Ogilvie 
The first two integrals in (1) represent the flow due toa 
line of 2-D_ sources on the negative z axis and a line of sinks sym- 
metrically located on the positive z axis. Together they cause only 
a vertical flow at the plane of the undisturbed free surface, z=0. 
The third integral in (1) can represent a flow with both vertical and 
horizontal components at the plane z=0. 
We now substitute the above potential function into the free- 
surface condition, [F] 
0 
I 
© 
ar —— 
yy Lanne poercarigest) Ss [Se 
¥ Weis us D3 2 r iz 
Co 
Rewritten slightly, this is an integro-differential equation for y (x, y) 
an v, (x, 7 ) SUK at ¢ b (x, 5) 
re eorace = ——$—$__—__—_—__——— . 3 
aL fesa ee - / gaye a 
-H(x) 
The next task is to solve this equation for wy (x,y) . When that has been 
done, we canuse (1) to express ¢ (x,y,z). 
The above equation applies to thin bodies of rather general 
shape ; there is not much restriction on the function b(x,z). Rather 
than try immediately to solve this general problem, I have decided 
that it was more important to determine first the degree of validity of 
the fundamental assumptions that were made. For this reason, I shall 
next concentrate on one special case, for which the solution is easily 
obtained. We can then compare the predictions of this analysis with 
the results of experiments and determine wether it is worthwhile to 
solve Equation (3) for more general shapes, 
III SOLUTION FOR A SPECIAL CASE : A WEDGE-SHAPED BOW 
We now restrict our attention to wedgelike bodies. In the bow 
near field, in which x =O(e2 ), we assume that the body shape is 
given by 
y = b&Sca(x)eio; -H<z<0 
1494 
