Developments in Theory of Bulbous Ships 



Krein (1955) proved in a rigorous manner the existence of a lower bound 

 for the Michell's resistance of ships with a given center plane, a given velocity, 

 a given displacement, and a given vertical distribution of volume. However he 

 concludes that the lower bound of the wave resistance due to a submerged ship 

 is obtained only with generalized functions (i.e., linear combinations of Dirac 

 delta functions) of a ship hull shape; and for floating bodies the wave resistance 

 achieves a lower bound but only for functions of hull shapes having integrable 

 singularities at the ends of the ship. 



In the Michell's ship hull representation (1), it is easy to see that the hull 

 shape f (x, z) is proportional to the doublet strength distributed on a given center 

 plane of the ship. Therefore, if we consider the body streamlines due to the 

 doublet distribution in the uniform stream instead of considering fCx, z) as a 

 hull shape, we may be readily convinced that the ship form of minimum wave 

 resistance has a bulbous bow. In addition, it is worthwhile to note here that, the 

 Dirac delta function of the distributed doublet at the bow is the concentrated line 

 doublet, and the integrable singularity of the doublet distribution at the bow may 

 also be interpreted as a doublet concentrated around the bow. 



ELEMENTARY WAVES AND THE WAVE 

 RESISTANCE FORMULA 



By Lord Kelvin (1887), it was found that the surface wave due to a point dis- 

 turbance in a uniform stream consists of two parts: the local disturbance which 

 is limited to the neighborhood of the point disturbance and the regular wave which 

 propagates far aft of the point, mainly restricted to the sector of \e\ < 19°30' . 

 This is a mathematical solution of the equation for the potential ^ perturbed by 

 the disturbance, 



v'0 = (2) 



with linear boundary conditions at the mean free surface z = o , considering the 

 wave height is small compared to the wave ler^th, 



^+ k M.o (3) 



Bx2 ° 3z 



where k^ = g,/v2 (g = acceleration of gravity) and at x ^-co and z ^-oo^ 



Vc;6 = . (4) 



Now it is well known that a point source of strength m located at a point 

 (xj,o,-z J, where z^ >o, produces a regular wave height ^ at a large x 



t, '^ 4k^ m exp(-k^Zj sec^^) sec^^ 



- 77/ 2 



X cos [k J sec^i^ {(x - Xj) cos 6 + y sin e}] dd (5) 



1069 



