Minimum Wave Resistance of Surface Pressure Distribution 



For example, the pressure distribution deduced from 



m(x,y) = (b2-y2)(l- x^)'* (18) 



belongs to case 1, and the pressure distribution deduced from 



m(x,y) = (b2-y2)'(l- x)Vl+x)3 (19) 



belongs to case 2. Figures 2 and 3 show these examples. 



When g is very small, the first term of Eq. (11) is dominant, but as g be- 

 comes larger and b smaller, the third term becomes dominant, where Eq. (10) 

 is to be remembered. 



The twin hull ship type for large g (Fig. 2) is especially interesting, for it 

 may be considered as a model of a broad flat stern of a displacement ship (12). 



Finally, the wave resistance R is (6) 



I 



^ / 2 



R = _il ( |F(g sec^ ^, ^)| sec5(9de, (20) 



-77/2 



where 



F(k,0) = p(x,y) exp[-ik(x cos + y sin 6)] dxdy , (21) 



s 



or, interchanging the order of integration, the wave resistance can be written 



R = -^ ||p(x,y) G'(x,y) dxdy , (22) 



s 



where 



G'(x,y) =1 jTp(x',y') P.5[g(x-x'), g(y-y'), 0] dx'dy', (23) 



s 



in which (13) 



P, ,1 f"/2 f-COS (X SeC0) COS^n+lfi"! 2o 



„'" ' (x,y,z)= (-1)" cos(ysec20sin0) . , ., 2n « \e-'^^''^d9. 



P I ^ ' -^ ' [ sin (x sec a) cos 6* J 



(24) 

 This function G' may be called the influence function. 



Putting Eq. (11) into Eq. (20), integrating partially, and making use of Eqs. 

 (12) and (15), it is found that the wave-free distribution has no wave resistance. 



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