Minimum Wave Resistance for Dipole Distributions 95 
eo gp = E90) 
&ND APPROXIMATION 
~~"-- ~~ EXACT SHAPE 
f=0o €=,.05 
ws. ws Tem 7g. /i2 38 x 
Fig. 6. Cyf( f), 6C,1-9( f) and 6C,9-5( £) Fig. 7. Graphs of the exact optimum shape, €4( x), 
vs f, Notice the changes of f-scale at for f = © and an approximation to the shape. The 
f = 0.7 and 1.0.C,£( f) is the universal min- figure shows the extent to which the approximate 
shape at the ends matched the linearized shape, and 
i . The fi i 
imum curve e figure shows that a dipole die carayisa watch Weg etied Bien 
distribution which is optimal for f = 1 per- 
forms well for 0.6< f<, but ceases to do 
so as f decreases. Likewise, a strut which 
is optimal for f = 0.5 performs well in the 
range 0.46< f <0.6. We now prove that the shape at each end is 
determined by the singularity of g(x) at that end 
in the following sense. Suppose we have a dis- 
tribution of x-directed dipoles whose strength 
is 0 for x’ >O and —9(x')/n(x')}/? for x'>0, where x' = x + L/2 and 
q(x') = a Ay Ae 1 ON an rey Be (54a) 
If we let x’ =7 cos 0 and Z' =7 sin O, the stream function of the two-dimensional flow is 
given by 
A 
SN ei aL 0 a a1/2 g a3/2 30 
é.= er sind (2, cos 5 + A,r cos 5 ~ A,r cos ~ si (55) 
Let us introduce dimensionless variables r=r/L, &= EL, so that the dipole density is 
q(x’) Lec ' 
- —— = - —— _ (B + Bx iy, ata 54b 
aie ner: ° la one 
where 
A 
Ba = 
cL 
1 
: AL /2 
