Ogi lvte 
There are the usual two boundary conditions to be satisfied 
on the free surface 
Zz 
a) ; 
(S] [3/2] [e3] [e2] [e2] es 
z= ¢(x,y). 
rat 
> 
c= 
° 
I 
ieee 2 
gf + Ud, fone. +a + ¢ 
[B] 0 = U q =F Piz ce ap ron) y by eG ? 
fe eben Ee iree Wes 
The orders of magnitude involving $ have been noted, but of course 
we have not yet reached any conclusions, even tentatively, about the 
order of magnitude of ¢ . In condition [A] we can clearly neglect 
all of the quadratic terms, and in condition [B] the second and third 
terms on the right side can be neglected. Thus we have reduced the 
number of terms to the following 
[A] 0 
[B] 0 
Boe Og: 
I 
Gq 
an 
1 
oe 
In [A] , the first term cannot be lower order than the second, be- 
cause we would then have the meaningless result: '=0. Thus, 
either the two terms are the same order of magnitude or the first 
term is higher order than the second, If the latter is the case, the 
first term in [B] is higher order than the second term in [B] , and 
this leads to an ill-posed potential problem. Therefore we must con- 
clude that ¢ =O(€3/2 ) , and the two conditions are consistent in orders 
of magnitude. Note that this order-of-magnitude estimate for § al- 
lows us to impose the boundary conditions at z= 0 with negligible 
error. 
Finally, we can combine the two conditions above into the 
following 
[F] OS %e + 7K 78h, on Zea OY 5 
1490 
