Variational Approaches to Steady Ship Wave Problems 



(a) SUBMERGED 



(b) VERYSUGHTLY SUBMERGED 



(c) SURFACE PIERCING 



Fig. 3. Slightly Submerged Ship 



This formula shows that there may be a thin layer of uniform flow 

 over the top of the submerged body. 



When this layer moves with the body. 



(}>x(x,y,0) = - gC(x,y) = - 1 on F, 



(4.19) 



and we clearly have the case of a surface -piercing body. 



On the other hand, the boundary value problem of a submerged 

 body is equivalent to the variational problem (4,10), After solving 

 this problem, we may calculate the surface elevation over the top 

 water plane by (4,18), but it will differ from (4,19), in general. 

 In this case, it might be necessary to introduce another potential 

 which satisfies condition (4,19), in addition to the above potential. 

 This procedure may not be practical because the treatment of the top 

 water plane is difficult. 



In this case, it would be more convenient to consider the follow- 

 ing two boundary value problems: Let us split the velocity potential 

 into two parts , 



+ <!»,» 



(4.20) 



with boundary conditions 



559 



