The Wave Generated by a Fine Shtp Bow 
As before, we assume that b =O(€) and also that b'(x) = db/dx = 
O(e) . A consequence is that in the bow near field we have 
bistaceawiO)s fa set G0) tx MKON Zhe 
[ 3/2] [<2] 
For a wedgelike entrance, b(0) = 0, and so we have, approximately, 
i a= t xa [1 + o(1)], Ho 2 ery § (4) 
as the description of the body, where @ = b'(0) , the wedge half-angle. 
This argument might have been used previously to justify the thin-body 
approximation, although one might question whether it would be more 
convincing than the simple statement of assumption made previously. 
However, now it serves a much more practical purpose : we can 
simplify the right-hand side of Equation (3). In the bow near field, the 
integro-differential equation becomes 
od 
xX us 
1 -y S = log qa > = e (5) 
At first sight, this equation appears rather formidable. But 
the integral can be considered as a convolution integral, a fact which 
suggests the use of Fourier transforms to eliminate the y dependence. 
In what follows, we manipulate some transforms which are nonsense 
in a classical analysis ; whenever necessary, integrals should be in- 
terpreted in the sense of generalized functions. We follow Lighthill 
[7] in such respects. 
Let the Fourier transform be defined as follows 
F {ay} = £*(f) = foo eify Egy Ss 
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