98 Samuel Karp, Jack Kotik, and Jerome Lurye 
6. COMPARISON OF VARIOUS FORMS 
In Section 4 we examined the resistance of some optimum dipole distributions. We now 
compare these with some others. Figure 9 shows the universal minimum Cf (f), C08 f) and 
C,,(f) for a parabolic dipole distribution vanishing at the ends (labeled C(f)). It is known 
that a dipole distribution —cg(x)/27, where 
- 2 sin 22 (1-4x2)”? 
&( x) = Se eS (61) 
(1+ 2x)4/? + (1- 2x)*/? - 2(1- 4x?) cos = 
generates (in two-dimensional flow) a lens given by 
ZC) = + (pore i - 4x? + cot ) ; (62) 
For p = 1.812 the full interior lens angle is (2 — p)# = 0.1887, while 
1/2 1/2 
| z(x) dx = 0.05 = and | &(x) dx = 0.057. 
-1/2 -1/2 
Figure 9 shows C,,(f) for a dipole distribution — cg(x)/2m (labeled chp) and also C,,(f) 
for a dipole distribution —cz(x)/27 (labeled CL5/(f)). This last quantity is Michell’s wave 
resistance for the shape 2(x), since —cz(x)/27 is the linearized dipole distribution defined 
by the shape 2(x). Figure 10 shows g(x) and z(x), and we see that even for these thin forms 
the approximation g(x) ~ z(x) is not very accurate. Comparing CcLD(f) with CLS/f) in Fig. 9 
we find a discrepancy of up to 25 percent. We notice that C2-5/f) is about half of CF (f), 
Chap): or CL5/f) over a considerable range of f. 
Some information on Pavlenko’s work was obtained from Refs. 4 (p. 115) and 12. The 
forms shown by Pavlenko are reminiscent of ours, and show the same trend as a function of 
f (for f < 0.43, the maximum f considered in these references). Shen and Kern [13] solved 
the integral equation approximately, by determining three coefficients in a polynomial ap- 
proximation to the solution. They carried out the computations for F = 0.1, 0.2, 0.4, 0.6 
Fig. 9. Graphs of 
: P L 127-LS 
efi, *cl'ey, chip, [C52 (p, andl 2h sig 
2 vs f. The upper curve and the points nearby show C.(f) for a 
parabolic dipole distribution 
fo) 
: (s[--I) 
27 L6 \4 ’ 
4 
2 for the dipole distribution generating a lens (in an infinite 
fluid), and for the linearized dipole distribution defined by the 
O lens. The lower curve and nearby points show the universal 
ee ‘ Mg Mite Ee ne “n “7 minimum C.(f) and also C,,(f) for the dipole distribution which 
SOC i atals Bini is optimum at f = 0.5. Areduction in C,,(f) by 50 percent has 
6.8 y “i 
xxx C,(#) C2) 446 Co) been achieved. 
