Ship Waves and Wave Resistance 



J. 



47T 



(^zG- 0G^) dS 



^ ^ It 



(^x^" 0G^) 



^ = x. 



f = x. 



dTJ 



4777, 



K^.OGdS 



^^-Xo J, 



(^xG-^G^)d77 , (2) 



where the line integral around the ship's load waterline Lp has to be understood 

 in counterclockwise sense when viewed from above (compare Eq. (19) of Ref. 3). 



The line integral has been thoroughly investigated by Yim (5). We shall 

 find, however, that it is pertinent to merge it with a similar term from the 

 wetted surface s^ , If now we assume the functions /; and i/^j^ uniformly bounded 

 for X < Xp, then, due to the finite size of s^ with conditions F, we may infer that 

 the contribution of s^ becomes insignificant as we let x^ tend to -00. Similarly, 

 if and ^^ ^^^^^ to zero as X->oo, then, due to boundedness of g and g^, the 

 contribution of S^ may be neglected with x^ becoming large. But the contribu- 

 tions of Sg and S^, must be independent of position x^ and X^ inasmuch as the 

 contribution of the defect 6(X,Y) may be neglected. Considering higher order 

 terms, however, we will see that independence from x^ cannot be assumed in 

 general. 



For the first integral in Eq. (2) over the wetted surface s^° , we shall make 

 the assumption that condition E for /' holds even for parts of the hull not included 

 in S^ up to the undisturbed free surface, so that we may substitute 



0^ dS = eF^/y/e- 



F^ + e'F^' ^ 1 dS,° 



(3) 



and observe that 



ds^° = ./I^VJTT^VIVI ds^' 



(4) 



For the second integral over s^° we substitute the actual components of the 

 normal vector to obtain (compare Eq. (18) of Ref. 3) 



(-Fx, ±1, -Fz)M + e^Fx' +e2F 



and thereby have 



^ f 



47T J_ 



0G dS' 



4t7 JJ 



>A k, ±eF(f,0. ^ 



6Fx(G^ + G") 



+ eFz(G^ + G-) - G; + G; 



d^d^ 



(5) 



653 



