Ship Form of Minimum Wave Resistance 



^^(•9') Y^ [y^Ccos 61' -cos d)]d9' = (p(e) (H) 



where y = gl/v' . Dorr has shown that 



i: 



^n\ ce^(e',q) Y^[2\fq (cos ^' - cos d)] dd' = ce^(e,q) (12) 



where ce^(6^,q) is the even Mathieu function of the integral order. As ce^ is 

 orthogonal in the interval (0,7t), the functions cr(e) and (p(0) can be expressed by 

 Fourier series of ce^. If cp(0) is expressed by 



00 



n = 



the solution of the integral equation becomes 



CO 



Then the optimum dipole distribution f(x) takes the form 



/i'2 2N-1/2 



(-i - x^) a cos 



-t 



Bessho [10] calculated the eigenvalues v^ and showed numerical results for the 

 solution of Eq. (9) at various Froude numbers. The function o-(0) does not van- 

 ish at (9= and -n, so that the singularity in the dipole distribution always ap- 

 pears, but becomes less remarkable at lower Froude numbers. The best form 

 has blunt cylindrical nose and tail, but the radius of the cylinder decreases rap- 

 idly according to the decreasing Froude number. Though the solutions at higher 

 Froude numbers show so to speak dog-bone shapes and are hardly regarded as 

 practical, the shape appears quite plausible at moderate and lower Froude num- 

 bers. It can be noted that negative ordinates which have appeared often at ap- 

 proximate solutions by Pavlenko and others never appear. Therefore the prob- 

 lem to minimize the wave resistance of infinite struts under a single condition 

 of constant sectional area always has a solution, if a slight deviation from 

 Michell's original assumption is allowed. The similar situation holds in the 

 case of elementary ships of finite draft. Though the kernel of the integral equa- 

 tion cannot be expressed by known functions and eigenfunctions which are given 

 in the case of infinite struts are not known, a numerical solution is possible. A 

 few results at Froude number 0.4 were putjlished by Kotik [11]. Weinblum's in- 

 vestigation has assumed not only the condition of constant volume but also other 

 side conditions such as the fixed beam. For elementary ships, the constant 

 beam together with the constant volume means a constant block coefficient. To 

 seek the best form among those of constant block coefficient seems to have 

 greater importance from the practical side because the solution under a single 

 condition of constant volume often presents a ship form of too small block 



1023 



