100 Samuel Karp, Jack Kotik, and Jerome Lurye 
ACKNOWLEDGMENT 
We wish to acknowledge the valuable contributions of Mr. V. Mangulis, who helped us 
in some of the analytical calculations, Mr. D. Cope, who heads the computer group at TRG, 
and Mr. G. Weinstein, who wrote the main digital computer programs. 
Appendix A 
PROOF OF THE WAVE RESISTANCE FORMULA 
We wigh to show that Eq. (29) gives the correct formula for the drag coefficient of a dis- 
tribution of horizontal dipoles in the half-strip —1/2 < x< 1/2, 0< y< oo, z=0, the density 
of the distribution being —(c/2m)(V/2L) f(x) with f(x) integrable but not necessarily bounded. 
In what follows, we revert temporarily to the unnormalized variables x and y. Consider 
first a finite number, n, of sources located in the above half-strip at the points (x., ¥ i, 0) 
(s=1, 2,..., m) and having strengths q,(s =1, 2,..., ). Then the wave resistance, R, of 
such an assemblage in a steady stream flowing in the x direction with velocity c, is shown 
by Lunde [2] to be 
oo 
2 2°. 2 2 
R= 167pv | (1; + J,)) cosh?u du (Al) 
0 
where p is the fluid density, vy = g/c?, and 
: 2 UR cosh’ u 
I= q, cos (vx! cosh u) e (A2) 
rs 
= : my -vyt cosh7u 
Jaz q, sin (vx, cosh u) e 3 (A3) 
a=l 
We now evaluate /, and J,, when the source celleceions is arranged to form a finite set 
of x-directed dipoles. For the | purpose, we must assume that n is even, and we locate the 
Hee at the points (x47 p> 0) (p =1,2,..., 2/2). The coordinates Bo9¥p are related to 
%.,Y, by the pauaions ae 
op = 1) x 
x, the x €s-= 2p) p= 1,2,..455- (A4) 
y, = yi (s = 2p - 1 or 2p) 
‘The quantity hisa emall distance which will eventually approach 0. 
