Yim 



7T <r 



C = Re 2 2 a^^ I | 



(k - kg sec^^) 

 k - kg sec^ 8 - ijj. sec 



(2n)!(-l)" 



k" 



. E 



k2"cos2n6' .^0 k"-J(kcos^)2i 



■kf / l^i^o , •'^ 



' " + e 



dkd^ , 



(27) 



where Wj = (x- l) cos ^ + y sin0. 



At the limit m ~* 0, this expresses the disturbance ^^k,, on z = o due to the 

 source distribution 



^ (2n)! V 



k. ;=n 



koi(z- f)"-J-^x^J(-l)J 

 (n- j- 1)! (2j)! 



(28) 



on o<x<l,f<z<oo in addition to the disturbance due to the double model of 

 ship and the doublet distribution (24) and (25) with its negative mirror image in 

 the infinite medium. For n > 2, this is not finite. The local disturbance of the 

 ideal zero regular wave of general ships may not be finite. This is because the 

 ideal bulb for the general body becomes large with depth and becomes infinite at 

 z = -03. Of course, in practice the effect of the bulb at shallow depth is so large 

 that we need not make the bulb so deep and big for approximate wave cancella- 

 tion (2). Besides, for low Froude numbers, which is rather usual, the influence 

 of the large n in Eqs. (23) and (24) is very small, both on the regular waves and 

 the local disturbance. This can be shown by taking one general term of Eq. (23). 

 The total wave height due to a x^", say, is 



TT J^ .1 k( i OJ -z , ) 



2n i2k sec a e 



"0 



a o„x 1 



-■no ^ 



k - kg sec^ 6 - ifj. sec 



dkd^dx 



2n! 



k cos 6)' 



- L 



(ik cos 6) ""^ .ikcos6 

 ,=0 (2n-j)! 



. k(ia; -Zj) 



le 



dkd0 



k - kg sec^ - ifj- sec 6 



If we replace k cos 9 by kk^ sec 0, 





2n ! 



k sec e)' 



^ (ik^k sec 0)2n-i .i^^^ ,^^ 



j = 



(2n- j)! 



. kgk sec 5(ia>Q-Zj) 

 le 



1 1 '■t^ 



k - 1 - - — cos 



kr 



'"■0 



dkd9 , 



(29) 



690 



