900 Samuel Karp, Jack Kotik, and Jerome Lurye 
O ——. 
w 2 3 4 a 6 
X= xX €= Var? = B/AL X 
Z-Z2/L 
Fig. 2. The optimum shape (at large depth) for large Froude number for various values of the thick- 
ness parameter € This form is obtained by considering the streamfunction of Eqs. (47) and (48). We 
have Eq. (48), where z = (1/2) sin 6 /(cosh € + cos @) and x = (1/2) sinh €/(cosh € + cos 9), 
Equation (28) implies that 
1/2 1/2 1/2 
| 2(x) dx = | o,(x) dx = | = E(x) dx = _ (50) 
-1/2 -*-1/2 -1/2 
or 
1/2 
| z(x) dx = € 
-1/2 
To see how well the approximate side condition specifies the volume we have compared 
max 
| ZC axe 
“x 
max 
computed by numerical integration, with €, for various values of €. The results are as 
follows: 
2(x) dx 
Expansion of the exact volume for small € yields 
1/2 
z(x) dx = « [1 - 0.44 <1/3| (50a) 
-1/2 
