86 Samuel Karp, Jack Kotik, and Jerome Lurye 
of Pavlenko’s theory in which g(x) represents the form of the strut. In the present theory 
g(x) represents a dipole density and the criticism does not apply. 
Now, according to the procedure developed in the previous sections, we can obtain the 
minimizing shape o by substituting g(x) from Eq. (33) into Eq. (25) and determining the 
closed streamlines of the potential defined by Eq. (27). However, this procedure can be by- 
passed if we desire only the limiting form assumed by o at large depths. At such depths, 
the flow is very nearly two-dimensional (horizontal) and can therefore be characterized by a 
complex velocity potential 
Q(T) = cLr + + wT) (34) 
where 7 = x + iz and w is the complex perturbation potential arising from the presence of the 
strut. 
In terms of its real and imaginary parts, 
wT) = w(x,z) + i€(x,z) (35) 
where pz is the real perturbation potential of Eqs. (25) and (27) and € is the associated real 
stream function. 
We now verify that w has the specific form 
ik 
wT) = ——.. 
1 36 
Vi- ma (36) 
In Eq. (36), K is a real constant to be determined. The definition of w is completed by 
stipulating that the 7 plane be slit along the x axis from —1/2 to +1/2, with the square root 
given the positive determination on the side of the slit for which z = 0+. 
Differentiating Eq. (36) with respect to z, we get 
iKT oT 
( 1 ) 3/2 02 (37) 
Fight 
Letting z + 0+ in Eq. (37) and introducing Eq. (35), we have 
ou( x,0+) pis 0&(x,0+) Bh cook wn ck 
oo Te (5 zh x2) ats (38) 
It follows that 
