Bow and Stern Waves and Snna 11 -Wave -Ship Singularities 



This indicates that on the free surface the x component of the perturbation ve- 

 locity is exactly the same as that due to the double model of the ship in the infi- 

 nite medium. The flow as a whole is the same as that due to the ship and bulb 

 plus the positive mirror image of the ship and the negative mirror image of 

 bulb with respect to the mean free surface (Fig. 1). 



In other words, for this special case of the uniform source distribution, if 

 we combine such a doublet distribution and the opposite sign of doublet images 

 and the same sign of source images, the free surface condition is exactly satis- 

 fied without any wave images. In the general case of ship shapes, it is not that 

 simple to express the local disturbance. 



For a body symmetric with respect to its midsection, the regular bow wave 

 is the same as the stern wave and the local disturbance is also symmetric with 

 respect to the midsection. 



When the analytic ship source distribution, antisymmetric with respect to 

 midsection, is represented by an even power series 



.(.,)- t a,„ "T (23) 



n = 



in < Xj < 1, Zj=f, y = 0, there exists a doublet distribution 



z:b„(z,-f)" (24) 



on f < Zj <oo, y=0, x=0 and y = , x = l by which the regular ship waves are 

 completely canceled (2) with 



n(2n)!^2n_ (25) 



n! k 



Due to the symmetry of the body with respect to the midsection the coefficients 

 of Eq. (23) have relations 



GO 



y — a, = -(2m)! a„ , m= 0. 1, 2, . . . . (26) 



^ (2n- 2m)! 2" v y 2m \ J 



n = m 



Using Eqs. (4), (8), and (25) we obtain formally the total wave height due to 

 Eqs. (23), (24), and (26): 



i - Re 



n = 





■2(2n)!a,„ 2b n! 



V ' 2n 5 „ n 



sec-* + 



(ik cos 6)' 



-kf kiojg kio), 



-! \^ 1^ i-dkd^ 



k - kg sec^^- ifi. sec 6 



(27) 

 (Cont.) 



689 



