Minimum Wave Resistance for Dipole Distributions 87 
Ou x,0+) Kx sail t, 
oz ia a/a 0 2 3 A 
(1 i <?) (39) 
dx (2- se ; (40) 
Now we set K=—Ac. Then, upon comparing Eqs. (39) and (40), we get 
Ou x,0+) _ mA dg 
0z dx ay 
which is what Eq. (20) becomes when ¢ is replaced by p and € by g. (The process z + 0- 
reproduces Eq. (41) with a negative sign on the right, in agreement with Eq. (20).) 
We conclude that for large Froude numbers and at large depths, the perturbation poten- 
tial, p, of Eqs. (25) and (27) is given by 
=) | 
U(x,Zz) = Re sc ele Im ae 
42 
Set ee 
The associated perturbation stream function, &, is therefore 
-Ac 
yi- of (43) 
We recall that the constant A appearing in Eqs. (42) and (43) is determined by the 
Froude number and the side condition Eq. (26). But 
E(x,z) = Re 
1/2 A 
| ————. dx = liimplies A= 7}. 
~-1/2 1/4 - x? 
From Eqs. (34) and (43) it follows that the stream function, I"(x,z), for the total flow at 
large depths and large Froude number has the form 
T(x,z) = cLz + M. Cx tay = "Lr" = Vovpe = 
2L 2L ; 44 
= 72 (44) 
