Free Surface Effects tn Hull Propeller Interactton 
1/2 
raf sec?Q exp [ (z+z') sec*O | cos[ (x-x') sec 9] 
P cos [(y-y') tan® secQ|d0 (B58) 
Since we are interested only in the flow induced by the propeller in its 
own plane (the so-called self-induced wake) , we confine further ana- 
lysis to the case x = x' = x,. The first three terms of (B58) then 
vanish and in the last we substitute 
u = secO@tan@, v = yin’ tats = (l+v)/2, du/d@= sv 
to get a (28) 
Gay yty’ 3, 2’) pan exp [(z+z')s“ ]cos[(y-y') u] tty du 
0 (B60) 
Now integrating over the propeller source distribution o (R) and 
taking advantage of its transverse symmetry , we get for the self - 
induced wake the following expression 
R 
P 7/2 
we (R, 0) ules dR bos G,(0,y-y',z,z') o(R') R'do 
) : oa 
=4_7 fi “7 }exp [ (zt+zp)s°] of R'o(R') I(sR')aR't 
0 Sie ieae (uy) du (Bé1) 
where the integral formula quoted after (B27) has been applied again. 
Since the function o(R') is in general not analytic , the R' integral 
must be evaluated by numerical quadrature (e.g. Simpson's rule) for 
suitable values s(u . such that the u integral can be approximated 
by the recurrence formulas for Fourier series , see Equations 
(B50-B52) . 
By proper choice of the fields points y = Rcos®, z= ZptRsin 0 
the self-induced free-surface wake wf(R,0) can be calculated at sui- 
table points (Rj , 9x) on the disk , from which the circumferential ave- 
rage wy (R) and the disk average wy can be obtained by numerical 
integration . 
1901 
