = 2, 

 the wave heights was comparable to the case h ^™ (Wigley, 1931). Therefore 

 the above estimation may be true even for the case of an infinite draft ship, and 

 the line integral I, in this case will be the most important contribution to the 

 higher order terms which have been previously neglected. Indeed, in (50), I is 

 the influence of the free surface on the potential. 



However, it is extremely difficult to understand the higher order effect just 

 by the formal estimation of the magnitude and without actual evaluation, since 

 the property of Green's function is very complicated particularly near the free 

 surface. As a simplest case for the evaluation of the line integral, Yim (1964) 

 considered a source distribution 



m = a in < X < a, v = 0, -oa < z < 



O _ _ _f _ 



on the forebody of a semi-infinite wedge shaped strut, 



y = X tan a in < x £ a 



-00 < z < 



y^tancL in < x < '^^ 



-o' 1 z < . 



For 'i or /.- inside the line integral (50) he used the first order solution obtained 

 by Havelock (1932), 



^1 



77 o 



k„ V 



k ^ ( n - ■^ ) 



j ° [H^(t) + 3YJt)] dt - I " [HJt) - Yjt)] dt 



1089 



