Minimum Wave Resistance of Surface Pressure Distribution 

 MINIMUM PROBLEM 



When the total pressure is given, namely, 



||p(x, 



y) dxdy ^ A , (25) 



if the influence function becomes constant over s, that is, if 



G'(x,y) = c > 0, (26) 



then the wave resistance is minimum. 



On the other hand, differentiating partially, G' satisfies the differential 

 equation 



ax" ^ ^ W' ^ ^2 

 assuming the existence of the integral. 



G'(x,y) = 0, 



(27) 



Accordingly, G' may be represented uniquely by some boundary conditions. 

 Hence, the integral equation given by Eq. (26) is regular in the domain, so that 

 it may have a unique solution. 



Since the existence of its derivative is assumed, the integral equation ob- 

 tained by the differentiation of both sides of Eq. (26) may also have a unique so- 

 lution, and this solution must be identically zero, because the right-hand side is 

 zero. 



Hence, the present minimum problem has no definite solution and no mini- 

 mum value exists for the wave resistance (1,14). This fact may mean that its 

 least value will be zero, because it may be possible to reduce the total wave 

 amplitude as small as necessary, adding the longitudinal and transversal distri- 

 bution to each other appropriately. However, the minimum solution exists in 

 elementary cases such as thin ship theory. 



Thus, the problem may be classified as follows: 



1. Twin hull ship type distribution. When the speed is very high and the 

 breadth narrow, this case nearly equals the next. As seen from the preceding 

 section, it is also interesting at low speed, but usually it seems more useful to 

 consider it in combination with case 3 (12), which means case 4. 



2. Slender ship. In this well-known case there exists a unique solution ex- 

 cept for arbitrary wave -free distributions which have no wave resistance but a 

 finite displacement (1,9). 



3. Transversal line distribution. This is the case to be studied in the fol- 

 lowing section (1,4,5,7). 



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