Ship Waves and Wave Resistance 



So far, we have not used any considerations regarding the smallness of 

 e = B/L. We should note that G is defined even for 17 = 0, i.e., for P' within the 

 ship's hull, and is well behaved there, if P is not too close to P' . By develop- 

 ment in a Taylor series regarding eF we therefore may infer that 



and 



G; = -eFC;^ + 0(e2) (9a) 



G; = +eFG;^ + 0(e2), (Qb) 



which shows that the factor of ^ in the last integral is small at least of order 

 e 2 . In general we have 



gI + GI = G^(^,0,O + 0(e2) (10a) 



and 



G^ + G^ = G^(^,0,O + 0(e2) . (10b) 



If we now assume an expansion 



cp = ecp^'^ + e2cp(2) + 0(e3) , (11a) 



S = eS(') + e2S(2) +0(e') , (lib) 



inserting into Green's formula and collecting terms of equal order in e, we find 

 that ^'•^^ is just the expression from Michell's theory with §( i) = o and no con- 

 tributions from Sg and S^. 



For examination of ^^ 2) ^g must go into the nature of §< 2)(x, Y). From 

 Ref. 2, page 464, Eq. (10.12), we find with p = const, and replacing 3/a^ by -B/^x 

 and -g by y^ for our nondimensional representation. 



y,4'> . ^i^' = 5<^.(X,V) = [grad ^^"^'^^ 4"[ro4" * tf], ■ 



(12) 



(It should be observed that only the local component of ^^ ^> contributes to the 

 expression brackets in the second term due to structure of G — see the appen- 

 dix.) Sisov's expression corresponding to Eq. (12) is incorrect. 



From the decay of G and its derivatives as 0(X-^) for x- +oo, causing the 

 same mode of decay for ^^ ^ ^ , it may be seen that s(^'> = 0(X' 2) for x -» +co, and 

 this means that -//^ 2) = o(X- ') ahead of the ship and that the contribution of s^ to 

 the Green's formula expression may be neglected. If now we can assume that 

 the potential 



655 



