Maruo 



the wave resistance becomes 



R = 8 



■n py'^ I 



P2 sec^e Ad 



(70) 



where 



-.00 



aj(x) sin (tx sec 6')dx + 2 sin (y^ sec i9) I ^^(z) exp (-yz sec^^)dz. ('-'■) 



J 1 



Substituting Eqs. (68) and (69) in (71) and carrying out the integration, one 

 obtains 





sin (y sec 



_y sec 



d0 



(72) 



if 



> 77. If there is a relation 



^l/To 



(73) 



the wave resistance vanishes. Though Krein and Bessho have shown that wave- 

 free distribution of sources gives zero linearized volume, the wave-free distri- 

 bution without negative ordinates does exist if the draft is allowed to be infinite. 

 The horizontal distribution corresponds to a half immersed body of revolution 

 with cross sectional area given by the equation 



A(x) = 



2l 



(74) 



The vertical distribution corresponds to a vertical strut of infinite depth, the 

 horizontal section of which is the Rankine oval. The resultant shape is a com- 

 bination of them and is so to speak a yacht shaped ship with infinite vertical 

 keel. As the infinite keel cannot be realized, it must be truncated at a finite 

 depth. The truncation invalidates the perfect cancellation of the waves gener- 

 ated by each system of sources. Figure 13 shows the results of calculation of 

 wave resistance when the vertical keel is truncated at a depth 0.25L and O.IL, 

 when the designed Froude number at which the wave resistance vanishes for the 

 infinite keel is 0.316 or 7^ = 5 . Though the truncation of the vertical keel does 

 not matter much at lower Froude numbers, it weakens the cancellation of the 

 wave at high Froude number especially when the vertical keel is truncated at 

 smaller depth because of the practical requirement. In order to compensate the 

 weakened effect of the vertical keel, the strength of the vertical distribution 

 should be augmented considerably. An investigation has been made so as to find 

 out the vertical distribution which makes the resultant wave resistance mini- 

 mum. According to the result, a remarkable peak appears at the bottom of the 

 vertical source distribution. This fact suggests that the best form has a con- 

 centration of the source at the bottom. Instead of pursuing the best distribution 

 along the vertical lines, a discrete source and a sink are assumed at the depth 



1042 



