Minimum Wave Resistance for Dipole Distributions 77 
ee 
FREE STREAM 
VELOCITY 
Fig. 1. Coordinate system fixedin athin strut. 
The X - Z plane (Y = 0) represents the upper 
BE AVA 
surface of the fluid, the acceleration of grav- LZ 
ity is downward and the flow velocity is in wy 
: 
the +x - direction. 
Nin 
Nis - 
“< 
Now let the strut be represented by the equation 
2S +0(%).* 
(2) 
The extremities of the strut are given by x= +L/2, so that c(t L/2) =0. Since we wish 
to perform a perturbation about the strut of zero thickness, we introduce V/2L as a pertur- 
bation parameter, where V is the volume-per-unit depth, and write 
ees Ve owc® (3) 
Since V is the cross-sectional area, V/2L is the average half-beam. The wave re- 
sistance of the strut is given by 
2 
R, = 5 00? (xr C (4) 
where C,, is the wave resistance coefficient. Michell’s theory gives [3, p. 115] 
mee y ES) 
Cae \ j SE SE Kl(e- 8] aa (5) 
-L/2 %-L/2 OX dx 
where v = g/c2 and the function K(t) is defined by 
(+) 
Kt) = 2 | COS NE (6) 
1 \2Vd? - 1 
*The strut is supposed symmetrical about the x - y plane. 
646551: O—62——7 
