Ship Waves and Wave Resistance 



2^ X-^0 



V^ COS 



V/o(Z+ O 



+ COS V7o(Z-0[ - U"*] cos 



v/o(z+^.; 



V.o(Z- 



+ 2U2V sin V7o(Z+ O 



e-"^"" dV 



4^ lim Re 

 ^ x-»o 



V+ iU^ iVr„(Z+0-Vynlx 



V- iU' 



dV + 



r 



J v = - < 



-UroX+iVyo(Z-0 



dV 



•^17 = - 



■uroixi _ ^-vrolxlj ^^^.^,^ ^iv.^cz.o 

 V - iU^ 



dV 



(A5) 



The second integral term in the last expression is zero for any finite X. The 

 last integral term is 0(X). The first integral term may be written 



4- x-!>0 1J,._ (V+iK„)[V-i(M,+ l)/2] " ^^r 



which shows poles of the integrand for v = i(M^, - 1)2 and V = -i(M^ + 1)2 = -iK^. 

 By shifting the path of integration downward in the complex plane we can make 

 the integral arbitrarily small after splitting off the residuum at v = -iK^; thus 

 we finally get 



277i Res 



(V-iK^[V+ i(M,+ l)/2] VroCZ+ol 1 _K,e 



K,ro(Z+0 



477 



(A7) 



V = -iK4 (V+ iKJ[V-i(M,+ l)/2] 

 To prove statement (b) we start with the representation 



e'"'^"^' COS u [(Y- t;) COS + (Z+ O sin e] dud(9 (A8a) 



•^U=0 •^d = 



l/Fj = 1/(277) 



and 



1/r = 1/(277) 



■ u I X- .f 



COS u [(Y- 77) COS 6) + (Z- O sin 0] dudt^ , (A8b) 



where Tj corresponds to r with ^ under negative sign. Introducing new varia- 

 bles of integration U, V, and U by 



U = u/7o , U = u cos S/y^ , V = u sin i^/y^ , 



(A9) 



671 



