88 ‘Samuel Karp, Jack Kotik, and Jerome Lurye 
The closed streamline defining the minimizing shape, a, is given by 
Cx.zy = 0. (45) 
To see this, note that the flow is symmetrical about the x-y plane, so that the part of 
the x axis outside the shape o belongs to the streamline in question. Now as x + © (which 
implies 7 + 0), ['(x,z) + cLz, as is evident from Eq. (44). But cLz=0 on the x axis, prov- 
ing that the desired streamline is defined by Eq. (45). 
Thus the limiting form assumed by the minimizing shape, o, at large depths and for large 
Froude number, is represented by 
cl inde igs any é 
cLz aL Re ; = 0 (7 = xX + 2 (46) 
REE 7) 
4 
If we define a thickness parameter € = V/2L? = B/2L,* where B is the average (full) 
- width, we can rewrite Eq. (46) as 
(47) 
In bipolar coordinates! € and 6 we have 
1 
; sin 0 2 sinh € 
“cosh £ + cose nae % > eoshwettteosnG 
letting the line 0 = 0 have length 1. Also 
= e&(cos 6 + i sin 6) 
and Kq. (47) becomes 
5 sin 6 ‘ aa g: 
cosh € + cos Bg 2) CEL PED Gy 9 
*€ is also equal to the horizontal-plane area coefficient multiplied by B/2L, where B is the usual max- 
imum beam. 
tSee, for example, Morse and Feshbach, “Methods of Theoretical Physics,” New York: McGraw-Hill, 
1953, p. 1210. 
