Minimum Wave Resistance for Dipole Distributions 81 
dt ,(%) 
Sr fe Co a a 
oz dx 
along the longitudinal center section of the strut, defined by the half-strip -L/2 < % < L/2, 
720,02 = 0. 
In terms of the dimensionless variables x,y,z, and the function $(x,y,z) = d(Lx,Ly,Lz), 
Eqs. (19) and (19a) become 
Ox? oy 
0d( x,y, 0+) dC( x) 1 1 i. 
Se eee ay cere My Sy eae pat Oe S205) 
¢ is thus determined by the potential Eqs. (20) and (20a), and the usual conditions at 
infinity. 
We introduce next a Green’s function G(x,y,z;x‘y',z "* representing the potential at 
(x,y,z) of a point sink at (x',y‘,z’) in a uniform flow under a free surface. Then G satisfies 
Eq. (20) in the variables x,y,z and x,y,z‘, and the potential 4(x,y,z) can be expressed in 
terms of G as follows: 
Cs) 1/2 
pve Benes ' ~20'(x! -x! y! / 
Yoxyne iw | dy Py 20'(x") G(x,y,z; x',y",0) dx’. (21) 
=b/2 
¢ as given by Eq. (21) satisfies Eq. (20) because G does. It can also be directly veri- 
fied to satisfy Eq. (20a). Thus the flow around a thin strut in Michell’s theory is charac- 
terized by the potential 
V 
b(xy.eg) = cLx + 57 $(X,Y,Z) (22) 
where ¢ is defined by Eq. (21). 
Note that the ship form associated with the potential ¢ is not really VC(x)/2L, which is 
only the approximate form to first order in V/2L; the actual ship form is defined by the sur- 
face of closed streamlines of the full flow whose potential is the w of Eq. (22). Thus if we 
denote the actual form by 
*The coefficient of the singularity in G is unity, i.e., G =[(x—x42 + (y—y')2 + (2—2')2]71/? plus a 
regular function. This coefficient is positive for a sink and negative for a source in contrast to the 
usual situation, because we take for the flow velocity the gradient of the potential, rather than the 
negative of the gradient. 
