Minimum Wave Resistance for Dipole Distributions 103 
According to (Al) and the above discussion, the wave resistance of a continuous dis- 
tribution of x-directed dipoles in the aforementioned half-strip is 
te) 
R = 167pv? | (I? + J?) cosh2u du (A20) 
0 
where / and J are given by (A18) and (A19). 
Squaring (A18) and (A19), we get 
1/2 1/2 
2 bE? ' : 
: | M(x) M(x") sin (Fx cosh u) 
sol faz “= 1/2 
* sin (Fx' cosh u) dxdx'  (A21) 
1/2 1/2 
L2 
de S Beet J M(x) M(x') cos (Fx cosh u) 
cosh“u Las 
* cos (Fx' cosh u) dxdx’. (A22) 
Thus 
L? 1/2 Wrz 
=) ft) HO) 
cosh“u wp aac age 
* cos [ Foxx") cosh u| dxdx'. (A23) 
ee y= 
Substituting from (A23) into (A20), we have 
Comey /22 1/2 
= 167pF? | | | M(x) M(x") 
0 “-1/2 “1/2 
“cos [ Fox-x") cosh | dxdx'du. (A24) 
We perform the u integration first, using the identity 
oe) 
| cos (t cosh u) du = a eet rel): (A25) 
0 
