Ogi lvte 
H 
(x,y,z) = ref Se f oF tog (y +12 - 1) (15) 
When we combine the two results above, we obtain a one-term ex- 
pansion of the potential, to be matched with the bow-near-field ex- 
pansion 
H 
U 4UaH 
o(x,y,z) ~ Re al dS log Crary - = log Cz 
-H 
3 
1 K 3 
—— see ny 
log C A log z 7 
This result should be compared with that in (14) : the matching is 
perfect, with the previously unknown constant C now fixed. The in- 
terpretation is, of course, different. In (14) , the potential was ap- 
proaching infinity logarithmically as x +00; here, the potential is 
approaching infinity logarithmically as x —0. 
The kinematic free-surface condition, [B] , does not mean 
that ¢, is precisely equal to zero on the plane z = 0 ; it means only 
that oo) z=0 = 0 for the leading-order term in the solution for® . 
The first approximation to ¢ is O( €) and the first approximation 
to ¢, is O(e€ ). Thus, the statement that $7 l,=0 = 9 really means 
that: zl, -~q9 = o(€). This remains true evenas x— 0. Thus, the 
first-order termin $, automatically has the correct behavior for 
matching with (11), which gave the behavior of ¢, in the bow near 
field, under the condition that x +o, 
Finally, we consider once more the wave-shape function, 
$(x, y) . From the dynamic boundary condition, [A] , combined with 
what we have found above concerning the slender-body potential for 
this problem, we can express $ in the following way 
[ey = -—-¢ = — F"(x), = =—@ ; eon %2->0 
For the wedge-shaped bow, with constant draft, we can find immedia- 
tely that 6, =>0. (See (15) ) For x very small, we also find easily 
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