Hydrofoils Running Near a Free Surface 173 
u(x,0) _ 
caer ae 1 (6) 
will be satisfied. 
By inserting the velocity components (5) into (4) the following integral equation for de- 
termining the pressure distribution f(€) results: 
+1 
Mig ats i 1 Ce a SE 9) 
Df" (x) = if Ke] = DEO dé. (7) 
The integral 
+1 
{ FE) ae 
be aS 
is the known relation for the downwash of a thin profile in an infinite medium. The part of 
the relation (7) dependent on the depth of submergence h* represents the influence of the 
surface, which results in a diminution cf the effective angle of incidence for the profile. 
For the solution of Eq. (7) the following function with unknown coefficients will be 
chosen: 
f(é) = 2a, +4 Se eae ee) ee (8) 
The coefficients a, will be found by complying with Eq. (7) for three points of the chord. 
For these points x = +c/4 and x = +0 may be chosen. By means of the a,, which are a func- 
tion of h*, the lift of the profile follows from the known relation 
C= €,th*)|= 2 |ag(h*) + a,(A*)] (9) 
If the distance h* is considerable, i.e., if the profile is deeply submerged, it then follows 
from (7) that 
+1 
-2nf'(x) = | ae 
-1 
ee 
(7a) 
that is, the Ackermann-Birnbaum integral equation for evaluating the pressure distribution of 
thin profiles in infinite flow. If the distance h* disappears, i.e., if the profile runs at the 
surface, it then follows from (7) that 
+1 
-nf'(x) = | f(E) = — sae (7b) 
= 
646551 O—62—_13 
