Developments in Theory of Bulbous Ships 



Namely the amplitude function of the elementary waves from the bow for all 

 angle (' in (28) can be made zero by attaching at the bow a concentrated doublet 

 line which extends to infinite depth. Since the deeply submerged part does not 

 influence too much the surface waves (Yim, 1963) this clarifies the mechanism 

 of the bulb and backs up the approach made by Inui (1962). 



SHIPS WITH ZERO BOW WAVE RESISTANCE 



Krein (1955) proved that there is no finite ship which has zero wave resist- 

 ance. Therefore it was essential for the latter to have an infinite doublet line. 

 Nevertheless, ships of zero resistance is not only of academic interest but also 

 gives us a good physical insight and directs us in practical usage. 



Although a doublet is good to cancel positive sine waves, it is not applicable 

 to cosine bow waves. Yim (1963) considered one step higher order singularities 

 than a doublet, which is called a quadrupole. The wave height due to a point 

 quadrupole with the strength \^ (in x direction) at x = o, y = 0, z = -Zj in the uni- 

 form flow V generates the wave heights 



77/ 2 



^q ■^"8^=0^ ^ exp(-k^z^ sec^0) sec^i9 cos (kjX sec 6) 



X cos (kjY sin e sec'^9) d0 (34) 



where 



k = \ /(H^LV) . (35) 



We notice here that (34) consists of cosine elementary waves with the same 

 sign, -k in all direction d. It was found that the cosine waves due to the source 

 distribution (7) of odd power series can be completely eliminated by a distribu- 

 tion of quadrupoles along the semi-infinite line x = o, y = o,-co>z>o with the 

 strength 



n + 2 



^ 2_^ ^n.l-^ 



^ - / b_, -r-J in < z, - -z < 1 



n = 



zL^-^^ ?rr2 ^" ii^ii- 



n = 



(36) 



and 



'n+l 



(-1)"^' k^ (2n+ 1)! 



' 2n + 3 

 (n+l)! kj 



(37) 



A quadrupole itself in a uniform stream does not produce a closed body, but it 

 may, when combined with the doublet line. Therefore these quadrupoles could 



1075 



221-249 O - 66 - 70 



