The Wave Generated by a Fine Ship Bow 
There seems to be little point in writing out the correspond- 
ing expression for @ (x,y,z), which could be done through use of 
Equation (1), In fact, we shall not even bother at this point to write 
out the inverse transform of the expression in (7), although we note 
that the latter can be expressed in terms of Fresnel integrals. Itis 
worthwhile to write at least the transforms of two related quantities, 
namely, ¢*, (x; f;0) and $* (x;L;0) 
o* (x: £0) = a (: - at) cos VK |d| x ; (8) 
aS VK|L| € - ities) sin VK |Z Kg 2c (9) 
o% (x; £50) 
The behavior of $*, at large distance from the bow will be interest- 
ing to note presently, and e”, is essentially the transform of the 
wave height, which can be seen from the dynamic free-surface condi- 
tion [A] ; 
IV. LIMIT BEHAVIOR OF THE SOLUTION FOR THE WEDGE BOW 
Behavior as _ |y|-. 00 . Since the potential and its deriva- 
tives on the plane z=0 are all given in terms of Fourier transforms 
with respect to y , it is nearly a trivial matter to determine how the 
inverse transforms act when y— > f°. We need only to examine the 
behavior of the transforms near their singularities. The only singu- 
larities occur at £=0. For example, ¢*,(x;2;0) can be expressed 
$ (x; £50) 2Ua {1 = H”|L| + M3 Kx’ +...) 
2UaH E kets Kx’ | + | 
Treating this transform as a generalized function, we can obtain the 
limit behavior of its inverse transform by using the methods describ- 
ed by Lighthill [7] . We find that 
2 
* (x; y, 0) ——— See as dar al, 
wy, 
This shows that, far off to the sides, the disturbance appears to be 
caused by a vertical dipole distribution. Such a result should not be 
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