The Wave Generated by a Fine Shtp Bow 
where K= afte i 
In finding the first approximation to ¢ , we have a boundary- 
value problem to solve inthe y-z plane. Thatis, we have a partial 
differential equation involving only the transverse rates of change. 
The body boundary condition is a simple Neumann condition, but the 
free-surface condition involves derivatives with respect to x , and 
soa 3-D aspect is introduced through this condition. The problem 
in the cross plane is illustrated in Figure l(a). 
For the moment, we shall confine our attention to a special 
case of this problem, namely, to narrow bodies which can be generat- 
ed approximately by a distribution of sources on the centerplane, 
y = 0. This special case is depicted in Figure 1(b). A modification 
of our method of solution has been worked out for more general cases, 
but we shall not consider such cases further in the present paper ; 
they would only distract us from the simple ideas which are being 
developed. 
(a) Zz 
Figure 1 Problem for the First Approximation 
(a) General body. (b) Thin body. 
In both cases, the potential satisfies :¢ t+@ = 0. 
Vy ZZ 
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