The Wave Generated by a Fine Shtp Bow 
0) 
¢(x,y,Zz) = Re Sah df b (x, £) log (y tiz - if) 
-H(x) 
H(x) 
U : ; 
-— df b_ (x, -%) log (y + iz -i¢) 
0 
1 anv (x, 7) 
“ri ea () 
The quantity in braces is a function of the complex variable (y + iz). 
The y and z components of velocity are obtained easily 
0 : (x) 
: doh gl ast ‘be (Ge, 6) 3) dS bu (x, =¢) 
- Oy hy, ps DM NO aE Mba i ee a 
%, ue 1 if - (ytiz) ia 1 [ i¢ ag er 
-H(x) 0 
; dav, (x, 7) 
ee ie n - (y + iz) 
If we require only that 
¥V(x,y) = v(x, -y) , 
the boundary condition on y= +0 is satisfied, for the last term above 
is purely imaginary on y = 0, and the next-to-last term represents a 
source distribution on the centerplane above z=0. Furthermore, if 
we approach the plane of the undisturbed free surface from below, we 
find that 
lim ¢ (x,y,z) = W(x,z) , (2) 
z To 
which will be an important fact in the analysis ahead. 
1493 
