Ship Form of Minimum Wave Resistance 



Weinblum considered a case in which the function f(x, z) is a product of a func- 

 tion of X only and that of z only, such as 



f(x,z) - X(x) Z(z) (2) 



and called it an elementary ship. The problem now becomes to choose the func- 

 tion X(x) in such a way that the wave resistance becomes minimum when the 

 function Z( z) is given. He published a number of numerical examples. 



The simplest case is the wall-sided ship of infinite draft or the infinite strut 

 for which Z(z) is constant throughout the whole range of the positive z. The 

 condition of the constant displacement is interpreted into the word "constant 

 area of the water plane." When the equation of the waterline is 



y = f(x) . (3) 



Michell's integral for the infinite strut of length 2I moving with a uniform speed 

 U becomes 



4pU^ f d\ f' r clf(x) df(x') 'M , ^ ' (4) 



R = — - — ^zr=r — A — T~T- cos L7X.(x- x )J dxdx ^ ' 



Ji X^y^^TTT J_(. }.i dx dx 



where > = g/u^. 



The area of the water plane is given by the integral 



r 



A^ = 2 I f(x)dx 



(5) 



r 



df(x) 



— ; x dx 



dx 



since f (+ P) = 0. 



The determination of the function f(x) so as to minimize Michell's integral 

 (4) for a fixed value of A^ leads to the equation derived by the theory of calculus 

 of variations, 



d^__ lifV^ COS [>Mx-x')]dx' + kx =0. (6) 



Sretenski [5] concluded that no solution could exist among square -integrable 

 functions, but there is some doubt in his reasoning as was pointed out by 

 Wehausen [6]. Karp, Kotikand Lurye [7] has proved explicitly that the integral 

 equation has really no solution except a trivial case df(x) dx = when k = 0. 

 Being integrated by parts with respect to x and x' remembering that f (±0 = 0, 

 Eq. (4) becomes 



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